Orbital Varieties, and Conormal Varieties to Schubert Varieties

by Rahul Singh

B.Sc. (Hons.) in Mathematics and Computer Science, Chennai Mathematical InstituteM.S. in Mathematics, Northeastern University

A dissertation submitted to

The Faculty ofthe College of Science ofNortheastern University

in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy

April 22, 2019

Dissertation directed by

Venkatramani LakshmibaiProfessor of Mathematics

1

Dedication

Dedicated to my grandfather, who dreamt of having a doctor in the family; and to the

Chennai Mathematical Institute, for giving me the opportunity to study mathematics when

I needed it most.

2

Acknowledgements

First and foremost, I would like to express my very great appreciation to Prof. Lakshmibai

for her guidance and support. Thank you for your patience and kindness, and for the many,

many hours of your time that you let me consume.

This work would not have been possible without the beautiful camaraderie that is a

centerpiece of our department. I am eternally grateful to my many colleagues and professors,

from whom I have learnt so much about mathematics, and even more about life. I would also

like to acknowledge the role of my family; their encouragement and support in instrumental

in keeping me going.

3

Abstract of Dissertation

Schubert varieties, being the foundational objects of enumerative geometry, have been stud-

ied by mathematicians for over a century. Their conormal varieties, and the closely related

orbital varieties, have come to play a key role in the representation theory of Lie algebras

and algebraic groups. While the combinatorics, and the associated representation theory, of

these objects is somewhat well-understood, little is known about their local geometry.

In this dissertation, we study primarily the conormal varieties of Schubert varieties in

a cominuscule Grassmannian, and the associated orbital varieties. We first show that a

compactification given by Lakshmibai, Ravikumar, and Slofstra, identifying the cotangent

bundle of the Grassmannian with an open subset of an affine Schubert variety, is the best

possible in type A, and does not extend to partial flag varieties. By studying the images

of conormal varieties under this compactification, we show that certain conormal varieties

can be identified as open subsets of certain affine Schubert varieties, and hence are normal,

Cohen-Macaulay, and compatibly Frobenius split. As a corollary, this allows us to compute

the projective dual of various determinantal varieties.

Next, we mimic the above compactification to show that the conormal varieties of certain

Schubert divisors (namely those corresponding to long roots) in a general flag variety can be

identified as open subsets of certain affine Schubert varieties. We also show that the same

is true for orbital varieties of minimal shape.

Finally, we construct a resolution of singularities, and in types A and C, compute a system

of defining equations, for the conormal variety of any Schubert subvariety of a cominuscule

Grassmannian. As a corollary, this recovers a known system of defining equations for two-

column orbital varieties, and also suggests a natural, type-independent, conjecture that could

be a first step towards obtaining a system of defining equations in other cominuscule cases.

4

Table of Contents

Dedication 2

Acknowledgments 3

Abstract of Dissertation 4

Table of Contents 5

Disclaimer 6

Introduction 7

Chapter 1: Simple Groups, Loop Groups, and Schubert Varieties 17

Chapter 2: Type Specific Descriptions 47

Chapter 3: A Compactification of the Cotangent Bundle in Type A 61

Chapter 4: Conormal Varieties on a Cominuscule Grassmannian: A Compactification 87

Chapter 5: Projective Duality for Determinantal Varieties 105

Chapter 6: Schubert Divisors and the Minimal Nilpotent Orbit 119

Chapter 7: A Resolution of Singularities 129

Chapter 8: Systems of Defining Equations 137

Bibliography 161

5

Disclaimer

I hereby declare that the work in this thesis is that of the candidate alone, except where

indicated in the text.

6

Introduction

Schubert varieties, being the foundational objects of enumerative geometry, have been stud-

ied by mathematicians for over a century. Their conormal varieties, and the closely related

orbital varieties, have come to play a key role in the representation theory of Lie algebras

and algebraic groups. While the combinatorics, and the associated representation theory, of

these objects are somewhat well-understood, little is known about their local geometry.

The key objects in this dissertation are Schubert varieties, their conormal varieties, and

orbital varieties; the latter two are closely related to the former via Springer theory. In many

cases, these varieties are indexed by classical combinatorial objects such as Weyl groups and

Young tableaux. A central motif of the subject is that identifying the closure of an affine

cell is often a difficult problem.

The techniques used can be broadly classified into two categories. The first is to provide

compactifications that identify these objects as certain open subsets of suitable affine Schu-

bert varieties, which are known to be normal, Cohen-Macaulay, and Frobenius split. This

idea has a rich history, see [LR08]. We also discuss some examples in Chapter 3. Given the

vast literature on Schubert varieties, this approach often yields strong results, by allowing

us to repurpose results on Schubert varieties to our particular needs.

The second idea is to present systems of defining equations for the varieties in question,

hence identifying the boundary explicitly. This can be a starting point for the development of

standard monomial theories, toric degenerations, and Grobner bases. Identifying a system of

7

defining equations, and the closely related problem of determining the primality of an ideal,

can often be quite challenging. However, when successful, this technique has the benefit of

being applicable in a more general setting than our first approach.

We present here an outline of the key objects used, and the results presented, in this

dissertation. For now, we mostly restrict ourselves to working over the field of complex

numbers. As explained in the body of the thesis, all the results presented here hold over

algebraically closed fields of good characteristic.

Conormal Varieties

Let V be a smooth variety, X ⊂ V a closed subvariety, and Xsm the smooth locus of X. The

conormal bundle of Xsm with respect to V is the vector bundle,

π : T ∗VXsm → Xsm,

whose fibre at a point x ∈ Xsm is precisely the annihilator of the tangent subspace TxX in

the cotangent space T ∗xV , i.e.,

(T ∗VXsm)x = {x ∈ T ∗xV |x(v) = 0, ∀ v ∈ TxX} .

The closure of T ∗VXsm in T ∗V is called the conormal variety of X with respect to V , see also

Section 1.6.5.

Projective Duality

Suppose V is a vector space, and that X ⊂ V is the affine cone over some subvariety

P(X) ⊂ P(V ). Let Y denote the fibre of the conormal variety T ∗VX over the point 0 ∈ V .

Then Y is naturally identified as a k∗-stable subvariety of V ∗, and its projectivization P(Y )

is precisely the projective dual of P(X) in P(V ), see Proposition 5.1.3.

8

The Springer Map

Let G be a simply connected, almost simple, algebraic group over C. A subgroup P ⊂ G is

called parabolic if the quotient space G/P is compact.

Let g denote the Lie algebra of G, and let N be the nilpotent cone of G, i.e.,

N = {x ∈ g | ad(x) is nilpotent} .

Let P be a parabolic subgroup of G, and let uP be the Lie algebra of the unipotent radical

of P . The cotangent bundle T ∗G/P of G/P has the following description,

T ∗G/P = G×P uP ,

Further, we have a proper finite map,

µG/P : T ∗G/P → N , (g, x) 7→ Ad(g)x,

called the Springer map. The image of µG/P is necessarily a G-orbit closure Nλ ⊂ N . In

many cases, see [Hes78], the induced map,

µG/P : T ∗G/P → Nλ,

is a resolution of singularities for Nλ ⊂ N .

Schubert Varieties

A subgroup B ⊂ G is called a Borel subgroup of G if it is minimal for the property that G/B

is compact. Let B (resp. P ) be a Borel (resp. parabolic) subgroup of G.

The group B acts on G/P by left multiplication. The B-orbits in G/P are called Schubert

cells, and their closures are called Schubert varieties. The Schubert cells, CP (w), and their

closures, XP (w), are indexed by certain elements of the Weyl group W of G, see Section 1.5.

9

Orbital Varieties

Let NP (w) denote the conormal variety of XP (w). The image µG/P (NP (w)) of a conormal

variety NP (w) under µG/P is called an orbital variety. In general, multiple conormal varieties

yield the same orbital variety.

For G = SLn, orbital varieties have a particularly pleasant description. Recall that the

SLn-orbits N ◦λ ⊂ N are indexed by partitions λ a n. The orbital varieties OT corresponding

to SLn are indexed by the standard Young tableaux T of size n. Further, if T is a standard

tableau of shape λ, then OT is a Lagrangian subvariety of Nλ, see Section 1.7.4.

Loop Groups and Their Schubert Varieties

The loop group LG = G (C[t, t−1]) is a Kac-Moody group; its combinatorics is controlled by

the extended Dynkin diagram of G. Much like with reductive groups, there exists a robust

theory of parabolic subgroups and Schubert varieties (also called affine Schubert varieties)

for loop groups. We discuss this in some detail in Section 1.3. For now, we only mention

that the affine Schubert varieties associated to loop groups are normal, Cohen-Macaulay,

and Frobenius split.

A Compactification Result in Type A

Let Gr(d, n) denote the Grassmannian of d-dimensional subspaces in an n-dimensional vec-

tor space. In [Lak16], Lakshmibai constructed an embedding φ of the cotangent bundle

T ∗Gr(d, n) into a partial flag variety associated to the loop group LSLn. The embedding φ

identifies the cotangent bundle with an open subset of an affine Schubert variety.

In Chapter 3, we extend Lakshmibai’s construction to partial flag varieties. For any

parabolic subgroup P ⊂ SLn, we define an embedding, φP , of T ∗SLn/P into a partial flag

variety of the loop group LSLn. We also compute the minimal affine Schubert variety X(κ)

10

containing φP (T ∗SLn/P), and show that X(κ) is a compactification of T ∗SLn/P if and only if

P is a maximal parabolic subgroup of G.

In Section 3.4, we consider a more general class of maps,

ψp : T ∗SL3/B → LSL3/B.

This class of maps is indexed by certain polynomials p with coefficients in k[t, t−1], with the

map φ corresponding to the polynomial (1− t−1x).

We show that even in this general class of maps, there is no choice of ψp which can

realize an affine Schubert variety as a compactification of the cotangent bundle T ∗SL3/B. This

suggests that an attempt to provide an equivariant compactification of cotangent bundle of

a flag variety into an affine Schubert variety is unlikely to succeed beyond the case of the

Grassmannian.

Compactification Results for Cominuscule Grassmannians

The compactification of T ∗Gr(d, n), presented in [Lak16], was generalized by Lakshmibai,

Ravikumar, and Slofstra [LRS16] to cominuscule Grassmannians (see Chapter 4 for a defi-

nition) of any type in characteristic 0.

In Chapter 4, we follow this work closely to construct a compactification, φ : T ∗G/P ↪→

X(τq), where G/P is a cominuscule Grassmannian over an algebraically closed field of good

characteristic, and X(τq) is an affine Schubert variety.

Further, we study the image of the conormal variety of a Schubert variety under φ to

obtain the following results.

Theorem 4.A. The closure of φ(N(w)) in LG/P is a Schubert variety if and only if the

opposite Schubert variety Xop(w) is smooth, see Section 1.5.13.

Theorem 4.B. Let X(w) be a Schubert subvariety of G/P fow which the corresponding oppo-

site Schubert variety Xop(w) is smooth. Then N(w) is normal, Cohen-Macaulay, and admits

11

a resolution of singularities via a Bott-Samelson variety. Further, the family of varieties,

{N(w) |Xop(w) is smooth}, is compatibly Frobenius split in positive characteristic.

Projective Duality for Determinantal Varieties

Let V the set of d × m matrices (resp. symmetric n × n matrices, skew-symmetric n × n

matrices), and let

Σr = {x ∈ V | rk(x) ≤ r} , Σr = {x ∈ V | co-rk(x) ≤ r} .

In Chapter 5, we use Theorem 4.A to recover, in Theorems 5.A to 5.C, the following

projective duality:

P(Σr)∨ = P(Σr).

The idea is the following: The space V is naturally identified as the opposite cell of a

cominuscule Grassmannian G/P of type A (resp. C, D), and Σr is naturally identified as the

opposite cell of some Schubert subvariety X(w) ⊂ G/P .

Given this w, we verify that the opposite Schubert variety Xop(w) is smooth, and apply

Theorem 4.A. This allows us to identify the conormal fibre of the determinantal variety at

the zero matrix as the intersection of two Schubert varieties. This intersection happens to

be a single Schubert variety, isomorphic to Σr, which we verify with a calculation in the

appropriate Weyl group.

Schubert Divisors and the Minimal Nilpotent Orbit

Recall that G is a simply connected, almost simple, algebraic group over C, and that N is

the nilpotent cone of G. Consider the element 0 ∈ N . The G-orbit {0}, called the zero

orbit, is the unique closed G-orbit in N . There exists a unique G-orbit closure, Nmin ⊂ N ,

which is contained in the closure of every non-zero G-orbit. By convention, Nmin is called

12

the minimal orbit of N . In [AH13], Achar and Henderson construct an open embedding,

ρ : Nmin ↪→ X(u), of the minimal orbit into a Schubert variety X(u) associated to the loop

group LG.

A Schubert divisor is a Schubert variety XP (w) which is of co-dimension 1 in its ambient

space G/P . The Schubert divisors in G/P are indexed by certain simple roots of G. Suppose

XP (w) is a Schubert divisor corresponding to a long root. In this case, it is known that

µG/P (NP (w)) is an orbital variety contained in Nmin.

In Chapter 6, we lift ρ along µG/P to obtain an embedding, φ : NP (w) ↪→ X(v).

Theorem 6.A. Suppose XP (w) is a Schubert divisor corresponding to a long root. Then

there exists a dense embedding, φ : NP (w) ↪→ X(v), where X(v) is a Schubert variety

corresponding to the loop group LG.

We also study the image of an orbital variety contained in Nmin under the map ρ to

obtain the following result.

Theorem 6.B. Let ρ be the dense embedding of the minimal nilpotent orbit described in

[AH13]. For O an orbital variety of minimal shape, we have,

ρ(O) = X(τβ∨).

Here X(τβ∨) is a Schubert variety corresponding to some short coroot β∨ of G.

A Resolution of Singularities

The Bott-Samelson varieties, X(w), are a class of smooth algebraic varieties that provide

resolutions of singularities for Schubert varieties, see Section 1.5.3.

Let N◦P (w) be the conormal bundle of a Schubert cell CP (w). The conormal variety,

NP (w), of the Schubert variety XP (w) is the closure of N◦P (w) in T ∗G/P . Given a resolution

13

of singularities, X(w) → XP (w), the fibre product X(w) ×G/P N◦P (w) is a vector bundle,

Z◦(w), on an open subset of X(w).

In general, describing the closure of Z◦(w) (in X(w)×G/P T∗G/P) seems to be not much

easier than determining the boundary of NP (w). However, when G/P is a cominuscule Grass-

mannian, we obtain the following result.

Theorem 7.A. Suppose G/P is a cominuscule Grassmannian. The closure Z(w) of Z◦(w) is

a vector bundle over the Bott-Samelson variety X(w). In particular, it yields a B-equivariant

resolution of singularities θw : Z(w)→ N(w).

Recall that an orbital variety is precisely the image of a conormal variety NP (w) under the

Springer map µG/P . When G = SLn, for any orbital variety OT, we may choose a parabolic

subgroup P , and an conormal variety NP (w), so that the induced map, µG/P : N(w)→ OT,

is birational. This yields the following corollary to Theorem 7.A.

Theorem 7.B. Let T be a two-column tableau, and let OT be the corresponding orbital

variety. There exists a B-equivariant resolution of singularities Z(w) → OT, for some

choice of word w.

Defining Equations

In Chapter 8, we restrict our attention to a cominuscule Grassmannian X corresponding to

either the simple linear group SLn, or the symplectic group Sp2d. These are precisely the

usual Grassmannians Gr(d, n), and the symplectic (often also called Lagrangian) Grassman-

nians SGr(d, 2d).

In Theorem 8.A, we present a system of equations identifying the conormal variety N(w)

as a subvariety of the cotangent bundle T ∗X. The key idea is that we can identify N(w)

as the image of the vector bundle Z(w). This allows us to argue that a point of T ∗X is in

N(w) if and only if it can be lifted to a point in Z(w).

14

The Geometric Order on Young Tableaux

Recall that the orbital varieties of SLn are indexed by the standard Young tableaux of size

n. The inclusion order on the set of orbital varieties induces a partial order, the so-called

geometric order, on the set of standard Young tableaux. It is an open problem to provide a

combinatorial description of the geometric order on Young tableaux.

Using Theorem 8.B, we obtain the following combinatorial description of the geometric

order, restricted to two-column Young tableaux.

Theorem 8.C. Let T be a two-column standard Young tableau. For integers 0 ≤ j < i ≤ n,

let xji be the square sub-matrix of x with corners (tj + 1, tj + 1) and (ti, ti), and let Tji denote

the rectification of the skew-tableau T\{1, · · · , j, i+ 1, · · · , n}, see [Ful97]. Then,

OT ={x ∈ N

∣∣ J(xji ) � Tji},

where J(xji ) denotes the Jordan type of xji , and � denotes the dominance order on the set of

partitions.

15

16

Chapter 1

Simple Groups, Loop Groups, and

Schubert Varieties

In this chapter, we recall the basics of the theory of finite type and extended Dynkin diagrams,

and the Kac-Moody algebras and groups associated to these Dynkin diagrams. Throughout,

we assume that the base field k is algebraically closed.

Primary references for the combinatorial results in this section are [Bou68, Kac90, Kum02,

Rem02]. For the section on algebraic groups, the reader may consult [Bor91, Mil17], and for

the section on Schubert varieties, the reader may consult [Fal03, Kum02, PR08, HR18].

1.1 Generalized Cartan Matrices and Dynkin Diagrams

Definition 1.1.1 (Generalized Cartan Matrices). A generalized Cartan matrix (GCM) is

an n× n matrix A = (aij)ni,j=1 satisfying:

• aii = 2 for all 1 ≤ i ≤ n.

• aij are non-positive integers for i 6= j.

• aij = 0 implies aji = 0.

17

Definition 1.1.2 (Type). For S ⊂ {1, · · · , n}, let AS denote the sub-matrix AS = (aij)i,j∈S.

We categorize generalized Cartan matrices into three families:

• If det(AS) > 0 for all subsets S ⊂ {1, · · · , n}, we say that A is of finite type.

• If detA = 0, and det(AS) > 0 for all proper subsets S ( {1, · · · , n}, we say that A is

of affine type.

• We say that A is of indefinite type if it is not of finite or affine type.

Definition 1.1.3 (Realization). Let A be some generalized Cartan matrix. A realization of

A is a quintuple (V, V ∨, S, S∨, 〈 , 〉), where:

• V and V ∨ are finite dimensional vector spaces over R.

• S ⊂ V \{0} and S∨ ⊂ V ∨\{0} are n-element spanning subsets of V and V ∨, along with

a bijection S → S∨, which we denote α 7→ α∨.

• 〈 , 〉 : V ∨ × V → R is a bilinear pairing satisfying 〈α∨i , αj〉 = aij, where α1, · · · , αn is

some enumeration of the elements of S.

Note that the enumeration α1, · · · , αn of the elements of S is not considered part of the

data describing a realization. In other words, a realization of A is invariant under a re-

indexing of α1, · · · , αn. Accordingly, we consider two GCMs A, B to be equivalent if there

exists some permutation σ of {1, · · · , n} such that ai,j = bσ(i),σ(j) for all i, j.

Lemma 1.1.4. A realization of a finite type GCM is unique up to isomorphism. In partic-

ular, if A is a GCM of finite type, we may speak of the realization of A.

Proof. Let A be a GCM of finite type, and (V, V ∨, S, S∨, 〈 , 〉) a realization of A. Recall

that the elements α1, · · · , αn (resp. α∨1 , · · · , α∨n) of S (resp. S∨) span V (resp. V ∨). Next,

observe that since

det(

(〈α∨i , αj〉)n

i,j=1

)= det(A) 6= 0,

18

the columns (resp. rows) of A are linearly independent. It follows that α1, · · · , αn (resp.

α∨1 , · · · , α∨n) are linearly independent, and hence form a basis of V (resp. V ∨). Further, the

pairing 〈 , 〉 is uniquely determined by the formula 〈α∨i , αj〉 = aij.

Corollary 1.1.5. Let (V, V ∨, S, S∨, 〈 , 〉) be a realization of a finite type GCM. Then, the

pairing 〈 , 〉 is non-degenerate, and induces an isomorphism V ∨∼−→ V ∗, where V ∗ is the dual

of V .

Definition 1.1.6 (Indecomposability). A generalized Cartan matrix A = (aij)ni,j=1 is said

to be indecomposable if there does not exist any partition {1, · · · , n} = S1 t S2 for which

i ∈ S1, j ∈ S2 implies aij = 0.

Definition 1.1.7 (Dynkin Diagrams). Let A be a GCM of finite or affine type. The Dynkin

diagram D of A is a graph with n vertices labeled 1, · · · , n, and edges determined by the

following rule: for every pair i, j, the vertices (i, j) are connected by max{|aij|, |aji|} lines,

and further, these lines are equipped with an arrow pointing towards i (resp. j) if |aij| ≥ 1

(resp. |aji| ≥ 1).

We see that A is indecomposable if and only if its Dynkin diagram is connected. The

importance of Dynkin diagrams stems from the following result.

Proposition 1.1.8 (cf. [Kac90]). A generalized Cartan matrix of finite or affine type is

uniquely determined (up to equivalence) by its Dynkin diagram.

Definition 1.1.9 (Duality). Following Definition 1.1.1, we see that the map A 7→ At induces

a duality on GCMs. We extend this duality to realizations as follows: the dual of a realization

(V, V ∨, S, S∨, 〈 , 〉) of A is the realization (V ∨, V, S∨, S, 〈 , 〉) of At.

Following Definition 1.1.2, we see that duality preserves type, i.e., the dual of a finite

(resp. affine, indefinite) type GCM is a finite (resp. affine, indefinite) type GCM.

19

Let A be GCM of finite or affine type, and D its Dynkin diagram. Following Defini-

tion 1.1.7, we see that the Dynkin diagram D∨ of the dual matrix At is obtained from D by

preserving the vertices and edges, and flipping the directions of the arrows. In other words,

duality on GCMs corresponds to flipping arrows on Dynkin diagrams.

Proposition 1.1.10. The Dynkin diagrams of all generalized Cartan matrices of finite type

are listed in Figure 1.1.11. We follow [Bou68] in our choice of labelling of the vertices.

An

1 d−1 d d+1 n

Bn

n≥3

1 2 n−2 n−1 n

Cn

n≥2

1 2 n−2 n−1 n

Dn

n≥4

1 2 3 n−3 n−2

n−1

n

E6

1 3 4 5 6

2

F4

1 2 3 4

E7

1 3 4 5 6 7

2

G2

2 1

E8

1 3 4 5 6 7 8

2

Figure 1.1.11: Finite type Dynkin diagrams with cominuscule simple roots marked in black,

see Chapter 4.

20

1.1.12 Weyl Groups

Let D be the Dynkin diagram of some generalized Cartan matrix A = (aij)ni,j=1 of finite or

affine type. With abuse of notation, we write α ∈ D to mean that α is a vertex of D. Given

α, β ∈ D, we denote by e(α, β) the number of edges between α and β.

The Weyl group WD of D is the Coxeter group given by generators {sα |α ∈ D}, and the

following relations: s2α = 1 for all α ∈ D, and further, for every pair α, β ∈ D,

(sαsβ)2 = 1 if e(α, β) = 0, (sαsβ)3 = 1 if e(α, β) = 1,

(sαsβ)4 = 1 if e(α, β) = 2, (sαsβ)6 = 1 if e(α, β) = 3.

The generators sα are called the simple reflections of W . It is clear from the construction

that the Weyl group is preserved under duality of GCMs.

Proposition 1.1.13 (cf. [Kac90]). Let (V, V ∨, S, S∨, 〈 , 〉) be a realization of a generalized

Cartan matrix A. The formulae

si(v) = v − 〈α∨i , v〉αi for v ∈ V,

si(v) = v − 〈v, αi〉α∨i for v ∈ V ∨,

induce actions of the Weyl group W on V and V ∨ respectively.

1.1.14 The Weyl Involution

The Weyl group W0 admits a unique longest element, denoted w0. The element w0 is an

involution, i.e., w20 = 1, and further satisfies w0

(∆+

0

)= ∆−0 . The linear map V0 → V0 given

by v 7→ −w0(v) induces an involution of D0, called the Weyl involution, see [Bou68].

The Weyl involution is trivial for Bn, Cn, F4, G2, E7, E8, and D2n. Labelling the roots

as in Figure 1.1.11, the Weyl involutions on the remaining diagrams have the following

descriptions:

21

• For D0 = An, we have −w0(αi) = αn+1−i for all i.

• For D0 = D2n+1, we have −w0(α2n) = α2n+1, −w0(α2n+1) = α2n, and −w0(αi) = αi for

all 1 ≤ i ≤ 2n− 1.

• For D0 = E6, we have −w0(α1) = α6, −w0(α6) = α1, −w0(α3) = α5, −w0(α5) = α3,

and w0(αi) = αi for i = 2, 4.

1.1.15 Symmetrizable GCMs

A generalized Cartan matrix A is called symmetrizable if there exists an invertible diagonal

matrix,

D =

ε1

. . .

εn

, εi ∈ Z,

such that D−1A is a symmetric matrix.

If A is symmetrizable, we may choose D such that all εi > 0, and further, for any matrix

D′ =

ε′1

. . .

ε′n

, ε′i ∈ Z>0,

such that D′−1A is symmetric, we have εi ≤ ε′i for all i. We say that D is the minimal

symmetrizing matrix.

1.1.16 The W -invariant Bilinear Form

Let (V, V ∨, S, S∨, 〈 , 〉) be a realization of a symmetrizable GCM A, let W be the Weyl group

of A, and let D be the minimal symmetrizing matrix as above. There exists a W -invariant

22

symmetric bilinear form, ( | ) : V × V → R, satisfying

(αi |αj) = 〈α∨i , αj〉 ε−1i .

Observe that the form ( | ) is not unique; rather, it is uniquely determined only on the span

of simple roots in V . Given a bilinear form ( | ) as above, we have,

α∨i (v) =2 (αi | v)

(αi |αi). (1.1.17)

Finite type and extended GCMs are symmetrizable, hence admit a W -invariant bilinear

form, see [Kac90].

1.2 Semisimple Groups and Root Systems

A connected algebraic group G is called almost simple if its Lie algebra is simple, see [Bor91].

A torus is an algebraic group which is isomorphic to (k∗)n for some n ≥ 0, where k∗ is the

multiplicative group of non-zero elements in k.

1.2.1 Maximal Tori

Let G be a reductive algebraic group, let T ⊂ G be a maximal torus in G, and let g and h

be the Lie algebras of G and T respectively. Let

X = Hom(T, k∗) X∨ = Hom(k∗, T ),

and let 〈 , 〉 denote the natural bilinear pairing,

X ×X∨ → Z, 〈f, g〉 = fg,

where we view fg as an integer via the canonical isomorphism, Hom(k∗, k∗) ∼= Z, see [Mil17].

23

1.2.2 The Root Space Decomposition

The group G, and hence the subgroup T , acts on g by the adjoint action. Consider the

eigenspace decomposition,

g = g0

⊕α∈X

gα,

where gα denotes the subspace on which T acts via the character α ∈ X. As a consequence

of the maximality of T , we have g0 = h.

The non-zero α occurring in this decomposition are called the roots of (G, T ), and the

corresponding subspaces gα are called root spaces. The roots of (G, T ) form a finite subset

∆0 ⊂ X.

1.2.3 The Weyl Group

Let NG(T ) denote the normalizer of T in G. The quotient group NG(T )/T is a finite group.

We denote this group by W (G, T ), and call it the Weyl group of (G, T ). The Weyl group

acts faithfully on the set of roots of (G, T ), see [Mil17].

1.2.4 Root Subgroups

For α a root of (G, T ), there exists a unique subgroup Uα of G such that, for any isomorphism

uα : k→ Uα, we have,

tuα(x)t−1 = uα(α(t)x) ∀ t ∈ T, x ∈ k.

The subgroup Uα is called a root subgroup of (G, T ).

1.2.5 Coroots of (G, T )

For α ∈ ∆0, let Tα be the connected component containing identity of the subgroup ker(α).

The centralizer Gα of Tα in G is a reductive group, and its Weyl group W (Gα, T ) contains

24

exactly one non-trivial element sα. Further, there is a unique α∨ ∈ X∨ such that

sα(x) = x− 〈α∨, x〉α ∀x ∈ X.

The elements α∨ are called the coroots of (G, T ); they from a finite subset ∆∨0 of X∨.

1.2.6 Borel Subgroups and Simple Roots

Maximal connected solvable subgroups B ⊂ G are called Borel subgroups of G. Observe

that every torus, being solvable, is contained in some Borel subgroup B ⊂ G. Hence, we

may choose a Borel subgroup B ⊂ G, such that T ⊂ B. We define the set ∆+0 (resp. ∆−0 ) of

positive (resp. negative) roots of the triple (G,B, T ),

∆+0 = {α ∈ ∆0 |Uα ⊂ B} , ∆−0 = {α ∈ X | −α ∈ ∆0} .

Every root is either positive or negative, i.e., ∆0 = ∆+0 t∆−0 , see [Bou68, Mil17].

1.2.7 Opposite Borels

Given B and T , there exists a unique Borel subgroup B− ⊂ G for which we have B∩B− = T .

We say that B and B− are opposite Borel subgroups with respect to T .

Let B− the Borel subgroup opposite to B. The elements of ∆−0 (resp. ∆+0 ) are precisely

the positive (resp. negative) roots of the triple (G,B−, T ).

Let U be the commutator of B, i.e., U = [B,B] = {xyx−1y−1 |x, y ∈ B}, and let u be

the Lie algebra of U . Then u = [b, b], and further, we have,

U =⟨Uα∣∣α ∈ ∆+

0

⟩, Lie(U) =

⊕α∈∆+

0

gα.

Proposition 1.2.8. The group T acts on U by conjugation, see Section 1.2.4. The map

U nT → B, given by (u, t) 7→ ut induces an isomorphism between the Borel subgroup B and

the wreath product U n T .

25

1.2.9 Simple Roots

There exists a unique minimal set S0 ⊂ ∆+0 such that U = 〈Uα |α ∈ S0〉. We call the

elements of S0 the simple roots of the triple (G,B, T ).

