DIFFERENTIAL OPERATORS ON HOMOGENEOUS SPACES

L. BARCHINI AND R. ZIERAU

Abstract. These are notes for lectures given at the ICE-EM Australian Grad-uate School in Mathematics in Brisbane, July 2-20, 2007.

Introduction

The real flag manifolds are a interesting family of smooth manifolds. Among thereal flag manifolds are the spheres, projective spaces and Grassmannians. Each isa compact homogeneous space G/P for reductive Lie group G; see Lecture 3 for adefinition. There are (at least) two reasons the real flag manifolds are particularlyimportant. One is that they are flat models for certain geometries, such as con-formal geometry and contact geometry. Another is that in representation theorythe standard representations of a reductive Lie group occur as spaces of sectionsof homogeneous vector bundles over G/P . In both situations differential operatorsbetween sections of bundles over G/P play an important role; typically systems ofsuch operators will have some G−equivariance properties. The goal of these lec-tures is to give some methods for constructing and studying interesting (families)of differential operators.

The course begins with four lectures on the structure of reductive Lie groupsand Lie algebras, and their finite dimensional representations. It is assumed thatthe student has had an introductory course on Lie groups and understands thebasics of manifold theory. The classical groups, such as GL(n,R) and SO(p, q), areexamples of reductive Lie groups. Using some linear algebra, much of the structuretheory presented is easily understood for these groups. In fact students may wishto focus entirely on the group GL(n,R) (except in Lectures 13 and 14).

Lectures 5-7 discuss homogeneous vector bundles over homogeneous spaces, anddifferential operators between sections of homogeneous bundles. Here the group G isarbitrary. Typically an interesting space of differential operators will be expressed interms of the enveloping algebra, where certain questions about differential operatorsare transformed into questions in algebra.

The remaining lectures are about differential operators between spaces of sectionsof homogeneous vector bundles over real flag manifolds. There are two importantpoints which are made. One is that there are explicit correspondences between (a)

July 31, 2007.

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G-invariant differential operators (on homogeneous bundles), (b) homomorphismsbetween Verma modules, and (c) conformally invariant systems of differential op-erators (as defined in Lecture 8). The second is that the existence of G-invariantdifferential operators (and therefore the existence of homomorphisms of Verma mod-ules and of conformally invariant systems) is somewhat special, occurring only forcertain bundles. Thus, constructing G-invariant differential operators is a nontrivialproblem. The lectures show invariant theory is useful in this construction.

Lectures 8 and 11 contain the definition of conformally invariant systems as wellas some of the theory relating these systems to the notions of G-invariant differ-ential operators and Verma modules. Lectures 10 and 12 work out examples ofconformally invariant systems when P is of abelian type, i.e., when P has abeliannilradical. Lecture 9 gives the necessary invariant theory for these examples. Lec-tures 13 ad 14 give an introduction to the construction of conformally invariantsystems when P is of Heisenberg type. The final lecture discusses a conjecture ofGyoja on reducibility of Verma modules (hence, on the existence of G-invariantdifferential operators and conformally invariant systems).

We thank the ICE-EM for the opportunity to present these lectures. We enjoyedworking with all of the students in the course.

LECTURE 1. Reductive groups

The groups we will consider are real and complex reductive Lie groups. The‘classical groups’, such as GL(n,C) and SO(n), are typical examples. We willbegin by giving the definitions of many of the classical groups and some of theirimportant properties. Some general facts about Lie groups and Lie algebras willbe recalled. Finally, a precise definition of a real reductive Lie group will be given.

1.1. Classical groups. Let us begin with the general linear group GL(V ). HereV is a real or complex vector space and GL(V ) is the group of invertible lineartransformations of V to itself. Suppose that the dimension of V is n. Choose abasis B = e1, . . . , en of V . Then the matrix of g ∈ GL(V ) with respect to B isthe matrix (gij) satisfying g(ej) =

∑i gijei. This provides a bijection

(1.1) φ : GL(V ) → GL(n,F) ≡ A ∈ Mn×n(F) : det(A) 6= 0,

(F = R or C, depending if V is a vector space over R or C). Of course, φ is agroup isomorphism. The bijection φ allows us to define a topology on GL(V ) asfollows. Give Mn×n(R) the topology from the norm

||X||2 =∑i,j

|xij |2.

Since the determinant function is continuous, GL(n,F) is an open set in Fn2. We

may define open sets in GL(V ) as those sets U for which φ(U) is an open set inFn2

. It follows (since matrix multiplication is continuous, in fact a polynomial)2

that GL(V ) is a topological group with respect to this topology. Similarly, GL(V )is a differentiable manifold for which multiplication is a smooth map. So GL(V ) isa Lie group. This definition of topology and differentiable structure is independentof the chosen basis (why?).

The Lie algebra may be identified with gl(V ), the set of linear transformationsof V to itself, with the bracket operation [A,B] = AB −BA. See Exercise 1.1.

The exponential map is the unique smooth function exp : gl(V ) → GL(V ) withthe property that for each X ∈ gl(V ), exp((t + s)X) = exp(tX) exp(sX) andddt exp(tX)|t=0 = X. Therefore,

exp(X) =∞∑

k=0

Xk

k!,

an absolutely convergent series in Mn×n(F).

When F = C, GL(V ) is identified with an open set in Cn2and multiplication is

holomorphic. Therefore, GL(V ) is a complex Lie group. Note that the Lie algebrais a complex Lie algebra.

In view of the fact that any closed subgroup H of a Lie group is a Lie group, manyother Lie groups are easily obtained from GL(V ). Furthermore, the Lie algebra ofH is

(1.2) Lie(H) ≡ h = X ∈ g : exp(tX) ∈ H, for all t ∈ R.

(See [6, Ch. II, §2].)

A simple instance of this is the case of the special linear group SL(V ) (resp.SL(n,F)). This is the group of linear transformations (resp. matrices) of deter-minant equal to one. The Lie algebra is sl(V ) (resp. sl(n,F)) consisting of lineartransformations (resp. matrices) of trace zero.

The orthogonal groups are defined in terms of symmetric bilinear forms on vectorspaces. Again let us consider a real or complex vector space V , let b be anynondegenerate symmetric bilinear form on V . Then the orthogonal group of b is

O(V, b) ≡ g ∈ GL(V ) : b(gv, gw) = b(v, w), for all v, w ∈ V .

The Lie algebra is

o(V, b) = X ∈ gl(V ) : b(Xv,w) + b(v,Xw) = 0, for all v, w ∈ V ,

with the bracket operation being the commutator. This may be checked by applying(1.2) as follows. Assuming exp(tX) ∈ O(V, b), taking the derivative (at t = 0) ofb(exp(tX)v, exp(tX)w) = b(v, w) gives b(Xv,w) + b(v,Xw) = 0. Conversely, if

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b(Xv,w) + b(v,Xw) = 0, then the curve c(s) = b(exp(sX)v, exp(sX)w) satisfies

c′(s) =d

dtb(exp((s + t)X)v, exp((s + t)X)w)|t=0

= b(X exp(tX)v, exp(tX)w) + b(exp(tX)v,X exp(tX)w)

= 0.

Therefore, c(s) is constant, so c(s) = c(0) = b(v, w). Therefore, exp(tX) ∈ O(V, b).

Recall that the matrix of a bilinear form with respect to a basis B = e1, . . . , enis the matrix B = (bij) where b(ei, ej) = bij . Since b is symmetric and nonde-generate, the matrix B is symmetric and has nonzero determinant. (Also, anysuch matrix determines a symmetric nondegenerate bilinear form by the formulab(ei, ej) = bij .) Under the identification of GL(V ) with matrices (using B) as in(1.1) the orthogonal groups (and Lie algebras) are realized as matrices by

O(n, B) = g ∈ GL(n,F) : gtBg = Bo(n, B) = X ∈ gl(n,F) : XtB + BX = 0.

It follows from this that the orthogonal groups are closed subgroups of GL(n,F).

A fact about bilinear forms ([4, §31],[8, §6.3]) is that for each symmetric bilinearform there is a basis so that the matrix B is

B = In, if F = C, and

B = Ip,q ≡(

Ip 00 −Iq

), with p + q = n, if F = R.

Here Im denotes the m×m identity matrix. The corresponding groups are denotedby O(n,C) and O(p, q), and similarly for the Lie algebras.

Imposing the further condition that the determinant be equal to one defines thespecial orthogonal groups SO(V, b), SO(n,C) and SO(p, q). The corresponding Liealgebras are so(V, b), etc. One may check (for example, by using the following blockform of the matrices in the Lie algebras that so(V, b) = o(V, b).

Example 1.3. In block form the Lie algebra so(p, q) is(A BBt D

): A ∈ Skewp(R), D ∈ Skewq(R) and B ∈ Mp×q(R)

.

Example 1.4. When (p, q) = (n, 0) the orthogonal group is denoted by O(n). Thisis a compact group. To see this note that gtg = In implies that ||g||2 =

∑g2

ij = n,

so O(n) is contained in the sphere of radius√

n in Rn2. Since any orthogonal group

is closed, O(n) is compact.

The definition of the symplectic group is similar to that of the orthogonal groups.For this, let ω be a nondegenerate antisymmetric (skew) bilinear form on V . Thedimension of V is necessarily even; we will denote it by 2n. Then the symplectic

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group is the following closed subgroup of GL(V ):

Sp(V, ω) = g ∈ GL(V ) : ω(gv, gw) = ω(v, w), for all v, w ∈ V .

The Lie algebra is

sp(V, ω) = X ∈ gl(V ) : ω(Xv,w) + ω(v,Xw) = 0, for all v, w ∈ V .

For either F = R or C there is a basis of V for which the matrix of ω is

J =(

0 In

−In 0

).

(See [8, §6.2].) Then the corresponding group and Lie algebra are

Sp(2n,F) = g ∈ GL(2n,F) : gtJg = Jsp(2n,F) = X ∈ gl(2n,F) : XtJ + JX = 0

= (

A BC −At

): A ∈ gl(n,F), B, C ∈ Symn(F).