If α =∑β∈S0

aββ is a positive (resp. negative) root, then aβ ≥ 0 (resp. aβ ≤ 0) for all

β ∈ S0.

1.2.10 Root and Coroot Lattices

Let Λ0 be the lattice in X spanned by the roots of (G, T ). The simple roots of the triple

(G,B, T ) form a basis of the root lattice, i.e.,

Λ0 =⊕

ZS0 =n⊕i=1

Zαi ⊂ X.

In fact, the choice of a Borel subgroup B containing T is in bijective correspondence with a

choice of basis S0 of the root lattice. The choice of basis also induces a partial order ≥ on

Λ0, given by

α ≥ β ⇐⇒ α− β ∈n⊕i=1

Z≥0αi.

Dually, S∨0 = {α∨ |α ∈ S0} is basis of the coroot lattice Λ∨0 , and induces a partial order,

x ≥ y ⇐⇒ x− y ∈n⊕i=1

Z≥0α∨i .

1.2.11 Structure Theorem for Reductive Groups

The abelian groups X and X∨ are torsion-free, hence we may view them as subsets of the

real vector spaces X∨R = R⊗X and XR = R⊗X∨ respectively. Further, we may also extend

the bilinear pairing 〈 , 〉 : X∨ × X → Z, see Section 1.2.1, to a bilinear pairing of vector

spaces,

〈 , 〉 : X∨R ×XR :→ R.

26

Let T ′ be a maximal torus in G. Then T ′ is conjugate to T , and further, this conjugation

induces an isomorphism between the root system of (G, T ) and the root system of (G, T ′).

Colloquially, we say that the root system of (G, T ) does not depend on choice of T .

Let S0 be the set of simple roots corresponding to some choice of Borel subgroup B

containing T . Then, (XR, X∨R , S0, S

∨0 , 〈 , 〉), is a realization of some finite type GCM A.

Further, the Weyl group of (G, T ) is isomorphic to the Weyl group of A, i.e., W (G, T ) = WA.

Conversely, given any finite type GCM A, there exists some almost simple group G for

which the above construction yields a realization of A. Further, we may chose G to be

simply connected. This induces a bijection between finite type GCMs and simply connected

almost-simple algebraic groups.

1.2.12 The Coxeter Number of G

Let D∨0 denote the Dynkin diagram dual to D0. Observe that (V ∗0 ,∆∨0 ,∆0) is precisely the

root system corresponding to D∨0 . Viewing the coroot latticen⊕i=1

Zα∨i as the root lattice of

the dual diagram D∨0 , we obtain a partial order on the coroot lattice.

Proposition 1.2.13. The set ∆0 admits a unique maximal (w.r.t ≥) element θ0, called the

highest root. Dually, ∆∨0 admits a unique maximal element θ∨0 , called the highest coroot.

The highest root and coroot are related by the formula 〈θ0, θ∨0 〉 = 2.

We write θ0 and θ∨0 as the sum of the simple roots and simple coroots respectively,

θ0 =∑α∈D0

aαα, θ∨0 =∑

α∨∈D∨0

aα∨α∨.

Then 1 +∑aα is called the Coxeter number of D0, and 1 +

∑aα∨ is called the dual Coxeter

number of D0.

27

1.3 Loop Groups

Let G be a simply connected, almost-simple algebraic group over k. Let T be a maximal

torus in G, let B and B− be a pair of opposite (with respect to T ) Borel subgroups of G

containing T , and let

O = k[t], O− = k[t−1], K = k[t, t−1].

We study the loop group LG = G(K). The group LG is ind-representable by an affine scheme

over k, see [Fal03]. Further, LG is a Kac-Moody group (in the sense of [Rem02]) with torus T

associated to D. Combinatorially, its structure has many similarities with with G, which can

be expressed naturally in the language of extended Dynkin diagrams and their root systems.

1.3.1 Borel Subgroups

Let L+G = G(O), and L−G = G(O−). We define evaluation maps,

π : L+G→ G, t 7→ 0, π− : L−G→ G, t−1 7→ 0,

and subgroups, B = π−1(B) and B− = π−1− (B−).

We call B and B− Borel subgroups of LG. Further, since B∩B− = T , we say that B and

B− are opposite with respect to T .

1.3.2 Extended Dynkin Diagrams

Let A0 = (aij)ni,j=1 be the GCM of G, and D0 the Dynkin diagram of A0. We denote the

set of roots (resp. corooots, simple roots, simple coroots, postive roots, negative roots) of

(G,B, T ) by ∆0 (resp. ∆∨0 , S0, S∨0 , ∆+0 , ∆−0 ), and we denote the Weyl group of (G, T ) by

W0. Let θ0 (resp. θ∨0 ) be the highest root (resp. coroot) in the root system of A0. We define

integers, a00 = 2, and

ai0 = −〈α∨i , θ0〉 for i ∈ {1, · · · , n}, a0j = −〈θ∨0 , αj〉 for j ∈ {1, · · · , n},

28

and construct the extended GCM A as follows: the rows and columns of A are indexed by

0, · · · , n, and the ijth entry of A is precisely aij.

Proposition 1.3.3. The matrix A is a GCM of affine type. The Dynkin diagram D of A

is obtained form D0 by adding an extra vertex, labeled 0, and some edges, as described in

Figure 1.3.4.

In contrast to Lemma 1.1.4, there are multiple non-isomorphic realizations of A. We

construct a particular realization (V, V ∨, S, S∨, 〈 , 〉), which will be useful to us in our study

of loop groups. Let

S = {α0, · · · , αn} = S0 t {α0},

and let V be the vector space with basis S, i.e.,

V = V0 ⊕ Rα0 =n⊕i=0

Rαi.

Next, let V ∨ = V ∨0 , extend 〈 , 〉 to a bilinear pairing on V ∨ × V via

〈α∨i , α0〉 = ai 0 = −〈α∨i , θ0〉 , for i 6= 0,

and set α∨0 = −θ∨0 . Then (V, V ∨, S, S∨, 〈 , 〉) is a realization of A.

Observe that the elements S∨ are not linearly independent. Further, the matrix A has

co-rank 1, and the induced map V → HomR(V ∨,R) given by v 7→ 〈 , v〉 admits an one

dimensional kernel, with basis δ = α0 + θ0. In fact, we have the following stronger result.

Proposition 1.3.5 (cf. [Kac90]). Let Λ =⊕α∈S

Zα. Matrix multiplication by A (with respect

to the basis α0, · · · , αn) induces a linear map Λ→ Λ, given by∑i

biαi 7→∑i,j

ajibiαj.

An element γ ∈ Λ satisfies Aγ = 0, if and only if γ = nδ for some n ∈ Z.

Corollary 1.3.6 (cf. [Kac90]). Let ( | ) be a W -invariant symmetric bilinear form on V

as defined in Section 1.1.16. Then, (δ | ) = 0, and further, any γ ∈ Λ satisfying (γ | ) = 0

is of the form γ = nδ, for some n ∈ Z.

29

A1

1 0

Ann≥2

1 d−1 d d+1 n

0

Bn

n≥3

1

0

2 n−2 n−1 n

Cnn≥2

0 1 2 n−2 n−1 n

Dn

n≥4

1

0

2 3 n−3 n−2

n−1

n F4

0 1 2 3 4

E6

1 3 4 5 6

2

0

G2

0 2 1

E7

0 1 3 4 5 6 7

2

E8

0 8 7 6 5 4 3 1

2

Figure 1.3.4: Extended Dynkin diagrams

30

1.3.7 Extended Root System

Set ∆∨ = ∆∨0 , and

∆ = {α + nδ |α ∈ ∆0, n ∈ Z} t {nδ |n ∈ Z, n 6= 0} .

We call (V, V ∨, S, S∨,∆,∆∨ 〈 , 〉) the root system of A. Let W denote the Weyl group of A.

A root α ∈ ∆ is called a real root if there exists a w ∈ W such that w(α) ∈ D; otherwise α

is called an imaginary root. The set ∆re of real roots, the set ∆im of imaginary roots, and

the set ∆+ of positive roots, have the following descriptions:

∆im = {nδ |n ∈ Z\{0}} ,

∆re = {α + nδ |α ∈ ∆0, n ∈ Z} ,

∆+ = {α + nδ |α ∈ ∆0 t {0} , n > 0} t∆+0 .

Consequently, we call δ the basic imaginary root.

Let ≥ denote the partial order on the root lattice⊕

ZS given by

α ≥ β ⇐⇒ α− β ∈n⊕i=1

Z≥0αi.

We say α ∈ ∆ is a positive (resp. negative) root if α ≥ 0 (resp. α ≤ 0), and we denote the

set of positive (resp. negative) roots by ∆+ (resp. ∆−).

1.3.8 The Root Space Decomposition of Lg

The Lie algebra of the group LG is Lg = g⊗K. We have a root space decomposition,

Lg = h⊕α∈∆

uα,

where

uα =

tngβ if α = β + nδ for some β ∈ ∆0, n ∈ Z,

tnh if β = nδ for some n ∈ Z6=0.

Observe that if α ∈ ∆re, then dim gα = 1. We call the uα root spaces in Lg.

31

1.3.9 The Weyl Group

The group LG acts on Lg via the adjoint action. Let N denote the normalizer of T in G.

The group N(K) acts on the set of root spaces of Lg. Since each root space is T -stable,

we have an induced action of N(K)/T on the set of root spaces. This action is faithful, and

identifies N(K)/T with the Weyl group W of the GCM A.

Recall the coroot lattice Λ∨0 of A0, and let W0 denote the Weyl group of A0. There exists

(cf. [Kum02, §13.1.7]) a group isomorphism W∼−→ W0 n Λ∨0 , given by

sα 7→ (sα, 0) for α ∈ D0,

sα0 7→ (sθ0 ,−θ∨0 ),

where θ∨0 is the highest coroot, i.e., the maximal element in ∆∨0 . For q ∈ Λ∨0 , we denote by

τq the element (1, q) ∈ W0 n Λ∨0 . The action of τq on ∆ is determined by the formulae,

τq(δ) = δ,

τq(α) = α− 〈q, α〉 δ ∀α ∈ ∆0.

(1.3.10)

1.3.11 Some Unipotent Subgroups of LG

Consider a finite set of real roots Ψ ⊂ ∆re. We say that Ψ is

• prenilpotent if there exist w,w′ ∈ W such that w(Ψ) ⊂ ∆+ and w′(Ψ) ⊂ ∆−.

• closed if for every pair of roots α, β ∈ Ψ satisfying α + β ∈ ∆, we have α + β ∈ Ψ.

• nilpotent if it is pre-nilpotent and closed.

If Ψ is nilpotent, then so is the Lie sub-algebra uΨ =⊕α∈Ψ

uα.

By a T -group scheme, we will mean a group scheme equipped with a group action of T .

For α a real root, let Uα be the T -group scheme over k isomorphic to Ga with Lie algebra uα.

To every nilpotent set of roots Ψ, Tits [Tit87] associates a T -group scheme UΨ that depends

only on Ψ, and is naturally a closed subgroup scheme of LG.

32

Proposition 1.3.12. For any ordering of Ψ, the product morphism∏α∈Ψ

Uα → UΨ, given by

(u1, · · · , uk) 7→ u1 · · ·uk,

is a T -equivariant scheme isomorphism. In particular, if Ψ ⊂ ∆0, then UΨ is precisely

the group generated by the root subgroups Uα. Consequently, we also have a T -equivariant

scheme isomorphism η : uΨ∼−→ UΨ.

1.4 Combinatorics of the Weyl Group

Let A be either a finite type GCM, or an extended GCM, and let D be the Dynkin diagram

of A. Let (V, V ∨, S, S∨) be the realization of A described in Section 1.2.11 (if A is of finite

type), or Section 1.3.7 (if A is an extended GCM), and let ∆ denote the corresponding set

of roots.

In this section, we study the Weyl group W of D.

1.4.1 Length

Recall the simple reflections sα, from Section 1.1.12. A word in W is a finite sequence of

simple reflections. Given a word w = (s1, · · · , sl), we call l the length of w, and we call the

product ev(w) = s1 · · · sl ∈ W the evaluation of w.

We say that a word w = (s1, · · · , sl) is reduced if it is of minimal length in the set

{v a word | ev(v) = ev(w)} .

For w ∈ W , we define the length l(w) of w to be equal to the length of any reduced word

w = (s1, · · · , sl) whose evaluation is w. Colloquially, we say that w = s1 · · · sl is a reduced

expression.

33

Proposition 1.4.2. The set ∆+ ∩ w−1(∆−) is called the inversion set of w. The length of

w is equal to the cardinality of its inversion set, i.e.,

l(w) = #{α ∈ ∆+

∣∣w(α) ≤ 0}.

1.4.3 The Bruhat Order

We define a partial order ≤, called the strong order, on the set of words. Given a word

w = (s1, · · · , sl), we say v ≤ w if and only if there exists some increasing sequence 1 ≤ i1 <

· · · < ik ≤ l such that v = (si1 , · · · , sik).

Proposition 1.4.4. The strong order on words induces a partial order ≤ on W ; we have

v ≤ w if and only if there exist reduced words v ≤ w with ev(v) = v and ev(w) = w. We call

≤ the strong order, or Bruhat order, on W .

Corollary 1.4.5. If v < w, then l(v) < l(w).

Proposition 1.4.6 (cf.[BB05]). Since the Weyl group acts faithfully on V , we may view W

as a subgroup of GL(V ). For α ∈ ∆, consider sα ∈ GL(V ), given by

sα(v) = v − 〈α∨, v〉α,

belongs to W . Further, if α ∈ ∆+, we have,

wsα > w ⇐⇒ w(α) > 0, ∀α ∈ ∆+,

sαw > w ⇐⇒ w−1(α) > 0, ∀α ∈ ∆+,

and in fact, the Bruhat order on W is precisely the transitive closure of the above relations.

1.4.7 Minimal Representatives

Let J be some proper sub-diagram of D. Following Definition 1.1.2, we see that J is

necessarily a Dynkin diagram of finite type, and hence the Weyl group WJ is a finite group.

34

Given an element w ∈ W , there exists a unique element wJ , which is of minimal length

in the coset wWJ . The element wJ is called the minimal representative of w with respect

to J , and the set of minimal representatives in W with respect to J is denoted WJ .

An element w has a unique decomposition w = wJ v, with v ∈ WJ . Given this decom-

position, we have l(w) = l(wJ ) + l(v).

Let wJ be the longest word in the Weyl group WJ . For any w ∈ WJ , the element wwJ

is the unique maximal element in the coset wWJ . We call wwJ the maximal representative

of w with respect to J .

1.4.8 Support

The support of an element w ∈ W , denoted Supp(w), is the smallest subset J ⊂ D satisfying

w ∈ WJ . For α =∑β∈D

aββ, we define the support of α to be

Supp(α) = {β ∈ D | aβ 6= 0} .

It follows from Proposition 1.4.6 that

WJ = {w ∈ W |w(α) > 0, ∀α ∈ J } . (1.4.9)

In particular, we have WJ ⊂ WD\J for any J ⊂ D. Observe that if α ∈ ∆+ and w(α) ∈ ∆−,

then Supp(α) ⊂ Supp(w).

1.5 Schubert Varieties

In this section, we recall some results about Schubert varieties. In particular, we study two

cases simultaneously.

1. Let G be a finite dimensional reductive group, whose Dynkin diagram D is connected.

We fix a maximal torus T ⊂ G, a pair of opposite Borel subgroups B, B−, and identify

35

the Weyl group W of D with N/T , where N is the normalizer of T in G. We will refer

to this as the finite type case.

2. Let G = LH be the loop group of an simply connected almost-simple algebraic group

H, and let D be the Dynkin diagram of G. Let T be a maximal torus in H, and BH , B−H

a pair of opposite Borel subgroups in H. Following Section 1.3, we set B = π−1(BH),

and B− = π−1(B−H). We also identify the Weyl group W of D with N(K)/T , where N is

the normalizer of T in H. We will refer to this as the extended/affine type case.

We denote the set of roots, positive roots, negative roots, simple roots, and simple coroots

of (G,B, T ) by ∆, ∆+, ∆−, D, and D∨.

1.5.1 Parabolic Subgroups

We call a subgroup P ⊂ G a standard parabolic subgroup if B ⊂ P . We call a subgroup

P ′ ⊂ G a parabolic subgroup, if P ′ is conjugate to a standard parabolic subgroup, i.e., there

exists some standard parabolic subgroup P ⊃ B, and some g ∈ G, such that P ′ = gPg−1.

For J a subset of D, recall the Weyl subgroup WJ = 〈sα |α ∈ J 〉. Then

PJ = BWJB = {b1wb2 | b1, b2 ∈ B, w ∈ WJ } ,

is a standard parabolic subgroup of G, and further, every standard parabolic subgroup of

G is of this form. In other words, the standard parabolic subgroups of G are in bijective

correspondence with subsets J ⊂ D via the map J 7→ PJ .

Given subsets J ,L of D, we have PJ ⊂ PL if and only if J ⊂ L. In particular, the set

of minimal standard parabolic subgroups of G is indexed by the set of simple roots D.

36

1.5.2 Partial Flag Varieties

Let J ( D be a proper subset of D. We study the quotient space G/PJ , also called the

partial flag variety. In the finite type case, G/PJ is naturally an algebraic variety, while in

the extended case, G/PJ has the structure of an ind-variety, see [Kum02].

1.5.3 Bott-Samelson Varieties

Let w = (s1, · · · , sr) be a word in W , i.e, a finite sequence of simple reflections, and set

Pi = B〈si〉B, where 〈si〉 is the Weyl subgroup generated by si. The Bott-Samelson variety,

X(w) = P1×B ···×BPr/B,

is smooth, proper algebraic variety of dimension r. Given a Bott-Samelson variety X(w) ,

and a subset J ⊂ D, we have a proper map,

ρJw : X(w)→ G/PJ ,

given by (p1, · · · , pr) 7→ p1 · · · pr(modPJ ).

1.5.4 Schubert Varieties

Let w = ev(w), the evaluation of w. The image XJ (w) of the map ρJw depends only on the

evaluation w of w. We call XJ (w) a Schubert variety, or a Schubert subvariety of G/PJ .

The Schubert varieties XJ (w) depends only on the element w ∈ W , and the underlying

Dynkin diagrams, J and D. In particular, two non-isomorphic groups with the same Dynkin

diagram yield isomorphic Schubert varieties.

37

1.5.5 The Bruhat Decomposition

For each w ∈ W , we fix a lift of w to N , which we also denote w. We have a Bruhat

decomposition,

G =⊔

w∈WJBwPJ ,

equivalently, a decomposition of the partial flag variety G/PJ ,

G/PJ =⊔

w∈WJBwPJ (modPJ ).

We call CJ (w) = BwPJ (modPJ ) a Schubert cell. Following Sections 1.4.3 and 1.5.4, we

have,

XJ (w) =⊔

v∈WJv≤w

CJ (v), (1.5.6)

where ≤ denotes the Bruhat order on W . We call CJ (w) the big cell of XJ (w).

Observe that Equation (1.5.6) yields the following geometric characterization of the

Bruhat order,

v ≤ w ⇐⇒ XJ (v) ⊂ XJ (w).

Let e denote the image of the identity element in the quotient G/PJ . The opposite cell

X−J (w), given by

X−J (w) = B−e⋂

XJ (w), (1.5.7)

is an open affine subvariety of XJ (w).

1.5.8 The Bott-Samelson Resolution

Let si be a simple reflection, and recall the standard parabolic subgroup Pi = B{1, si}B.

Let P ◦i = BsiB ⊂ Pi, and consider the induced map,

ρJw : P1×B ···×BPr/B → XJ (w).

38

Suppose w is a reduced word for w, and further, that w = ev(w) ∈ WJ . In this case, ρJw is

a resolution of singularities, inducing an isomorphism,

X◦(w) = P ◦1×B ···×BP ◦r/B∼−→ CJ (v).

Consequently, it also follows that if w ∈ WJ , then dim(XJ (w)) = l(w), the length of w.

1.5.9 The Demazure Product

There exists a unique associative product ? on W , called the Demazure product, satisfying

sα ? w =

w if sαw < w,

sαw if sαw > w.

(1.5.10)

We call the double coset BwB a Bruhat cell. Suppose v = s1 . . . sk is a reduced presentation

for v ∈ W . Then v = s1 ? . . . ? sk, and

BvB = Bs1B . . . BskB.

More generally, for v ∈ W , w ∈ WJ , we have

BvB BwPJ = B(v ? w)PJ .

Further, we have an equivalence

l(vw) = l(v) + l(w) ⇐⇒ v ? w = vw. (1.5.11)

In particular, given v ∈ WJ and w ∈ WJ , we have v ? w = vw, see Section 1.4.7.

1.5.12 Homogeneous Schubert Varieties

Let L ( D be a proper subset of D, let wL be the longest element in WL, and let v = wJL

denote the minimal representative of wL with respect to J . We have PL ∩PJ = PL∩J , and

39

further, XJ (v) = PL/PL∩J . In particular, the Schubert variety XJ (v) is PL-homogeneous,

hence smooth. The opposite cell of XJ (v) is isomorphic to the affine space Ak, for

k = l(v) = #(∆−L\∆J ),

where ∆−L denotes the set of negative roots supported on L. Let α1, . . . , αk be an enumeration

of ∆−L\∆J . Following Proposition 1.3.12, we see that the map,

Uα1 × . . .× Uαk −→ G/PJ , (u1, . . . , uk) 7→ u1 . . . uk (modP),

is an open immersion onto the opposite cell X−J (v).

1.5.13 Opposite Schubert Varieties

Let◦w be a lift of w ∈ WJ to NG(T ), and let w denote the image of

◦w in G/P . The closure

XopJ (w) of the B−-orbit B−w ⊂ G/P is called an opposite Schubert variety. The codimension

of the opposite Schubert variety XopJ (w) in G/P is precisely l(w), the length of w.

Suppose G is of finite type, and let w0 (resp. wJ ) denote the maximal element of W

(resp. WJ ). In this case, we have an isomorphism, XopJ (w)

∼−→ XJ (w0wwJ ).

1.6 Orbital Varieties and Conormal Varieties

Let G be a simply connected almost simple algebraic group, and D the Dynkin diagram of

G. A prime p is called a good prime for G if p does not divide the Coxeter number of D.

Throughout this section, we assume that the characteristic of k is a good prime.

Let T be a maximal torus in G, let B and B− be a pair of opposite Borel subgroups with

respect to T , and let g, h, b, and b− be the Lie algebras of G, T , B, and B− respectively.

40

1.6.1 The Nilpotent Cone

An element x ∈ g is called nilpotent if the linear map,

ad(x) : g→ g, y 7→ [x, y],

is nilpotent. Let N be the set of nilpotent elements in g, i.e.,

N = {x ∈ g | ad(x) is nilpotent} .

We call N the nilpotent cone of g. The group G acts on N (via the adjoint action) with

finitely many orbits. We denote the G-orbits N ◦λ , and their closures Nλ.

1.6.2 The Cotangent Bundle of a Partial Flag Variety

The cotangent bundle π : T ∗G/P → G/P is the vector bundle whose fibre T ∗p G/P at any point

p ∈ G/P is precisely the cotangent space of G/P at p. We call π the structure map defining

the cotangent bundle.

Proposition 1.6.3 (cf. [BK05]). Let UP denote the unipotent radical of P , and let uP denote

the Lie algebra of UP . We have a G-equivariant isomorphism,

T ∗G/P = G×P uP = (G×uP )/P ,

where the quotient is with respect to the adjoint action of P on uP .

1.6.4 The Springer Resolution

The Springer map, µG/P : T ∗G/P → N , given by

µG/P (g, x) = Ad(g)x,

is a proper map. The image µG/P (T ∗G/P), being a closed, irreducible, G-stable subvariety of

N , is necessarily a G-orbit closure Nλ ⊂ N .

41

The induced map µG/P : T ∗G/P → Nλ is a proper, finite map. In this case, we say that

P is a polarization of Nλ. In general, not every orbit closure in the nilpotent cone admits

a polarization. For a comprehensive discussion of polarizations, the reader may consult

[Hes78].

If the degree of µG/P is one, we call the induced map, µG/P → Nλ, a Springer resolution.

In particular, if G = SLn, the Springer map µG/P is a resolution for every parabolic subgroup

P ⊂ G, and conversely, every G-orbit Nλ ⊂ N admits a Springer resolution.

1.6.5 The Conormal Variety of a Schubert Variety

Let C(w) (resp. X(w)) be a Schubert cell (resp. Schubert variety) in G/P , corresponding to

some w ∈ WJ . The conormal bundle of C(w) in G/P is the vector bundle,

π◦w : N◦(w)→ C(w),

whose fibre at a point p ∈ C(w) is precisely the annihilator of the tangent subspace TpC(w)

in T ∗p G/P , i.e.,

N◦p (w) ={x ∈ T ∗p G/P

∣∣x(v) = 0, ∀ v ∈ TpC(w)}.

The conormal variety N(w) of X(w) in G/P is the closure (in T ∗G/P) of the conormal

bundle N◦(w). The restriction of the structure map π : T ∗G/P → G/P to the conormal

variety induces a structure map, πw : N(w)→ X(w).

Proposition 1.6.6. Let R ={α ∈ ∆+\∆J

∣∣w(α) > 0}

, and let uw =⊕α∈R

gα. Given a point

(bw, x) ∈ T ∗G/P , we have (bw, x) ∈ N◦(w) if and only if x ∈ uw.

Proof. The tangent space of G/P at identity is g/p. Consider the action of P on g/p induced

from the adjoint action of P on g. The tangent bundle TG/P is the fibre bundle over G/P

associated to the principal P -bundle G → G/P , for the aforementioned action of P on g/p,

42

i.e., TG/P = G×P g/p. Let

R′ = ∆+\(∆+J ∪R

)={α ∈ ∆+

∣∣α ≥ αd, w(α) < 0},

Uw = 〈Uα | −α ∈ R′〉 .

For any point b ∈ B, we have (see, for example [Bor91]):

BwP (modP ) = bBwP (modP )

= b(wUww−1)wP (modP )

= bwUwP (modP ).

It follows that the tangent subspace at bw of the big cell C(w) is given by

TwC(w) =

{(bw,X) ∈ G×P g/p

∣∣∣∣∣x ∈ ⊕−α∈R′

gα/p

},

where gα/p denotes the image of a root space gα under the map g→ g/p.

Recall from Proposition 1.6.3 the symmetric bilinear G-invariant form on g identifying

the dual of a root space gα with the root space g−α. We see that a root space gα ⊂ u

annihilates TbwC(w) if and only if α ∈ ∆+\∆+J and α 6∈ R′, or equivalently, α ∈ R.

1.6.7 Orbital Varieties

Let Nλ be a G-orbit closure in N , and let uB be the unipotent radical of b. The irreducible

components of Nλ ∩ uB are called orbital varieties of shape of λ. For OT an orbital variety

of shape of λ, we have dimOT = 1/2 dimNλ.

Proposition 1.6.8 (cf. [DR09]). The image of N(w) under µG/P is an orbital variety.

Conversely, every orbital variety is of the form µG/B(N(w)), where N(w) is the conormal

variety of some Schubert variety XB(w) ⊂ G/B.

For more details on the relationship between conormal varieties and orbital varieties, the

reader may consult [DR04, Spa82, CG97].

43

1.7 Double Flag Varieties

Let G be a simply connected, almost simple algebraic group, let B be a Borel subgroup of

G, and let P be any parabolic subgroup of G. We further assume the characteristic of k is

a good prime.

The group G acts diagonally on the product G/B × G/P , with finitely many orbits. We

have a G-equivariant isomorphism,

G×B G/P → G/B × G/P , (g, hP ) 7→ (gB, ghP ).

In particular, the G-orbits are precisely of the form G×B C(w), for w ∈ WJ , and the orbit

closures, called G-Schubert varieties, are of the form G×B X(w).

1.7.1 Partial Flag Varieties - A Second Look

A parabolic subgroup P ⊂ G is its own normalizer, i.e., StabG(P ) = P for the conjugation

action of G on itself. Consequently, we can identify the G/P with the set of all parabolic

subgroups conjugate to P , i.e.,

G/P ={P ′ ⊂ G

∣∣P ′ = gPg−1, for some g ∈ G}.

For P ′ ⊂ G a parabolic subgroup, let UP ′ denote the unipotent radical of P ′, and let uP ′

denote the Lie algebra of UP ′ . The product map,

(π, µG/P ) : T ∗G/P → G/P ×N , (gP, x) 7→ (g, µG/P (g, x)),

is a closed immersion, yielding the identification,

T ∗G/P = {(P ′, x) ∈ G/P ×N |x ∈ uP ′} . (1.7.2)

44

1.7.3 The Steinberg Variety

The Steinberg variety, ZJ ⊂ G/P × G/B ×N , is given by

ZJ = {(B′, P ′, x) ∈ G/B × G/P ×N |x ∈ uB′ ∩ uP ′} .

The irreducible components of ZJw are precisely the conormal varieties of the G-Schubert

varieties G×B X(w).

1.7.4 Symplectic Structures

Suppose k = C. In this case, the cotangent bundle T ∗G/P has a canonical symplectic structure,

for which the conormal varieties N◦(w) are Lagrangian. In particular, we have,

dimN(w) = 1/2 dimT ∗G/P = dim G/P .

Observe that a conormal variety N◦(w) is stable under the scaling action of C∗ on the cotan-

gent fibres. Every B-stable, C∗-stable, closed Lagrangian subvariety in T ∗G/P is necessarily

a conormal variety N(w), see [CG97].

Every G-orbit Nλ ⊂ N has a symplectic structure, the so-called Kirillov-Kostant form.

Let OT be an orbital variety of shape λ. Then OT is a Lagrangian subvariety of Nλ with

respect to the Kirillov-Kostant form.

Suppose P polarizes Nλ. In this case, the Springer map µG/P : T ∗G/P → Nλ is finite,

and induces a local (in the analytic topology) isomorphism of symplectic manifolds on the

unramified locus.