The unitary groups are the symmetry groups of the hermitian forms. Recall thata hermitian form on a complex vector space V is a map h : V × V → C which islinear in the first variable and satisfies h(v, w) = h(w, v). Then

U(V, h) = g ∈ GL(V ) : h(gv, gw) = h(v, w), all v, wu(V, h) = g ∈ gl(V ) : h(Xv,w) + h(v,Xw) = 0, all v, w.

It is a fact that for each hermitian form there is a basis of V so that the matrixis Ip,q, for some p and q (with p + q = n). It follows that any unitary group isisomorphic to one of

U(p, q) = g ∈ GL(n,C) : gtIp,qg = Ip,q.

The Lie algebra is

u(p, q) = X ∈ gl(n,C) : XtIp,q + Ip,qX = 0.

The special unitary groups (resp. Lie algebras) have the additional requirementthat the determinant is one (resp. trace is zero). These are denoted by SU(p, q)and su(p, q).

In each of the examples above the Lie algebras are nearly simple. Recall thata Lie algebra g is simple means that g has no ideals other that g and 0, anddim(g) > 1. For example, gl(n,F) = sl(n,F) ⊕ z, where z = zIn : z ∈ F is thecenter and sl(n,F) is simple.

Definition 1.5. A Lie algebra g is semisimple if and only if g is the direct sum ofideals each of which is a simple Lie algebra. When g is a direct sum of ideals gss

and z, with gss semisimple and z abelian (and thus z equals the center of g), theng is called reductive.

Although it is not immediate, each of the classical Lie algebras is reductive.5

1.2. Cartan involutions and maximal compact subgroups. Of crucial im-portance in the representation theory of reductive Lie groups is the notion of amaximal compact subgroup. One reason for this is that the representation theoryof compact Lie groups is fairly simple. In particular, each representation (finiteor infinite dimensional) of a compact group is the direct sum of irreducible repre-sentations and every irreducible representation is finite dimensional. Furthermore,the irreducible representations of compact groups are easily parameterized and de-scribed. This is far from the situation for noncompact groups.

The maximal compact subgroups of the classical groups are easily described. Todo this we use the notion of Cartan involutions of the group G and its Lie algebrag. Define Θ : G → G by Θ(g) = (gt)−1 for any of the groups GL(n,F), SL(n,F),O(p, q), SO(p, q), Sp(n,F), U(p, q) and SU(p, q). (Note that for these groups Θ(g)is indeed in G when g ∈ G. Check this.) Now let θ : g → g be the differential of Θ.That is, θ(X) = −X

t. We refer to Θ and θ as Cartan involutions.

Since Cartan involutions have order two, g is the direct sum of the +1 and −1eigenspaces of θ. Let us write this decomposition as

g = k⊕ s, with k ≡ X ∈ g : θ(X) = X and s ≡ X ∈ g : θ(X) = −X.

Here are some easily checked facts.

(1) Θ is a group homomorphism and θ is a Lie algebra homomorphism.(2) k is a subalgebra of g.(3) [k, s] ⊂ s and [s, s] ⊂ k.(4) K ≡ g ∈ G : Θ(g) = g is a Lie subgroup with Lie algebra k.(5) K is compact.

For G = GL(n,R), K = O(n) and s = Symn(R). A fact from linear algebrais that any invertible matrix may be written in a unique way as the product ofan orthogonal matrix and a positive definite matrix (with orthogonal matrix first).This is called the polar decomposition of the matrix. See [4, §32] for a proof. Aslightly stronger statement is that

φ : O(n)× Symn(R) → GL(n,R)

φ(k, X) = k exp(X)

is a diffeomorphism.

Corollary 1.6. O(n) is a maximal compact subgroup of GL(n,R) in the sense thatit is not contained in another compact subgroup of GL(n,R).

Proof. Suppose that O(n) ⊂ K ′, with K ′ compact. Let k′ ∈ K ′ \ O(n). Writek′ = k exp(X), with X ∈ Symn(R), X 6= 0. Then exp(X) ∈ K ′. We claim thatexp(mX) cannot be in a compact set for all m ∈ Z. This will give a contradiction.To prove the claim, note that X has real eigenvalues (as X is symmetric), at least

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one of which is nonzero. Let λ be a nonzero eigenvalue and vλ to correspondingeigenvector. The eigenvalues of exp(mX) are emλ. Since λ 6= 0, emλ → ∞ asm →∞ (or as m → −∞). Now exp(mX)vλ →∞ in Rn. But by the compactnessof K ′, gvλ : g ∈ K ′ is bounded, and we arrive at a contradiction.

Continuing with the groups GL(n,F), SL(n,F), O(p, q), SO(p, q), Sp(n,F),U(p, q) and SU(p, q), we may define a bilinear form on g by

T (X, Y ) = Trace(XY ).

This form may be referred to as the trace form of g. Then T is Ad-invariant inthe sense that T (Ad(g)X, Ad(g)Y ) = T (X, Y ), for all X, Y ∈ g. Also, for X 6= 0,T (θ(X), X) = −Trace(X

tX) 6= 0, so T is nondegenerate. Furthermore, we have

the following facts.

(1) k and s are orthogonal with respect to T

(2) T |k×k is negative definite and T |s×s is positive definite.

Each of these may easily be checked.

1.3. Real reductive groups. We will now give a general definition of a realreductive group.

Definition 1.7. A real reductive group is a quadruple (G, K, θ, κ) with G a Liegroup, K a compact subgroup, θ : g → g a Lie algebra homomorphism with θ2 = I

and κ a nondegenerate, symmetric Ad invariant bilinear form, satisfying

(1) g is the Lie algebra of G and is reductive,(2) K has Lie algebra k and g = k ⊕ s (vector space direct sum of the ±1

eigenspaces with respect to θ),(3) k and s are orthogonal with respect to κ, κ|k×k is negative definite and κ|s×s

is positive definite,(4) φ : K × s → G, φ(k, X) = k exp(X) is a diffeomorphism.

EXERCISES

(1.1) Prove the claim that the Lie algebra of SL(n,F) is sl(n,F). You will need tocompute d

dt det(exp(tX))|t=0.

(1.2) Suppose that B and B′ are symmetric nondegenerate matrices and there isan invertible matrix Q so that QtBQ = B′. Prove that O(n, B) and O(n, B′) areisomorphic Lie groups (so the Lie algebras so(n, B) and so(n, B′) are isomorphic).When

B =(

0 In

In 0

),

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for which p, q is O(2n, B) isomorphic to O(p, q)? Write the Lie algebra so(2n, B) inblock form as in Example 1.3 and explicitly write down the isomorphism so(2n, B) 'so(p, q).

(1.3) Check that U(n) ≡ U(n, 0) is compact.

(1.4) Let g be a reductive Lie algebra and let κ be an Ad-invariant symmetricform as in the definition a real reductive group. Define 〈X ,Y 〉 = −κ(X, θ(Y )).Prove that 〈 , 〉 is positive definite on g, is invariant under Ad(K), and ad(X) is asymmetric (resp. skew symmetric) operator when X ∈ s (resp. X ∈ k).

(1.4) Let T be the trace form on one of the classical Lie algebras. Define 〈Y , Z〉 =−T (Y, θ(Z)). Show that 〈 , 〉 is positive definite and invariant under Ad(k) for allk ∈ K. Show that the linear transformation ad(X) : g → g is skew if X ∈ k and issymmetric if X ∈ s.

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LECTURE 2.Structure of reductive Lie algebras

Let g be a reductive Lie algebra over either R or C. The structure of g isdescribed in terms of ‘root systems’. In this lecture we will describe root systemsand how they give us much information about the structure of the Lie algebra.

First let us recall some basic linear algebra. Suppose that T : V → V is alinear transformation and V is a complex vector space. Then T is said to bediagonalizable if and only if there is a basis of V consisting of eigenvectors of T .Then, of course, with respect to such a basis the matrix of T is a diagonal matrix,with the eigenvalues as the diagonal entries. A diagonalizable linear transformationis also called semisimple. A matrix A is diagonalizable (i.e., semisimple) if thereis an invertible matrix Q so that QAQ−1 is a diagonal matrix. Thus, a lineartransformation is diagonalizable if and only if the matrix (with respect to any basis)is diagonalizable. The Jordan Decomposition Theorem states that T = S+N , withS semisimple, N nilpotent and [S, N ] = 0. Furthermore, such an expression for T

is unique. These facts may be found in [4, §24] and [7, §4.2]

Now consider a complex semisimple Lie algebra g. (We will return to reduc-tive Lie algebras shortly.) Consider ad(X) : g → g, for any X ∈ g. Since g issemisimple, the center of g is zero. Therefore, ad : g → gl(g) is one-to-one. Anelement X ∈ g is called semisimple (resp. nilpotent) if ad(X) is semisimple (resp.nilpotent). The Jordan decomposition of X ∈ g is defined as follows. Write theJordan decomposition ad(X) = S + N . One may prove ([7, §4.2]) that both S andN lie in the image of ad, that is there exist XS and XN in g so that S = ad(XS)and N = ad(XN ). Since ad is one-to-one, we may conclude that X = XS + XN .Furthermore, it is easy to check that [XS , XN ] = 0. Then X = XS + XN is calledthe Jordan decomposition of X. Therefore, the Jordan decomposition of X is theunique decomposition of X into a sum of semisimple and nilpotent elements whichcommute.

Suppose that π is any finite dimensional representation of g, i.e., π : g → gl(V )is a Lie algebra homomorphism with V a finite dimensional complex vector space.Then it is a fact that if π is injective, then X is semisimple (resp. nilpotent) if andonly if π(X) is a semisimple (resp. nilpotent) linear transformation of V . For aproof of this fact see [7, §6.4]. We may conclude that if g is realized as complexmatrices, then an element is semisimple in g precisely when it is semisimple as amatrix. Similarly for nilpotent.

When g is reductive then g = gss ⊕ z, where z is the center of g. Then, since thekernel of ad (equal to the center) may not be zero, we need to adjust the definitionof semisimple and nilpotent. In this case we say that X is semisimple if and onlyif ad(X) is semisimple, and X is nilpotent if and only if X is in gss and ad(X) is

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nilpotent. Now the Jordan decomposition of X is the unique decomposition of X

into a sum of commuting semisimple and nilpotent elements.

Definition 2.1. For a complex semisimple Lie algebra g, a Cartan subalgebra is amaximal subalgebra of commuting semisimple elements of g.

Example 2.2. For g = gl(n,C), a convenient Cartan subalgebra is the subalgebraof diagonal matrices:

(2.3) h =

a1

a2

. . .an

: ai ∈ C

.

For g = sp(2n,C),

(2.4) h =

a1

. . .an

−a1

. . .−an

: ai ∈ C

is a Cartan subalgebra.

Two key facts about complex semisimple Lie algebras ([7, §16.4 and §8.2]) are:

(a) Any two Cartan subalgebras are conjugate. (By this we mean that thereis some g in the identity component of Aut(g) so that Ad(g)h1 = h2. Theidentity component of Aut(g) is denoted by Int(g).)

(b) The centralizer of a Cartan subalgebra h is h.

A fact from linear algebra is that any commuting set of diagonalizable lineartransformations is simultaneously diagonalizable. In other words there is a basis ofV consisting of eigenvectors for all of the linear transformations in the set. Applyingthis fact to the set of all ad(H), for H in a Cartan subalgebra h, we may concludethat there is a basis of g so that each vector in this basis is an eigenvector for eachad(H).

Let X be such a common eigenvector. Then ad(H)X = [H,X] = α(H)X forsome scalar α(H). It follows easily that α : h → C is a linear function, that isα ∈ h∗. When α 6= 0 we say that X is a root vector and α is the correspondingroot. The set of roots is denoted by ∆ = ∆(h, g). Since the centralizer of h is h,the decomposition of g into common eigenspaces for ad(h) may be written as

g = h +⊕α∈∆

g(α), g(α) ≡ X ∈ g : [H,X] = α(H)X, for all H ∈ h

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Facts: (a) Suppose X ∈ g(α) and Y ∈ g(β). Then there are three possibilities for[X, Y ]:

[X, Y ] ∈ g(α+β), if α + β ∈ ∆

[X, Y ] ∈ h, if α + β = 0

[X, Y ] = 0, otherwise.

In the first two cases [X, Y ] 6= 0.(b) dim(g(α)) = 1 for each root α.(c) If α is a root, then the only multiples of α which are roots are ±α.

Example 2.5. Consider g = gl(n,C) and let h be the Cartan subalgebra of diag-onal matrices as in (2.3). Let

H =

a1

a2

. . .an

.

Let Eij be the matrix with 1 in the ij-place and 0′s elsewhere. Then (of course)H =

∑akEkk and it is easy to check that [H,Eij ] = (ai−aj)Eij . Therefore Eij,

along with a basis of h, is a basis of simultaneous eigenvectors for ad(H),H ∈ h.We conclude from this that if one writes εk(H) = ak, then ∆ = εi − εj : i 6=j, i, j = 1, 2, . . . , n.

Example 2.6. For g = sp(2n,C) and h as in (2.4), let

H =

a1

. . .an

−a1

. . .−an

.

Then an easy matrix calculation shows that

[H,Ei,j − En+j,n+i] = (ai − aj)(Ei,j − En+j,n+i)

[H,Ei,n+j + Ej,n+i] = (ai + aj)(Ei,n+j + Ej,n+i)

[H,En+i,j + En+j,i] = −(ai + aj)(En+i,j + En+j,i).

It follows that we have found a basis of simultaneous eigenvectors and

∆ = ±(εi ± εj) : 1 ≤ i < j ≤ n ∪ ±2εi : 1 ≤ i ≤ n.

Let g be a semisimple Lie algebra with a Cartan subalgebra h and correspondingroots ∆. We make a few definitions.

Definition 2.7. A subset Π = α1, . . . , αr ⊂ ∆ is called a simple system of rootsif and only if

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(a) Π is a basis of h∗ and(b) every root is of the form

∑i miαi, with all mi ≥ 0 or all mi ≤ 0.

If Π is a system of simple roots then the corresponding system of positive rootsconsists of all roots of the form

∑i miαi with all mi ≥ 0.

Example 2.8. In the case of g = sl(n,C), one may take Π = ε1 − ε2, ε2 −ε3, . . . , εn−1 − εn and the corresponding set of positive roots is ∆+ = εi − εj :1 ≤ i < j ≤ n. For g = sp(2n,C), a system of simple roots is Π = ε1 − ε2, ε2 −ε3, . . . , εn−1 − εn ∪ 2εn with corresponding positive roots ∆+ = (εi ± εj) : 1 ≤i < j ≤ n ∪ 2εi : 1 ≤ i ≤ n.

One more important feature of root systems is that κ, the Killing form on g,defines an inner product on the real span of the roots. Since κ is nondegenerate onh (see Exercise (2.3)), κ defines a nondegenerate symmetric form on h∗ as follows.First, for λ ∈ h∗ let Hλ ∈ h be defined by κ(Hλ,H) = λ(H),H ∈ h. Now set〈λ1, λ2〉 = κ(Hλ1 ,Hλ2). For the following properties see [7, §8.4,§8.5]:(a) 〈 , 〉 is positive definite on h∗R = spanR∆.

(b) For α, β ∈ ∆,2〈α , β〉〈β , β〉

are integers (called the Cartan integers).

The Weyl group is the group W which is generated by all reflections about planesorthogonal to the roots. Suppose that α is a root. Then the corresponding reflectionis given by the formula

σα(λ) = λ− 2〈λ , α〉〈α , α〉

, λ ∈ h∗.

It is a fact that each element of W permutes ∆.

In the two examples above the inner product described here is the inner productfor which εi is an orthonormal basis. The Cartan integers turn out to be 0,±1and ±2. For sl(n,C) the Weyl group is isomorphic to the symmetric group Sn.Elements of W are the linear transformations defined by permutations of the basisεi : i = 1, . . . , n. For sp(2n,C), W ' Zn × Sn.

We make a few comments on useful ways to describe positive systems of roots.Fix Λ ∈ h∗R so that Λ is regular in the sense that 〈Λ , α〉 6= 0 for all roots α. Then∆+

Λ ≡ α ∈ ∆ : 〈Λ , α〉 > 0 is a positive set of roots. In both of the examplesgiven, Λ =

∑mεm defines the positive system described. It is clear that there are

many positive systems. How many are there for each of the two examples?

Another way to describe positive sets of roots is to use a lexicographic order.For this let Hi be a basis of spanRHα : α ∈ ∆. Define α ∈ ∆+ if and onlyif α(H1) > 0, or α(Hi) = 0, i = 1, . . . , k and α(Hk+1) > 0 for some k. Then ∆+

is a positive system of roots. This method has the advantage that an ordering on12

hR = span∆ is determined by

λ ≥ µ if and only if λ(H1) > µ(H1), or

λ(Hi) = µ(Hi), i = 1, . . . , k and λ(Hk+1) > µ(Hk+1).(2.9)

A third way is to apply an element of W to any known positive system; w∆+ ispositive if ∆+ is positive. In fact, the action of W on the set of positive systems issimply transitive.

It is useful to define an abstract root system as follows.

Definition 2.10. An abstract root system is a pair (V,∆) where V is a real vectorspace with an inner product 〈 , 〉 and ∆ is a finite subset of V \ 0 so that thefollowing four properties hold.

(1) ∆ spans V .(2) If α ∈ ∆, then the only multiples of α which are in ∆ are ±α.(3) For each α ∈ ∆, the reflection σα permutes ∆.

(4) For α, β ∈ ∆,2〈α , β〉〈β , β〉

are integers.

The notions of simple and positive systems of roots, as well as the Weyl group,are defined as above. The Dynkin diagram of a root system is defined as follows.Choose a system of simple roots Π. There is a node for each α ∈ Π. For any pair

α, β ∈ Π connect the nodes for α and β by2〈α , β〉〈β , β〉

2〈β , α〉〈α , α〉

edges. In addition,

if one of the two simple roots α and β is longer than the other then place either> or < along the edges pointing in the direction of the longer root. The Dynkindiagrams may be classified. It is a fact that for each root system there is a uniqueDynkin diagram and two Dynkin diagrams are the same if and only if the rootsystems are isomorphic (in the appropriate sense). It is also a fact that there is aone to one correspondence between Dynkin diagrams and semisimple Lie algebras(up isomorphism) established as follows. Given a semisimple Lie algebra g choosea Cartan subalgebra h and form the root system ∆(h, g) and choose a set of simpleroots Π. This gives a Dynkin diagram as above. (It is not obvious, but the Dynkindiagram is independent of the choices of Cartan subalgebra and of simple systemof roots Π.) One can check that g is simple if and only if the corresponding Dynkindiagram is connected.

We have discussed root systems for complex semisimple Lie algebras. Now let usturn to real Lie algebras. A real Lie algebra g0 is semisimple if and only if g0⊗C is acomplex semisimple Lie algebra. So we may consider any real semisimple Lie algebraas a real subalgebra of a complex semisimple Lie algebra having complexificationequal to g.

Definition 2.11. A real form of a complex Lie algebra g is a real subalgebra g0 ofg so that g = g0 ⊕ ig0.

13

So let us fix a complex semisimple Lie algebra g. A conjugation of g is anconjugate linear homomorphism τ : g → g of order two. Another way to say thisis that τ is a real Lie algebra homomorphism so that τ(aX) = aX, for a ∈ C andX ∈ g, and τ2 = I.

For g = sl(n,C), typical conjugations are

X 7→ X

X 7→ −Xt

X 7→ −Ip,qXtIp,q, where p + q = n

X 7→ JXJ−1, with J =(

0 Im

−Im 0

), when n = 2m.

(2.12)

Proposition 2.13. Given a conjugation τ , g0 = X ∈ g : τ(X) = X is a realform. Conversely, if g0 is a real form then τ(X + iY ) = X − iY, for X, Y ∈ g0,defines a conjugation of g. This establishes a one-to-one correspondence betweenconjugations and real forms.