45

46

Chapter 2

Type Specific Descriptions

In this chapter, we collect some common concrete realizations of the root systems, Weyl

groups, and algebraic groups associated to the Dynkin diagrams of types A,B,C,D. The

primary reference is [LR08].

2.1 Type A

For n ≥ 2, the GCM (aij)n−1i,j=1 corresponding to the Dynkin diagram An−1 is given by

aij =

2 if i = j,

−1 if |i− j| = 1,

0 otherwise.

2.1.1 The Group SLn

The special linear group G = SLn is the unique simply connected almost-simple group with

Dynkin diagram An−1. Let T be the group of diagonal matrices in G, and let B (resp. B−)

be the group of upper (resp. lower) triangular matrices in G. Then T is a maximal torus in

G, and B, B− are opposite Borel subgroups with respect to T .

47

2.1.2 The Root system

Let E be the vector space of real valued diagonal matrices of size n× n, and let E∗ be the

dual of E. Let ε1, · · · , εn be the linear functionals on E given by εi(Ejj) = δij, and set

αi = εi − εi+1, α∨i = Eii − Ei+1 i+1.

Then (E∗, E, {α1, · · · , αn−1}, {α∨1 , · · · , α∨n−1}, 〈 , 〉) is a realization of An−1, and the corre-

sponding root system is given by,

∆0 = {εi − εj | 1 ≤ i 6= j ≤ n} , ∆∨0 = {ei − ej | 1 ≤ i 6= j ≤ n} ,

∆+0 = {εi − εj | 1 ≤ i < j ≤ n} , ∆∨+

0 = {ei − ej | 1 ≤ i < j ≤ n} ,

∆−0 = {εi − εj | 1 ≤ j < i ≤ n} , ∆∨−0 = {ei − ej | 1 ≤ j < i ≤ n} .

Writing the root εi − εj as a sum of simple roots, we have,

εi − εj =

αi + · · ·+ αj−1 if i < j,

−αi−1 − · · · − αj if i > j.

The highest root θ is given by θ = ε1 − εn = α1 + · · ·+ αn−1, and the Coxeter number is n.

2.1.3 Root Spaces

The Lie algebra g of G = SLn is precisely,

g = sln = {x ∈ Matn×n(k) |Tr(x) = 0} .

Let Ei,j denote the elementary n×n matrix with 1 in the (i, j) position and 0 elsewhere;

and let [Ei,j] be the one-dimensional subspace of g spanned by Ei,j. Then [Ei,j] is precisely

the root space corresponding to the root εi − εj.

48

2.1.4 The Weyl Group

The normalizer NG(T ) of T in G is the set of all matrices in G which contain precisely one

non-zero entry in each row and each column. Hence, we identify the Weyl group W0∼= NG(T )/T

with the group Sn of permutation matrices of size n× n.

For α = εi− εj, the reflection sα ∈ W corresponds to the transposition (i, j). The action

of W on E∗ is given by

w(εi − εj) = εw(i) − εw(j).

We may identify the Weyl group Sn with the group of n × n permutation matrices with

entries 1 via the map w 7→ Ew, where Ew =∑Ei w(i). Under this identification, the maximal

element w0 ∈ W0 is the permutation iw07−→ n−i, and the action of W on E is given by matrix

conjugation, i.e., Mw7−→ EwME−1

w .

2.1.5 The Nilpotent Cone

The nilpotent cone N is precisely the variety of nilpotent n × n-matrices. For any x ∈ N ,

the sequence,

λ = (dim(ker x), dim(kerx2/kerx), dim(kerx3/kerx2), · · · ),

is a partition of n. The conjugate partition λT is called the Jordan type of x. We will denote

the Jordan type of x by J(x).

For µ ∈ Parn, let

N ◦µ = {x ∈ N | J(x) = µ} .

The G-orbit decomposition of the nilpotent cone N of g is given by

N =⊔

λ∈Par

N ◦λ.

49

For ν ∈ Parn, let Nν be the closure of N ◦ν . The inclusion order on the Nµ is given by

Nλ ⊂ Nµ ⇐⇒∑j≤i

λj ≤∑j≤i

µj ∀ i.

This inclusion order translates to the so-called dominance order � on Parn,

λ � µ ⇐⇒∑j≤i

λj ≤∑j≤i

µj ∀ i.

2.1.6 Partial Flag Varieties

Let p = (p0, · · · , pr) be an integer-valued sequence satisfying 0 = p0 ≤ p1 ≤ · · · ≤ pr = n,

and let P be the parabolic subgroup of SLn corresponding to the subset

J = {αj | j 6= pi, 1 ≤ i ≤ r} .

For 0 ≤ i ≤ n, we denote by E(i), the subspace of E(n) with basis e1, · · · , ei, and by

E(p), the partial flag E(p0) ⊂ · · · ⊂ E(pr). The partial flag variety G/P is precisely the

variety of flags of shape p, i.e.,

G/P = Fl(p) = {F (p0) ⊂ · · · ⊂ F (pr) | dimF (pi) = pi} .

For brevity, we will denote a partial flag F (p0) ⊂ · · · ⊂ F (pr) of shape p by F (p).

Following Proposition 1.6.3 and Equation (1.7.2), we have the following description of

the cotangent bundle:

T ∗G/P ={

(F (p), x) ∈ XJ ×N∣∣x ∈ N , xF (pi) ⊂ F (pi−1)

}.

Recall the identification W0 = Sn from Section 2.1.4. The set of minimal representatives

with respect to J is given by,

SJn = {w ∈ Sn |w(pi + 1) < w(pi + 2) < · · · < w(pi+1),∀ 0 ≤ i ≤ r} . (2.1.7)

In other words, w ∈ Sn is minimal with respect to J if and only if the function j 7→ w(j) is

increasing on every interval (pi, pi+1].

50

2.1.8 Polarizations

Recall from Section 1.6.4 that a polarization of Nλ is a parabolic subgroup P for which

µG/P (T ∗G/P) = Nλ.

Let J , P be as in Section 2.1.6, and for 1 ≤ i ≤ r, let li = pi − pi−1. Observe that since

n = l1 + · · ·+ lr, rearranging the li in decreasing order yields a partition µ of n. In this case,

P is a polarization of Nλ if and only if λ is conjugate to µ, see [Hes78].

2.1.9 Standard Young Tableau

The Young diagram of a partition λ a n is a collection of boxes, arranged in left-justified

rows, with the ith row containing precisely λi boxes. The Young lattice is the partially

ordered set Par =⊔n

Parn, ordered by inclusion of their Young diagrams.

A standard Young tableau of shape λ is a filling of the Young diagram of λ with the

integers {1, · · · , n}, such that each integer is used once, and each row and each column is

increasing.

Let 1 be the sequence (1, 2, · · · , n). To any point (F (1), x) ∈ T ∗G/B, we associate a

standard Young tableau T in the following fashion: For 1 ≤ i ≤ n, we have xFi ⊂ Fi−1 ⊂ Fi.

This yields nilpotent maps x|Fi : Fi → Fi, and a corresponding chain in the Young lattice,

J(x|F1) ≤ · · · ≤ J(x|Fn),

where J(x|Fi) denotes the Jordan type of x|Fi. Then T is the tableau of shape λ, with the

integer i filling the unique box of λ which occurs in the Young diagram of J(x|Fi), but not

in the Young diagram of J(x|Fi+1).

51

2.1.10 Orbital Varieties

For T = λ(1) ≤ · · · ≤ λ(n) a standard Young tableau of shape λ, let

N ◦T ={

(F (1), x) ∈ T ∗G/B∣∣∣ J(x|Fi) = λ(i)

},

and let OT be the closure of O◦T in T ∗G/B.

Let E(1) = E1 ⊂ · · · ⊂ En be the flag stabilized by B, and let OT be the fibre over E(1)

of the first projection OT → G/B. Then OT is an orbital variety of shape λ.

The above construction induces a bijection between the orbital varieties of shape λ and

the standard Young tableaux of shape λ. As a consequence, we deduce that the orbital

varieties of SLn are indexed by the standard Young tableaux of size n.

2.1.11 The Robinson-Schensted Correspondence

Consider w ∈ W . Let N(w) be the conormal variety of the Schubert variety X(w) ⊂ G/B,

and let S (resp. T) be the left (resp. right) Robinson-Schensted tableau of w, see [Ful97]. In

[Ste88], Steinberg provides the following description of N(w),

N(w) ={

(F (1), x)∣∣∣x ∈ OT, (F (1), x) ∈ OS

}.

In particular, we have µG/P (N(w)) = OT. Following [Ste88], we interpret this formula as a

geometric characterization of the Robinson-Schensted correspondence.

52

2.2 Type A

We extend An−1 to An−1 by attaching the extra root α0, see Fig. 1.3.4. The GCM (aij)n−1i,j=0

corresponding to the Dynkin diagram An−1 is given by

aij =

2 if i = j,

−1 if |i− j| = 1,

0 otherwise.

2.2.1 Affine Permutation Matrices

An affine permutation matrix is an n× n matrix, with coefficients in Z[t, t−1], satisfying:

• Every row and every column has a unique non-zero matrix.

• Every non-zero entry is of the form ti for some i ∈ Z.

The set of affine permutation matrices form a group under matrix multiplication. We can

identify the Weyl group W of An−1 with the group of affine permutation matrices via the

map,

sαi 7→ Ei,i+1 + Ei+1,i +∑

j 6=i,i+1

Ejj, for 1 ≤ i ≤ n− 1,

sα0 7→

t−1

1

. . .

1

t

.

For w ∈ W , we denote by Ew the matrix corresponding to w under the above identification,

and we call Ew the affine permutation matrix of w.

53

2.2.2 Semi-Direct Product Decomposition of the Weyl Group

Recall the simple coroots ei − ei+1 of An−1. The coroot lattice Λ∨0 of An−1 is precisely

Λ∨0 ={∑

aiei

∣∣∣ ai ∈ Z,∑

ai = 0}.

Given q =∑aiei ∈ Λ∨0 , set

t−q =∑

t−aiEii.

Let W0 be the Weyl group of An−1. The isomorphism W0 n Λ∨0∼−→ W is given by

(w,q) 7→ Ewt−q,

where Ew denotes the permutation matrix of w, and Ewt−q is the affine permutation matrix

of the appropriate element in W .

2.2.3 The Root system

Let ∆0 (resp. ∆) denote the set of roots of An−1 (resp. An−1). Let θ0 be the highest root in

∆0, and let δ = α0 + θ0 be the basic imaginary root. Following Section 1.3.7, we have,

∆+ = {kδ + α | k > 0, α ∈ ∆0}⊔

∆+0

⊔{kδ | k > 0} ,

∆− = {kδ + α | k < 0, α ∈ ∆0}⊔

∆−0⊔{kδ | k < 0} .

2.2.4 The Weyl Group

Let S∞ denote the group of permutations of Z, and let t : Z→ Z denote the map t(x) = x+n.

The Weyl group W of An−1 acts faithfully on Z via the formula:

si(j) =

j + 1 if j = i (mod n),

j − 1 if j = i+ 1 (mod n),

j otherwise.

54

Let St∞ be the subgroup of permutations of Z that commute with t, i.e.,

St∞ = {σ ∈ S∞ |σ(x+ n) = σ(x) + n, ∀x ∈ Z} .

Using the action of W on Z, we have,

W =

{σ ∈ St∞

∣∣∣∣∣n∑i=1

(σ(i)− i) = 0

}, (2.2.5)

W0 = {σ ∈ W |σ({1, · · · , n}) = {1, · · · , n}} . (2.2.6)

2.2.7 Action of W on the Root System

Let ∼ be the equivalence relation on Z× Z given by

(i, j) ∼ (i+ kn, j + kn) ∀ k ∈ Z.

We identify ∆ with Z×Z/∼ via the map:

kδ 7→ (0, kn),

(εi − εj) + kδ 7→ (i, j + kn), for (i, j) ∈ ∆0.

Under this identification, we have (i, j) ∈ ∆+ if and only if i < j, and further, the action of

W on ∆ is precisely the quotient of the diagonal action of W on Z× Z by ∼.

2.2.8 Minimal Representatives

Recall the identification W = St∞ from Equation (2.2.5). The set of minimal representatives

with respect to J is given by,

WJ ={w ∈ St∞

∣∣w(pi + 1) < w(pi + 2) < · · · < w(pi+1),∀ 0 ≤ i ≤ r}.

In other words, w ∈ St∞ is minimal with respect to J if and only if the function j 7→ w(j)

is increasing on every interval (pi, pi+1].

55

2.3 Types C and C

Let E(2n) be a 2n–dimensional vector space with a privileged basis {e1, · · · , e2n}. For

1 ≤ i ≤ 2n, we define,

µ(i) = 2n+ 1− i.

Consider the non-degenerate skew-symplectic bilinear form ω on E(2n) given by,

ω(ei, ej) =

δi,µ(j) if i ≤ n,

−δi,µ(j) if i > n.

For V a subspace of E(2n), we define,

V ⊥ = {u ∈ E(2n) |ω(u, v) = 0, ∀ v ∈ V } .

2.3.1 The Symplectic Group Sp2n

Let G = StabSL2n(ω), i.e.,

G = {g ∈ SL2n |ω(gu, gv) = ω(u, v), ∀u, v ∈ E(2n)} .

The group G is the symplectic group Sp2n. Its Lie algebra g is given by

g = {x ∈ sl2n |ω(xu, v) + ω(u, xv) = 0, ∀u, v ∈ E(2n)} .

Let T ′ (resp. t′) be the set of diagonal matrices, and B′ (resp. b′) the set of upper triangular

matrices in SL2n (resp. sl2n). The subgroup T = T ′ ∩ G is a maximal torus in G, and the

subgroup B = B′ ∩G is a Borel subgroup of G.

2.3.2 The Root System of Sp2n

The group G is a simple group with Dynkin diagram Cn. Recall from Section 2.1.2, the

linear functionals ε1, · · · , ε2n on t′. By abuse of notation, we also denote by εi, the restriction

56

εi|t. Following [LR08], we present the root system of G with respect to (B, T ). The simple

root αi ∈ Cn is given by

αi =

εi − εi+1 for 1 ≤ i < n,

2εn for i = n.

The set of roots ∆, and the set of positive roots ∆+, are given by

∆ = {±εi ± εj | 1 ≤ i 6= j ≤ n} t {±2εi | 1 ≤ i ≤ n} ,

∆+ = {εi ± εj | 1 ≤ i < j ≤ n} t {2εi | 1 ≤ i ≤ n} .

The corresponding root spaces are given by g2εi = [Ei,µ(i)], g−2εi = [Eµ(i),i],

gεi+εj = [Ei,µ(j) + Ej,µ(i)], g−εi−εj = [Eµ(i),j + Eµ(j),i], gεi−εj = [Ei,j − Eµ(j),µ(i)].

2.3.3 The Weyl Group of Cn

We label the simple roots of the extended diagram Cn as in Fig. 1.3.4, and write si for

the simple reflection sαi . The Weyl group W of Cn is generated by the simple reflections

s0, · · · , sn, along with the relations,

sisj = sjsi if |i− j| ≥ 2,

sisi+1si = si+1sisi+1, for 1 ≤ i ≤ n− 2,

sn−1snsn−1sn = snsn−1snsn−1,

s1s0s1s0 = s0s1s0s1.

Let r0, · · · , r2n−1 be the simple reflections of the Weyl group W (A2n−1) of A2n−1. We

have an embedding W ↪→ W (A2n−1), given by

si 7→ rir2n−i, for 1 ≤ i ≤ n− 1,

sn 7→ rn,

s0 7→ r0.

(2.3.4)

57

Following Equation (2.3.4) and Equation (2.2.5), we have,

W = W (A2n−1)µ ={w ∈ W (A2n−1)

∣∣∣w(2n+ 1− i) = 2n+ 1− w(i)}.

2.3.5 Minimal Representatives

We identify the elements of W with permutations of Z via Equation (2.3.7).

Let 0 = p1 < · · · < pr = n be an increasing sequence, and let J = {αi | i 6= pj, 1 ≤ j ≤ r}.

An element w ∈ W (Cn) is minimal with respect to J if and only the function w : Z→ Z is

increasing on the intervals (pi, pi+1] for all i, i.e.,

W (Cn)J = {w ∈ W |w(pi + 1) < w(pi + 2) < · · · < w(pi+1), ∀ 1 ≤ i ≤ r − 1} .

2.3.6 The Weyl Group of Cn

The Weyl group W0 of Cn is the subgroup of W generated by s1, · · · , sn. Equation (2.3.4)

restricts to an embedding W (Cn) ↪→ W (A2n−1). This yields the identification,

W (Cn) = Sµ2n = {w ∈ S2n |w(2n+ 1− i) = 2n+ 1− w(i)} . (2.3.7)

Remark 2.3.8. Observe that since the Dynkin diagrams Bn and Cn are dual to each other in

the sense of Definition 1.1.9, their Weyl groups are isomorphic.

58

2.4 The Weyl Groups of Dn and Dn

We label the simple roots of Dn as in Fig. 1.3.4, and write si for the simple reflection sαi .

The Weyl group W of Dn is generated by s0, · · · , sn, along with the relations,

sisj = sjsi for 1 ≤ i, j ≤ n− 1, with |i− j| ≥ 2,

sisi+1si = si+1sisi+1, for 1 ≤ i ≤ n− 2,

sn−2snsn−2 = snsn−2sn,

snsi = sisn, for i 6= n− 2,

s2s0s2 = s0s2s0,

s0si = sis0, for i 6= 2,

and the Weyl group W0 of Dn is the subgroup of W generated by s1, · · · , sn. Let t0, · · · , tn be

the simple reflections of the Weyl group W (Cn) of Cn. We have an embedding, W ↪→ W (Cn),

given by,

si 7→ ti, for 1 ≤ i ≤ n− 1,

sn 7→ tntn−1tn,

s0 7→ t0t1t0.

(2.4.1)

This embedding identifies W with an index 2 subgroup of W (Cn), and W0 with an index 2

subgroup of the Weyl subgroup of Cn.

59

60

Chapter 3

A Compactification of the Cotangent

Bundle in Type A

We work over an algebraically closed field k of good characteristic. Let G be the special

linear group SLn, and let T (resp. B, B−) be subgroup of diagonal (resp. upper triangular,

lower triangular) matrices in G. Finally, let LG be the loop group SLn(k[t, t−1]), and let π,

π−, B, and B− be as in Section 1.3.

Let D0 (resp. D = D0 t {α0}) be the set of simple roots of (G,B, T ) (resp. (LG,B, T )),

and let P be the standard maximal parabolic subgroup of G corresponding to some subset

J ⊂ D0. Recall that the parabolic subgroup P ⊂ LG corresponding to J ⊂ D is precisely

P = π−1(P ).

Let P be a maximal parabolic subgroup in G, and set P = π−1(P ). In [Lak16], Laksh-

mibai has constructed an embedding, φ : T ∗G/P ↪→ LG/P, that identifies the cotangent bundle

T ∗G/P with an open subset of an affine Schubert variety.

In this chapter, we first recall some classical constructions identifying various algebraic

varieties as open subsets of certain Schubert varieties. We then extend the above construction

61

to all parabolic subgroups P ⊂ G. In particular, we define an embedding,

φP : T ∗G/P ↪→ LG/P,

and compute, in Theorem 3.A, the minimal κ ∈ WJ such that φP (T ∗G/P) is contained in the

Schubert variety XJ (κ). We also compute the dimension of XJ (κ), and show that XJ (κ) is

a compactification of T ∗G/P if and only if P is a maximal parabolic subgroup of G.

Finally, in Section 3.4, we consider a more general class of maps, ψp. We show, in

Theorem 3.B, that even in this general class of maps, there is no map which can realize

an affine Schubert variety as a compactification of the cotangent bundle T ∗SL3/B. This

suggests that an attempt to provide an equivariant compactification of cotangent bundle of

a flag variety into an affine Schubert variety is unlikely to succeed beyond the case of the

Grassmannian.

3.1 Special Open Subsets of Schubert Varieties

Many naturally occurring algebraic varieties have been identified as open subsets of Schubert

varieties. We exhibit some examples in Sections 3.1.1, 3.1.2 and 3.1.4. In particular, we

observe that the conormal varieties of determinantal varieties are isomorphic to open subsets

of certain affine Schubert varieties. This suggests a connection between conormal varieties

to Schubert varieties in the (finite-dimensional) flag variety and affine Schubert varieties.

In [Lak16], Lakshmibai explored this possibility by constructing an open embedding,

φ : T ∗Gr(d, n) ↪→ X(κ),

of the cotangent bundle of a Grassmannian into a Schubert subvariety of the affine Grass-

mannian, and relating it with Lusztig’s embedding of the nilpotent cone of sln into an affine

Schubert variety. We discuss this work in Sections 3.1.5 and 3.1.8.

62

Inspired by this work, we construct, in Section 3.1.9, an embedding,

φP : T ∗G/P ↪→ LG/P,

and show how it relates to Lusztig’s embedding of the nilpotent cone of sln.

3.1.1 Determinantal Varieties

Let V1 and V2 be finite dimensional vector spaces of dimensions d1 and d2 respectively, and

let r be an integer satisfying r ≤ min{d1, d2}. The variety

Σrd1,d2

= {x ∈ Hom(V1, V2) | rk(x) ≤ r} ,

is called a determinantal variety. Let V = V1 ⊕ V2. For x ∈ Σrd1,d2

, we define x ∈ SL(V ) via

the formula,

x(v1, v2) = (v1, v2 + xv1).

Let n = d1 + d2, and consider w ∈ W , given by

w = (r, d], (n− r, n] = [r + 1, · · · , d1, n− r + 1, · · · , n],

in the sense of Section 2.1.6. The embedding

f : Σrd1,d2

↪→ Gr(d1, V ), x 7→ x(V1),

identifies Σrd1,d2

as the opposite cell of the Schubert variety X(w) ⊂ Gr(d1, V ), see [LS78,

LR08].

3.1.2 Nilpotent Orbits of the Ar Quiver

Consider the cyclic quiver Ar with r vertices and dimension vector d = (d1, · · · , dr), and set

Rep(d, Ar) = Hom(V1, V2)× · · · × Hom(Vr, V1).

63

f1

fr

f2

fr−1

1

2

r r−1

Figure 3.1.3: The Quiver Ar marked with dimension vector d.

We study the action of the group,

SLd ={

(g1, · · · , gr) ∈ GL(V1)× · · · ×GL(Vr)∣∣∣∏ det(gi) = 1

},

on Rep(d, Ar), given by the formula

(g1, · · · , gr) · (f1, · · · , fr) = (g2f1g−11 , g3f2g

−12 , · · · , g1frg

−1r ).

We say that a SLd-orbit in Rep(d, Ar) is nilpotent if and only if it is contained in the

SLd-stable subvariety

Z ={

(f1, · · · , fr) ∈ Rep(d, Ar)∣∣∣ fr ◦ fh−1 ◦ · · · ◦ f1 : V1 → V1 is nilpotent

}.

For 0 ≤ i ≤ r, set di = d1 + · · · + di. Let n = dr, and let Q be the standard parabolic

subgroup of LG corresponding to the subset

J = D\{α0, αd1 , αd2 , · · · , αdr−1}.

Lusztig (cf. [Lus90]) has constructed an SLd-equivariant open embedding ψ : Z ↪→ LG/Q,

with SLd viewed as a block-diagonal subgroup of G. The embedding ψ identifies every

SLd-orbit closure in Z as an open subset of some Schubert subvariety of LG/Q.

64

3.1.4 Conormal Varieties of Determinantal Varieties

When r = 2, the SLd orbit closures in Rep((d1, d2), A2) are precisely the determinantal

varieties,

Σrd1,d2

= {x ∈ Hom(V1, V2) | rk(x) ≤ r} , 1 ≤ r ≤ min{d1, d2}.

Further, we have a canonical identification,

Rep((d1, d2), A2) = T ∗Rep((d1, d2), A2).

Strickland (cf. [Str82]) has shown that the subvariety

Z0 = {(f1, f2) ∈ Hom(V1, V2)× Hom(V2, V1) | f2 ◦ f1 = 0, f1 ◦ f2 = 0},

is precisely the union of the conormal varieties of the SLd-orbit closures in Rep((d1, d2), A2),

i.e., the determinantal varieties Σrd1,d2

.

3.1.5 Lusztig’s Compactification of the Nilpotent Cone

Let (ν1, · · · , νs) be a partition of n, and let v ∈ W be given by the affine permutation matrix

τq =s∑i=1

tνi−1Ei,i +n∑

i=s+1

t−1Ei,i. (3.1.6)

Observe that v = τq, where q ∈ Λ∨0 is given by

q =s∑i=1

(1− νi)Ei,i +n∑

i=s+1

Ei,i.

Recall the opposite cell X−0 (τq) of the Schubert variety X0(τq). There exists a G-

equivariant isomorphism ψ : Nν ↪→ X−0 (τq), given by,

ψ (x) =(1− t−1x

)(modL+G), for x ∈ Nν . (3.1.7)

A proof of this statement can be found in [AH13, §4.1]. A variant of the map ψ was first

introduced in [Lus81].

65

3.1.8 Lakshmibai’s Compactification

We fix a simple root αd ∈ D0. Let P (resp. P) be the standard parabolic subgroup of G

(resp. LG) corresponding to the subset J = D0\{αd}. Recall from Proposition 1.6.3 the

identification T ∗G/P = G ×P u, where u is the Lie algebra of the unipotent radical of P .

Lakshmibai ([Lak16]) has constructed a G-equivariant embedding,

φ : T ∗G/P → LG/P,

φ(g, x) = g(1− t−1x)(modP ), for g ∈ G, x ∈ u.

The embedding φ identifies T ∗G/P as an open subset of some Schubert subvariety of LG/P.

Further, we have a commutative diagram,

T ∗G/P LG/P

N LG/L+G,

φ

µG/P pr

ψ

where µ is the Springer map, and pr is the natural projection map.

3.1.9 The Map φP

Let d = (d0, · · · , dr) be an integer-valued sequence satisfying

0 = d0 ≤ d1 ≤ · · · ≤ dr = n,

and let P (resp. P) be the standard parabolic subgroup of G (resp. LG) corresponding to

J = {αi | i 6= d1, · · · , dr−1} ⊂ D0.

For 1 ≤ i ≤ r, let λi = di − di−1, and set λ = (λ1, · · · , λr). Following Section 1.6.4, we have

the Springer resolution µ : T ∗G/P → Nν , where ν is as in Section 2.1.8.

We define a map,

φP : T ∗G/P → LG/P,

φP (g, x) = g(1− t−1x)(modP), for g ∈ G, x ∈ u.

66

For g ∈ G, p ∈ P , and x ∈ u, we have,

φP(gp, p−1xp

)= gp

(1− t−1p−1xp

)(modP)

= g(p− t−1xp

)(modP)

= g(1− t−1x

)(modP)

= g φP (1, x) .

It follows that φP is well-defined and G-equivariant.

Lemma 3.1.10. The map φP is injective.

Proof. Suppose φP (g, y) = φP (g1, y1), i.e.,

g(1− t−1y

)= g1

(1− t−1y1

)(modP)

=⇒ g(1− t−1y

)= g1

(1− t−1y1

)x,

for some x ∈ P . Denoting h = g−11 g and y′ = hyh−1, we have,

h(1− t−1y

)=(1− t−1y1

)x

=⇒ x =(1− t−1y1

)−1h(1− t−1y

)=(1− t−1y1

)−1 (1− t−1y′

)h

=⇒ xh−1 =(1 + t−1y1 + t−2y2

1 + · · ·) (

1− t−1y′).

Now since x ∈ P , and h ∈ G, the left hand side is integral, i.e., does not involve negative

powers of t. Hence both sides must equal identity. It follows that x = h ∈ P = P⋂G, and

y1 = y′ = hyh−1. In particular, we have (g1, y1) = (gh−1, hyh−1) ∼ (g, y).

Corollary 3.1.11. The maps φP and ψ from Section 3.1.9 and Equation (3.1.7) sit in the

following commutative diagram:

T ∗G/P LG/P

Nν LG/L+G,

φP

µ pr

ψ

67

3.2 The Weyl group

Recall that the Dynkin diagram of G is An−1, and the Dynkin diagram of LG is An−1. Let W

denote the Weyl group of LG, and W0 ⊂ W the Weyl group of G. In this section, we present

a formula to compute the length of an element w ∈ W . We also give a characterization of

the Bruhat order on W in terms of affine permutation matrices.

3.2.1 Affine Permutation Matrices

Recall from Section 2.2.1, the identification of the Weyl group elements with affine permu-

tation matrices. Recall also, from Section 1.3.9, the identification,

W = N(K)/T ,

where N is the normalizer of T in G. Given an element n ∈ N (K), the affine permutation

matrix of its image in W is the image of n under the homomorphim,

n∑i=1

tiEσ(i),i 7−→n∑i=1

tord(ti)Eσ(i),i.

Proposition 3.2.2. For α ∈ ∆+0 , and q ∈ Λ∨0 , we have,

τqsα > τq if and only if 〈q, α〉 ≤ 0,

sατq > τq if and only if 〈q, α〉 ≥ 0.

Proof. We have α = εa − εb for some 1 ≤ a < b ≤ n. The (not necessarily simple) reflection

sα is given by the affine permutation matrix,

sα = Eab + Eba +∑

1≤i≤ni 6=a,b

Eii.

Following Equation (1.3.10), we have,

τq(α) = α− 〈q, α〉 δ.

68

Writing τq =∑tiEii for the affine permutation matrix of τq, we have,

〈q, α〉 = ord(tb)− ord(ta).

It follows that τq(α) > 0 if and only if 〈q, α〉 ≤ 0, and τ−q(α) > 0 if and only if 〈q, α〉 ≥ 0.

The result now follows from Proposition 1.4.6.

Proposition 3.2.3. Let w ∈ W be given by the affine permutation matrix,

w =∑

tciEσ(i),i

The length l(w) of w is given the formula,

l(w) =∑

1≤i<j≤n

|ci − cj − fσ(i, j)| ,

where fσ(i, j) =

0 if σ(i) < σ(j),

1 otherwise.