Proof. This (easy) proof is left to the reader. However, observe that for any X ∈ g

X =12(X + τ(X)) +

12(X − τ(X)) ∈ g0 + ig0.

For the example of sl(n,C), the real forms corresponding to the conjugationslisted in (2.12) are sl(n,R), su(n), su(p, q) and when n = 2m

sl(m,H) ≡( A B

−B A

).

(The last Lie algebra may be identified with the ‘trace’ zero linear transformationsof Hm, H the quaternions.)

It turns out that for a real form g0 of g (a complex semisimple Lie algebra),the structure may be described in terms of roots, however the situation is a bitmore complicated than for complex Lie algebras. This is essentially because alinear transformation of a real vector V space may have complex eigenvalues; theeigenvectors lie in the complexification of V , but not in V . An example of thisoccurs in g0 = su(n) where eigenspaces of ad(H), H a diagonal matrix, are not insu(n). Instead of root systems, the appropriate notion is that of ‘restricted roots’.

Let us fix a real Lie algebra g0 and a Cartan involution with decompositiong0 = k0 + s0. Fix a maximal abelian subspace a0 of s0. This will play the role ofCartan subalgebra. Note that (by Exercise (1.4)) when X ∈ a0, ad(X) : g0 → g0

is symmetric. Therefore, ad(X) is diagonalizable (as a real linear transformationof a real vector space). Since all ad(X) for X ∈ a0 mutually commute, they aresimultaneously diagonalizable.

14

Example 2.14. Let g0 = su(2, 4). Then we may choose a0 to be

a0 =

a1

a2

00

a2

a1

: ai ∈ R

.

The common eigenspaces for ad(A), A ∈ a0 are called restricted root spaces andthe eigenvalues are given by elements of a∗0, which we call restricted roots (whennonzero).

Here is the reason the term restricted roots is used. Consider zk0(a0), the subalge-bra of elements of k0 which commute with a0, and let t0 be a maximal commutativesubalgebra of zk0(a0). Then h = (t0)C + (a0)C is a Cartan subalgebra of g. (Checkthis.) Then we have the following.

Proposition 2.15. The set of restricted roots, denoted by Σ(a0, g0), is

β : β = α|a0 , α ∈ ∆(h, g) and β 6= 0.

The following example illustrates that ∆(a0, g0) is not a root system since part(2) of the definition fails. Also, the dimension of restricted root spaces need notbe one. Note that the (complexification of) the restricted root space for β is∑α∈∆(h,g),α|a0=β

g(α).

Example 2.16. Consider g0 = u(2, 4) with a0 as in the previous example. Thenwe may choose t0 so that

h0 = t0 + a0 =

it1 a1

it2 a2

it3 00 it4

a2 it2a1 it1

: ai ∈ R

.

This is conjugate to

a1 + it1a2 + it2

it3it4

−a2 + it2−a1 + it1

: ai ∈ R

,

therefore we have15

Roots Restricted roots Multiplicity±(a1 ± a2)± i(t1 − t2) ±(a1 ± a2) 2

±aj ± i(tj − tk) ±aj 4±2aj ±2aj 1

±i(t3 − t4) − −

The multiplicity is the dimension of the restricted root space.

The above discussion immediately gives the following decomposition of g0.

Proposition 2.17. Let a0 be as above and fix a system of positive roots Σ+ inΣ(a0, g0). Let

n =∑

β∈Σ+

g(β)0 , n =

∑β∈Σ+

g(−β)0 and m = zk0(a0).

Then g0 = m0 + a0 + n + n.

EXERCISES

(2.1) Consider the complex Lie algebra defined by the bilinear form

B =(

0 In

In 0

)(a Lie algebra isomorphic to so(2n,C)). Determine a convenient Cartan subalgebra.Do the same for so(2n + 1,C). (Hint: use the bilinear form

B =

0 0 In

0 1 0In 0 0

to define the Lie algebra.)

(2.2) Write down the root systems for the orthogonal Lie algebras using the Cartansubalgebras found in the preceding problem.

(2.3) Use the root space decomposition of g to show that sl(n,C), (n ≥ 2), so(n,C),(n ≥ 3) and sp(2n,C), (n ≥ 1) are simple Lie algebras.

(2.4) Find the Weyl group for the orthogonal groups.

(2.5) Suppose that κ is an invariant symmetric bilinear form on a semisimple Liealgebra g and that h is a Cartan subalgebra. Prove that the g(α) and g(β) areorthogonal with respect to κ unless α+β = 0. Prove that κ is nondegenerate on h.

(2.6) Prove that if Λ ∈ h∗R then

q ≡ h +⊕

〈Λ ,α〉>0

g(α)

is a subalgebra of g.16

(2.7) Prove that for h = a⊕ t, for each α ∈ ∆(h, g)

α(H) ∈ R, if H ∈ a0

α(H) ∈ iR, if H ∈ t0

holds.

17

LECTURE 3.Parabolic subgroups

Consider G = GL(n,R). We have already seen that G = K exp(s), the polardecomposition of G. There are two other decompositions which we know fromlinear algebra. One is called the ‘L-U’ decomposition. It says that most matricesg ∈ G may be written as g = LU with L a lower triangular matrix with 1’s onthe diagonal and U is an upper triangular matrix (with arbitrary nonzero diagonalentries). Such an expression is easily seen to be unique. The set of invertiblematrices with such a decomposition is open and dense in GL(n,C), but this is notimmediately clear. In order to find the L and U for a given matrix g, one mayperform row operations in the form of adding a multiple of a row to a later row. Aslong as no pivots are 0 one ends up with a matrix U , i.e., for some L (of the properform) L−1g = U . It may be seen that the pivots will never be zero if and only if allprincipal minors are nonzero. In other words, the subspace of all g = LU is openand dense in GL(n,R). This decomposition applies equally well for GL(n,C).

The other decomposition referred to above is called the Iwasawa decomposition.This states that each element of GL(n,R) may be written uniquely as g = kan

with k, a and n in the following groups.

K = O(n),

A =

a1

a2

. . .an

: aj > 0

, and

N =

1 ∗ ∗ ∗0 1 ∗ ∗

0 0. . . ∗

0 0 0 1

.

This is essentially a restatement of the Gram-Schmidt orthogonalization process([4, §15]) applied to the columns of g. Let us see how this happens. Write the usualdot product as 〈 , 〉. Letting gj be the j-th column, write

u1 = g1/||g1||,u2 = (g2 − 〈u1 , g2〉u1)/||g2 − 〈u1 , g2〉||,

...

un = (gn −n−1∑j=1

〈uj , gn〉uj)/||gn −n−1∑j=1

〈uj , gn〉uj ||,

18

an orthonormal basis of Rn. For each m

um = a−1m (gm −

m−1∑j=1

〈uj , gj〉um),with am > 0,

= a−1m (gm +

m−1∑j=1

bjmgj),

for some scalars bij . Now take n−1 to be the matrix with (i, j)-entry equal to 1 ifi = j, 0 if i > j and bij if i < j. Then it follows that g = kan where the columnsof k are the column vectors u1, . . . , un. Note that since ui is an orthonormal set,the matrix k is orthogonal.

All three of these decompositions, if properly stated, hold for general reductivegroups. In this lecture we will carefully state each.

The polar decomposition is referred to as the Cartan decomposition and is partof the definition of a real reductive group. In Exercise (3.2) you will prove thefollowing for the classical groups introduced in Lecture 1.

Suppose that G is a classical group defined in Lecture 1. Let K be the fixedpoints of Θ and let s be the −1-eigenspace of θ.

Proposition 3.1. φ : K × s → G, φ(k, X) = k exp(X), is a diffeomorphism.

Corollary 3.2. The group G is connected (resp. simply connected) if and only ifK is connected (resp. simply connected).

For the other two decompositions we will need some preparation. In particularwe will need the definition of and some information about parabolic subgroups. Letus first assume that g is a complex reductive Lie algebra.

A Borel subalgebra is a maximal solvable subalgebra of g. Let b be a Borelsubalgebra. Then b contains a Cartan subalgebra h, and b is the direct sum ofcertain root spaces. It can be shown that there is a positive system of roots ∆+ sothat

(3.3) b = h +∑

α∈∆+

g(α).

All Borel subalgebras of g are conjugate under Int(g). A parabolic subalgebra isany subalgebra which contains a Borel subalgebra. By the conjugacy of Borelsubalgebras, each parabolic subalgebra is conjugate to one containing a fixed Borelsubalgebra. Therefore to describe the parabolic subalgebras it suffices to fix aCartan subalgebra h, a positive system ∆+ and describe the parabolic subalgebrascontaining the Borel subalgebra of (3.3). Call Λ ∈ h∗ dominant if 〈Λ , α〉 ≥ 0, forall α ∈ ∆+. Then for each dominant Λ ∈ h∗ set

(3.4) p(Λ) = h +∑

α∈∆,〈Λ ,α〉≥0

g(α).

19

Then p(Λ) contains b, so is a parabolic subalgebra. Conversely, each parabolicsubalgebra containing b is of this form for some dominant Λ. Furthermore,

p(Λ) = l(Λ) + n(Λ),

where

l(Λ) = h +∑

〈Λ ,α〉=0

g(α), a reductive subalgebra, and

n(Λ) =∑

〈Λ ,α〉>0

g(α), the maximal nilpotent ideal in p(Λ).

We will call this decomposition the Levi decomposition.

It is useful to refine this description slightly as follows. Let Π be the system ofsimple roots (for our fixed ∆+). Define the fundamental weights λα, α ∈ Π by

2〈λα , β〉〈β , β〉

=

1 if β = α

0 if β 6= α,

for β ∈ Π.

Fact 3.5. The set of parabolic subalgebras containing b are in one-to-one correspon-dence with the subsets of Π. This one-to-one correspondence is given by S 7→ p(ΛS),where ΛS =

∑α∈S

λα.

The parabolic subalgebra p(ΛS) may be thought of as coming from the Dynkindiagram by crossing out the simple roots in S. Then the root system generatedby the remaining roots is ∆(l), and n is spanned by the root vectors for the otherpositive roots.

For example, when g = sp(2n,C) and

Π = αi = εi − εi+1 : i = 1, 2, . . . , n− 1 ∪ αn = 2εn,

the maximal parabolic subalgebras (that is, those not properly contained in anyother parabolic subalgebras other than g itself) correspond to those S containingjust one (simple) root. For m = 1, . . . , n− 1 and S = αm,

∆(l) = εi − εj : 1 ≤ i, j ≤ m ∪ ±(εi ± εj) : n + 1 ≤ i, j ≤ n,l ' gl(m,C)⊕ sp(2(n−m),C).

In these cases n is 2-step nilpotent (i.e., n is nonabelian and [n, [n, n]] = 0). WhenS = αn, l ' gl(n,C) and n is abelian.

If G is a connected complex Lie group with Lie algebra g, define any subgroupP which is the normalizer of some parabolic subalgebra to be a parabolic subgroup.Thus each parabolic subgroup is of the form P = NG(p) = g ∈ G : Ad(g)p ⊂ p.It may be checked that the Lie algebra of P is p. It is also a fact that P is connected.

Now consider a real reductive group G with Lie algebra g. Choose a maximalabelian subspace in s and call it amin. (The reason for this notation will become

20

clear in a moment.) Let tmin be a maximal abelian subalgebra of mmin = zk(amin).Then, as in Lecture 2, h = (tmin)C ⊕ (amin)C is a Cartan subalgebra of g. Fix apositive system of roots ∆+ ⊂ ∆(h, g) with the property that β : β = α|amin , α ∈∆+, β 6= 0 is a positive system of restricted roots, which we will call Σ+. Writenmin =

∑β∈Σ+

g(β). Then

pmin = mmin + amin + nmin

is called a minimal parabolic subalgebra. A subalgebra p of g is a parabolic subalgebraif p is conjugate (under Int(g)) to a subalgebra of g containing pmin. Note that(pmin)C is a parabolic subalgebra of gC, since it contains all root spaces for α ∈ ∆+.Therefore the complexification of each parabolic subalgebra in g is a parabolicsubalgebra in gC. The converse does not hold, the extreme case being when G iscompact (where amin = 0 and pmin = g).

Suppose p ⊃ pmin. We wish to describe the Levi decomposition p = l + n. Sincep contains amin,

p = mmin + amin +∑β∈Γ

g(β),

for some set Γ of restricted roots. The restricted roots in Γ fall into two types,those β for which −β ∈ Γ, and the others. Therefore, Γ = Γ0 ∪ Γ1, Γ0 = Γ ∩ (−Γ)and Γ1 = Γ \ Γ0. We will call

p =l + n, with(3.6)

l = mmin + amin +∑

β∈Γ0

g(β) and n =∑

β∈Γ1

g(β),

the Levi decomposition of p.

For the real reductive group G we define a parabolic subgroup to be a subgroupwhich is the normalizer of a parabolic subalgebra in g. Suppose the P = NG(p), pa parabolic subalgebra of g. Then, as for complex groups, the Lie algebra of P is p.Then there is a real reductive group L so that P = LN,N = exp(n). The group L

may be chosen so that Lie(L) = l, however L is typically not connected. Explicitly,if Le is the connected subgroup of G with Lie algebra l, then L = NK(p)Le. So,the normalizer in K meets each connected component of L.

A simple example is when G = GL(n,R) and P = Pmin is the normalizer ofpmin. Then NK(pmin) is the group of diagonal matrices with ±1 in the diagonalentries, Le is the group of all diagonal matrices with positive diagonal entries andL is the group of all invertible diagonal matrices.

For a parabolic subalgebra p ⊃ pmin with Levi decomposition p = l + n, theopposite parabolic subalgebra is

p = l + n, with n =∑β∈Γ

g(−β).

21

For the parabolic subgroup P = NK(p), the opposite parabolic subgroup is

(3.7) P = NK(p), so P = LN, with N = exp(n).

The opposite parabolic is defined similarly for a complex Lie algebra (or group).

The Iwasawa decomposition of a real reductive group is given in the followingtheorem.

Theorem 3.8. ([6, Ch. VI,§3]) The map

φ : K ×Amin ×Nmin → G

(k, a, n) 7→ kan

is a diffeomorphism.

Definition 3.9. A real flag manifold is any homogeneous space G/P where G isreal reductive group and P is a subgroup having Lie algebra which is a parabolicsubalgebra. Note that P need not be a parabolic subgroup; P lies between aparabolic subgroup and its identity component.

The Iwasawa decomposition implies that a real flag manifold is compact.

The generalization of the ‘L-U’ decomposition is the following.

Proposition 3.10. For any parabolic subgroup P = LN of a real reductive group,NP is a dense open subset of G. The expression for any g ∈ NP as g = n(g)p(g),with n(g) ∈ N and p(g) ∈ P is unique, and n and p are smooth functions of g.

We will refer to

(3.11) g = n(g)p(g)

as the Bruhat decomposition of g.

EXERCISES

(3.1) Prove that each parabolic subalgebra is conjugate to one of the form definedin Equation (3.4).

(3.2) Prove the fact stated in 3.5.

(3.3) Prove that if p is a parabolic subalgebra of a complex reductive Lie algebra,then there exists H0 ∈ h so that α(H0) ∈ Z, ∆(l) = α : α(H0) = 0 and∆(n) = α : α(H0) > 0

(3.4) Determine all parabolic subalgebras of sl(n,C) and so(n,C) for which n isabelian. Determine all parabolic subalgebras of sl(n,C) and so(n,C) for which n

is 2-step nilpotent.

(3.5) Let g be a real reductive Lie algebra. Show that pmin contains no subal-gebra having complexification which is a parabolic subalgebra of gC. (Thus theterminology ‘minimal parabolic subalgebra’ for pmin.)

22

(3.6) For each maximal parabolic subalgebra of sp(2n,R) determine the group L

of the Levi decomposition.

(3.7) Prove the Iwasawa decomposition and Prop. 3.10 for the group Sp(2n,R) byusing the corresponding statement for GL(2n,R).

23

LECTURE 4.Complex groups and Finite dimensional representations

In this lecture we will describe finite dimensional representations of complexreductive Lie algebras. The main theorem is Theorem 4.5, called the Theoremof the Highest Weight. See [7, §6,20 and 21] for details and proofs of most ofthe statements made in this Lecture. We will also make some comments on finitedimensional representations of real Lie algebras and Lie groups.

Suppose that g is any real or complex Lie algebra. Then a (finite dimensional)representation of g is a pair (σ,E) with E a finite dimensional complex vector spaceand σ a Lie algebra homomorphism

σ : g → gl(E).

Sometimes we refer to σ or E as the representation. We also say that (σ,E) is arepresentation on E. Note that σ is a real or complex homomorphism depending onwhether g is a real or complex Lie algebra, but in either case E is a complex vectorspace. (Sometimes it is useful to consider representations on real vector spaces,however we will not do this.)

Fix for a moment a Lie algebra g and a representation (σ,E) of g. There are anumber of standard definitions to be made. First, if dimC(E) = n then we say σ isn-dimensional. A subspace F of E is a g-invariant (or just invariant) subspace ifσ(X)v ∈ F for all v ∈ F and X ∈ g. We say that the representation is irreducibleif there are no invariant subspaces other than 0 and E. If F is an invariantsubspace, then we refer to F as a subrepresentation of (σ,E); more precisely, thesubrepresentation is the representation (σF , F ) defined by σF (X) = σ(X)|F , foreach X ∈ g. In the case that E = F1 ⊕ · · · ⊕ Fm with each Fi an invariantsubspace, we say that E is direct sum of the subrepresentations F1, . . . , Fm. IfE may be written as the direct sum of irreducible subrepresentations, then σ issaid to be completely reducible. Given two representations (σi, Ei), i = 1, 2, of g, anintertwining map (also referred to a g-homomorphism) is a linear map T : E1 → E2

so that σ2(X)T = T σ1(X), for all X ∈ g. The space of all such maps is denotedby Homg(E1, E2). The representations σ1 and σ2 are equivalent if there is aninvertible g-homomorphism T : E1 → E2.

A simple observation is that for any intertwining map from E1 to E2, bothKer(T ) and Im(T ) are invariant subspaces. It follows immediately that if E1 andE2 are irreducible, then T is either invertible or identically zero.

Lemma 4.1. (Schur’s Lemma) If (σ,E) is an irreducible representation of E, thenHomg(E,E) = cIE : c ∈ C.

24

Proof. Suppose that T ∈ Homg(E,E). Then T − λI is also a g-homomorphism.Therefore T − λI is either invertible or zero. Choosing λ to be an eigenvalue of T

ensures that T − λI is not invertible. Then T = λI holds.

Theorem 4.2. (Weyl’s Theorem) Every finite dimensional representation of asemisimple Lie algebra is completely reducible.

Proof. See [7, §6] for a proof.

An example of a finite dimensional representation is the adjoint representation(ad, g) on g. This is defined by ad(X)(Y ) = [X, Y ]. An invariant subspace isan ideal and the adjoint representation is irreducible if and only if g is simple orone-dimensional.

There are several natural constructions of representations from given represen-tations. We will give the definitions of a few of these below. Let (σ,E), (σj , Ej)and (ρ, F ) be representations of a Lie algebra g.

(a) The dual of σ is the representation (σ∗, E∗) defined by (σ∗(X)µ)(v) = −µ(σ(X)v),for X ∈ g, µ ∈ E∗ and v ∈ E.(b) There are three representations on HomC(E,F ). Let X ∈ g, v ∈ E andT ∈ HomC(E,F ). Then three different representations are defined by

(X · T )(v) = T (σ(X)v)− ρ(X)T (v).

(X · T )(v) = ρ(X)T (v),

(X · T )(v) = −T (σ(X)v),(4.3)

(c) The direct sum of (σj , Ej) is the representation on ⊕Ej is defined by

X · (v1 + · · ·+ vm) = σ1(X)v1 + · · ·+ σm(X)vm.