Proof. Recall from Section 2.2.7 the description,

∆+ = {(i, j) | 1 ≤ i ≤ n, i < j} .

69

For i ∈ Z, we have w(i) = σ(i)− cin. Following Proposition 1.4.2, we compute,

l(w) = #{α ∈ ∆+

∣∣w(α) < 0}

= # {(i, j) | 1 ≤ i ≤ n, i < j, w(i) > w(j)}

=∑

1≤i≤n

∑1≤j≤n

# {(i, j′) | j′ = j modn, i < j′, w(i) > w(j′)}

=∑

1≤i<j≤n

# {(i, j + kn) | k ≥ 0, w(i) > w(j + kn)}

+ # {(j, i+ kn) | k ≥ 1, w(j) > w(i+ kn)}

=∑

1≤i<j≤n

#

{k

∣∣∣∣ 0 ≤ k <σ(i)− σ(j)

n+ cj − ci

}+ #

{k

∣∣∣∣ 1 ≤ k <σ(j)− σ(i)

n− cj + ci

}=

∑1≤i<j≤n

|ci − cj − fσ(i, j)| .

Proposition 3.2.4. Suppose w ∈ W is given by the affine permutation matrix,∑tiEσ(i),i =

∑Eσ(i),i

∑tiEi,i = στq,

with σ ∈ Sn and τq =∑tiEi,i ∈ Λ∨0 . Consider integers 1 ≤ a < b ≤ n, and let

α = εa − εb ∈ ∆+0 , sr = sα, sl = σsrσ

−1 = sσ(α).

The set {w, slw,wsr, slwsr} admits a unique minimal element u, which we describe:

Case 1 Suppose ord(ta) = ord(tb). Then slw = wsr, and

u =

w if σ(a) < σ(b),

slw, if σ(a) > σ(b).

Case 2 Suppose ord(ta) 6= ord(tb) and σ(a) < σ(b). Then slw 6= wsr, and

u =

wsr if ord(ta) < ord(tb),

slw if ord(ta) > ord(tb).

70

Further, we have u < slu < slusr and u < usr < slusr.

Case 3 Suppose ord(ta) 6= ord(tb) and σ(a) > σ(b). Then slw 6= wsr, and

u =

slwsr if ord(ta) < ord(tb),

w if ord(ta) > ord(tb).

Further, we have u < slu < slusr and u < usr < slusr.

Proof. Recall from Equation (1.3.10) that 〈q, α〉 = ord(tb)− ord(ta).

Case 1 Suppose first that ord(ta) = ord(tb), which from Equation (1.3.10) is equivalent to

τq(α) = α. It follows that

wsr = στqsα = σsατq = sσ(α)στq = slw.

Recall that for α ∈ ∆+0 , we have wsα > w if and only if w(α) > 0. Now, since

w(α) = στq(α) = σ(α)

is positive if and only if σ(a) < σ(b), it follows that w < wsr if and only if σ(a) < σ(b).

This is precisely the description of u given in Case 1.

Case 2 Suppose ord(ta) 6= ord(tb) and σ(a) < σ(b). In particular, σ(α) is a positive root.

Set k = −〈q, α〉 = ord(ta) − ord(tb). Following Equation (1.3.10), we have τq(α) =

α + kδ, hence

w(α) = στq(α) = σ(α + kδ) = σ(α) + kδ,

w−1σ(α) = τ−1q σ−1σ(α) = τ−q(α) = α− kδ.

Since k 6= 0, we see that w(α) is positive if and only if k > 0, while w−1σ(α) is positive

if and only if k < 0. Hence, we have,

slw < w < wsr if ord(ta) > ord(tb), (3.2.5)

wsr < w < slw if ord(ta) < ord(tb).

71

In particular, we have slw 6= wsr. Next, observe that

slwsr =∑i 6=a,b

tiEσ(i),i + tbEσ(a),a + taEσ(b),b.

Further, the affine permutation matrix of slwsr is obtained by interchanging ta and tb

in the affine permutation matrix of w. Applying Equation (3.2.5) to w′ = slwsr, we

have

slw = w′sr < w′ < slw′ = wsr, if ord(ta) > ord(tb), (3.2.6)

wsr = slw′ < w′ < w′sr = slw, if ord(ta) < ord(tb).

From Equations (3.2.5) and (3.2.6), we deduce the description of u as claimed in Case

2 of the Proposition.

Case 3 Suppose ord(ta) 6= ord(tb), and σ(a) > σ(b). Then Case 2 applies to the element

w′ = slw. Hence, we deduce the answers in this case by replacing w with slw every-

where in Case 2.

3.3 The Element κ

We now return to the setting of Section 3.1.9. Let d = (d0, · · · , dr) be an integer-valued

sequence satisfying,

0 = d0 ≤ d1 ≤ · · · ≤ dr = n,

and let P (resp. P) be the standard parabolic subgroup of G (resp. LG) corresponding to

J = {αi | i 6= d1, · · · , dr−1} ⊂ D0.

For 1 ≤ i ≤ r, let λi = di − di−1, and set λ = (λ1, · · · , λr), and let ν be as in Section 2.1.8.

72

In this section, we compute various properties of the element κ ∈ W , and the associated

Schubert variety XJ (κ). For convenience, we write the Schubert subvarieties XD0(w) of

LG/L+G as X0(w).

3.3.1 The Tableau

We draw a composition tableau with r rows, with the ith row from top having λi boxes, and

we fill the boxes of the tableau in increasing order from left to right and top to bottom. Let

Row(i) be the set of entries in the ith row of the tableau.

Observe that the number of boxes in the ith column from the left is νi. The Weyl group

WP is the set of elements in Sn that preserve the partition {1, . . . , n} =⊔i

Row(i).

We define a co-ordinate system χ(•, •) on {1, . . . , n} as follows: For 1 ≤ i ≤ r, 1 ≤ j ≤ νi,

let χ(j, i) denote the jth entry (from the top) of the ith column. Note that χ(j, i) need not

be in Row(j).

Finally, let F ij,k denote the elementary matrix Eχ(j,i),χ(k,i). Observe that χ(b, i) = χ(c, j)

if and only if i = j and b = c. In particular, we have,

F ia,bF

jc,d = δijδbcF

ia,d. (3.3.2)

3.3.3 Red and Blue

We partition the set {1, . . . , n} into disjoint subsets Red and Blue. Let

S1 = {χ(1, i) | 1 ≤ i ≤ s}

be the set of entries which are topmost in their column, and set

S1(i) = S1

⋂Row(i), S2(i) = Row(i)\S1(i), S2 =

⋃i

S2(i).

The set Red(i) is the collection of the #S1(i) smallest entries in Row(i), i.e.,

Red(i) := {j | di−1 < j ≤ di −max {λk | k < i}} .

73

For each i, let Blue(i) = Row(i)\Red(i), and let

Red =⋃i

Red(i), Blue =⋃i

Blue(i).

Observe that given any x ∈ Red(i) and y ∈ Blue(i), we have x < y.

We enumerate the elements of Red in increasing order as l(1), . . . , l(s). We enumerate

the elements of Blue, written row by row from bottom to top, each row written left to right,

as m(1), . . . ,m(n− s).

Example 3.3.4. Let n = 17 and the sequence (di) be (1, 5, 9, 11, 17). The corresponding

tableau is

12 3 4 56 7 8 910 1112 13 14 15 16 17

• r = 5, s = 6.

• The sequence (λi) is (1, 4, 4, 2, 6).

• The sequence (νi) is (5, 4, 3, 3, 1, 1).

• Row(3) = {6, 7, 8, 9}.

• χ(4, 1) = 10, χ(3, 4) = 15, χ(1, 6) = 17 etc.

• F 12,4 = E2,10, F

33,3 = E14,14 etc.

• S1 = {1, 3, 4, 5, 16, 17}.

• The sequence l(i) is (1, 2, 3, 4, 12, 13).

• The sequence m(i) is (14, 15, 16, 17, 10, 11, 6, 7, 8, 9, 5).

74

3.3.5 The element κ

Let κ ∈ W be given by the affine permutation matrix

s∑i=1

tνi−1Ei,l(i) +n−s∑i=1

t−1Ei+s,m(i).

Recall the element τq ∈ W from Equation (3.1.6). The primary result of this chapter is that

the commutative diagram in Corollary 3.1.11 can be refined to the following:

T ∗G/P XJ (κ)

Nν X0(τq).

φP

µ pr

ψ

The only additional part is the claim φP (T ∗G/P) ⊂ XJ (κ), which we prove in Theorem 3.A.

Proposition 3.3.6. The Schubert variety XJ (κ) is stable under left multiplication by L+G.

Proof. Consider the affine permutation matrix of κ. We will show, for 1 ≤ i < n, either

siκ = κ(modWJ ), or siκ < κ. We split the proof into several cases, and use Proposition 3.2.4

with sl = si:

1. i < s and νi = νi+1 : We deduce from νi = νi+1 that the entries l(i) and l(i + 1)

appear in the same row of the tableau. In particular, αl(i) ∈ J . The non-zero entries

of the ith and (i + 1)th row of κ are tνi−1Ei,l(i) and tνi+1−1Ei+1,l(i+1) respectively. We

see siκ = κsl(i) = κ(modWJ ).

2. i < s and νi > νi+1 : The non-zero entries of the ith and (i+1)th row of κ are tνi−1Ei,l(i)

and tνi+1−1Ei+1,l(i+1) respectively. Applying Case 2 of Proposition 3.2.4 with a = l(i)

and b = l(i+ 1), we deduce siκ < κ.

3. i = s : The non-zero entries of the ith and (i + 1)th row of κ are tνs−1Es,l(s) and

t−1Es+1,m(1) respectively. Since m(1) ∈ Blue(r), it follows from Section 3.3.3 that

l(s) < m(1). Case 2 of Proposition 3.2.4 applied with a = l(s) and b = m(1) yields

siκ < κ.

75

4. i > s and αm(i−s) ∈ J : The non-zero entries of the i and (i + 1)th row of κ are

t−1Ei,m(i−s) and t−1Ei+1,m(i+1−s) respectively. It follows siκ = κsm(i−s) = κ(modWJ ).

5. i > s and αm(i−s) /∈ J : It follows from αm(i−s) /∈ J that if m(i − s) ∈ Row(j) then

m(i+1−s) ∈ Row(j−1). In particular, m(i+1−s) < m(i−s). The non-zero entries

of the ith and (i+ 1)th row of κ are t−1Ei,m(i−s) and t−1Ei+1,m(i+1−s) respectively. Case

1 of Proposition 3.2.4 applied with a = m(i+ 1− s) and b = m(i+ s) yields siκ < κ.

Definition 3.3.7. An element in x ∈ u is called a Richardson element of u if the P -orbit of

x is dense in u, or equivalently, the G-orbit of (1, x) is dense in T ∗G/P = G×P u.

Lemma 3.3.8. The following matrix Z is a Richardson element of u,

Z =s∑i=1

νi−1∑j=1

F ij,j+1.

Proof. It is clear that Z ∈ u. Following Section 2.1.8, we see that Z is a Richardson element

of u if and only if Z ∈ N ◦ν , i.e., the Jordan type of Z is ν.

Consider the action of Z act on the vector space kn, given by matrix multiplication with

respect to the basis e1, · · · , en. Then{eχ(νi,i)

∣∣ 1 ≤ i ≤ s}

is a minimal generating set for kn

as a module over k[Z] (the k-algebra generated by Z). Consequently, the Jordan type of Z

is ν, i.e., Z ∈ N ◦ν .

Lemma 3.3.9. There exist b, c ∈ B, such that b(1− t−1Z)c =∼$, where

∼$ =

s∑i=1

(tνi−1F i

νi,1−

νi∑j=2

t−1F ij−1,j

).

76

Proof. For 1 ≤ i ≤ s, let

bi :=

νi∑j=1

νi∑k=j

tk−jF ik,j =

νi∑j=1

νi−j∑k=0

tkF ij+k,j,

ci :=

νi∑j=1

F ij,j +

νi∑j=2

tj−1F ij,1,

Zi :=

νi∑j=1

F ij,j − t−1

νi−1∑j=1

F ij,j+1.

We compute

biZici =

(νi∑j=1

νi−j∑k=0

tkF ij+k,j

)(νi∑j=1

F ij,j − t−1

νi−1∑j=1

F ij,j+1

)ci

=

(νi∑j=1

νi−j∑k=0

tkF ij+k,j −

νi−1∑j=1

νi−j∑k=0

tk−1F ij+k,j+1

)ci

=

(νi∑j=1

νi−j∑k=0

tkF ij+k,j −

νi∑j=2

νi−j∑k=−1

tkF ij+k,j

)ci

=

(νi−1∑k=0

tkF i1+k,1 −

νi∑j=2

t−1F ij−1,j

)(νi∑j=1

F ij,j +

νi∑j=2

tj−1F ij,1

)

=

νi−1∑k=0

tkF ik+1,1 −

νi∑j=2

t−1F ij−1,j −

νi∑j=2

tj−2F ij−1,1

= tνi−1F iνi,1−

νi∑j=2

t−1F ij−1,j.

It follows from Equation (3.3.2) that for i 6= j, we have biZj = 0 and Zjci = 0. Observe

further that

1− t−1Z =∑

1≤i≤s

Zi.

Setting b =∑i

bi, c =∑i

ci, we have b, c ∈ B, and

b(1− t−1Z)c =∑i

biZici =s∑i=1

(tνi−1F i

νi,1−

νi∑j=2

t−1F ij−1,j

)= $.

77

Lemma 3.3.10. Let B$B be the Bruhat cell containing 1 − t−1Z. There exists wg ∈ W0

and wp ∈ WJ such that $ = wgκwp. Further, we have L+G ·XJ ($) ⊂ XJ (κ).

Proof. Recall the partition {1, . . . , n} = S1 t S2 from Section 3.3.3. Consider the bijection

ι : S2 → {χ(j, i) | 1 ≤ j ≤ νi − 1}

given by ι(χ(j, i)) = χ(j − 1, i). We reformulate Lemma 3.3.9 as

∼$ =

s∑i=1

tνi−1Eχ(νi,i),χ(1,i) −∑i∈S2

Eι(i),i.

It follows from Section 3.2.1 that the affine permutation matrix of $ is given by

$ =s∑i=1

tνi−1Eχ(νi,i),χ(1,i) +∑i∈S2

Eι(i),i.

Observe that for 1 ≤ k ≤ r, we have

#Red(k) = #S1(k), #Blue(k) = #S2(k).

Since both (l(i))1≤i≤s and (χ(1, i))1≤i≤s are increasing sequences, χ(1, i) and l(i) are in the

same row for each i. Furthermore, there exists an enumeration t(1), . . . , t(n− s) of S2 such

that t(i) is in the same row as m(i) for all i. We define wg ∈ W and wp ∈ WP via their affine

permutation matrices:

wg =s∑i=1

Ei,χ(νi,i) +n−s∑i=1

Ei+s,ι(t(i)),

wp =s∑i=1

Eχ(1,i),l(i) +n−s∑i=1

Et(i),m(i).

A simple calculation shows $ = wgκwp. Consequently, we have,

B$P ⊂ BwgBκBwpP =⇒ XJ ($) ⊂ L+G ·XJ (κ) =⇒ XJ ($) ⊂ XJ (κ),

where the final implication follows from Proposition 3.3.6.

Theorem 3.A. We have φP (T ∗G/P) ⊂ XJ (κ). Further, κ ∈ WJ is minimal for the property

Im(φP ) ⊂ XJ (κ).

78

Proof. Let O denote the G-orbit of (1, Z) ∈ G ×P u. It follows from Lemma 3.3.8 that

T ∗G/P = O, and from Lemma 3.3.9 that φP (1, Z) ∈ XJ ($). Since φP is G-equivariant, it

follows that

φP (O) ⊂ G ·XJ ($) ⊂ L+G ·XJ ($) = XJ (κ)

=⇒ φP (T ∗G/P) = φP(O)⊂ φP (O) ⊂ XJ (κ),

Observe that the image φP (T ∗G/P), being a closed subvariety of XJ (κ), is compact.

To prove the minimality of κ, we show that there exists an element a ∈ G such that

φP (a, Z) ∈ CJκ , the Bruhat cell in LG/P corresponding to κ. Since κ is maximal in the right

coset W0$, see Proposition 3.3.6, there exists w ∈ W such that κ = w$. It follows from

Sections 1.4.7 and 1.5.9 that

l(κ) = l(w) + l($) =⇒ κ = w ? $ =⇒ BwB$P = BκP .

Hence, for any lift a of w to the normalizer of T in G, we have φP (a, Z) ∈ CJκ .

Proposition 3.3.11. The element κ is minimal with respect to WJ , i.e., κ ∈ WJ . In

particular, we have dimXJ (κ) = l(κ), the length of κ.

Proof. We show that κsi > κ, for all αi ∈ J . Recall from Section 3.3.3 the partition,

{1, . . . , n} = Red t Blue.

Note that we have αi ∈ J if and only if i and i + 1 appear in the same row of the tableau.

In particular, given i ∈ Blue(j) such that αi ∈ J , we have i+ 1 ∈ Blue(j).

1. Suppose i ∈ Blue. Then i = m(k) and i + 1 = m(k + 1) for some k. The non-zero

entries in the ith and (i+ 1)th columns of κ are t−1Ek+s,i and t−1Ek+s+1,i+1. We apply

Case 1 of Proposition 3.2.4 with a = i, b = i+ 1.

2. Suppose i ∈ Red and i + 1 ∈ Blue. The non-zero entries in the ith and (i + 1)th

columns of κ are tνk−1Ek,i and t−1Ej,i+1. Since k ≤ s < j, we can apply Case 2

79

of Proposition 3.2.4 with a = i, b = i + 1, νk − 1 = ord(ta) > ord(tb) = −1 and

k = σ(a) < σ(b) = j to get κ < κsi.

3. Suppose i, i+ 1 ∈ Red. Since i and i+ 1 are in the same row of the tableau, we have

i = l(k) and i + 1 = l(k + 1) for some k. The non-zero entries in the ith and (i + 1)th

columns of κ are tνk−1Ek,i and tνk+1−1Ek+1,i+1. If νk = νk+1, Case 1 of Proposition 3.2.4

applies with a = i, b = i+ 1 and k = σ(a) < σ(b) = k+ 1 to give κ < κsi. If νk > νk+1,

Case 2 of Proposition 3.2.4 applies with a = i, b = i+ 1, k = σ(a) < σ(b) = k + 1 and

νk − 1 = ord(ta) > ord(tb) = νk+1 − 1 to give κ < κsi.

Lemma 3.3.12. The length of κ is given by the formula

l(κ) = 2 dim G/P +∑k′<k

#Row(k)#Blue(k′).

Proof. Note that κ = τqσ, where τq is given by Equation (3.1.6), and σ ∈ W is given by the

permutation matrix

σ =s∑i=1

Ei,l(i) +n−s∑i=1

Ei+s,m(i). (3.3.13)

Viewing σ as an element of Sn, we have

σ−1(i) =

l(i) i ≤ s,

m(i− s) i > s.

It follows that

l(σ) =#{

(i, j)∣∣ 1 ≤ i < j ≤ n, σ−1(i) > σ−1(j)

}=# {(i, j) | i < j ≤ s, l(i) > l(j)}

+ # {(i, j) | i ≤ s < j, l(i) > m(j − s)}

+ # {(i, j) | s < i < j, m(i− s) > m(j − s)} .

80

Recall that l(i) is an increasing sequence, i.e., i < j < s =⇒ l(i) < l(j), and so,

l(σ) =# {(i, j) | i ≤ s, j ≤ n− s, l(i) > m(j)}

+ # {(i, j) | i < j ≤ n− s, m(i) > m(j)}

=# {(i, j) | i ∈ Red, j ∈ Blue, i > j}

+ # {(i, j) | i < j ≤ n− s, m(i) > m(j)}

=∑k′<k

# {(i, j) | i ∈ Red(k), j ∈ Blue(k′)}

+∑k′<k

# {(i, j) | i ∈ Blue(k), j ∈ Blue(k′)}

=∑k′<k

# {(i, j) | i ∈ Row(k), j ∈ Blue(k′)}

=∑k′<k

#Row(k)#Blue(k′).

Now, it follows from Equation (1.3.10) and Proposition 3.2.2 that τq ∈ W 0. In particular,

l(τq) = dimX0(τq) = dimNν = 2 dim G/P .

Consequently, we have,

l(κ) = l(τq) + l(σ) = 2 dim G/P +∑k′<k

#Row(k)#Blue(k′).

Corollary 3.3.14. The Schubert variety XJ (κ) is a compactification of T ∗G/P if and only

if P is a maximal parabolic subgroup.

Proof. The parabolic subgroup P is maximal if and only if J = D0\ {αd} for some d, equiv-

alently, the corresponding tableau has exactly 2 rows. In this case Blue ⊂ Row(2), (recall

from Section 3.3.3 that Blue =⊔i

Blue(i)). It follows that the second term in Lemma 3.3.12

is an empty sum, which implies l(κ) = 2 dim G/P .

Conversely, suppose that the tableau has r ≥ 3 rows. In this case, both Blue(2) and

Row(r) are non-empty, and so the second term in Lemma 3.3.12 is strictly positive.

81

3.4 Generalizations of φP

Let G be the group SL3, and B the Borel subgroup of upper triangular matrices in G. In

this section, we generalize the map φ by replacing the matrix 1 − t−1x in Section 3.1.9 by

general polynomial p. We show that even in this general class of maps, there is no map ψp

which can realize an affine Schubert variety (in LG/B) as a compactification of the cotangent

bundle T ∗G/B.

3.4.1 The Maps ψp

Let p be a polynomial in one variable, with coefficients in K, and constant term 1,

p(y) = 1 +∑i≥1

pi(t)yi

Lemma 3.4.2. There exits a G-equivariant map, ψp : T ∗G/B → LG/B, given by,

ψp(g, y) = gp(y)(mod B), g ∈ G, y ∈ uB.

Proof. For y ∈ uB, we have p(y) ∈ LG. Consider g ∈ G, b ∈ B, and y ∈ uB. We have,

ψp((gb, b−1yb)) = gb

(Id+ p1(t)b−1yb+ p2(t)b−1y2b+ · · ·

)(modB)

= g(Id+ p1(t)yb+ p2(t)y2b+ · · ·

)(modB)

= g(Id+ p1(t)y + p2(t)y2 + · · ·

)(modB)

= ψp(g, y).

Further, for any h ∈ G, we have,

hψp(g, y) = hgp(y)(mod B) = ψp(gh, y).

It follows that ψ is G-equivariant.

Theorem 3.B. Let p be a polynomial as in Section 3.4.1. Suppose that the associated map,

ψp : T ∗G/B → LG/B, is injective. Then there exist g ∈ G, w ∈ W , and y ∈ uB, such that

ψp(g, y) ∈ BwB and l(w) > 6.

82

Proof. Following Section 3.4.1, we write

p(y) = 1 +∑i≥1

pi(t)yi.

We first claim that p1(t) /∈ O. Assume the contrary. For

z =

0 1 0

0 0 0

0 0 0

we see that z2 = 0, and so p(z) = 1 + p1(t)z ∈ B. In particular, ψp(z) = ψp(0), contradicting

the injectivity of ψp.

We now write p(y) = 1− t−aqy − t−bry2, where,

• q, r ∈ O.

• q(0) 6= 0.

• a ≥ 1.

• Either r = 0 or r(0) 6= 0.

Let

y =

0 1 0

0 0 1

0 0 0

, g =

0 0 −1

0 −1 0

−1 0 0

, g p(y) =

0 0 −1

0 −1 t−aq

−1 t−aq t−br

.

Our strategy is to find elements c, d ∈ B such that cgp(y)d ∈ N(K). We can then identify

the Bruhat cell containing gp(y), and so identify the minimal Schubert variety containing

ψp(g, y). The choice of c, d depends on the values of certain inequalities, which we divide

into 4 cases. We draw a decision tree showing the relationship between the inequalities and

the choice c, d.

83

Case 1

r = 0

Case 1

b ≤ a

Case 2

a < b < 2a Case 3q2 + r = 0

Case 4q2 + r 6= 0

2a = b

Case 4

2a< b

r 6= 0

We identify a rational function in t with its Laurent power series at 0. In particular, a

rational function f belongs to O if and only if f has no poles at 0, i.e. its denominator is

not divisible by t.

1. If r = 0 or b ≤ a, let

c =

rt2a−b + q2 qta t2a

0 − q

rt2a−b + q2− ta

rt2a−b + q2

0 0 −1

q

,

d =

1 0 0

qta

rt2a−b + q21 −rt

a−b

qt2a

rt2a−b + q20 1

.

We compute,

cgp(y)d =

−t2a 0 0

0 0 t−a

0 t−a 0

.

It follows that gp(y) ∈ Bτqs2B, where q = −2aα∨1 − aα∨2 . Following Proposition 3.2.3,

we have,

l(τq) = 3a+ 0 + 3a = 6a.

It follows from Equation (1.3.10) and Proposition 1.4.6 that l(τqs2) > l(τq) = 6a ≥ 6.

84

2. Suppose a < b < 2a. In particular, a ≥ 2, b ≥ 3. Let

c =

−rt2a−b + q2 qta t2a

0 − r

rt2a−b + q2

−qtb−a

rt2a−b + q2

0 01

r

d =

1 0 0

taq

q2 + t2a−br1 0

t2a

q2 + t2a−br−qt

b−a

r1

.

We compute,

cgp(y)d =

t2a 0 0

0 tb−2a 0

0 0 t−b

.

It follows that gp(y) ∈ BτqB, where q = −2aα∨1 − bα∨2 . Following Proposition 3.2.3, we

have

l(τq) = (4a− b) + (2b− 2a) + (2a+ b)

= 4a+ 2b ≥ 14.

3. If b = 2a and q2 + r = 0, let

c =

q ta 0

0 q ta

0 01

q2

, d =

1 0 0

0 1 0

−t2a

q2

ta

q1

.

We have,

cgp(y)d =

0 −ta 0

−ta 0 0

0 0 −t−2a

.

85

It follows that gp(y) ∈ Bτqs1B, where q = −aα∨1 − 2aα∨2 . Similar to the first case, we

see that l(τqs1) > 6.

4. Suppose either b > 2a, or b = 2a and r + q2 6= 0. In particular, r + q2tb−2a 6= 0 and

b ≥ 2. Let

c =

−r − q2tb−2a −qtb−a −tb

0 − r

r + q2tb−2a

qtb−a

r + q2tb−2a

0 01

r

d =

1 0 0

qtb−a

q2tb−2a + r1 0

tb

q2tb−2a + r−qt

b−a

r1

.

We compute,

cgp(y)d =

tb 0 0

0 1 0

0 0 t−b

.

It follows that gp(y) ∈ BτqB, where q = −bα∨1 − bα∨2 . Following Proposition 3.2.3, we

have,

l(τq) = b+ b+ 2b

= 4b ≥ 8.

86

Chapter 4

Conormal Varieties on a Cominuscule

Grassmannian: A Compactification

Let G be a simply connected, almost-simple algebraic group with Dynkin diagram D0, and

let θ0 be the highest root in the root system ∆0 of G. A simple root αd ∈ D0 is called

cominuscule if the coefficient of αd in θ0 is 1. For a list of cominuscule roots in various

Dynkin diagrams, see Fig. 1.1.11.

Let αd ∈ D0 be a cominuscule root, and let P be a parabolic subgroup of G corre-

sponding to the subset J = D0\{αd}. In this chapter, we study the conormal varieties of

Schubert subvarieties XJ (w) ⊂ G/P . We will denote the Schubert varieties XJ (w) and the

corresponding conormal varieties NJ (w) as simply X(w) and N(w).

Following [LRS16], we construct in Theorem 4.3.4 a compactification, φ : T ∗G/P ↪→ X(τq),

of the cotangent bundle T ∗G/P into an affine Schubert variety X(τq). Our main results are

the following:

Theorem 4.A. Let N(w) be the conormal variety of X(w) in G/P , see Section 1.6.5. The

closure of φ(N(w)) in LG/P is a Schubert subvariety if and only if the opposite Schubert

variety Xop(w) is smooth.

87

Theorem 4.B. Let w ∈ WJ0 be such that Xop(w) is smooth. Then N(w) is normal, Cohen-

Macaulay, and admits a resolution of singularities via a Bott-Samelson variety. Further, the

family {N(w) |Xop(w) is smooth} is compatibly Frobenius split in positive characteristic.

4.1 The Loop Group

We fix a maximal torus T in G, and a pair of opposite Borel subgroups B and B− corre-

sponding to T . Let LG, L+G, B, B−, and π be as in Section 1.3. Recall that the Dynkin

diagram D of LG is precisely the extended Dynkin diagram corresponding to D0. We will

will denote by α0 the extra root that is in D, but not in D0.

If P is the standard parabolic subgroup of G corresponding to some subset L ⊂ D0, then

the standard parabolic subgroup P of LG corresponding to L ⊂ D is precisely P = π−1(P ).

In particular, the standard parabolic subgroup corresponding to D0 ⊂ D is L+G.

4.2 An Involution of the Extended Dynkin Diagram

We see from Figure 1.3.4 that a simple root αd is cominuscule if and only if there exists an

automorphism ι of D such that ι(α0) = αd. In this section, we provide a canonical choice of

the involution ι, with a type independent definition.

Let Dd = D\{αd}. We write ∆d, ∆+d , ∆−d , θd, and Wd (resp. ∆0, ∆+

0 , ∆−0 , θ0, and W0)

for the set of roots, positive roots, negative roots, highest root, and Weyl group respectively

of the root system associated to Dd (resp. D0). We have

θd = δ − αd, θ0 = δ − α0,

where δ is the basic imaginary root, see Proposition 1.3.5.

88

Lemma 4.2.1. Let wJ be the longest element of WJ . Then, we have,

wJ (αd) = θ0, wJ (α0) = θd.

Proof. Observe first that wJ preserves the sets ∆0 and ∆J , and that the inversion set of wJ

is precisely ∆+J . In particular, we have,

wJ (∆+0 \∆+

J ) = ∆+0 \∆+

J .