(d) The tensor product of (σj , Ej) is the representation on E1 ⊗ · · · ⊗ Em definedby

X · (v1 ⊗ · · · ⊗ vm) = (σ1(X)v1)⊗ v2 ⊗ · · · ⊗ vm

+ v1 ⊗ (σ2(X)v2)⊗ · · · ⊗ vm + · · ·+ v1 ⊗ · · · ⊗ vm−1 ⊗ (σm(X)vm).

(e) A representation on the exterior product∧m(E) is defined by

X · (v1 ∧ · · · ∧ vm) = (σ(X)v1) ∧ v2 ∧ · · · ∧ vm

+ v1 ∧ (σ(X)v2) ∧ · · · ∧ vm + · · ·+ v1 ∧ · · · ∧ vm−1 ∧ (σ(X)vm).

(f) A representation on Sm(E), the symmetric tensors of degree m, is defined as in(d).

Now let us turn to representations of complex semisimple Lie algebras. So letg be a complex semisimple Lie algebra and fix a representation (σ,E) of g. A keyto understanding σ is to understand the weights. Fix a Cartan subalgebra h of g.

25

A weight vector is a nonzero vector v ∈ E so that v is a common eigenvector forσ(H),H ∈ h. So there is a µ ∈ h∗ so that σ(H)v = µ(H)v, for all H ∈ h. Inthis case µ is called a weight. The set of weights is denoted by ∆(E), and givenµ ∈ ∆(E)

Eµ = v ∈ E : σ(H)v = µ(H)v, for all H ∈ h ∪ 0is the space of weight vectors (along with 0) corresponding to µ. As mentioned inLecture 2, each σ(H) is semisimple for H ∈ h. Therefore,

E =⊕

µ∈∆(E)

Eµ.

An example which we have already encountered is the adjoint representation.The weights are the roots and 0. The dimension of each weight space is one, exceptfor the weight 0 which has dimension equal to the dimension of h.

Another example is the ‘standard representation’ of sp(2n,C) on C2n. Withrespect to the Cartan subalgebra of Example 2.6, the weights are ±εj : i =1, . . . , n.

An important observation is that

v ∈ Eµ, X ∈ g(α) implies σ(X)v ∈ Eµ+α.

This holds because when H ∈ h we have σ(H)σ(X)v = σ([H,X])v + σ(X)σ(H)v= (α(H) + µ(H))σ(X)v.

Now fix a positive system of roots ∆+. Recall that b = h + n, n =∑

α∈∆+ g(α),is a Borel subalgebra of g. A vector v ∈ E is called a highest weight vector if andonly if σ(X)v = 0 for all X ∈ n. The space of highest weight vectors is denoted byEn. Since E is finite dimensional, En 6= 0.

The following statements are proved in [7, §20,21].

Proposition 4.4. Let (σ,E) be an irreducible finite dimensional representation ofa semisimple Lie algebra g. Let b = h + n be a Borel subalgebra corresponding to apositive system ∆+. Then the following hold.

(1) dim(En) = 1 and the weight λ of a highest weight vector (called a highestweight) is the unique weight satisfying λ + α /∈ ∆(E) for all α ∈ ∆+.Furthermore, dim(Eλ) = 1.

(2) Each weight is of the form λ−∑

α∈∆+ mαα, with mα ∈ Z≥0.(3) The Weyl group W acts on ∆(E) and dim(Eµ) = dim(Ewµ).(4) All weights are contained in the convex hull of wλ : w ∈ W.

The set of dominant integral elements of h∗ is

Λ+ = µ ∈ h∗ :2〈µ , α〉〈α , α〉

∈ Z≥0, for all α ∈ ∆+

With the hypothesis of the proposition we have the following, which is often referredto as the Theorem of the Highest Weight.

26

Theorem 4.5. The highest weight of E is in Λ+. The map taking the irreduciblerepresentation E to its highest weight defines a one-to-one correspondence

Λ+ ↔ irreducible finite dimensional representations of g.

Now we turn to complex reductive Lie algebras. Therefore we will let g be ofthe form g = gss ⊕ z, that is, g is the direct sum of its center z and a semisimpleLie algebra gss. Fix a Cartan subalgebra h = hss⊕ z, with hss a Cartan subalgebraof gss. If (σ,E) is an irreducible finite dimensional representation of g, then bySchur’s Lemma for each Z ∈ z there is a constant c so that σ(Z) = cIE . Thereforethe restriction to gss is irreducible, so there is a unique highest weight vector andhighest weight. On the other hand, given λ ∈ h∗ which is dominant and integralthere is an irreducible representation (σss, E) of gss with highest weight λ|hss .Defining σ(X + Z) = σss(X) + λ(Z)IE for X ∈ gss and Z ∈ z gives an irreduciblerepresentation of g with highest weight λ. Therefore the Theorem of the Highestweight holds for complex reductive Lie algebras.

Let g now be a real Lie algebra, and let gC be its complexification. Given arepresentation (σ,E) of g there is a representation (σC, E) defined by σC(X +iY ) = σ(X) + iσ(Y ) for X, Y ∈ g. Then σC is irreducible (resp. completelyreducible) if and only if σ is irreducible (resp. completely reducible). It followsthat the irreducible representations of a real reductive Lie algebra are in one-to-onecorrespondence with those of gC.

If G is a Lie group, a finite dimensional representation is a pair (ρ,E) with E

a complex vector space and ρ : G → GL(E) is a smooth group homomorphism.The definitions of invariant subspace, irreducible, etc are as already defined forrepresentations of Lie algebras. The constructions given for representation of Liealgebras are similar. Let g ∈ G.(a) The dual E∗: (σ∗(g)µ)(v) = µ(g−1v).

(b) HomC(E,F ):

(g · T )(v) = ρ(g−1)T (σ(g)v).

(g · T )(v) = ρ(g)T (v),

(g · T )(v) = T (σ(g−1)v),

(c) The direct sum of (σj , Ej) is the representation on ⊕Ej is defined by

g · (v1 + · · ·+ vm) = σ1(g)v1 + · · ·+ σm(g)vm.

(d) Tensor product:

g · (v1 ⊗ · · · ⊗ vm) = (σ1(g)v1)⊗ (σ2(g)v2)⊗ · · · ⊗ (σm(g)vm).

(e) Exterior product

g · (v1 ∧ · · · ∧ vm) = (σ(g)v1) ∧ (σ(g)v2) ∧ · · · ∧ (σ(g)vm).27

(f) Sm(E): defined as in (d).

(g) P (E): (g · p)(v) = p(g−1v).

Given a representation (µ,E) of a Lie group G, a representation of its Lie algebrais defined by

dµ(X)v =d

dtµ(exp(tX))v|t=0, for X ∈ g.

A basic fact is that for a connected Lie group G with Lie algebra g, a represen-tation µ of G is irreducible if and only if dµ is an irreducible representation of g.When G is not connected such a statement fails.

EXERCISES

(4.1) Give an example of a 2-dimensional Lie algebra g and 2-dimensional repre-sentation of g which is not completely reducible.

(4.2) Show that given a representation (σ,E) of G, the representations Sm(E) andP (E∗) are equivalent.

(4.3) Show that for a representation (σ,E) of g, E∗ ⊗ E is equivalent to End(E).

(4.4) Let g be a semisimple Lie algebra and b = h + n a Borel subalgebra. Showthat for any finite dimensional representation E of g, En 6= 0.

(4.5) Write down the highest weight of the adjoint representation of sl(n,C),so(n,C) and sp(2n,C).

28

LECTURE 5.Homogeneous spaces, differential operators

and the enveloping algebra

5.1. Homogeneous spaces. For any group G we say that a group action of G ona set M is a map

(5.1) G×M → M, written as (g,m) 7→ g ·m

satisfying g1 · (g2 ·m) = (g1g2) ·m and e ·m = m, for all g1, g2 ∈ G and m ∈ M . Anaction is called transitive if and only if whenever m1,m2 ∈ M , there is some g ∈ G

so that g ·m1 = m2.

When G is a topological group and M a topological space, we say that an actionis a continuous action if and only if (5.1) is continuous. If G is a Lie group andM is a differentiable manifold, a continuous action is called a smooth action if andonly if (5.1) is smooth.

Suppose G is a topological group and H a subgroup. Then the coset space maybe given the quotient topology. Letting π : G → G/H be the natural quotientmap (π(g) = gH), the quotient topology is defined by the property that U ⊂ G/H

is open if and only if π−1(U) ⊂ G is open. Note that π is continuous and open.Furthermore, G×G/H → G/H is continuous and for each g ∈ G, left translationτg : G/H → G/H (xH 7→ gxH) is a homeomorphism.

Proposition 5.2. If a continuous action of a locally compact, second countabletopological group on a locally compact Hausdorff space M is transitive, then thefollowing hold.(a) For each p ∈ M , Gp ≡ StabG(p) ≡ g ∈ G : g · p = p is a closed subgroup ofG.(b) gGp 7→ g · p is a homeomorphism G/Gp ' M .

For a proof see [6, Ch. II,§3].

Proposition 5.3. For any closed subgroup H of a Lie group G, the topologicalspace G/H has the unique structure of smooth manifold having the property thatG × G/H → G/H, (g, xH) 7→ gxH is smooth. For each X ∈ g a vector field onG/H is defined by

(Xf)(gH) =d

dtf(exp(−tX)gH)|t=0.

This gives an isomorphism TgH(G/H) ' g/Ad(g)h, for each g ∈ G.

Local coordinates for G/H may be described as follows. It is a fact about theexponential map that if g = V1⊕V2 (as vector spaces) then there are neighborhoodsUi of 0 in Vi so that

U1 ⊕ U2 → exp(U1) exp(U2)

(X1, X2) 7→ exp(X1) exp(X2)(5.4)

29

is a diffeomorphism. To construct local coordinates on G/H, choose a vector spacecomplement m to h = Lie(H) in g. Then there are neighborhoods U1 of 0 ∈ m andU2 of 0 ∈ h so that (5.4) is a diffeomorphism. Then a basis X1, . . . , Xm of m

determines local coordinates on the neighborhood gπ(exp(U1)) of gH ∈ G/H by

(5.5) ϕ(x1, . . . , xm) = g exp(x1X1 + · · ·xmXm).

For details, and a proof of the following proposition, see [6, Ch II, §4].