Next, observe that for any α ∈ ∆+0 \∆+

J , the coefficient of αd in α is precisely 1. Therefor,

for any such α, we have the following chain of implications:

α− αd ∈ Z≥0J =⇒ wJ (α− αd) ≤ 0 =⇒ wJ (α) ≤ wJ (αd).

We see that wJ (αd) is maximal in ∆+0 \∆+

J . Since θ0 is the unique maximal root in ∆+0 , see

Proposition 1.2.13, we deduce that wJ (αd) = θ0.

The formula wJ (α0) = θd follows similarly, by showing that wJ (α0) is maximal in

∆+d \∆

+J .

Definition 4.2.2 (The Involution ι). Let V be the real vector space with basis D. We define

ι to be the linear involution of V given by

ι(α) =

αd for α = α0,

α0 for α = αd,

−wJ (α) for α ∈ J .

Lemma 4.2.3. The bilinear form ( | ) : V ×V → R, see Section 1.1.16, is invariant under

ι ∈ GL(V ).

Proof. It is sufficient to verify the claimed invariance on the basis J ∪ {α0, αd}. Recall that

( | ) is W -invariant, and further, that (δ |x) = 0 for all x ∈ V , see Corollary 1.3.6. Given

89

α, β ∈ J , we have,

(ι(α) | ι(β)) = (−wJ (α) | −wJ (β)) = (α | β) ,

(ι(α0) | ι(β)) = (αd | −wJ (β)) = (wJ (θ0) | −wJ (β))

= (−θ0 | β) = (α0 − δ | β) = (α0 | β) ,

(ι(αd) | ι(β)) = (α0 | −wJ (β)) = (wJ (θd) | −wJ (β))

= (−θd | β) = (αd − δ | β) = (αd | β) ,

(ι(α0) | ι(αd)) = (αd |α0) = (α0 |αd) .

Proposition 4.2.4. The map ι induces an involution of the Dynkin diagram D.

Proof. Recall form Section 1.1.14 that −wJ induces an involution on the Dynkin diagram

J . We see from Definition 4.2.2 that ι preserves the set of simple roots D. Further, we

see from Equation (1.1.17) that the GCM (α∨i (αj))ij is preserved under ι. It follows that ι

preserves the Dynkin diagram structure on D.

Corollary 4.2.5. We have the equality ι(δ) = δ.

Proof. Recall from Corollary 1.3.6 that

{γ ∈ Λ | (γ | ) = 0} = Zδ.

It follows from Lemma 4.2.3 that (ι(δ) | ) = 0, hence ι(δ) = kδ for some k ∈ Z . Further, it

follows from Proposition 4.2.4 that k > 0, and k2 = 1. Consequently, we deduce that k = 1,

i.e., ι(δ) = δ.

4.2.6 Induced Action on the Weyl Group

We define an involution ι of W , given by

ιsα = sι(α), ∀α ∈ D.

90

It is clear that ι preserves the length and the Bruhat order on W , and further,

ιWJ = WJ ,ιWJ = WJ , ιwJ = wJ ,

ιW0 = Wd,ιW 0 = W d, ιwd = w0.

Next, we show that the action of ι on W is the same as conjugation by ι, i.e.,

ιw = ι ◦ w ◦ ι, (4.2.7)

where both w and ι are viewed as elements of GL(V ). Using Equation (1.1.17), Defini-

tion 4.2.2, and Lemma 4.2.3, we have, for any β ∈ D,

(ι ◦ sα ◦ ι)(β) = ι(sα(ι(β))) = ι

(ι(β)− 2

(α | ι(β))

(α |α)α

)= β − 2

(ι(α) | β)

(ι(α) | ι(α))ι(α)

= sι(α)(β).

Remark 4.2.8. Note also that since Schubert varieties depend only on the underlying Dynkin

diagrams, there exists an isomorphism X(ιw) ∼= X(w) for any w ∈ W .

4.2.9 The Coroot q

Let w0, wd be the maximal elements in W0, Wd respectively, and set

q = w0($∨d )−$∨d ∈ Λ∨0 , (4.2.10)

where $∨d is the fundamental co-weight dual to αd.

Proposition 4.2.11. For all α ∈ D0, we have,

w0wJwdwJ (α) = α + 〈q, α〉 δ. (4.2.12)

Proof. Recall from Section 1.1.14 that−w0 induces an involution onD0. Set β = −w0(αd), so

that−w0($∨d ) = $∨β , the fundamental co-weight dual to β. It follows from Equations (1.3.10)

91

and (4.2.10) that

〈q, α〉 = 〈$∨d − w0($∨d ), α〉 =⟨$∨d +$∨β , α

⟩. (4.2.13)

Next, it follows from β = −w0(αd) that

ι(β) = −ι(w0(αd)) = −ιw0(ι(αd)) using Equation (4.2.7),

=⇒ ι(β) = −wd(α0) using Section 4.2.6. (4.2.14)

Further, we have,

wdwJ (αd) = wd(θ0) = wd(δ − α0) using Lemma 4.2.1

= δ − wd(α0) = δ + ι(β) using Equation (4.2.14), (4.2.15)

w0wJ (α0) = ι(wdwJ )(ι(αd)) using Section 4.2.6

= ι(δ + ι(β)) = δ + β using Corollary 4.2.5. (4.2.16)

We are now ready to prove that Equation (4.2.12) holds for α ∈ {αd, β}.

Case 1 Suppose β = αd. Then ι(β) = α0, $∨d = $∨β , and q = 2$∨d . We have,

w0wJwdwJ (αd) = w0wJ (δ + ι(β)) using Equation (4.2.15)

= w0wJ (δ + α0) using β = αd

= w0(δ + θd) using Lemma 4.2.1

= w0(2δ − αd) = 2δ + β using β = −w0(αd)

= αd + 〈2$∨d , αd〉 δ using Equation (4.2.13).

Case 2 Suppose β 6= αd. Then β, ι(β) ∈ J , and $∨β (αd) = $∨d (β) = 0. It follows from

Definition 4.2.2 that ι(β) = −wJ (β), hence wJ (ι(β)) = −β. In particular, we have

92

〈q, αd〉 = 〈q, β〉 = 1. We compute:

w0wJwdwJ (αd) = w0wJ (δ + ι(β)) using Equation (4.2.15)

= w0(δ − β) = δ − w0(β)

= δ + αd = αd + 〈q, αd〉 δ using Equation (4.2.13)

w0wJwdwJ (β) = w0wJwd(−ι(β))

= w0wJwdwd(α0) using Equation (4.2.14)

= w0wJ (α0) = δ + β using Equation (4.2.16)

= β + 〈q, β〉 δ using Equation (4.2.13).

Finally, we prove that Equation (4.2.12) holds for any α ∈ D0\ {αd, β} = J \ {β}. Since

α ∈ D0, we have −w0(α) ∈ D0. Observe further that since α 6= β, we have −w0(α) 6= αd,

and so −w0(α) ∈ J . Following Definition 4.2.2, we have,

ι(−w0(α)) = wJw0(α) =⇒ wJ ιw0(α) = −w0(α). (4.2.17)

We see from Equation (4.2.13) that 〈q, α〉 = 0. We compute:

w0wJwdwJ (α) = −w0wJιw0ι(α) using α 6∈ J , and Definition 4.2.2

= −w0wJ ιw0(α) = w20(α) using Equation (4.2.7)

= α = α + 〈q, α〉 δ using Equation (4.2.17).

Corollary 4.2.18. We have q ∈ Λ∨0 , and further,

τq = w0wJwdwJ = wJ0 wJd ,

where τq is an element of W in the sense of Section 1.3.9.

93

4.3 An Embedding of the Cotangent Bundle

Lemma 4.3.1. Recall from Section 1.3.11 the notion of a nilpotent set of roots. Set

Ψ = ∆−d \∆J .

All subsets of Ψ are nilpotent. Further, for every γ ∈ D0 ∪±J , the set Ψ∪ {γ} is nilpotent.

Proof. First observe that

Ψ = ∆−d \∆J = {α ∈ ∆ | −θd ≤ α ≤ −α0} .

Consider α, β ∈ Ψ. We have α, β ≤ −α0, hence α + β ≤ −2α0. Further, since α0 is

cominuscule in Dd, and Supp(α + β) ⊂ Dd, the element α + β is not a root. It follows that

every subset of Ψ is closed.

Next, we prove that Ψ ∪ {γ} is closed.

1. Suppose γ = αd. Consider α ∈ Ψ. The coefficient of α0 in α+ γ is −1, and the coefficient

of αd is 1. Hence, α+ γ is not a root, since all coefficients of a root are either positive or

negative.

2. Suppose γ ∈ J . Suppose further that α+γ ∈ ∆ for some α ∈ Ψ. Since Supp(α+γ) ⊂ Dd,

we see that α + γ ∈ ∆d. Further, since the coefficient of α0 in α + γ is −1, we have

α + γ 6∈ ∆J . Consequently, we have α + γ ∈ ∆−d \∆J = Ψ.

3. Suppose γ ∈ −J . Suppose further that α + γ ∈ ∆ for some α ∈ Ψ. Then α + γ ≤ −α0,

and α + γ ∈ ∆d. It follows that α + γ ∈ Ψ.

Finally, consider u+, u− ∈ W given by

u+ =

wd if γ = αd,

wdsγ if γ ∈ J ,

wd if γ ∈ −J ,

u− =

sγ if γ ∈ D0,

1 if γ ∈ −J .

The nilpotency of Ψ ∪ {γ} now follows from the observation that u± (Ψ ∪ {γ}) ⊂ ∆±.

94

Recall from Section 1.3.11 the Lie sub-algebra uΨ =⊕α∈Ψ

uα, and from Proposition 1.3.12

the T -equivariant isomorphism η : uΨ → UΨ for some ordering on Ψ.

Proposition 4.3.2. There exists a P -equivariant isomorphism φ : uP → X−(wJd ) given by

φ(X) = η(t−1X) modP ,

where X−(wJd ) is the opposite cell in X(wJd ), see Equation (1.5.7).

Proof. The map X 7→ t−1X is G-equivariant (hence also P -equivariant), and takes the root

space uα to uα−δ. Observe that uP = u∆+0 \∆J

, and further,

∆+0 \∆J − δ =

{α− δ

∣∣α ∈ ∆0\∆J}

= {α− δ |α ∈ ∆, αd ≤ α ≤ θ0}

= {α ∈ ∆ | −θd ≤ α ≤ −α0} = ∆−d \∆J .

(4.3.3)

Hence, we have a map t−1 : uP → uΨ, where Ψ = ∆−d \∆J . It follows from Section 1.5.12

that η ◦ t−1 is an isomorphism from uP to X−(wJd ).

It remains to show that η is P -equivariant. For γ ∈ D0∪−J , it follows from Lemma 4.3.1

that the group scheme UΨ∪{γ} is well-defined. The action of Uγ on UΨ (resp. uΨ) being the

restriction of the adjoint action of UΨ∪{γ} on itself (resp. uΨ∪{γ}), the map η is Uγ-equivariant.

Now, since P = 〈T, Uγ | γ ∈ D0 ∪ −J 〉, and since η is T -equivariant, it follows that η is

P -equivariant.

Let e denote the image of the identity element of G in the quotient LG/P, and set

YJ (τq) = L−Ge⋂

X(w),

YD0(τq) = L−Ge⋂

XD0(w).

Theorem 4.3.4. The map φ : uP → X−(wJd ) extends to a G-equivariant isomorphism

φ : T ∗G/P → YJ (τq), given by,

φ(g, x) = g φ(x) for g ∈ G, x ∈ uP .

95

Let C ⊂ N be the G-orbit such that µG/P (T ∗G/P) = C, and let pr denote the quotient map

LG/P → LG/L+G. We have a commutative diagram,

T ∗G/P YJ (τq)

C YD0(τq),

φ

µG/P pr

ψ

where µG/P is the Springer map, see Section 1.6.4. The maps φ and ψ are isomorphisms.

Proof. The proof of [LRS16, Theorem 1.3] applies to show that φ is well-defined and G-

equivariant. We see from Equation (4.2.10) that

φ(T ∗G/P) ⊂ X(wJ0 wJd ) = X(τq).

Further, since φ is G-equivariant and φ(1, uP ) = X−(wJd ), we have,

φ(T ∗G/P) = GX−J (wJd ) = GB−X−J (wJd ) = YJ (τq).

Next, we observe that for X ∈ uP , we have,

pr(φ(1, X)) = φ(X) (modL+G) = ψ(µ(X)).

Consequently, the diagram commutes, and further, ψ(C) = Y0(τq). Finally, the induced map

ψ : C → Y0(τq) is an isomorphism because it admits an inverse map, see [AH13].

Remark 4.3.5. The G-orbits Nλ ⊂ N are naturally indexed by certain co-weights, see, for

example, [Car85]. The orbit C ⊂ N is precisely the orbit corresponding to the co-weight q.

4.4 The Image of a Conormal Variety Under φ

We fix an element w ∈ WJ , and set v = ι(w0wwJ ). We will use the following notation:

W d0 = W0 ∩W d, and W 0

d = Wd ∩W 0.

96

Lemma 4.4.1. We have the following equalities:

W0 ∩WJ = W0 ∩W d, Wd ∩WJ = Wd ∩W 0.

Proof. It follows from Section 1.4.8 that W0 ⊂ W {α0}, hence

W0 ∩WJ ⊂ W0 ∩W {α0} ∩WJ = W d0 .

Conversely, since W d ⊂ WJ , we have W d0 = W0 ∩W d ⊂ W0 ∩WJ . Consequently,

W0 ∩WJ = W d0 .

Applying ι to this equality, we have Wd ∩WJ = W 0d .

Lemma 4.4.2. The element v belongs to W d0 , and further, we have,

l(wv) = l(w) + l(v) = dim G/P .

Proof. Since w0, w, wJ ∈ W0, we have w0wwJ ∈ W0, and hence v = ι(w0wwJ ) ∈ Wd, see

Section 4.2.6. Next, observe that,

v(∆+J ) = ιw0wwJ (ι(∆+

J )) using Equation (4.2.7),

=⇒ v(∆+J ) = ιw0wwJ (∆+

J ) using Section 4.2.6,

=⇒ v(∆+J ) = ιw0w(∆−J ) using Section 1.1.14. (4.4.3)

Since Supp(w) ⊂ D0, we have w(∆−J ) ⊂ ∆0, see Section 1.4.8. Further, since w ∈ WJ , it

follows from Proposition 1.4.6 that w(∆−J ) ⊂ ∆−0 . Equation (4.4.3) now yields

v(∆+J ) ⊂ ιw0(∆−0 ) = ι(∆+

0 ) ⊂ ∆+.

We deduce from Proposition 1.4.6 that v ∈ WJ , and from Section 4.2.6 that ιv ∈ WJ . In

particular, we have,

l(w0w) = l(w0wwJ ) + l(wJ )

=⇒ dimG− l(w) = l(v) + dimP

=⇒ dim G/P = l(w) + l(v) = l(wv),

97

where the last equality follows from the observations w ∈ W0 and v ∈ W 0.

Lemma 4.4.4. Let u ∈ W 0d . Then Supp(u) is a connected sub-graph of Dd.

Proof. Suppose Supp(u) is not connected. Let L1 be the connected component of Supp(u)

containing αd, and let L2 = Supp(u)\L1. Now, since L1 and L2 are disconnected, we have,

sαsβ = sβsα ∀ sα ∈ L1, sβ ∈ L2.

Let s1 . . . sl be a reduced word for u, and let k be the largest index such that sk ∈ L2. Then

sksm = smsk for all m > k. It follows that

usk = (s1 . . . sl)sk = (s1 . . . sk−1sk+1 . . . slsk)sk

= s1 . . . sk−1sk+1 . . . sl.

Now, since Supp(u) ⊂ Dd, we have α0, αd 6∈ L2. In particular, sk ∈ WJ . Hence usk =

u (modWJ ) and usk < u, contradicting the assumption u ∈ W 0d ⊂ WJ .

Proposition 4.4.5. For u ∈ W 0d , the following are equivalent:

1. X(u) is smooth.

2. X(u) is PL-homogeneous, i.e., X(u) = PL/(PL ∩ P) for some connected sub-graph

L ⊂ Dd.

3. l(u−1 ? uwJ ) = l(uwJ ), where ? is the Demazure product, see Section 1.5.9.

4. (uwJ )−1(α) < 0 for all α ∈ Supp(u).

5. {α ∈ ∆+ |u(α) < 0} = ∆+Supp(u)\∆

+J .

6. u = wLwL∩J , where L = Supp(u), and wL, wL∩J denote the maximal elements in WL,

WL∩J respectively.

98

Proof. The claim (1) ⇐⇒ (2) is [BM10, Theorem 1.1]. Suppose (2) holds, i.e., X(u) =

PL/PL∩J for some L. We have the following Cartesian square:

XB(uwJ ) LG/B

PL/PL∩J X(u) LG/P.

Since X(u) is PL-stable, the same is true of its pull-back XB(uwJ ). Further, since u ∈ WL,

any lift of u−1 to N(K) (see Section 1.3.9) is in PL. Consequently, XB(uwJ ) is u−1 stable,

and so u−1 ? uwJ = uwJ . Hence we obtain (2) =⇒ (3).

It is clear from Section 1.5.9 that (3) is equivalent to u−1 ? uwJ = uwJ , which holds if

and only if sα ? uwJ = uwJ for all α ∈ L. This is equivalent to (4) from Proposition 1.4.6.

Hence we obtain (3) ⇐⇒ (4).

Suppose (4) holds. Then (uwJ )−1(α) < 0 for all α ∈ ∆+Supp(u). It follows from Sec-

tion 1.4.8 and Equation (1.4.9) that{α ∈ ∆+

∣∣u(α) < 0}⊂ ∆+

Supp(u)\∆+J .

Consider α ∈ ∆+Supp(u)\∆

+J satisfying u(α) > 0. We have u(α) ∈ ∆+

Supp(u), since u preserves

∆Supp(u). Item (4), applied to u(α), yields

wJu−1(u(α)) = wJ (α) < 0.

It follows that α ∈ ∆+J , contradicting the assumption α ∈ ∆+

Supp(u)\∆+J . Hence, we obtain

the implication (4) =⇒ (5).

Suppose (5) holds. We verify that{α ∈ ∆+

∣∣wLwL∩J (α) < 0}

= ∆+L\∆

+J .

Now since u ∈ W is uniquely determined by the set {α ∈ ∆+ |u(α) < 0}, see Proposi-

tion 1.4.2, we deduce that u = wLwL∩J . Hence, we obtain the implication (5) =⇒ (6).

Finally, suppose (6) holds. Since wL∩J ⊂ WJ , we have u = wL (modWJ ). It follows

that X(u) = X(wL) = PL/(PL∩J ). Hence, we obtain the implication (6) =⇒ (2).

99

Lemma 4.4.6. For α ∈ ∆+0 \∆J , we have v(α− δ) = −ιw0w(α).

Proof. Since α ∈ ∆+0 \∆J , we have α ≥ αd. Set γ = α − αd. Since αd is cominuscule,

Supp(γ) ⊂ J . In particular, ι(γ) = −wJ (γ). We compute,

ι(α) = ι(αd) + ι(γ) = α0 − wJ (γ)

=⇒ ι(α− δ) = −θ0 − wJ (γ). (4.4.7)

Recall the formula v = ι(w0wwJ ). We now compute v(α− δ):

v(α− δ) = ι(w0wwJ )(α− δ) = ιw0wwJ (ι(α− δ)) using Section 4.2.6,

= −ιw0wwJ (θ0 + wJ (γ)) using Equation (4.4.7),

= −ιw0w(αd)− ιw0w(γ) using Lemma 4.2.1,

= −ιw0w(αd + γ) = −ιw0w(α).

Lemma 4.4.8. The map α 7→ α− δ induces a bijection,

{α ∈ ∆+

0

∣∣α ≥ αd, w(α) > 0} ∼−→

{α ∈ ∆−d

∣∣ v(α) > 0}.

Proof. Observe that

−ιw0(∆±0 ) = ι(−w0(∆±0 )) = ι(∆±0 ) using Section 1.1.14,

= ∆±d using Section 4.2.6.

Now, since v(α− δ) = −ιw0w(α), see Lemma 4.4.6, the claim w(α) > 0 is equivalent to the

claim v(α− δ) > 0, for all α ∈ ∆+0 \∆J . The result now follows from Equation (4.3.3), which

states that α 7→ α− δ induces a bijection from ∆+0 \∆J to ∆−d \∆J .

Proposition 4.4.9. Recall the map φ from Proposition 4.3.2. Let

R ={α ∈ ∆+

0

∣∣α ≥ αd, w(α) > 0}.

Then φ(uR) is dense in v−1X(v).

100

Proof. Let Φ = {α− δ |α ∈ R}. It follows from Lemma 4.4.8 that

Φ ={α ∈ ∆−d

∣∣ v(α) > 0}

= v−1(∆+) ∩∆−.

In particular, #Φ = l(v), see Proposition 1.4.2, and so,

dim uR = dimUΦ = #Φ = l(v) = dimX(v) = dim v−1X(v).

Following the proof of Proposition 4.3.2, we deduce that η(t−1uR) = UΦ. Consequently,

we have

φ(uR) = η(t−1uR) (modP) ⊂ v−1Bv (modP) ⊂ v−1X(v).

The result now follows from the injectivity of φ and the irreducibility of v−1X(v).

Theorem 4.A. The closure of φ(N(w)) in LG/P is a Schubert variety if and only if the

opposite Schubert variety Xop(w) is smooth.

Proof. Let (bw, x) be a generic point of N◦(w), the conormal bundle of the Schubert cell

C(w) in G/P . Recall from Chapter 1 that N◦(w) = BwP ×P uR. We deduce,

φ(bw, x) = bw φ(x) ∈ Bwv−1BvP (modP).

In particular, the minimal Schubert variety containing φ(N(w)) is X(wv−1 ? v), and hence

the closure φ(N(w)) is a Schubert variety if and only if

dimX(wv−1 ? v) = dimN(w) = dim G/P . (4.4.10)

Consider the following Cartesian diagram,

XB(wv−1 ? v ? wJ ) LG/B

X(wv−1 ? v) LG/P.

Since XB(wv−1 ? v ? wJ ) is the pullback of X(wv−1 ? v) to LG/B, and the generic fibre of the

right projection is P/B, Equation (4.4.10) is equivalent to

dimXB(wv−1 ? v ? wJ ) = dim G/P + dim P/B = dim G/B. (4.4.11)

101

We see from Lemma 4.4.2 and Equation (1.5.11) that wv−1 = w ? v−1 and v ? wJ = vwJ .

Hence, we have:

wv−1 ? v ? wJ = w ? v−1 ? vwJ . (4.4.12)

Observe that since v, wJ ∈ Wd, we have v−1 ?vwJ ∈ Wd. Recall also from Lemma 4.4.2 that

w ∈ WJ ∩W0 ⊂ W d. It follows that

w ? v−1 ? vwJ = w(v−1 ? vwJ ) using Equation (1.5.11),

=⇒ l(wv−1 ? v ? wJ ) = l(w(v−1 ? vwJ )) using Equation (4.4.12),

= l(w) + l(v−1 ? vwJ ) using w ∈ W d, v−1 ? vwJ ∈ Wd. (4.4.13)

Now l(v−1?vwJ ) ≥ l(vwJ ), with equality holding if and only if v−1?vwJ = vwJ . Continuing

Equation (4.4.13), we have,

dimXB(wv−1 ? v ? wJ ) ≥ l(w) + l(vwJ )

= l(w) + l(v) + l(wJ ) using v ∈ WJ , wJ ∈ WJ ,

= l(wv) + l(wJ ) using w ∈ W d, v ∈ Wd,

= dim G/B.

Hence, Equation (4.4.11) holds if and only if v−1 ? vwJ = vwJ , which is equivalent to

X(v) being smooth, see Proposition 4.4.5. Further, X(v) is isomorphic to X(w0wwJ ), see

Remark 4.2.8, and X(w0wwJ ) is isomorphic to Xop(w), see Section 1.5.13. Hence, the

closure φ(N(w)) is a Schubert variety if and only if the opposite Schubert variety Xop(w) is

smooth.

Theorem 4.B. Let w ∈ WJ0 be such that Xop(w) is smooth. Then N(w) is normal, Cohen-

Macaulay, and admits a resolution of singularities via a Bott-Samelson variety. Further, the

family {N(w) |Xop(w) is smooth} is compatibly Frobenius split in positive characteristic.

102

Proof. These are standard results for Schubert varieties. One can find details in [Fal03,

Lit98, MR85, PR08, HR18].

Let $i denote the fundamental weight associated to the simple co-root αi. Consider a

dominant weight λ =∑ai$i, ai ∈ Z≥0, satisfying ai = 0 if and only if αi ∈ J . Let L(λ) be

the line bundle on G/P associated to the weight λ, and set

V (λ) = H0(G/P , L(λ)),

the dual Weyl module, see [Wey03]. Recall from [Lit98, Theorem 2], the monomial basis of

V (λ), whose elements uπ are indexed by LS paths π of shape λ. For an LS path π, we shall

denote its initial point by i(π), as in [Lit98].

Theorem 4.4.14. Let w ∈ WJ0 be such that Xop(w) is smooth. The ideal sheaf of the

conormal variety N(w) in T ∗G/P is φ−1I, where I is the ideal generated by the monomials,

{uπ | i(π) ≤ τq, i(π) 6≤ wv} .

Proof. The ideal sheaf of X(wv) in X(τq) is generated by the monomials

{uπ | i(π) ≤ τq, i(π) 6≤ wv} ,

see [Lit98, Theorem 6]. Since N(w) is closed in T ∗G/P , we have,

φ(N(w)) = φ(N(w)) ∩ φ(T ∗G/P)

= X(wv) ∩ L−G · e ∩X(τq) = YJ (wv).

It follows that the ideal sheaf of N(w) in T ∗G/P is the pull-back (via φ) of the restriction of

I to YJ (τq), i.e., the ideal sheaf is φ−1I.

103

4.5 The Conormal Fibre at Identity

Proposition 4.5.1. Let w ∈ WJ0 be such that Xop(w) is smooth, and let Ne(w) denote the

fibre at identity of the conormal variety N(w). Then

φ(Ne(w)) =⋃u∈S

X−(u),

where S ={u ∈ W 0

d

∣∣u ≤ (wv)D0}

, and (wv)D0 is the minimal representative of wv with

respect to D0.

Proof. Recall from Proposition 4.3.2 that φ(T ∗eG/P) = X−(wJd ). It follows that

φ(Ne(w)) = X−(wJd ) ∩X(wv) =⋃

X−(u),

where the union runs over{u ∈ WJ

∣∣u ≤ wJd , u ≤ wv}

.

Observe that since wJd is maximal in WJd , we have,

{u ∈ WJ ∣∣u ≤ wJd

}= Wd ∩WJ = W 0

d ,

where the last equality is from Lemma 4.4.1. Consequently, we have,

{u ∈ WJ ∣∣u ≤ wJd , u ≤ wv

}={u ∈ W 0

d

∣∣u ≤ wJd , u ≤ wv}.

Finally, consider u ∈ W 0. If u ≤ wv, then u ≤ (wv)D0 . It follows that

{u ∈ WJ ∣∣u ≤ wJd , u ≤ wv

}={u ∈ W 0

d

∣∣u ≤ wJd , u ≤ wv}

={u ∈ W 0

d

∣∣u ≤ (wv)D0}

= S.

104

Chapter 5

Projective Duality for Determinantal

Varieties

Projective duality is a classical notion closely related to conormal varieties. We give a

definition in Section 5.1, and explain one way in which projective duality relates to conormal

varieties.

Let V the set of d×m matrices (resp. symmetric n× n matrices, skew-symmetric n× n

matrices), and let

Σr = {x ∈ V | rk(x) ≤ r} , Σr = {x ∈ V | co-rk(x) ≤ r} .

In this chapter, we use the results of Chapter 4 to recover, in Theorems 5.A to 5.C the

following projective duality:

P(Σr)∨ = P(Σr).

In other words, the projective dual of P(Σr) in P(V ) is P(Σr).

The idea is the following: The space V is naturally identified as the opposite cell of a

cominuscule Grassmannian G/P of type A (resp. C, D), and Σr is naturally identified as the

opposite cell of some Schubert subvariety XJ (w) ⊂ G/P .

105

Given this w, we verify that the opposite Schubert variety XopJ (w) is smooth, and apply

Proposition 4.5.1. We show that the union of the various Schubert varieties in Proposi-

tion 4.5.1 is equal to a single Schubert variety XJ (u), and we verify that the opposite cell

X−J (u) is isomorphic Σr. The claimed duality then follows from Proposition 5.1.3.

5.1 Projective Duality and Conormal Varieties

Let X ⊂ Pn be a closed irreducible algebraic subvariety. A hyperplane H ⊂ Pn is said to be

tangent to X if there exists a smooth point x ∈ X such that x ∈ H, and the tangent space

to H at x contains the tangent space to X at x. Denote by X∨ ⊂ Pn the closure of the set

of all hyperplanes tangent to X. We call X∨ the projective dual of X.

Lemma 5.1.1 (cf. [GKZ08]). For every projective variety X ⊂ Pn, we have X∨∨ = X.

Lemma 5.1.2 (cf. [GKZ08]). If X is irreducible, then X∨ is irreducible.

Proposition 5.1.3. Let I be a homogeneous ideal in the polynomial ring R = k[x1, · · · , xn].

Let V = Spec(R), P(V ) = Proj(R), X = Spec(R/I), P(X) = Proj(R/I), and let N0(X)

denote the fibre at the point 0 ∈ V of the conormal variety T ∗VX. The projective dual of

P(X) in P(V ) is then precisely the projectivization of N0(X), i.e.,

P(N0(X)) = P(X)∨,

Proof. Let π2 : T ∗V → V ×V ∗ be the projection to the second coordinate under the canonical

identification T ∗V = V × V ∗. We have,

T ∗VXsm = {(x, α) ∈ V × V ∗ |α(TxXsm) = 0} ,

and P(V )∨ = P(π2(T ∗VX)). It remains to show that

N0(X) = π2(T ∗VX) = {α ∈ V ∗ | ∃x ∈ Xsm, such that α(TxX) = 0} .

106

Now, since the ideal I is homogeneous, the variety X is k∗-stable. In particular, for

k ∈ k∗, we have,

(x, α) ∈ T ∗VX ⇐⇒ (kx, α) ∈ T ∗VX.

Finally, since T ∗VX is a closed subvariety in T ∗V , we have (0, α) ∈ T ∗VX if and only if

(x, α) ∈ T ∗VX for some x ∈ X, i.e., N0(X) = π2(T ∗VX).

5.2 The Determinantal Variety

Let Matd,m be the variety of matrices of size d×m. The rank r determinantal variety Σd,mr

is the subvariety of Matd,m given by

Σd,mr = {x ∈ Matd,m | rk(x) ≤ r} .

Let N = d + m, and let D0 = AN−1 denote the Dynkin diagram of G = SLN . We label

the simple roots of G as in Figure 1.3.4. Let P be a parabolic subgroup of G corresponding

to omitting the cominuscule root αd, i.e., corresponding to the subset

J = {α1, · · · , αN−1} \{αd}.

For a ≤ b, let (a, b] denote the contiguous integer sequence a+1, · · · , b. Following [LR08],

we identify Matd,m with the opposite cell in G/P . Under this identification, the zero matrix

in Matd,m corresponds to e ∈ G/PJ , and Σd,mr = X−J (ur), where ur ∈ WJ can be described

as a concatenation of contiguous sequences,

ur = (r, d], (N − r,N ], (0, r], (d,N − r]. (5.2.1)

107

5.2.2 The Involution ι for AN−1

The Dynkin diagram J = D0\{αd} is isomorphic to Ad−1 × Am−1. Following Chapter 2

and Definition 4.2.2, we see that the involution ι on D is given by

ι(αi) =

αd−i for 0 ≤ i ≤ d,

αN+d−i for d+ 1 ≤ i ≤ N − 1.

Consequently, the involution ι on W is given by

ιsi =

sd−i for 0 ≤ i ≤ d,

sN+d−i for d+ 1 ≤ i ≤ N − 1.

Recall from Equation (2.2.5), the identification,

W =

{σ ∈ St∞

∣∣∣∣∣n∑i=1

(σ(i)− i) = 0

},

where t(i) = i+N . Consider the involution σ : Z→ Z, given by

σ(i) = d+ 1− i,

and observe that tσ = σt−1. Consequently, for w ∈ W , we have

σwσ−1t = σwt−1σ−1 = σt−1wσ−1 = tσwσ−1.

It follows that conjugation by σ induces an involution on St∞. A simple computation yields,

σsασ−1 = ιsα, ∀α ∈ D. (5.2.3)

Hence, we see that the map ι : W → W is precisely the map w 7→ σwσ−1.

Lemma 5.2.4. Suppose d ≤ m, let ur be as in Equation (5.2.1), and let vr = ι(w0urwJ ).

We have urvr = ιud−rιvd−r, and (urvr)

D0 = ιud−r.

108

Proof. We use Equation (5.2.3) to compute,

ιur = (d−N, d+ r −N ], (0, d− r], (d+ r,N ], (d− r +N, d+N ],

Recall from Chapter 2 that the longest elements w0 ∈ W0, and wJ ∈ WJ , are given by

w0 = [N,N − 1, · · · , 1],

wJ = [d, · · · , 1, N, · · · , d+ 1].

Using Equation (5.2.1), we have,

ιvr = w0urwJ = (0, r], (N − d,N − r], (r,N − d], (N − r,N ],

ι(urvr) = (d−N, d+ r −N ], (N − d+ r,N ], (0, d− r], (d+ r,N − d+ r], (d− r +N, d+N ]

= (d−N, d+ r −N ], (N − d+ r,N ], (0, d− r], (d+ r,N − d+ r], (d− r +N, d+N ].

Next, we use the substitution d 7→ d− r, to compute

ud−r = (d− r, d], (N − d+ r,N ], (0, d− r], (d,N − d+ r],

ιvd−r = (0, d− r], (N − d,N − d+ r], (d− r,N − d], (N − d+ r,N ]

=⇒ vd−r = (2d−N − r, 2d−N ], (−r, 2d− r], (2d−N, r].

Finally, we compute,

ud−rvd−r = (d−N, d+ r −N ], (N − d+ r,N ], (0, d− r], (d+ r,N − d+ r], (d− r +N, d+N ]

= ι(urvr).

Applying ι to this equality yields urvr = ιud−rιvd−r. Now, since ιvd−r ∈ W0, we have

(urvr)D0 = ιud−r(modW0).

Further, following Section 2.2.8, we have ud−r ∈ W d0 , and hence, ιud−r ∈ W 0

d ⊂ WD0 .

Corollary 5.2.5. The set{u ∈ W 0

d

∣∣u ≤ (wrvr)D0}

contains a unique maximal element,

which is ιwd−r.

109

Theorem 5.A. For d ≤ m, the projective dual of P(Σd,mr ) is P(Σd,m

d−r).

Proof. For L a sub-diagram of D0, we write wL for the longest element in W supported

on L, and wJL for its minimal representative with respect to J . Let ur be as defined in

Equation (5.2.1), and set vr = ι(w0urwJ ). Then,

w0urwJ = (0, r], (N − d,N − r] = wJL ,

where L = {αr+1, · · · , αN−r−1}. In particular, XJ (w0urwJ ) is smooth, see Proposition 4.4.5.

It follows from Proposition 4.5.1 and Corollary 5.2.5 that

φ(Ne(w)) = X−J (ιud−r) ∼= X−J (wd−r) ∼= Σd,md−r.

Consequently, we have P(Σd,mr )∨ = P(Σd,m

d−r), see Chapter 5.

5.3 The Symmetric Determinantal Variety

Let Matsymn be the variety of symmetric matrices of size n × n. The rank r symmetric

determinantal variety Σsymn,r is the subvariety of Matsymn given by

Σsymn,r = {x ∈ Matn | rk(x) ≤ r} .

Let D0 = Cn denote the Dynkin diagram of G = Sp2n. We label the simple roots of

G as in Figure 1.3.4. Let P be a parabolic subgroup of G corresponding to omitting the

cominuscule root αn, i.e., corresponding to the subset J = {α1, · · · , αn−1}.

Following [LR08], we identify Matsymn with the opposite cell in G/P . Under this identifi-

cation, the zero matrix Matsymn corresponds to e ∈ G/PJ , and Σsymn,r = X−J (uJr ), where

ur = (r, n], (n− r, n], (0, r], (n, n− r], (5.3.1)

in the sense of Equation (2.1.7).

110

5.3.2 The Involution ι for Cn

The Dynkin diagram J = {α1, · · · , αn−1} is isomorphic to An−1. Following Chapter 2

and Definition 4.2.2, we see that the involution ι on D is given by,

ι(αi) = αn−i, for 1 ≤ i ≤ n,

and that the involution induced on W is given by,

ιsi = sn−i, for 1 ≤ i ≤ n.

Recall from Section 2.3.3, the identification

W = St,µ∞ = {w ∈ S∞ |wt = tw, wµ = µw} ,

where t(i) = i + 2n, and µ(i) = 2n + 1 − i. Let σ : Z → Z be the involution given by

σ(i) = n+ 1− i. Observe that

tσ = σt−1, σµ = µσ,

and hence, for any w ∈ St,µ∞ , we have,

σwσ−1µ = σwµσ−1t = σµwtσ−1 = µσtwtσ−1 = µσwσ−1.

It follows that conjugation by σ induces an involution on St,µ∞ . A simple computation yields,

σsασ−1 = ιsα, ∀α ∈ D.

We see that the map ι : W → W is precisely the map w 7→ σwσ−1.

Lemma 5.3.3. Let ur be as in Equation (5.3.1), and let vr = ι(w0urwJ ). We have urvr =

ιun−rιvn−r, and (urvr)

D0 = ιun−r.

Proof. The equality urvr = ιun−rιvn−r follows from Lemma 5.2.4 applied with d = n, and

N = 2n. Further, since ιvd−r ∈ W0, we have,

(urvr)D0 = ιud−r(modW0).

Finally, we see from Section 2.3.5 that ud−r ∈ W d0 , and hence, ιud−r ∈ W 0

d ⊂ WD0 .

111

Corollary 5.3.4. The set{u ∈ W 0

J∪{α0}

∣∣∣u ≤ (wrvr)D0

}contains a unique maximal ele-

ment, which is ιwn−r.

Theorem 5.B. The projective dual of P(Σsymn,r ) is P(Σsym

n,n−r).

Proof. For L a sub-diagram of D0, we write wL for the longest element in W supported

on L, and wJL for its minimal representative with respect to J . Let ur be as defined in

Equation (5.3.1), and set vr = ι(w0urwJ ). Then,

w0urwJ = (0, r](n, · · · , 2n− r](r, n](2n− r, 2n] = wJL ,

where L = {αr+1, · · · , αn}. In particular, XJ (w0urwJ ) is smooth, see Proposition 4.4.5. It

follows from Proposition 4.5.1 and Corollary 5.3.4 that

φ(Ne(w)) = X−J (ιun−r) ∼= X−J (wn−r) ∼= Σn,rsym.

Consequently, we have P(Σsymn,r )∨ = P(Σsym

n,n−r), see Equation (5.3.1).

5.4 The Skew-Symmetric Case

Let Matskn be the variety of skew-symmetric n × n matrices. The rank r skew-symmetric

determinantal variety Σsk,nr is the subvariety of Matskn given by

Σsk,nr =

{A ∈ Matskn

∣∣ rk(A) ≤ r}.

Recall that the rank of a skew-symmetric matrix is necessarily even. Hence, we assume

without loss of generality that r is even.

5.4.1 The Weyl Group of Dn

We set D0 = Dn, D = Dn, and label the simple roots as in Figure 1.3.4. Let W0 (resp. W )

be the Weyl group of D0 (resp. D). For 1 ≤ i ≤ n, we denote by si the simple reflection sαi .

112

The relations in W0 are

sisj = sjsi if |i− j| ≥ 2,

sisjsi = sjsisj if |i− j| = 1.

The latter are called braid relations. The remaining relations are snsi = sisn for i 6= n− 2,

and the braid relation

snsn−2sn = sn−2snsn−2.

Let µ be the involution on {1, · · · , 2n} given by µ(i) = 2n+ 1− i. Following [LS78, LR08],

we embed the Weyl group W0 into the symmetric group S2n via the homomorphism,

si 7→ rir2n−i for i 6= n,

sn 7→ rnrn−1rn+1rn,

where ri denotes the transposition (i, i+ 1) in S2n. Under this embedding, we have,

W0 = {w ∈ S2n |wµ = µw, sgn(w) = 1} . (5.4.2)

It is clear that w ∈ W0 is uniquely determined by its value on 1, · · · , n. Accordingly, we

represent w by the string [w(1), · · · , w(n)].

5.4.3 The Involution ι for Dn

The simple root αn is cominuscule in D0 = Dn. Let

J = D0\{αn} = {α1, · · · , αn−1} ,

and let ι be the involution defined in Definition 4.2.2. Observe that since J is isomorphic

to An−1, the Weyl involution −wJ on J is given by −wJ (αi) = αn−i, for 1 ≤ i ≤ n− 1, see

Chapter 2. Further, since ι interchanges α0 and αn, for 1 ≤ i ≤ n, we have,

ι(αi) = αn−i, and ιsi = sn−i.

113

5.4.4 The Schubert Variety

Let G be the simply connected, almost-simple group with Dynkin diagram Dn, and let P ⊂ G

be the parabolic group corresponding to J = {α1, · · · , αn−1}, see Figure 1.1.11. Following

[LR08], we identify Matskn with the opposite cell in G/P . Under this identification, the zero

matrix corresponds to e ∈ G/P , and Σsk,nr = X−J (wr), where

wr = [r + 1, · · · , n, 2n− r + 1, · · · , 2n], (5.4.5)

in the sense of Equation (5.4.2). Observe that

wr ∈ WJ ∩W0. (5.4.6)

Following Lemma 4.4.2, (1), we have wr ∈ WJ∪{α0}0 .

Remark 5.4.7. In [LR08], the skew-symmetric variety is identified with a Schubert variety

corresponding to the group SO(2n), which is not simply connected. This is not a problem

however, since Schubert varieties depend only on the underlying Dynkin diagram, and not

on the group per se, see Section 1.5.4.

Theorem 5.C. The conormal fibre of Σsk,nr at 0 is isomorphic to Σsk,n

n−r, where

n =

n if n is even,

n− 1 if n is odd.

Proof. For L a sub-diagram of D0, we write wL for the longest element in W supported on

L, and wJL for its minimal representative with respect to J . The longest elements w0 ∈ W0

and wJ ∈ WJ are given by

w0 = wn = [2n, · · · , n+ 2, n+ 1],

wJ = [n, · · · , 1],

114

in the sense of Equation (5.4.2), see [LR08]. Let wr be as defined in Equation (5.4.5), and

set vr = ι(w0wrwJ ). We have

w0wrwJ = [1, · · · , r, n+ 1, n+ 2, · · · , 2n− r] = wJL , (5.4.8)

where L = {αr+1, · · · , αn}. Hence XJ (w0wrwJ ) is smooth, see Proposition 4.4.5. It now

follows from Proposition 4.5.1 and Proposition 5.4.12 that

φ(N∗0XJ (wr)) = X−J (ιwn−r) ∼= X−J (wn−r) ∼= Σsk,nn−r .

It remains to prove Proposition 5.4.12. The proof is obtained as a consequence of the

following two lemmas.

Lemma 5.4.9. Consider xi ∈ W0 defined inductively as

xi =

sn for i = n− 1,

si+1sixi+1 for 1 ≤ i < n− 1.

(5.4.10)

Then si+2si+3xi = xisisi+1 for 1 ≤ i ≤ n− 4.

Proof. Observe that for j ≤ i − 2, we have sjxi = xisj, from which we deduce the equality

sisi+1xi+3 = xi+3sisi+1. Now,

si+2si+3xi = si+2si+3si+1sisi+2si+1si+3si+2xi+3 using Equation (5.4.10)

= si+2si+1si(si+3si+2si+3)si+1si+2xi+3

= (si+2si+1si+2)si+3si(si+2si+1si+2)xi+3 using Braid relations

= si+1si+2si+3(si+1sisi+1)si+2si+1xi+3 using Braid relations

= si+1si+2sisi+1si+3si+2sisi+1xi+3 using Braid relations

= si+1sisi+2si+1si+3si+2xi+3sisi+1

= xisisi+1 using Equation (5.4.10).

115

Lemma 5.4.11. Let xi be given by Equation (5.4.10). For 3 ≤ j, k ≤ n− 1, we have

ιxn−k xk = xk−2ιxn−k+2,

ιxn−k xkxk−2 · · ·xj = xk−2xk−4 · · ·xj−2ιxn−j.

Proof. The second equality follows from repeated applications of the first. Observe first

that xk ∈ 〈sj | j ≥ k〉, or equivalently, ιxn−k ∈ 〈sj | j ≤ k〉. Consequently, the braid relations

yields ιxixj = xjιxi whenever i+ j ≥ n+ 2. Now,

ιxn−k xk = ιsn−k+1ιsn−k

ιsn−k+2ιsn−k+1

ιxn−k+2 xk using Equation (5.4.10)

= sk−1sksk−2sk−1ιxn−k+2xk using Section 5.4.3

= sk−1sk−2sksk−1xkιxn−k+2

= xk−2ιxn−k+2 using Equation (5.4.10).

This proves the claim when n is even. Suppose n is odd, so that n = n− 1 and k ≤ n− 2.

Then, we have,

ιxn−k−1 xk = ιsn−kιsn−k−1

ιxn−k xk using Equation (5.4.10)

= sksk+1 xk−2ιxn−k+2 using Section 5.4.3

= xk−2sk−2sk−1ιxn−k+2 using Lemma 5.4.9

= xk−2ιxn−k+1 using Equation (5.4.10).

This proves the claim when n is odd.

Proposition 5.4.12. For wr given by Equation (5.4.5), and vr = ι(w0wrwJ ), we have

(wrvr)D0 = wrvr

ιv−1n−r = ιwn−r ∈ W 0

J∪{α0}.

Consequently, ιwn−r is the unique maximal element in{u ∈ W 0

J∪{α0}

∣∣∣u ≤ (wrvr)D0

}.

116

Proof. Let xi be as in Equation (5.4.10). We have the following formulae, which are easily

verified inductively,

xi = [1, · · · , i− 1, i+ 2, · · · , n− 2, 2n− i, 2n− i+ 1]

wr = xr−1xr−3 · · ·x1

w0wrwJ = xn−1xn−3 · · · xr+1.

Now,

ιwrιvr = ιxr−1

ιxr−3 · · · ιx1xn−1xn−3 · · ·xr+1

= xn−r−1xn−r−3 · · ·x1ιxn−1

ιxn−3 · · · ιxn−r+1 using Lemma 5.4.11.

= wn−rvn−r.

It follows that ιwrιvrv

−1n−r = wn−r, hence wrvr

ιv−1n−r = ιwn−r. Next, Equation (5.4.6) yields

wn−r ∈ WJ∪{α0}0 =⇒ ιwn−r ∈ W 0

J∪{α0} ⊂ WD0 .

Further, since ιvn−r ∈ W0 (see Equation (5.4.8)), we have wrvr = ιwn−r mod W0. Together,

we deduce (wrvr)D0 = ιwn−r.

117

118

Chapter 6

Schubert Divisors and the Minimal

Nilpotent Orbit

Let G be an almost simple group over an algebraically closed field k of good characteristic,

let D0 be the Dynkin diagram of G, and let N be the nilpotent cone of G.

Throughout, we fix a Borel subgroup B ⊂ G, and set B as in Section 1.3. We also fix a

proper subset J ⊂ D0, and let P (resp. P) be the standard parabolic subgroup of G (resp.

LG) corresponding to J .

In this chapter, we denote the highest root of G by θ, and we denote the Schubert varieties

XJ (w) and their conormal varieties NJ (w) by X(w) and N(w) respectively. For Schubert

varieties in G/B (resp. LG/B), we use XB(w) (resp. XB(w)), and for the conormal variety of

XB(w) in G/B, we use NB(w). Finally, we denote by W0 (resp. W ) the Weyl group of G

(resp. LG).

Given a root α ∈ D0\J , the Schubert varietyX(w0sα) is a divisor in G/P , see Lemma 6.3.1.

The main results of this chapter are the following:

Theorem 6.A. Assume α ∈ D0\J is a long root, let w = (w0sα)J , and let u = (w0τ−α∨)J ,

where τ−α∨ is an element of W in the sense of Section 1.3.9. Then there exists a dense

119

embedding, φ : N(w)→ X(u), where X(u) is a Schubert subvariety of LG/P.

Theorem 6.B. Let ρ be the dense embedding of the minimal nilpotent orbit into LG/L+G

described in [AH13]. For O an orbital variety of minimal shape, we have,

ρ(O) = XD0(τβ∨),

for some short coroot β∨ ∈ D∨0 .

6.1 Long and Short Roots

Let W0 (resp. W ) be the Weyl group of D0 (resp. D). We may view the co-root lattice Λ∨0

as a subgroup of W via the isomorphism W ∼= W0 n Λ∨0 , see Section 1.3.9.

Recall from Section 1.1.16 the symmetric bilinear form ( | ) on the root lattice Λ0 of G.

The cardinality of the set {(α |α) |α ∈ ∆0} is either 1 or 2, see [Bou68].

We say that α ∈ ∆0 is a long root if

(α |α) = max {(α |α) |α ∈ ∆} ,

and we say that α ∈ ∆0 is a short root if

(α |α) = min {(α |α) |α ∈ ∆} .

In particular, if # {(α |α) |α ∈ ∆0} = 1, then every root is both short and long.

Lemma 6.1.1 (cf. [Bou68]). The highest root θ of G is a long root.

Lemma 6.1.2 (cf. [Bou68]). The Weyl group W0 acts transitively on the set of long (resp.

short) roots.

Consider a pair of simple roots α and β which are adjacent in the Dynkin diagram D0.

We have (α |α) = (β | β) if and only if α and β are connected by a single edge, and we have

(α |α) > (β | β) if and only if the arrow connecting α and β points from α to β.

120

6.2 The Minimal Orbit in the Nilpotent Cone

The G-orbit closures Nλ ⊂ N in the nilpotent cone N are indexed by certain elements in

the co-root lattice Λ∨0 of G, see [Car85]. Let γ∨ be the highest short co-root of G, i.e., the

highest short root of the dual Dynkin diagram D∨0 . Then, there exists a G-orbit closure Nγ∨

corresponding to γ∨, and further, Nγ∨ is the unique minimal non-zero orbit closure in N .

Let L−G = G(k[t−1]), let π− : L−G→ G be the map given by t 7→ 0, and set

U− ={g ∈ G(k[t−1])

∣∣ π−(g) = 1}.

Proposition 6.2.1 (cf. [AH13]). There exists a G-equivariant embedding ρ : Nγ∨ ↪→ U−.

Let us identify U− as the opposite cell of LG/L+G. The closure of ρ (Nγ∨) is then precisely

the Schubert variety XD0 (τγ∨) ⊂ LG/L+G.

Lemma 6.2.2. Let gθ be the root space corresponding to the highest root θ. Then gθ ⊂ N ◦γ∨.

Proof. Since θ is the highest long root, its dual in ∆∨ is precisely the highest short co-root

γ∨. Hence, we have gθ ⊂ N ◦γ∨ .

6.3 The Conormal Variety of a Schubert Divisor

We fix a maximal torus T ⊂ G, and a Borel subgroup B ⊂ G containing T , and denote

by ∆0 the root system of G with respect to (B, T ). Let B ⊂ LG be as in Section 1.3, and

denote ∆ the root system of LG with respect to (B, T ).

We fix a proper subset J ⊂ D0, and a long simple root α ∈ D0\J . Let P ⊂ G (resp.

P ⊂ LG) be the standard parabolic subgroup corresponding to J , let wJ0 be minimal

representative of the longest element in W0 with respect to J , and let w = (w0sα)J .

Lemma 6.3.1. The Schubert variety X(w) is a divisor in G/P .

121

Proof. Since α 6∈ J , we have w0sα 6= w0(modWJ ). It follows that,

dimX(w) ≤ dim G/P − 1.

On the other hand, we have,

l(wJ0 ) ≤ l(w) + l(sα).

We deduce that

dimX(w) = dim G/P − 1,

and hence, X(w) is a divisor in G/P .

Lemma 6.3.2. There exists v ∈ W0 such that v(α) = θ. Further, for any such v, we have,

l(w0sαv−1) = l(w0)− l(v)− 1.

Proof. The highest root θ is a long root, see Lemma 6.1.1. It follows from Lemma 6.1.2

that there exists v ∈ W0 such that v(α) = θ. Further, since v(α) = θ is positive, we have

sαv−1 > v−1, and hence, following Proposition 1.4.6, we have,

l(sαv−1) = l(v−1) + 1.

It follows that

l(w0sαv−1) = l(w0)− l(sαv−1)

= l(w0)− l(v)− 1.

Proposition 6.3.3. Let s0 be the simple reflection corresponding to the simple root α0 ∈ D.

For v as in Lemma 6.3.2, we have sαv−1s0v = τ−α∨.

122

Proof. Recall from Section 1.2.5 the (non-linear) map ∆ → ∆∨, given by α 7→ α∨. This map

is W0-equivariant, see [Bou68]. Applying this map to the equation v(α) = θ, we obtain,

v(α∨) = θ∨.

Furthermore, since s0 = τθ∨sθ, see Section 1.3.9, we have,

sαv−1s0v = sαv

−1τθ∨sθv = sατv−1(θ∨)sv−1(θ)

= sατα∨sα = τsα(α∨) = τ−α∨ .

Lemma 6.3.4. We have µG/P (N(w)) = Nγ∨, where µG/P : T ∗G/P → N is the Springer map,

see Section 1.6.4.

Proof. Following Section 1.6.5, the conormal variety N(w) is the closure of

N◦(w) = {(bw, x) | b ∈ B, x ∈ gα} , (6.3.5)

in T ∗G/P . Recall from Lemma 6.2.2 that gθ ⊂ N ◦γ∨ . Now, since gα is conjugate to gθ,

see Lemma 6.3.2, we have Ad(bw)gα ⊂ N ◦γ∨ , and hence µG/P (N(w)) ⊂ Nγ∨ . The equality

µG/P (N(w)) = Nγ∨ now follows from the observations that µG/P (N(w)) 6= 0, and that Nγ∨ is

the minimal non-zero orbit-closure in N .

6.3.6 The Embedding φ

Recall the map ρ : Nγ∨ ↪→ LG/L+G from Proposition 6.2.1. We define,

φ′ : G×P Nγ∨ ↪→ LG/P, (g, x) 7→ g ρ(x)P . (6.3.7)

Lemma 6.3.8. The map φ′ is injective.

123

Proof. Consider (g, x) and (g′, x′) in G×P Nγ∨ , such that gρ(x)P = g′ρ(x′)P . We have,

gρ(x)L+G = g′ρ(x′)L+G,

=⇒ ρ(Ad(g)x)g L+G = ρ(Ad(g′)x′)g′ L+G,

=⇒ ρ(Ad(g)x)L+G = ρ(Ad(g′)x′)L+G.

Since ρ is injective, we have Ad(g)x = Ad(g′)x′. Consequently, we have,

gρ(x)P = g′ρ(x′)P ,

=⇒ ρ(Ad(g)x)gP = ρ(Ad(g′)x′)g′P ,

=⇒ gP = g′P .

Let h = g−1g′. We see that h ∈ P . Further, since g, g′ ∈ G, we deduce that h ∈ G∩P = P .

Consequently, we have,

(g, x) = (gh,Ad(h−1x)) = (g′, x′) ∈ G×P Nγ∨ ,

and hence, the map φ′ is injective.

Following Equation (6.3.5) and Lemma 6.2.2, we have,

N◦(w) ⊂ G×P (u ∩Nγ∨) ⊂ G×P Nγ∨ .

Restricting φ′ to N(w), we obtain the embedding φ : N(w) ↪→ LG/P.

We are now ready to prove Theorem 6.A. We first prove the result for P = B, and then

extend the result to the general case.

Proposition 6.3.9. The embedding φ restricts to a dense embedding N(w) ↪→ X(w0τ−α∨).

Proof. Following Equation (6.3.5), a generic point of the conormal variety N(w) may be

written as (bw, x) ∈ G×B uB for some b ∈ B, x ∈ gα. For any such point, we have,

φ(bw0sα, x) = bw0sαρ(x) modB,

= bw0sαv−1ρ(Ad(v)x)v modB.

124

Let P0 be the standard parabolic subgroup of LG corresponding to the subset {α0} ⊂ D.

Since Ad(v)x ∈ gθ, we have ρ(Ad(v)x) ∈ g−α0 , and hence ρ(Ad(v)x) ∈ P0. Consequently,

we have,

φ(bw, x) ∈ Bw0sαv−1Bs0BvB ⊂ XB(w0sαv

−1 ? s0 ? v),

where ? is the Demazure product, see Section 1.5.9. Next, since φ is injective, we have,

dimXB(w0sαv−1 ? s0 ? v) = l(w0sαv

−1 ? s0 ? v)

≥ dimN(w) = l(w0).

On the other hand, we see from Lemma 6.3.2 that

l(w0sαv−1 ? s0 ? v) ≤ l(w0sαv

−1) + l(sα) + l(v−1) = l(w0).

It follows that

w0sαv−1 ? s0 ? v = w0sαv

−1s0v,

= w0τ−α, see Proposition 6.3.3,

and further, that the induced map φ : N(w) ↪→ X(w0τ−α∨) is a dense embedding.

Theorem 6.A. Assume α ∈ D0\J is a long root, let w = (w0sα)J , and let N(w) be the

conormal variety of X(w). There exists a dense embedding,

φ : N(w)→ X(u),

where u is the minimal representative of w0τ−α∨ with respect to J .

Proof. The quotient map,

pr : LG/B→ LG/P,

is a P/B−fibre bundle, and the induced map,

pr : XB(w0sα)→ X(w),

125

is precisely the restriction of this fibre bundle to X(w). Consequently, NB(w0sα) is a fibre

bundle over N(w); the fibre bundle structure is simply the restriction of the quotient map,

pr : G×B Nγ∨ → G×P Nγ∨

to N(w). Therefore, we have a commutative diagram,

XB(w0sα) NB(w0sα) LG/B

X(w) N(w) LG/P.

pr

pr

pr

φ

pr

pr φ

It follows that φ(N(w)) ⊂ pr(XB(w0τ−α∨)) = X(u), see Proposition 6.3.9.

Next, we see from Equation (6.3.7) that the embeddings,

φ : NB(w) ↪→ LG/B, φ : N(w) ↪→ LG/P,

commute with the fibre bundle structure of LG/B→ LG/P, i.e, we have,

φ ◦ pr = pr ◦φ.

In particular, φ(NB(w0sα)) is a fibre bundle over φ(N(w)), and hence,

dimX(u) = dimXB(w0τ−α)− dim P/B

= dim G/B − dim P/B

= dim G/P = dimN(w).

From this, we deduce that the map φ : N(w) ↪→ X(u) is a dense embedding.

Corollary 6.3.10. The variety N(w) is normal, Cohen-Macaulay, Frobenius split, and ad-

mits a resolution of singularities via (an open subvariety of a) Bott-Samelson variety.

Proof. These are standard results for Schubert varieties, and remain true for open subsets

of Schubert varieties. One can find details in [Fal03, Lit98, MR85, PR08, HR18].

126

Theorem 6.B. Recall the embedding ρ : Nγ∨ ↪→ LG/L+G from Proposition 6.2.1. For O an

orbital variety of minimal shape, we have,

ρ(O) = XD0(τβ∨),

for some short coroot β∨ ∈ D∨0 .