Proposition 5.6. If a smooth action of a Lie group G on a manifold M is tran-sitive, then gGp 7→ g · p is a diffeomorphism G/Gp ' M .

When H is a closed subgroup of a Lie group G, we call G/H a homogeneousspace.

An example is that real projective space RP(n) is a homogeneous space. Theaction of GL(n + 1,R) by g · [v] = [gv] is smooth (why?). The stabilizer of thecolumn vector p = [1, 0, 0, . . . , 0]t is

P =(

a b0 D

): a ∈ R×, b ∈ Rn, D ∈ GL(n,R)

.

The action is transitive, so RP(n) is diffeomorphic to GL(n + 1,R)/P .

5.2. Invariant differential operators. Let M be a smooth manifold of dimen-sion m. Let (U,ϕ) be a coordinate system in a neighborhood of a point p ∈ M .

Consider the partial derivatives∂

∂xidefined on U . It will be useful for us to use the

multi-index notation for higher derivatives. For this let α = (α1, . . . , αm) ∈ Zm≥0

and write∂α

∂xα=

∂α1

∂xα1. . .

∂αm

∂xαm. Then a differential operator on M may be defined

to be a linear map from C∞(M) to itself which, in any local coordinate system,has the form

(5.7) Df =∑

aα∂αf

∂xα,

with aα ∈ C∞(M). The space of differential operators on a manifold M is denotedby D(M). The space D is an algebra and (hence) a module over C∞(M).

When G is a Lie group of dimension n and H a closed subgroup of G, a differentialoperator on M = G/H is said to be G-invariant if `g(Df) = D(`gf), where `g isleft translation by g (defined by (`gf)(x) = f(g−1x)). The algebra of G-invariantdifferential operators is denoted by DG(G/H).

Let us first consider the case when H = e, i.e., M = G. For X ∈ g define righttranslation by X by the formula

(R(X)f)(x) =d

dtf(x exp(tX))|t=0, for f ∈ C∞(G).

30

Then R(X) is a left invariant vector field on G. In particular, R(X) is in DG(G).In the local coordinates (5.5) on gU ,

(R(Xj)f)(g) =d

dtg(g exp(tXj))|t=0

=d

dtf(ϕ(0, . . . , 0, t, 0, . . . , 0))|t=0

=∂f

∂xj

∣∣∣x=0

.

When X ∈ gC the above formula for R(X) does not make sense. In this caseright translation may be defined by R(X1 + iX2) = R(X1) + iR(X2), X1, X2 ∈ g.Therefore, for X ∈ gC, R(X) ∈ DG(G). Therefore we may conclude the following.

Corollary 5.8. If Y1, . . . YN ∈ gC, then R(Y1) · · ·R(YN ) is a G-invariant differen-tial operator on G.

We now show that the differential operators R(Y1) · · ·R(YN ), Yi ∈ gC, spanDG(G). In fact, we show a little more. For any basis X1, . . . , Xn of gC writeR(Xα) for R(X1)α1 · · ·R(Xn)αn for any multi-index α. We will show that R(Xα) :α ∈ Zn

≥0 is a basis of DG(G). The technical fact we use is in the following lemma.

Lemma 5.9. Let M be any manifold with a set of vector fields ξ1, . . . , ξm with theproperty that for each x ∈ M , ξ1,x, . . . , ξm,x is a basis of the complexified tangentspace Tx(M). Then ξα ≡ ξα1 · · · ξαm : α ∈ Z≥0 is a basis over C∞(M) of D(M).

Proof. The proof is in [15, §1.1].

It is immediate from the Lemma that R(Xα) : α ∈ Z≥0 is independent. Italso follows from the lemma that each differential operator in D(G) is of the form∑

aαR(Xα), aα ∈ C∞(G).

For D ∈ DG(G), the G-invariance gives

(Df)(g) = (D(`g−1f)(e)

=∑

aα(e)(R(Xα)(`g−1f)(e)

=∑

aα(e)(R(Xα)f)(g).

Therefore, D =∑

AαR(Xα), Aα = aα(e) ∈ C. We have shown the following.

Proposition 5.10. R(Xα) : α ∈ Z≥0 is a basis (as complex vector space) ofDG(G).

5.3. The enveloping algebra. The description of DG(G) given above suggeststhe definition of the enveloping algebra U(gC).

31

We will give the construction for Lie algebras over C, any base field may beconsidered in the same manner. Consider the tensor algebra of gC,

T (gC) = C⊕ gC ⊕ (gC ⊗ gC)⊕ · · ·

and the ideal I generated by

X ⊗ Y − Y ⊗X − [X, Y ] : X, Y ∈ gC.

Definition 5.11. The enveloping algebra of gC is U(gC) = T (gC)/I.

Let π : T (gC) → U(gC) be the natural quotient map. Then one typically writesY1Y2 · · ·YN for π(Y1 ⊗ Y2 ⊗ · · · ⊗ YN ), for Yi ∈ gC.

Theorem 5.12. (Poincare-Birkoff-Witt Theorem, see [7, §17.3]) For any basisX1, . . . , Xn of gC, Xα = Xαi

1 · · ·Xαnn : α ∈ Zn

≥0 is a basis of U(g).

We may now restate what the characterization of D given earlier.

Proposition 5.13. As algebras, DG(G) ' U(gC).

EXERCISES

(5.1) Give an example for which the conclusion of Prop. 5.2 fails if G is not secondcountable.

(5.2) Let G = H2n+1 be the Heisenberg group defined as R2n+1 with multiplicationgiven by (x, y, z) · (x′, y′, z′) = (x+x′, y + y′, z + z′ +x · y′) (where x · y′ is the usualdot product on Rn). Find generators of DG(G) in terms of ∂

∂xi, ∂

∂yiand ∂

∂t

(5.3) Prove that there is a one-to-one correspondence between complex representa-tions of a complex Lie algebra and modules over the enveloping algebra.

(5.4) Suppose that G is a Lie group with Lie algebra g. Show that the adjointaction of G on g extends to a representation on U(g) (by Ad(g)(Y1Y2 · · ·YN ) =Ad(g)(Y1)Ad(g)(Y2) · · ·Ad(g)(YN )). Show that this representation is G-finite inthe sense that spanAd(g)u : g ∈ G is finite dimensional for any u ∈ U(g).Describe the differential of the Ad-representation on U(g). Show that if G is areductive group then under the Ad-representation U(g) decomposes into a directsum of irreducible finite dimensional representations. (Hint: you may wish toconsider the filtration Un = spanY1Y2 · · ·Yk : Yi ∈ g, k ≤ n.)

(5.5) Suppose that h is a Lie subalgebra of the complex Lie algebra g and m isa vector space complement of h in g. Fix a basis Y1, . . . , Ym of m. Show thatU(g) = spanY β : β ∈ Zm

≥0 ⊕ U(g)h.

32

LECTURE 6.Differential operators between sections of bundles, I

Suppose that E and F are complex vector spaces and C∞(Rn, E) is the space ofsmooth E-valued function in Rn. Then a linear map D : C∞(Rn, E) → C∞(Rn, F )is a differential operator if it is of the form

(6.1) D =∑

aα∂α

∂xα

where aα ∈ C∞(Rn,Hom(E,F )).

Now suppose that M is a smooth manifold and πE : E → M and πF : F → M

are smooth vector bundles over M . Then for any p ∈ M , there is a coordinateneighborhood U of p so that we have local trivializations π−1

E (U) ' U × E andπ−1

F (U) ' U ×F . A linear map D from smooth sections of E to smooth sections ofF is a differential operator if in each local trivialization D is in the form of (6.1).The smallest integer k with

∑αj ≤ k, for all α for which aα 6= 0, is called the order

of D.

Example 6.2. Let M be any smooth manifold and E (resp. F) be the bun-dle of p-forms (resp. p + 1-forms) on M . Locally a p-form is expressed as ω =∑

j1<···<jpaj1,...,jpdxj1 ∧ . . . dxjp . Then exterior differentiation is an example of a

first order differential operator: dω =∑

j1<···<jp

∑nk=1

∂aj1,...,jp

∂xkdxk∧dxj1∧. . . dxjp .

Now let H be a closed subgroup of a Lie group G, and let σ : G → GL(E)be a (smooth) representation. Then a vector bundle over G/H with fiber E overthe identity coset may be constructed as follows. Define an equivalence relation onG×E by (gh, v) ∼ (g, σ(h)v), for g ∈ G, h ∈ H and v ∈ E. Then set E = G×E/ ∼.The notation G×

HE is often used for E . Let π : E → G/H be the map (g, v) 7→ gH.

Proposition 6.3. E has a unique structure of smooth manifold so that π : E →G/H is a smooth vector bundle. The space of smooth sections may be identifiedwith

C∞(G/H, E) ≡ f : G → E | f is smooth and

f(gh) = σ(h−1)f(g), for all h ∈ H, g ∈ G.

Exercise (6.2) asks for a proof of this proposition.

A homogeneous vector bundle over G/H is a finite rank bundle π : E → G/H

with an action of G satisfying

(1) π(gv) = gπ(v) for all g ∈ G and v ∈ E, and(2) the action of H on π−1(eH) is by linear transformations.

It is a fact that each homogeneous vector bundle is equivalent to G ×H

E for the

representation of H on E ≡ π−1(eH).33

Example 6.4. Here are a few standard examples.

E Eg/h Tangent bundle

(g/h)∗ Cotangent bundle∧p(g/h)∗ Bundle of p-forms

Example 6.5. Let Z = CP(n), complex projective n-space. Therefore, Z =complex lines in Cn+1 through the origin. We will denote the line through v ∈Cn+1 by [v]. Then Z is a smooth manifold in a natural way. Gl(n + 1,C) actstransitively on Z, so Z ' GL(n + 1,C)/P where

P =(

a b0 d

), a ∈ C×, b ∈ Cn and d ∈ GL(n,C)

(the stabilizer of the column vector e1 ' [(1, 0, . . . , 0)t]). The family of one dimen-sional representations

χm

((a b0 d

))= am

define the powers of the canonical bundle L1. Let’s check that L1 is the homoge-neous bundle for the character χ1. The canonical bundle is defined as the disjointunion of all lines through the origin of Cn+1. The fiber over a point Z in projectivespace is the line Z. It is enough to see that L1 is a homogeneous vector bundle andthat the action of P on the fiber π−1([e1]) = Ce1 is by χ1:(

a b0 d

)ze1 = aze1 = χ1

((a b0 d

))(ze1).