Proof. Let O be an orbital variety of minimal shape, i.e., an irreducible component of the

variety uB∩Nγ∨ . Following Proposition 1.6.8, we can realizeO as the image of some conormal

variety NB(w) under the moment map µG/B, i.e., O = µG/B(N(w)) for some w ∈ W . Now,

since O ⊂ Nγ∨ , we see that a generic element of the fibre of NB(w) → XB(w) belongs to

Nγ∨ . Hence, we must have w = w0sα, for some long root α.

Next, the commutative diagram,

NB(w) LG/B

O LG/L+G,

µG/B

φ

pr

ρ

yields,

ρ(O) = ρ ◦ µG/B(NB(w)) = pr ◦φ(NB(w)).

Since φ(NB(w)) is a dense subvariety of XB(w0τ−α∨), it follows that ρ(O) is a dense

subvariety of XD0(w0τ−α∨). Finally, we have β∨ = −w0(α∨), and hence,

w0τ−α∨ = τβ∨w0 = τβ∨ (modW0).

Consequently, we see that φ(NB(w)) is a dense subvariety of XB(w0τβ∨), where β∨ is the

image of α∨ under the Weyl involution on D∨0 . Since, α is a long root, its dual α∨, and hence

also β∨, is a short coroot.

Corollary 6.3.11. Orbital varieties of minimal shape are normal, Cohen-Macaulay, and ad-

mit a resolutions of singularities via (open subvarieties of) Bott-Samelson varieties. Further,

the orbital varieties of minimal shape are compatibly Frobenius split in Nγ∨.

127

Proof. These are standard results for Schubert varieties. One can find details in [Fal03, Lit98,

MR85, PR08, HR18]. The compatible splitting is a consequence of the fact that ρ(Nγ∨) is

itself an open subset of a Schubert variety.

128

Chapter 7

A Resolution of Singularities

Let G be a simply connected, almost-simple algebraic group with Dynkin diagram D, let

T be a maximal torus in G, and let B be a Borel subgroup in G containing T . We fix a

cominuscule root γ ∈ D, and set P to be the standard parabolic subgroup corresponding to

the subset J = D\{γ}.

In this chapter, we will denote the Schubert varieties XJ (w) as simply X(w), the Bruhat

cells CJ (w) as simply C(w), and the conormal varietyNJ (w) (resp. conormal bundleN◦J (w))

as simply N(w) (resp. N◦(w)).

Recall from Section 1.5.8 the Bott-Sameslon resolutions X(w) → X(w) of Schubert

varieties, and from Section 2.1.10 the indexing of orbital varieties corresponding to G = SLn

by standard Young tableaux of size n. The main results of this chapter are the following.

Theorem 7.A. There exists a B-equivariant resolution of singularities θw : Z(w)→ N(w),

where Z(w) is a particular vector bundle over a Bott-Samelson variety X(w).

Theorem 7.B. Let T be a two-column tableau, and let OT be the corresponding orbital

variety. There exists a B-equivariant resolution of singularities Z(w) → OT, for some

choice of word w.

129

7.1 Resolving the Conormal Variety

Let ∆ (resp. D) denote the set of roots (resp. simple roots) of G with respect to (B, T ), let

W denote the Weyl group of (G, T ), and let u (resp. uB) be the Lie algebra of the unipotent

radical of P (resp. B).

We fix a w ∈ WJ , and set uw = u ∩ Ad(w−1)uB.

Lemma 7.1.1. Let R = {α ∈ ∆ |α ≥ γ, w(α) > 0}. For any α ∈ R and β ∈ D, such that

α + β ∈ ∆, we have α + β ∈ R.

Proof. Suppose first that β = γ. For any α ∈ R, we have γ ≤ α, hence 2γ ≤ α+ β. Since γ

is cominuscule, it follows that α + β 6∈ ∆.

Next, suppose β ∈ D\{γ}. Since w ∈ WJ , it follows from Equation (1.4.9) that w(β) > 0.

Now, for any α ∈ R, we have w(α) > 0, hence,

w(α + β) = w(α) + w(β) > 0.

It follows from the definition of R that if α + β ∈ ∆, then α + β ∈ R.

Corollary 7.1.2. The subspace uw is B-stable for the adjoint action.

Proof. Let R be as in Lemma 7.1.1. Recall that for α ∈ ∆, we have Ad(w−1)gα = gw−1(α).

Further, we have,

∆+0 \∆J = {α ∈ ∆0 |α ≥ γ} ,

and hence, u =⊕α≥γ

gα. It follows that

uw =⊕α≥γ

gα⋂⊕

α∈∆0

gw−1(α) =⊕α∈R

gα.

Following Lemma 7.1.1, we see that uw is Uα-stable for all α ∈ D.

Next, the subspaces u and Ad(w−1)uB being T -stable, their intersection uw is also T -

stable. It follows that uw is B-stable, since B is generated by the torus T and the root

subgroups Uα, for α ∈ D.

130

Proposition 7.1.3. The conormal bundle N◦(w) is isomorphic to the vector bundle,

BwB ×B uw → C(w),

given by (bw, x) 7→ bw(modP ).

Proof. Let pr : XB(w) → X(w) be restriction of the quotient map G/B → G/P to XB(w).

Since w ∈ WJ , the map pr restricts to an isomorphism of Schubert cells CB(w)∼−→ C(w),

see Section 1.5.8. The claim now follows from Section 1.6.5.

7.1.4 The Subgroup Q

We see from Corollary 7.1.2 that StabG(uw) is a standard parabolic subgroup. Let Q be any

standard parabolic subgroup contained in P that stabilizes uw, i.e.,

Q ⊂ StabG(uw) ∩ P. (7.1.5)

We define a vector bundle πQw : ZQ(w)→ XQ(w), where,

ZQ(w) = BwQ×Q uw, πQw (g, x) = g(modQ).

Let w = (s1, · · · , sr) be a minimal word for w. Recall the Bott-Samelson variety X(w)

from Section 1.5.3. We lift ZQ(w) to a vector bundle, πw : Z(w)→ X(w), given by,

Z(w) = P1 ×B · · · ×B Pr ×B uw,

and πw(p, x) = p (modB) for p ∈ P1 ×B · · · ×B Pr.

Proposition 7.1.6. Let τ be the quotient map G ×Q u → G ×P u. Viewing ZQ(w) as a

subvariety of G×Q u, we have τ(ZQ(w)) ⊂ N(w). Let τw : ZQ(w)→ N(w) denote the map

induced by restricting τ to N(w). We have a commutative diagram,

Z(w) ZQ(w) N(w)

X(w) XQ(w) X(w)

πw

θQw τw

πQw πw

ρQw pr

131

Here pr : XQ(w) → X(w) is the restriction of the natural quotient map, G/Q → G/P , to

XQ(w), and θQw : Z(w)→ ZQ(w) is the map given by θQw (p1, · · · , pr, x) = (p1 · · · pr, x).

Proof. Let Z◦Q(w) be the restriction of of the vector bundle πQw : ZQ(w) → XQ(w) to the

Schubert cell CQ(w). The quotient map G/B → G/Q induces an isomorphism CB(w)∼−→ CQ(w)

of Schubert cells. Consequently, the quotient map,

Z◦w = BwB ×B uw −→ BwQ×Q uw, (bw, x) 7→ (bw, x), (7.1.7)

is an isomorphism. Observe that this map is the inverse of τ |Z◦Q(w), and so,

τ(Z◦Q(w)) = T ∗XC(w) ⊂ N(w).

Now, since N(w) is a closed subvariety, it follows that τ(ZQ(w)) ⊂ N(w).

Finally, the commutativity of the diagram is a simple verification based on the formulae

defining the various maps.

Before we prove Theorem 7.A, let us recall some standard results about proper maps,

which the reader can find, for example, in [Har77, Ch.2].

Proposition 7.1.8. The following are true:

1. Closed immersions are separated and proper.

2. The composition of proper maps is proper.

3. If g : X → Y is a proper map, then g × idZ : X × Z → Y × Z is proper.

4. Let f : Y ↪→ Z be a closed immersion. A map g : X → Y is proper if and only if f ◦ g

is proper.

Theorem 7.A. The maps θQw and τw are proper and birational, and the composite map

θw = τw ◦ θQw is a B-equivariant resolution of singularities θw : Z(w)→ N(w). The map θw

is independent of the choice of Q.

132

Proof. The birationality of τw is a consequence of Equation (7.1.7). Recall from Section 1.5.8

that ρQw induces an isomorphism X◦(w)∼−→ CQ(w). Consequently, θQw induces an isomorphism

Z◦(w)∼−→ Z◦Q(w), and hence θQw is birational.

Consider now the commutative diagram,

Z(w) ZQ(w) N(w)

X(w)×N XQ(w)×N X(w)×N ,

f

θQw τw

g h

ρQw×idN pr× idN

(7.1.9)

where f, g, h are the closed immersions given by

f(p1, · · · , pr, x) = (πw(p1, · · · , pr, x), Ad(p1 · · · pr)x),

g(a, x) = (πQw (a, x), Ad(a)x),

h(a, x) = (πw(a, x), Ad(a)x).

Observe that the map,

(pr× idN ) ◦ (ρQw × id) = ρw × idN ,

is independent of the choice of Q, and therefore, the map θw = τw ◦ θQw is also independent

of the choice of Q.

Next, the maps ρQw and pr are proper; hence ρQw × idN and X(w)×N are proper. Con-

sequently, θQw and τw are proper.

Finally, observe that Zw, being a vector bundle over the smooth variety X(w), is itself a

smooth variety. Therefore, the map θw is a resolution of singularities.

7.2 Two-Column Orbital Varieties

Proposition 7.2.1. Let T be a standard Young tableau of size n with two columns. There

exists a maximal parabolic subgroup P ⊂ SLn, and a Schubert variety X(w) ⊂ G/P , such

that the Springer map, µG/P : T ∗G/P → N , induces a proper birational map, N(w)→ OT.

133

Proof. Let λ denote the shape of T, and let P be the standard parabolic subgroup of G

corresponding to An−1\{αk}, where k is the number of boxes in the first column of λ. The

maximal element wP of WP is given by,

wP (i) =

k + 1− i for i ≤ k,

n+ 1− k for i > k.

Let a1, · · · , ak be the entries in the first column of T, written in increasing order, i.e.,

top to bottom; and b1, · · · , bn−k the entries in the second column, also written in increasing

order. We consider the element w ∈ Sn, given by,

w(i) =

ai for i ≤ k,

bi−k for i > k.

Let v = wwP . Since w ∈ SPn , the Schubert variety XB(v) is a fibre bundle over XP (w) with

fibre B/P , and we have a Cartesian diagram,

XB(v) G/B

XP (w) G/P ,

pr

where pr is the restriction of the quotient map G/B → G/P to XB(v). Consequently, we have

a B/P -fibre bundle, NB(v)→ NP (w), which is precisely the restriction of the map,

pr× idN : XB ×N → X ×N ,

to NB(v) ⊂ T ∗X ⊂ G/B ×N . In particular, this yields µG/B(NB(w)) = µG/P (NP (w)).

We verify that T is the left RSK-tableau of v, thus obtaining,

OT = µG/B(NB(w)) = µG/P (NP (w)).

Next, the map µG/P is birational, see [Hes78]. In particular, for x ∈ N ◦λ , the fibre

µ−1G/P (x) contains precisely one point. It follows that the map, NP (w)→ OT, induced by the

restriction of µG/P to N(w), is birational.

134

Finally, observe the maps OT ↪→ Nλ and NP (w) ↪→ T ∗G/P are closed immersions. Fol-

lowing Section 1.6.4 and Proposition 7.1.8, we deduce that the induced map NP (w) → OT

is proper.

Theorem 7.B. Let T be a two-column tableau, and let OT be the corresponding orbital

variety. There exists a B-equivariant resolution of singularities Z(w) → OT, for some

choice of word w.

Proof. This is a direct consequence of Theorem 7.A and Proposition 7.2.1.

135

136

Chapter 8

Systems of Defining Equations

Let X be either the Grassmannian Gr(d, n), i.e., the variety of d-dimensional subspaces in

kn, or the symplectic Grassmannian SGr(d, 2d), i.e., the variety of Lagrangian subspaces in

a symplectic vector space of dimension 2d. Let X(w) be a Schubert variety in X, and let

N(w) be the conormal variety of X(w) in X.

In this chapter, we present a system of defining equations for the inclusion N(w) ↪→ T ∗X.

Theorem 8.A. Consider (V, x) ∈ T ∗X. Then (V, x) ∈ N(w) if and only if V ∈ X(w), and

further, for all 1 ≤ j < i ≤ l + 1, we have,

dim(xE(ti)/E(tj)) ≤

ri−1 − rj,

ci − cj+1.

(8.0.1)

The integers ri and ci depend on w, as described in Section 8.1.11.

The key idea in the proof of Theorem 8.A is that we can identify N(w) as the image

of ZQ(w) under the map τw, see Proposition 7.1.6. This allows us to argue that a point

(V, x) ∈ T ∗X is in N(w) if and only if it can be lifted to a point in ZQ(w).

As a corollary to Theorem 8.A, we also determine a system of defining equations for the

orbital variety µX(N(w)). This provides a new proof of some earlier results in [BM17, BM17].

137

Theorem 8.B. Let G, B, P , X, w, and µX be as in Theorem 8.A. Then,

µX(N(w)) =

x ∈ uB

∣∣∣∣∣∣∣∣x2 = 0, dim(xE(ti)/E(tj)) ≤

ri−1 − rj,

ci − cj+1,

∀ 1 ≤ i < j ≤ l

.

Recall from Section 2.1.10 that the orbital varieties corresponding to SLn are indexed

by the standard Young tableaux of size n. The inclusion order on the set of orbital varieties

induces a partial order on the set of standard Young tableaux, called the geometric order.

Using Theorem 8.B, we obtain the following combinatorial description of the geometric order,

restricted to two-column Young tableaux.

Theorem 8.C. Let T be a two-column standard Young tableau. Consider integers 0 ≤ j <

i ≤ n, and the skew-tableau T\{1, · · · , j, i + 1, · · · , n}. Let Tji denote the tableau obtained

from this skew-tableau via ‘jeu de taquin’. Then,

OT ={x ∈ N

∣∣ J(xji ) � Tji},

where xji is the square sub-matrix of x with corners (tj + 1, tj + 1) and (ti, ti), J(xji ) denotes

the Jordan type of xji , and � denotes the dominance order on the set of partitions, see

Section 2.1.5.

In Section 8.1, we present concrete descriptions of Schubert varieties of type A, and

present a choice of Q satisfying Equation (7.1.5). In Section 8.2, we present concrete descrip-

tions of Schubert varieties of type C, and present a choice of Q satisfying Equation (7.1.5).

Our choices of Q in Sections 8.1 and 8.2 allow us to present a uniform proof of Theorem 8.A

in Section 8.3. The primary reference for this chapter is [LR08].

8.1 The Type A Grassmannian

Let E(n) be a n–dimensional vector space with privileged basis {e1, · · · , en}. The group

G = SLn acts on E(n) by left multiplication with respect to the basis e1, · · · , en.

138

The variety of d–dimensional subspaces of E(n) is called the usual Grassmannian variety,

Gr(d, n) = {V ⊂ E(n) | dimV = d} .

It is a cominuscule Grassmannian corresponding to the group G = SLn, and the cominuscule

root αd ∈ An−1, see Fig. 1.1.11.

8.1.1 Schubert Varieties

For Q a parabolic subgroup of G, the variety G/Q is a partial flag variety, in the sense of

Section 2.1.6. For notational convenience, we denote Gr(d, n) as Xd, and we denote the

Schubert subvarieties of Xd as Xd(w).

Recall from Section 2.1.4 that the Weyl group of G is isomorphic to Sn, the symmetric

group on n elements. For w ∈ Sn, let mw(i, j) be the number of non-zero entries in the top

left i× j sub-matrix of the permutation matrix∑Ew(k),k, i.e.,

mw(i, j) = # {w(1), · · · , w(j)} ∩ {1, · · · , i}

= # {(k, w(k)) | k ≤ j, w(k) ≤ i} .(8.1.2)

The Schubert cells, CQ(w) ⊂ XQ and Cd(w) ⊂ X, are given by,

CQ(w) ={F (q) ∈ XQ

∣∣ dim(F (qi) ∩ E(j)) = mw(j, qi), 1 ≤ j ≤ n, 1 ≤ i ≤ l},

Cd(w) = {V ∈ Xd | dim(V ∩ E(j)) = mw(j, d), 1 ≤ j ≤ n} ,

while the Schubert varieties, XQ(w) ⊂ XQ and Xd(w) ⊂ Xd, are given by,

XQ(w) ={F (q) ∈ XQ

∣∣ dim (F (qi) ∩ E(j)) ≥ mw(j, qi), 1 ≤ j ≤ n, 1 ≤ i ≤ l},

Xd(w) = {V ∈ Xd | dim (V ∩ E(j)) ≥ mw(j, d), 1 ≤ j ≤ n} .(8.1.3)

In particular, we have F (q) ∈ XQ(w), if and only if F (qi) ∈ Xqi(w) for all i.

139

8.1.4 The Projection Map

Suppose d = qi for some i, and consider the projection map,

prd : XQ → Xd, F (q) 7→ F (d).

For any w ∈ Sn, we have prd(XQ(w)) = Xd(w). Further, we have pr−1d (Xd(w)) = XQ(w), if

w is maximal with respect to An−1\{αd}, i.e.,

w(1) > · · · > w(d), and w(d+ 1) > · · · > w(n), (8.1.5)

The following lemmas are easy consequences of standard results on Schubert varieties. They

are used repeatedly in the proofs of Propositions 8.3.6 and 8.3.9.

Lemma 8.1.6. Suppose we have integers 0 ≤ k ≤ d ≤ n, a permutation w ∈ Sn, and a

k-dimensional subspace U ⊂ E(n). If U ∈ Xk(w), then

dim(U ∩ E(i)) ≥ mw(i, d)− (d− k) ∀ 1 ≤ i ≤ n.

Conversely, suppose the above inequalities hold, and further, w(1) > · · · > w(d). Then

U ∈ Xk(w).

Proof. Observe that for 1 ≤ i ≤ n, we have,

mw(i, k) ≥ max{0,mw(i, d)− (d− k)},

with equality holding for all i if and only if w(1) > · · · > w(d).

Lemma 8.1.7. Consider integers 0 ≤ k ≤ d ≤ n, and a permutation w ∈ Sn, satisfying

w(k + 1) > · · · > w(n). Given U ∈ Xd, we have U ∈ Xd(w), if and only if,

dim(U ∩ E(i)) ≥ mw(i, k) ∀ 1 ≤ i ≤ n.

Proof. There exists 1 ≤ i ≤ n such that dim(U ∩ E(i)) = k. Set V = U ∩ E(i). It is clear

that V ∈ Xk(w). Now, since the statement of the lemma only involves w via the integers

mw(i, k) and mw(i, d), we may assume without loss of generality that w(1) > · · · > w(k).

140

Let Q be the parabolic group corresponding to the sequence q = (k, d). Then, we have

prk(V ⊂ U) = V and prd(V ⊂ U) = U . It follows from Section 8.1.4 that,

pr−1k (Xk(w)) = XQ(w) =⇒ prd(pr−1

k (Xk(w))) = Xd(w).

Consequently, we obtain U ∈ Xd(w).

Proposition 8.1.8. Consider integers 0 ≤ k ≤ d ≤ m ≤ n, and a permutation w ∈ Sn.

Given subspaces U ⊂ V ⊂ E(n) satisfying dimU = k, and

dim(U ∩ E(i)) ≥ mw(i, d)− (d− k) ∀ 1 ≤ i ≤ n,

dim(V ∩ E(i)) ≥ mw(i,m) ∀ 1 ≤ i ≤ n,

there exists U ′ ∈ Xd(w) satisfying U ⊂ U ′ ⊂ V .

Proof. Set l = dimV . Observe that

l = dim(V ∩ E(n)) ≥ mw(n,m) = m.

Let Q′ be the parabolic group corresponding to the sequence (k, l), and Q the parabolic

group corresponding to the sequence (k, d, l). We have a projection map pr : XQ → XQ′ ,

given by F (k, d, l) 7→ F (k, l).

Since the statement of the proposition only involves w via the integers mw(i, d) and

mw(i,m), we may replace w by any permutation v satisfying

{v(1), · · · , v(d)} = {w(1), · · · , w(d)},

{v(d+ 1), · · · , v(m)} = {w(d+ 1), · · · , w(m)},

{v(m+ 1), · · · , v(n)} = {w(m+ 1), · · · , w(n)},

without changing the statement. In particular, we may assume that

w(1) > w(2) > · · · > w(d),

w(d+ 1) > · · · > w(m),

w(m+ 1) > · · · > w(n).

(8.1.9)

141

Using Lemmas 8.1.6 and 8.1.7, we deduce that (U ⊂ V ) ∈ XQ′(w). Further, it follows from

Equation (8.1.9) that the projection map XQ(w) → XQ′(w) is surjective, see [LR08]. In

particular, there exists a partial flag F (k, d, l) ∈ XQ(w), for which F (k) = U and F (l) = V .

This yields the required subspace F (d), satisfying U ⊂ F (d) ⊂ V , and F (d) ∈ Xd(w).

Corollary 8.1.10. Consider subspaces U1 ⊂ U2 ⊂ E(n), satisfying dimU1 = k′, and

dim(U1 ∩ E(i)) ≥ mw(i, k′) ∀ 1 ≤ i ≤ n,

dim(U2 ∩ E(i)) ≥ mw(i,m) ∀ 1 ≤ i ≤ n,

for some integers 0 ≤ k′ ≤ m ≤ n. Then, there exists a subspace U ⊂ E(n), satisfying

dimU = m, U1 ⊂ U ⊂ U2, and

dim(U ∩ E(i)) ≥ mw(i,m), ∀ 1 ≤ i ≤ n.

Proof. Let t = m− k′. Observe that

mw(i, k′) = mw(i,m− t) ≥ mw(i,m)− t.

In particular, we have,

dim(U1 ∩ E(i)) ≥ mw(i,m)− t ∀ 1 ≤ i ≤ n.

The result now follows from Proposition 8.1.8 applied with k = m.

8.1.11 The Numbers ri, ci

For integers a, b, recall that (a, b] denotes the sequence,

a+ 1, a+ 2, · · · , b.

We fix w ∈ SPn . Following Equation (2.1.7), we have,

w(1) < w(2) < . . . < w(d), w(d+ 1) < . . . < w(n).

142

Consequently, w ∈ SPn is uniquely identified by the sequence w(1), · · · , w(d), which we now

write as the following concatenation of contiguous sequences,

(t′1, t1], (t′2, t2], · · · , (t′l, tl].

Here the ti, t′i are certain integers satisfying

∑(ti − t′i) = d, and

0 ≤ t′1 < t1 < t′2 < · · · < tl−1 < t′l < tl ≤ n,

For convenience, we set t0 = 0 and t′l+1 = n. Observe that the sequence w(d+ 1), · · · , w(n)

is precisely

(t0, t′1], (t1, t

′2], · · · , (tl, · · · , t′l+1].

We define the numbers r0, · · · , rl, and c0, · · · , cl+1, as the following partial sums,

ri =∑

1≤j≤i

(tj − t′j), ci =∑

1≤j≤i

(t′j − tj−1). (8.1.12)

For 1 ≤ i ≤ l, we have ti = ri + ci. Further, we have rl = d, cl+1 = n− d, and

mw(ti, rj) = min{ri, rj} = rmin{i,j},

mw(ti, d+ cj) = ri + min{ci, cj} = ri + cmin{i,j},

(8.1.13)

for all 1 ≤ i, j ≤ l. The permutation matrix of w is described in Fig. 8.1.14.

Proposition 8.1.15. Consider the sequence q = (q0, · · · , q2l+1), given by,

qi =

ri for 0 ≤ i ≤ l,

d+ ci−l for l < i ≤ 2l + 1.

Let Q be the standard parabolic subgroup associated to the sequence q, in the sense of Sec-

tion 2.1.6. Then Q satisfies Equation (7.1.5), i.e., Q ⊂ StabG(uw) ∩ P .

Proof. Observe that since ql = d, we have Q ⊂ P . We show that Q stabilizes uw. Recall the

set R and the subspace uw from Proposition 1.6.6. It follows from Section 8.1.11 that

R = {εi − εj | ∃k such that i ≤ qk, j > qk+l} ,

uw = {x ∈ g |xE(ql+i) ⊂ E(qi−1), ∀ 1 ≤ i ≤ l + 1} .(8.1.16)

Now, since Q stabilizes the flag E(q), it also stabilizes uw.

143

. . .

. . .

. . .

. . .

. . .

. . .

. . .

Figure 8.1.14: The permutation matrix of w. The empty boxes are zero matrices, while the

dotted cells are identity matrices of size r1, r2 − r1, · · · , rl − rl−1, c1, c2 − c1, · · · , cl+1 − cl,

going left to right.

8.2 The Symplectic Grassmannian

Consider the non-degenerate skew-symplectic bilinear form ω on E(2d) given by,

ω(ei, ej) =

δi,j if i ≤ d,

−δi,j if i > d,

where i = 2d+ 1− i. For V a subspace of E(2d), let

V ⊥ = {u ∈ E(2d) |ω(u, v) = 0, ∀ v ∈ V } .

Observe that for any subspace V ⊂ E(2d), we have (V ⊥)⊥ = V . Observe also that for

1 ≤ i ≤ 2d, we have E(i)⊥ = E(2d− i).

A subspace V ⊂ E(2d) is called Lagrangian if V = V ⊥. The variety of Lagrangian

subspaces in E(2d),

SGr(2d) ={V ⊂ E(2d)

∣∣V = V ⊥},

is called the symplectic Grassmannian. It is a cominuscule Grassmannian corresponding to

the group G = Sp2d, see Section 2.3, and the cominuscule root αd ∈ Cd, see Fig. 1.1.11.

144

8.2.1 The Weyl Group

Let s1, · · · , sd denote the simple reflections in Weyl group W of G, and let r1, · · · , r2d−1

denote the simple reflections of S2d−1. Following Section 2.3.3, we have an embedding,

W ↪→ S2d, given by,

si 7→

rir2d−i for 1 ≤ i < d,

rd for i = d.

Via this embedding, we have,

W ={w ∈ S2d

∣∣∣w(i) = w(i), 1 ≤ i ≤ d}.

The Bruhat order on S2d induces a partial order on W . This induced order is precisely

the Bruhat order on W , see [LR08]. Further, by virtue of being a subgroup of S2d, the group

W acts on sl2d. One obtains the action of W on g by restricting this action.

8.2.2 Standard Parabolic Subgroups

Let q = (q0, · · · , qr) be any integer-valued sequence satisfying 0 = q0 ≤ q1 ≤ · · · ≤ qr = 2d,

and further, qi + qr−i = 2d for 1 ≤ i ≤ r. Suppose Q′ is the standard parabolic subgroup of

SL2d corresponding to the subset,

{αj ∈ A2d−1 | j 6= qi, 1 ≤ i ≤ r − 1} .

Then Q = Q′ ∩G is the parabolic subgroup of G corresponding to the subset,

{αj ∈ Cd | j 6= qi, 1 ≤ i ≤ dr/2e} .

The variety XQ = G/Q is precisely the variety of isotropic flags of shape q, i.e.,

XQ ={F (q) ∈ SL2d/Q′

∣∣F (qi)⊥ = F (qr−i)

}.

The non-degeneracy of ω implies that for any subspaces U, V ⊂ E(2d), we have,

dimV + dimV ⊥ = 2d, U⊥ ∩ V ⊥ = (U + V )⊥. (8.2.3)

145

Let P ′ be the standard parabolic subgroup of SLn corresponding to the subsetA2d−1\{αd},

and let P = P ′∩G. Then P is the standard parabolic corresponding to Cd\{αd}, and further,

X = G/P ={V ⊂ E(2d)

∣∣V ⊂ V ⊥}.

We see from Equation (8.2.3) that for a d-dimensional subspace V , the condition V ⊂ V ⊥ is

equivalent to the condition V = V ⊥. It follows that G/P = SGr(d, 2d), and hence, SGr(d, 2d)

is cominuscule Grassmannian corresponding to the Dynkin diagram Cd.

8.2.4 Schubert Varieties

Consider an element w ∈ W . By viewing w as an element of S2d, see Section 8.2.1, we define

the numbers mw(i, k) precisely as in Equation (8.1.2). The Schubert cells, C(w) and CQ(w),

are then given by,

CQ(w) ={F (q) ∈ XQ

∣∣ dim(F (qi) ∩ E(j)) = mw(j, qi), 1 ≤ i ≤ l, 1 ≤ j ≤ n},

C(w) = {V ∈ X | dim(V ∩ E(j)) = mw(j, d), 1 ≤ j ≤ n} ,

and the Schubert varieties, X(w) and XQ(w), are given by,

XQ(w) ={F (q) ∈ XQ

∣∣ dim (F (qi) ∩ E(j)) ≥ mw(j, qi), 1 ≤ i ≤ l, 1 ≤ j ≤ n},

X(w) = {V ∈ X | dim(V ∩ E(i)) ≥ mw(i, d), 1 ≤ i ≤ n} .(8.2.5)

In particular, any Schubert subvariety of XQ can be identified (set-theoretically) as the

intersection of a Schubert subvariety of SL2d/Q′ with Sp2d/Q ⊂ SL2d/Q′.

8.2.6 Numerical Redundancy

By viewing w as an element of S2d, we define the numbers ti, t′i, ri, ci exactly as in Sec-

tion 8.1.11 and Equation (8.1.12). Observe that since w(i) = w(i) for all 1 ≤ i ≤ 2d, the

permutation matrix of w is symmetric across the anti-diagonal, see Fig. 8.1.14. Consequently,

146

for any 0 ≤ i ≤ l, we have,

ri + cl−i = d, ti + tl−i = 2d. (8.2.7)

In particular, we have E(ti)⊥ = E(tl−i).

The conditions defining the Schubert variety XQ(w) ⊂ XQ, described in Equation (8.2.5),

are not minimal. We describe some of this redundancy in the next lemma. For a more

comprehensive discussion of this matter, the reader may consult [And18].

Lemma 8.2.8. Consider F (q) ∈ XQ. Then F (q) ∈ XQ(w) if and only if

dim (F (qi) ∩ E(j)) ≥ mw(j, qi), 1 ≤ i ≤ l, 1 ≤ j ≤ 2d.

Proof. Since the permutation matrix of w is symmetric across the anti-diagonal, the number

of non-zero entries in the top left i × j corner of w equals the number of entries in the

bottom right i × j corner. Further, since each row and column of this matrix has precisely

one non-zero entry, we have,

mw(i, j) =# {(k, w(k) | k > 2d− j, w(k) > 2d− i}

=2d−# ({(k, w(k)) | k ≤ 2d− j} ∪ {(k, w(k)) |w(k) ≤ 2d− i})

=2d− ((2d− j) + (2d− i)−mw(2d− i, 2d− j)).

Hence, for 1 ≤ i, j ≤ 2d, we have the formula,

2d− (i+ j −mw(i, j)) = mw(2d− i, 2d− j).

Consider some F (q) ∈ XQ satisfying the inequalities of the lemma. Given 1 ≤ i ≤ l and

147

1 ≤ j ≤ 2d, we have,

dim(F (qi) ∩ E(j)) ≥ mw(j, qi)

=⇒ dim(F (qi) + E(j)) ≤ qi + j −mw(j, qi)

=⇒ dim((F (qi) + E(j))⊥) ≥ 2d− (qi + j −mw(j, qi))

=⇒ dim(F (qi)⊥ ∩ E(2d− j)) ≥ mw(2d− j, 2d− qi).

The final inequality follows from the penultimate as a consequence of Equation (8.2.3). We

see that F (q) satisfies Equation (8.2.5), and hence obtain F (q) ∈ XQ(w).

8.2.9 The Subspace uw

Let v be the Lie algebra of the unipotent radical of P ′, and uthe Lie algebra of the unipotent

radical of P . We have

v =⊕i≤d<j

[Ei,j], u =⊕

1≤i<j≤d

gεi+εj =⊕

1≤i<j≤d

[Ei,j + Ej,i].

In particular, we have u = v ∩ g. Recall the subspace uw from Proposition 1.6.6. Since g is

stable under the action of Ad(w−1), we have,

uw = u ∩ Ad(w−1)b = (v ∩ g) ∩ Ad(w−1)(b′ ∩ g) = v ∩ Ad(w−1)b′ ∩ g. (8.2.10)

Let q0, q1, · · · , q2l+1 be the sequence defined by

qi =

ri for 0 ≤ i ≤ l,

d+ ci−l for l < i ≤ 2l + 1.

It follows from Equation (8.2.7) that qi + q2l−i = 2d for all 1 ≤ i ≤ 2l.

Proposition 8.2.11. Let Q be the standard parabolic subgroup of G associated to the se-

quence q = (q0, · · · , q2l), in the sense of Section 8.2.2. Then Q satisfies Equation (7.1.5),

i.e., Q ⊂ StabG(uw) ∩ P .

148

Proof. It follows from Equation (8.2.7) that cl+1 = cl = d, hence q2l+1 = q2l = 2d. Therefore,

the standard parabolic subgroup Q′ ⊂ SL2d associated to (q0, · · · , q2l+1) is the same as the

standard parabolic subgroup of SL2d associated to (q0, · · · , q2l).

Next, it follows from Equations (8.1.16) and (8.2.10) that

uw = {x ∈ g |xE(ql+i) ⊂ E(qi−1), ∀ 1 ≤ i ≤ l + 1} . (8.2.12)

Now, since Q′ stabilizes uw, and since Q = Q′ ∩ G, we have Q ⊂ StabG(uw). Finally, since

ql = d, we have Q ⊂ P , hence Q ⊂ StabG(uw) ∩ P .

8.3 Defining Equations for the Conormal Variety in

Types A and C

Fix integers d < n. Let G be either SLn or Sp2d, let B be the subgroup of upper triangular

matrices in G, and let P be the standard parabolic subgroup of G corresponding to omitting

the simple root αd. As discussed in Sections 8.1 and 8.2, the variety G/P is either the usual

Grassmannian Gr(d, n), or the symplectic Grassmannian SGr(2d).

We fix a Schubert variety X(w) ⊂ G/P corresponding to some w ∈ W P . In this section,

we prove Theorem 8.A, which gives a system of defining equations for the conormal variety

N(w) as a subvariety of T ∗G/P .

Let π : T ∗G/P → G/P be the structure map, and µG/P : T ∗G/P → N the Springer map.

Theorem 8.A states that a point p ∈ T ∗G/P is in N(w) if and only if π(p) ∈ X(w) and µG/P (p)

satisfies Equation (8.3.13).

Recall the commutative diagram from Proposition 7.1.6. We show in Proposition 8.3.5

that for any point in ZQ(w), its image under µG/P ◦ τw satisfies Theorem 8.A. Conversely, we

show in Propositions 8.3.6 and 8.3.9 that any point in T ∗G/P lying over X(w), and further

satisfying Equation (8.3.13), belongs to τw(ZQ(w)) = N(w).

149

8.3.1 Combinatorial Description of X(w)

Fix w ∈ W P . Let the integers mw(i, j), ri, and ci be as in Equations (8.1.2) and (8.1.12)

respectively. It follows from Equations (8.1.3), (8.1.13) and (8.2.5) that F (q) ∈ XQ(w) if

and only if

dim(F (qi) ∩ E(tj)) ≥ min{ri, rj} = rmin{i,j}, ∀ 1 ≤ i, j ≤ l,

dim(F (qi+l) ∩ E(tj)) ≥ rj + min{ci, cj} = rj + cmin{i,j}, ∀ 1 ≤ i, j ≤ l.

In particular, when i = j, this yields F (qi) ⊂ E(ti) ⊂ F (qi+l).

8.3.2 The Cotangent Bundle

Following Equation (1.7.2), we identify the cotangent bundle T ∗G/P with its image under the

closed embedding (π, µG/P ) : T ∗G/P ↪→ G/P ×N ,

T ∗G/P = {(V, x) ∈ G/P ×N |xE(n) ⊂ V, xV = 0} .

8.3.3 The Variety ZQ(w)

For G = SLn, let q and Q be as in Proposition 8.1.15. For G = Sp2d, let q and Q be as in

Proposition 8.2.11. Recall the variety ZQ(w) from Section 7.1.4, and the descriptions of uw

from Equations (8.1.16) and (8.2.12). Using the closed embedding f from Equation (7.1.9),

we obtain,

ZQ(w) ={

(F (q), x) ∈ XQ(w)×N∣∣xF (qi+l) ⊂ F (qi−1), ∀ 1 ≤ i ≤ l + 1

}. (8.3.4)

Theorem 8.A states that given (V, x) ∈ T ∗G/P , we have (V, x) ∈ N(w), if and only if V ∈

X(w), and x satisfies Equation (8.3.13). The purpose of the following proposition is to show

that Equation (8.3.13) is necessary, i.e., if (V, x) ∈ N(w), then x satisfies Equation (8.3.13).

150

Proposition 8.3.5. For any point (F (q), x) ∈ ZQ(w), we have, for 1 ≤ j < i ≤ l,

dim(xE(ti)/E(tj)) ≤

ri−1 − rj,

ci − cj+1.

Proof. Consider (F (q), x) ∈ ZQ(w), and integers 1 ≤ j < i ≤ l. We see from Section 8.3.1

that E(ti) ⊂ F (qi+l), and from Equation (8.3.4) that xF (qi+l) ⊂ F (qi−1). Consequently, we

have xE(ti) ⊂ F (qi−1), and hence,

dim(xE(ti)/E(tj)) ≤ dim(F (qi−1)/E(tj))

= dimF (qi−1)− dim(F (qi−1) ∩ E(tj))

≤ ri−1 − rj,

where the final inequality follows from Section 8.3.1. Next, we see from Section 8.3.1

and Equation (8.3.4) that xF (qj+l+1) ⊂ F (qj) ⊂ E(tj). In particular, F (qj+l+1) is con-

tained in the kernel of the map,

F (qi+l)→ E(n)/E(tj), v 7→ xv(modE(tj)).

Since the image of this map is precisely xF (qi+l)/E(tj), we have,

dim(xF (qi+l)/E(tj)) ≤ dimF (qi+l)− dimF (qj+l+1)

= qi+l − qj+l+1 = ci − cj+1.

Finally, since E(ti) ⊂ F (qi+l), we deduce that dim(xE(ti)/E(tj)) ≤ ci − cj+1.

The following two propositions lay the groundwork required to prove that Equation (8.3.13)

is sufficient.

Proposition 8.3.6. Consider (V, x) ∈ X(w)×N satisfying Imx ⊂ V ⊂ kerx, and

dim(xE(ti)/E(tj)) ≤ ri−1 − rj, 0 ≤ j < i ≤ l + 1. (8.3.7)

151

Then, there exists a sequence of subspaces V0 ⊂ · · · ⊂ Vl = V , satisfying,

dimVi = qi,

xE(ti+1) ⊂ Vi ⊂ E(ti),

dim(Vi ∩ E(tj)) ≥ min{ri, rj} = mw(tj, qi),

(8.3.8)

for all 1 ≤ i, j ≤ l.

Proof. Since V ∈ X(w), it follows from Section 8.3.1 that Vl = V satisfies Equation (8.3.8).

We construct the subspaces Vi inductively. In particular, given subspaces Vi, · · · , Vl satisfying

Equation (8.3.8), we construct Vi−1.

Applying Equation (8.3.7) with j = i − 1, we have xE(ti) ⊂ E(ti−1). Further, Equa-

tion (8.3.8) yields xE(ti) ⊂ xE(ti+1) ⊂ Vi. Hence, we have,

xE(ti) ⊂ Vi ∩ E(ti−1).

Set U1 = xE(ti), and U2 = Vi ∩ E(ti−1). Applying Equation (8.3.7) with j = 0, we see

that dimU1 ≤ ri−1. Let k = ri−1 − dimU1.

Observe that U1 ∩ E(tj) is the kernel of the quotient map U1 → U1/E(tj). Now, since

dim(xE(ti)/E(tj)) ≤ ri−1 − rj, we have, for 1 ≤ j ≤ l,

dim(U1 ∩ E(tj)) = dimU1 − dim (U1/E(tj))

≥ (ri−1 − k)− (ri−1 − rj) = rj − k

≥ min{ri−1, rj} − k

= mw(tj, qi−1)− k.

152

On the other hand, observe that,

U2 ∩ E(tj) =

Vi ∩ E(ti−1) if i ≤ j,

Vi ∩ E(tj) if i > j,

=⇒ dim(U2 ∩ E(tj)) ≥

ri−1 if i ≤ j,

rj if i > j,

= min{ri−1, rj} = mw(tj, qi−1).

It now follows from Proposition 8.1.8 that there exists a subspace Vi−1 satisfying xE(ti) ⊂

Vi−1 ⊂ U2 ⊂ E(ti−1), and Equation (8.3.8).

Proposition 8.3.9. Consider (V, x) ∈ X(w)×N satisfying Imx ⊂ V ⊂ kerx, and

dim(xE(ti)/E(tj)) ≤ ci − cj+1, ∀ 0 ≤ j < i ≤ l + 1. (8.3.10)

Then, there exists a sequence of subspaces V = Vl ⊂ · · · ⊂ V2l+1, satisfying,

dimVl+i = ql+i,

Vl+i ⊂ kerx+ E(ti),

dim(Vl+i ∩ E(tj)) ≥ rj + min{ci, cj} = mw(tj, ql+i).

(8.3.11)

for all 1 ≤ i, j ≤ l.

Proof. Since V ∈ X(w), it follows from Section 8.3.1 that Vl = V satisfies Equation (8.3.11).

We construct the subspaces Vl+i inductively. In particular, given subspaces Vl, · · · , Vl+i−1

satisfying Equation (8.3.11), we construct Vl+i.

We see from Equation (8.3.11) that Vl+i−1 ⊂ kerx + E(ti). Set U = kerx + E(ti). We

first prove that,

dim(U ∩ E(tj)) ≥ rj + min{ci, cj} = mw(tj, ql+i), ∀ 1 ≤ j ≤ l + 1. (8.3.12)

153

For j ≤ i, we have E(tj) ⊂ U , hence U ∩ E(tj) = E(tj). It follows that,

dim(U ∩ E(tj)) = tj = rj + cj = rj + min{ci, cj}.

For j > i, consider the map,

E(tj)→ xE(tj)/E(ti−1), v 7→ xv(modE(ti−1)).

Since xE(ti) ⊂ E(ti−1), the subspace U ∩ E(tj) is contained in the kernel of this map.

Further, Equation (8.3.10) states that dim(xE(tj)/E(ti)) ≤ cj − ci+1, hence

dim(U ∩ E(tj)) ≥ tj − (cj − ci)

= rj + ci = rj + min{ci, cj}.

This finishes the proof of Equation (8.3.12). It now follows from Equation (8.3.12), Lemma 8.1.6,

and Proposition 8.1.8 that there exists a subspace Vl+i containing Vl+i−1, and further satis-

fying Equation (8.3.11).

Theorem 8.A. Consider (V, x) ∈ T ∗G/P . Then (V, x) ∈ T ∗XX(w) if and only if V ∈ X(w),

and further, for all 1 ≤ j < i ≤ l + 1, we have,

dim(xE(ti)/E(tj)) ≤

ri−1 − rj,

ci − cj+1.

(8.3.13)

Proof. Recall the map τw : ZQ(w)→ T ∗XX(w) from Proposition 7.1.6, given by,

τw(F (q), x) = (F (d), x).

The map τw is proper and birational (see Theorem 7.A), hence surjective. It follows that

(V, x) ∈ T ∗XX(w) if and only if there exists F (q) ∈ XQ(w) such that F (d) = V , and

(F (q), x) ∈ ZQ(w).

Consider (V, x) ∈ T ∗XX(w). It follows from Theorem 7.A that V ∈ X(w), and from

Proposition 8.3.5 that Equation (8.3.13) holds. Conversely, consider (V, x) ∈ T ∗G/P satisfying

154

V ∈ X(w), and Equation (8.3.13). We will construct F (q) ∈ XQ(w) such that (F (q), x) ∈

ZQ(w), and τw(F (q), x) = (V, x).

Using Proposition 8.3.6, we construct subspaces V0, · · · , Vl = V satisfying Equation (8.3.8).

Similarly, we use Proposition 8.3.9 to construct subspaces Vl+1, · · · , V2l+1 satisfying Equa-

tion (8.3.11).

Suppose first that G = SLn. We set,

F (q) = V0 ⊂ V1 ⊂ · · · ⊂ V2l+1.

Observe that F (d) = Vl = V . It follows from Equations (8.3.8) and (8.3.11) that F (q) ∈

XQ(w). Further, for 1 ≤ i ≤ l + 1, we have,

F (ql+i) ⊂ kerx+ E(ti) =⇒ xF (ql+i) ⊂ xE(ti) ⊂ F (qi−1).

This is precisely the condition for (F (q), x) to belong to ZQ(w).

Suppose next that G = Sp2d. Let F (q) be the partial flag given by,

F (qi) =

Vi for i ≤ l,

V ⊥2l−i for l < i ≤ 2l.

In particular, we have F (d) = Vl = V . It follows from Lemma 8.2.8 and Proposition 8.3.6

that F (q) ∈ XQ(w). It remains to show that (F (q), x) ∈ ZQ(w).

For 0 ≤ i ≤ l, we have xE(ti+1) ⊂ Vi, hence ω(xE(ti+1), V ⊥i ) = 0. It follows from

Section 2.3.1 that ω(E(ti+1), x(V ⊥i )) = 0, hence,

xF (q2l−i) = x(V ⊥i ) ⊂ E(ti+1)⊥ = E(tl−i−1).

The final equality is a consequence of Equation (8.2.7). Substituting i 7→ l − i yields

xE(ql+i) ⊂ E(ti−1) for all 0 ≤ i ≤ l. It follows that (F (q), x) ∈ ZQ(w), and hence,

(F (d), x) ∈ N(w).

155

8.4 A Type Independent Conjecture

In this section, we assume that X is a cominuscule Grassmannian corresponding to some

Dynkin diagram. We conjecture, for any Schubert variety X(w) ⊂ X, the following equality,

N(w) = µ−1G/P (µG/P (N(w))) ∩ π−1(X(w)). (8.4.1)

The question is well-posed in both set-theoretic and scheme-theoretic settings.

Suppose X(w) ⊂ X is a smooth Schubert subvariety. We prove in Proposition 8.4.2 that

Equation (8.4.1) holds set-theoretically in this case.

Next, let w0 denote the longest element in the Weyl group W . We show in Proposi-

tion 8.4.4 that if X(w) ⊂ X is a Schubert variety such that the opposite Schubert variety

Xw0w is smooth, then Equation (8.4.1) holds scheme-theoretically. This is a straightforward

corollary to Theorem 4.A.

When X is the usual Grassmannian or the symplectic Grassmannian, the set-theoretic

version is a consequence of Theorems 8.A and 8.B. In type B, the only cominuscule Grass-

mannian is the one corresponding to the cominuscule root α1. In this case, one easily verifies

that for each w ∈ W P , either X(w) is smooth, or X(w0w) is smooth, hence settling the

set-theoretic version of our conjecture for all cominuscule Grassmannians in types A, B, and

C.

We would like to know in which of these cases Equation (8.4.1) holds scheme-theoretically,

and also whether Equation (8.4.1) holds for types D and E. If it does, can we find a uniform,

type independent proof?

Proposition 8.4.2. Suppose X(w) is smooth. Then the conormal variety N(w) satisfies

Equation (8.4.1) set-theoretically.

Proof. A Schubert variety X(w) in a cominuscule Grassmannian X is smooth if and only if

X(w) is homogeneous for some standard parabolic subgroup L, see Proposition 4.4.5.

156

Suppose X(w) is homogeneous for some standard parabolic subgroup L; let SL be the

corresponding subset of S, and wL the longest word of W supported on SL. Then w is the

minimal representative of wL in W P . Further, the subspace uw ⊂ u from Proposition 1.6.6

is precisely,

uw =⊕

α≥γ, Supp(α)6∈SL

gα

In particular, uw is L-stable.

The quotient map G/B → G/P induces an isomorphism L/B∼−→ X(w), and the conormal

variety N(w)→ X(w) is simply the vector bundle L×B uw → L/B. Consequently, we have,

µG/P (N(w)) = {Ad(l0)x0 | l0 ∈ L, x0 ∈ uw} . (8.4.3)

Now, consider some (l, x) ∈ G×P u, satisfying

π(l) ∈ X(w), and µG/P (l, x) ∈ µG/P (N(w)).

We may assume, without loss of generality, that l ∈ L. As a consequence of Equation (8.4.3),

there exist l0 ∈ L, and x0 ∈ uw, such that,

µG/P (l, x) = Ad(l)x = Ad(l0)x0

=⇒ x = Ad(l−1l0)x0.

Now, since uw is L-stable, we have x ∈ uw, hence (l, x) ∈ N(w).

Proposition 8.4.4. Suppose the opposite Schubert variety X(w0w) is smooth for some w ∈

W P . Then N(w) satisfies Equation (8.4.1) scheme-theoretically.

Proof. Let D0 denote the Dynkin diagram of G, and let D be the corresponding extended

Dynkin diagram. The loop group LG is an affine Kac-Moody group corresponding to the

extended Dynkin diagram D. Let G0, Gd, and P be parabolic subgroups of LG corresponding

to the subsets D\{α0},D\{αd}, and D\{α0, αd} respectively.

157

Following Theorem 4.A, there exists an embedding φ : N(w)→ LG/P such that φ(N(w)) is

an open subset of some Schubert subvariety of LG/P. Further, we can identify the structure

map π and the Springer map µG/P as the restrictions to φ(N(w)), of the quotient maps

πd : LG/P → LG/Gd and π0 : LG/P → LG/G0 respectively.

Now, for any Schubert variety Y ⊂ LG/P, we have the scheme-theoretic equality,

Y = π−10 (π0(Y )) ∩ π−1

d (πd(Y )).

From this, we deduce that Equation (8.4.1) holds for N(w) scheme-theoretically.

8.5 Orbital Varieties

As discussed in Chapter 1, providing a combinatorial description of the inclusion order on

orbital varieties, and providing the defining equations for an orbital variety viewed as a

subvariety of uB, are both open problems in general. For certain orbital varieties in types A,

B, C (those corresponding to the nilpotent orbits satisfying x2 = 0), these problems were

solved in [Mel05, BM17].

Suppose G is either SL2n or Sp2d, and P is the standard parabolic group corresponding

to S\{αd}. We derive, in Theorem 8.B, a system of defining equations for orbital varieties

of the form µX(N(w)). This is an easy consequence of Theorem 8.A, and recovers some of

the results of [Mel05, BM17].

Theorem 8.B. Let G, B, P , X, w, and µX be as in Theorem 8.A. Then,

µX(N(w)) =

x ∈ uB

∣∣∣∣∣∣∣∣x2 = 0, dim(xE(ti)/E(tj)) ≤

ri−1 − rj,

ci − cj+1,

∀ 1 ≤ i < j ≤ l

.

158

Proof. Consider x ∈ uB satisfying x2 = 0, and

dim(xE(ti)/E(tj)) ≤

ri−1 − rj,

ci − cj+1,

∀ 1 ≤ j < i ≤ l. (8.5.1)

Substituting j = 0 in Equation (8.5.1), we obtain,

dim(Im(x|E(ti)) = dim(xE(ti)) ≤ ci − c1

=⇒ dim(kerx ∩ E(ti)) = dim(ker(x|E(ti)))

≥ ti − (ci − c1)

= ri + c1 ≥ ri.

Let k = d− dim(Imx). Substituting i = l in Equation (8.5.1) yields,

dim(Imx/E(tj)) ≤ rl−1 − rj ≤ d− rj

=⇒ dim(Imx+ E(tj)) ≤ (d− rj) + tj

=⇒ dim(Imx ∩ E(tj)) = dim(Im x) + dimE(tj)− dim(Imx+ E(tj))

≥ (d− k) + tj − (d− rj + tj) = rj − k.

Observe that since x2 = 0, we have Imx ⊂ kerx. It now follows from Proposition 8.1.8

and Section 8.3.1 that there exists V ∈ Xw such that,

Imx ⊂ V ⊂ kerx,

i.e., (V, x) ∈ N(w). Consequently, we have x ∈ µX(N(w)).

8.5.2 The Geometric Order on Young Tableaux

The geometric order on the set of all Young tableaux of size n is defined by

T ≤ S ⇐⇒ OT ⊂ OS.

159

It is an open problem to provide a combinatorial description of the geometric order on

Young tableaux. We present here a description of the restriction of the geometric order to

two-column orbital varieties.

Theorem 8.C. Let T be a two-column standard Young tableau. Consider integers 0 ≤ j <

i ≤ n, and the skew-tableau T\{1, · · · , j, i + 1, · · · , n}. Let Tji denote the tableau obtained

from this skew-tableau via ‘jeu de taquin’. Then,

OT ={x ∈ N

∣∣ J(xji ) � Tji},

where xji is the square sub-matrix of x with corners (tj + 1, tj + 1) and (ti, ti), J(xji ) denotes

the Jordan type of xji , and � denotes the dominance order on the set of partitions, see

Section 2.1.5.

Proof. This statement is proved in [Mel05]. We explain here how it also follows as a conse-

quence of Theorem 8.B and Section 8.5.2.

Since x2 = 0, we have (xji )2 = 0 for all i, j. Consequently, the inequality J(xji ) � T ji

is equivalent to the inequality rk(xji ) ≤ f ji , where f ji is the number of boxes in the second

column of Tji . On the other hand, it follows from Theorem 8.B and Section 8.5.2 that

OT ={x ∈ N

∣∣ rk(xji ) ≤ gji},

for certain integers gji . It is a simple exercise to verify that the integers f ji and gji defined

here are equal.

160

Bibliography

[AH13] Pramod N. Achar and Anthony Henderson, Geometric Satake, Springer correspon-

dence and small representations, Selecta Math. (N.S.) 19 (2013), no. 4, 949–986.

MR 3131493

[And18] David Anderson, Diagrams and essential sets for signed permutations, Electron. J.

Combin. 25 (2018), no. 3, Paper 3.46, 23. MR 3853898

[BB05] Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate

Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266

[BK05] Michel Brion and Shrawan Kumar, Frobenius splitting methods in geometry and

representation theory, Progress in Mathematics, vol. 231, Birkhauser Boston, Inc.,

Boston, MA, 2005. MR 2107324

[BM10] Sara C. Billey and Stephen A. Mitchell, Smooth and palindromic Schubert varieties

in affine Grassmannians, J. Algebraic Combin. 31 (2010), no. 2, 169–216. MR

2592076

[BM17] Nurit Barnea and Anna Melnikov, B-orbits of square zero in nilradical of the sym-

plectic algebra, Transform. Groups 22 (2017), no. 4, 885–910. MR 3717217

[Bor91] Armand Borel, Linear algebraic groups, second ed., Graduate Texts in Mathemat-

ics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012

161

[Bou68] N. Bourbaki, Elements de mathematique. Fasc. XXXIV. Groupes et algebres de

Lie. Chapitre IV: Groupes de Coxeter et systemes de Tits. Chapitre V: Groupes

engendres par des reflexions. Chapitre VI: systemes de racines, Actualites Scien-

tifiques et Industrielles, No. 1337, Hermann, Paris, 1968. MR 0240238

[Car85] Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New

York), John Wiley & Sons, Inc., New York, 1985, Conjugacy classes and complex

characters, A Wiley-Interscience Publication. MR 794307

[CG97] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry,

Birkhauser Boston, Inc., Boston, MA, 1997. MR 1433132

[DR04] J. Matthew Douglass and Gerhard Rohrle, The geometry of generalized Steinberg

varieties, Adv. Math. 187 (2004), no. 2, 396–416. MR 2078342

[DR09] , The Steinberg variety and representations of reductive groups, J. Algebra

321 (2009), no. 11, 3158–3196. MR 2510045

[Fal03] Gerd Faltings, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math.

Soc. (JEMS) 5 (2003), no. 1, 41–68. MR 1961134

[Ful97] William Fulton, Young tableaux, London Mathematical Society Student Texts,

vol. 35, Cambridge University Press, Cambridge, 1997, With applications to rep-

resentation theory and geometry. MR 1464693

[GKZ08] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resul-

tants and multidimensional determinants, Modern Birkhauser Classics, Birkhauser

Boston, Inc., Boston, MA, 2008, Reprint of the 1994 edition. MR 2394437

[Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg,

1977, Graduate Texts in Mathematics, No. 52. MR 0463157

162

[Hes78] Wim H. Hesselink, Polarizations in the classical groups, Math. Z. 160 (1978), no. 3,

217–234. MR 0480765

[HR18] T. J. Haines and T. Richarz, On the normality of schubert varieties: remaining

cases in positive characteristic, arXiv:1806.11001 (2018), 23.

[Kac90] Victor G. Kac, Infinite-dimensional Lie algebras, third ed., Cambridge University

Press, Cambridge, 1990. MR 1104219

[Kum02] Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory,

Progress in Mathematics, vol. 204, Birkhauser Boston, Inc., Boston, MA, 2002.

MR 1923198

[Lak16] V. Lakshmibai, Cotangent bundle to the Grassmann variety, Transform. Groups

21 (2016), no. 2, 519–530. MR 3492046

[Lit98] Peter Littelmann, Contracting modules and standard monomial theory for sym-

metrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (1998), no. 3, 551–567.

MR 1603862

[LR08] Venkatramani Lakshmibai and Komaranapuram N. Raghavan, Standard monomial

theory, Encyclopaedia of Mathematical Sciences, vol. 137, Springer-Verlag, Berlin,

2008, Invariant theoretic approach, Invariant Theory and Algebraic Transformation

Groups, 8. MR 2388163

[LRS16] V. Lakshmibai, Vijay Ravikumar, and William Slofstra, The cotangent bundle of

a cominuscule Grassmannian, Michigan Math. J. 65 (2016), no. 4, 749–759. MR

3579184

163

[LS78] V. Lakshmibai and C. S. Seshadri, Geometry of G/P . II. The work of de Concini

and Procesi and the basic conjectures, Proc. Indian Acad. Sci. Sect. A 87 (1978),

no. 2, 1–54. MR 490244

[Lus81] G. Lusztig, Green polynomials and singularities of unipotent classes, Adv. in Math.

42 (1981), no. 2, 169–178. MR 641425

[Lus90] , Canonical bases arising from quantized enveloping algebras, J. Amer.

Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415

[Mel05] Anna Melnikov, The combinatorics of orbital varieties closures of nilpotent order

2 in sln, Electron. J. Combin. 12 (2005), Research Paper 21, 20. MR 2134184

[Mil17] J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol.

170, Cambridge University Press, Cambridge, 2017, The theory of group schemes

of finite type over a field. MR 3729270

[MR85] V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing

for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27–40. MR 799251

[PR08] G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties,

Adv. Math. 219 (2008), no. 1, 118–198, With an appendix by T. Haines and

Rapoport. MR 2435422

[Rem02] Bertrand Remy, Groupes de Kac-Moody deployes et presque deployes, Asterisque

(2002), no. 277, viii+348. MR 1909671

[Spa82] Nicolas Spaltenstein, Classes unipotentes et sous-groupes de Borel, Lecture Notes

in Mathematics, vol. 946, Springer-Verlag, Berlin-New York, 1982. MR 672610

[Ste88] Robert Steinberg, An occurrence of the Robinson-Schensted correspondence, J. Al-

gebra 113 (1988), no. 2, 523–528. MR 929778

164

[Str82] Elisabetta Strickland, On the conormal bundle of the determinantal variety, J.

Algebra 75 (1982), no. 2, 523–537. MR 653906

[Tit87] Jacques Tits, Uniqueness and presentation of Kac-Moody groups over fields, J.

Algebra 105 (1987), no. 2, 542–573. MR 873684

[Wey03] Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in

Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690

165