Our interest will be in homogeneous vector bundles. The space of sectionsC∞(G/H, E) is a representation of G. The action is by left translation of sections:(`gf)(x) = f(g−1x). It is also the case that the space of sections is a representationof the Lie algebra g of G: (X · f)(x) = d

dtf(exp(−tX)x)|t=0.

So let (σ,E) and (%, F ) be finite dimensional representation of H and let E →G/H and F → G/H be the corresponding homogeneous vector bundles. Thenthe space of differential operators from E → G/H to F → G/H will be denoted byD(E ,F). Such a differential operator D is G-invariant means that D(`gf) = `gD(f)for all g ∈ G and all smooth sections f of E . The space of G-invariant differentialoperators will be denoted by DG(E ,F).

We will identify the space DG(E ,F) in terms of the enveloping algebra. Thestatement is contained in Prop. 6.11. It is a generalization of Prop. 5.13. Thestatement and proof require a little preparation.

Consider the Lie algebra h. Then the representation of h on E gives E a U(h)-module structure defined by Y1 · · ·YNv = σ(Y1) · · ·σ(YN )v, for Y1, . . . , YN ∈ g. (SeeExercise 5.3.) Consider the representation of h on HomC(E,F ) as defined in (4.3).Then HomC(E,F ) is a U(h)-module as follows. There is an antiautomorphism ofU(h) defined (Y1Y2 . . . YN ) = (−1)NYN . . . Y2Y1, whenever Yj ∈ h, and extending

34

linearly to all of U(h). Then for T ∈ HomC(E,F ), u ∈ U(h) the module actionis given by (u · T )(v) = T (uv). Let J be the submodule of U(g) ⊗ HomC(E,F )generated by

uh⊗ T − u⊗ (T σ(h)), for u ∈ U(g), h ∈ U(h) and T ∈ Hom(E,F ).

Then we define1

U(g) ⊗U(h)

HomC(E,F ) = U(g)⊗HomC(E,F )/J .

This is the usual tensor product of the right U(h)-module U(g) with the left U(h)-module HomC(E,F ) (left action defined by h · T = T σ(h)).

Now consider any finite dimensional vector space F . Let C∞(G, F ) be the spaceof smooth functions from G to F . We will temporarily use the notation of DG(E , F )for the vector space of linear maps from C∞(G/H, E) → C∞(G, F ) with the fol-lowing two properties. (1) In any local coordinate system

(6.6) D =∑α

aα∂α

∂xα,

with aα a smooth function with values in HomC(E,F ), and (2) for each g ∈ G,

D(`gf) = `gD(f).

For each∑

uj ⊗ Tj ∈ U(g)⊗HomC(E,F ), define an element of DG(E , F ) by

(∑

uj ⊗ Tj ) f ≡∑

j

Tj(R(uj)f)

for all f ∈ C∞(G/H, E). For Y ∈ h, u ∈ U(g) and T ∈ HomC(E,F ),

((uY ⊗ T ) f)(g) = T ((R(u)R(Y )f)(g))

= T (d

dtσ(exp(−tY ))(R(u)f)(g))|t=0

=((u⊗ T dσ(Y )) f

)(g).

Therefore J ˜ = 0, so there is a well-defined map U(g) ⊗U(h)

HomC(E,F ) into

DG(E , F ).

The following lemma describes DG(E , F ) in terms of the enveloping algebra.

Lemma 6.7. The map∑

uj ⊗ Tj 7→ (∑

uj ⊗ Tj ) is an isomorphism

U(g) ⊗U(h)

HomC(E,F ) → DG(E , F ).

1The definition here is that of the usual tensor product for modules over an algebra. If M is aright module over an algebra R and N is a left module over R , then M⊗RN is M⊗CN/J , whereJ is the subspace generated by all mr⊗n−m⊗ rn. If M is also a left module for another algebraS, then M⊗RN is a left S module by s · (m⊗ n) = (sm)⊗ n.

35

Proof. We need to see that the map is surjective and injective. For surjectivity,let D : C∞(G/H, E) → C∞(G, F ) be G-invariant and satisfy (6.6). By applying

Lemma 5.9 we may write , in a neighborhood of e, each∂α

∂xαas a linear combination

(over C∞) of R(Xgb) for various multi-indices β. Therefore,

(Df)(g) =∑α

aα(g)(R(Xα)f)(g)

with aα ∈ C∞(G, Hom(E,F )) and g in a neighborhood of e. By G-invariance

Df(g) = (`g−1D(f))(e)

= D(`g−1f)(e)

=∑α

aα(e)R(Xα)(`g−1f)(e)

=∑α

aα(e)(R(Xα)f)(g)

=∑α

((Xα ⊗ aα(e))˜f)(g).

For injectivity we will use two fairly easy facts. The proofs are left as exercises.Let Y1, . . . , Ym be a basis of m (with m any vector space complement of h in g).

Fact 6.8. If V is any U(h)-module, then

U(g) ⊗U(h)

V ' spanY β : β ∈ Zm≥0 ⊗

CV

as vector spaces, and if vj is a basis of V , then Xβ⊗vj is a basis of U(g) ⊗U(h)

V .

(The proof is similar to the proof in Exercise (5.5).)

Fact 6.9. There is a neighborhood U of 0 in m so that

exp : U → π(exp(U))

is a diffeomorphism, and

C∞(G/H,E) → C∞(exp(U), E)

f 7→ f |exp(U)

is onto.

(The idea of the proof is to begin with a coordinate neighborhood V of 0 in m.Take U to be an open ball about 0 in m (with respect to any metric) containedon V . Now let F ∈ C∞(exp(U), E). Then F extends to a smooth function F0 :exp(U) → E (why?). Now set f0(exp(X)h) = σ−1(h)f0(exp(X)), X ∈ V, h ∈ H.Apply the C∞-Urysohn Lemma to extend to a function f n all of G.)

Now we return to injectivity of our map∑

uj ⊗ Tj 7→ (∑

uj ⊗ Tj ) . Lemma 5.9says that R(Y β) is independent over C∞(exp(U)) (as a set of vector fields on

36

exp(U)). Suppose that (∑

uj ⊗ Tj ) f = 0, for all f ∈ C∞(G/H, E). For injectivitywe need to conclude that

∑β Y β ⊗ Tβ = 0 in U(g) ⊗U(h) Hom(E,F ). By Fact 6.9

we know that ∑Tβ(R(Y β)f)(g) =

∑(R(Y β)(T F ))(g) = 0,

for all F ∈ C∞(exp(U), E) and all g ∈ exp(U). By the independence of R(Y β)(applied to each coordinate of Tβ F ), Tβ(F (g)) = 0 for all F ∈ C∞(exp(U), E),so Tβ = 0 for all gb. This completes the proof.

The group H acts on U(g)⊗HomC(E,F ) by

(6.10) h · (u⊗ T ) = Ad(h)u⊗ %(h)Tσ(h−1), h ∈ H.

For h ∈ H,Y ∈ h, u ∈ U(g) and T ∈ HomC(E,F ),

h · (uY ⊗ T − u⊗ (T dσ(Y ))) ∈ J

so that action of H preserves J . Thus, (6.10) is well-defined on U(g) ⊗U(h)

HomC(E,F ).

Denote by U(g) ⊗U(h)

HomC(E,F )H the space of all elements in U(g) ⊗U(h)

HomC(E,F )

which are fixed by this action of H.

Proposition 6.11. ([11, Prop. 1.2]) For homogeneous vector bundles E and Fover G/H

DG(E ,F) ' U(g) ⊗U(h)

HomC(E,F )H .

Proof. In view of Lemma 6.7, it is enough to show that (∑

uj⊗Tj ) f ∈ C∞(G/H,F)for all f ∈ C∞(G/H, E) if and only if

∑uj ⊗ Tj ∈ U(g) ⊗

U(h)HomC(E,F )H .

Let f ∈ C∞(G/H, E). For X ∈ g and T ∈ HomC(E,F ),((X ⊗ T ) f

)(gh) = T (

d

dtf(gh exp(tX))|t=0)

= T (d

dtf(g exp(tAd(h)X)h)|t=0)

= T (σ(h−1)(R(Ad(h)X)f)(g))

= %(h−1)((h · (X ⊗ T )) f

)(g).

It follows that for uj ∈ U(g) and Tj ∈ HomC(E,F )(∑(uj ⊗ Tj ) f

)(gh) = %(h−1)

(∑(h · (uj ⊗ Tj)) f

)(g).

Therefore,∑

(uj ⊗ Tj ) f ∈ C∞(G/H,F), for all f ∈ C∞(G/H, E) if and only if∑(uj ⊗ Tj ) = h ·

∑(uj ⊗ Tj ) , which is equivalent to

∑(uj ⊗ Tj) = h ·

∑(uj ⊗ Tj)

in U(g) ⊗U(h)

HomC(E,F ).

EXERCISES

(6.2) Fill in the details and prove the statements given in Example 6.5.

(6.2) Prove Prop. 6.3.37

LECTURE 7.Differential operators between sections of bundles, II

There are many interesting special cases of Prop. 6.11. In this lecture we willdiscuss a few.

Suppose that G/H is a reductive homogeneous space. This means that h has anH-invariant complement in g. Let us write this as g = m⊕h, with Ad(h)m = m, forall h ∈ H. Then one may show that U(g) ' S(m) ⊕ U(g)h, as H-representations,where S(m) is the symmetric algebra of m. This decomposition of the envelopingalgebra is closely related to the one of Exercise 5.5. See [6] for details.

When E = F = C, the trivial H-representations, the submodule J is justU(g)h⊗C. Therefore,

Corollary 7.1. When G/H is reductive,

DG(G/H) ' S(m)H .

Example 7.2. The n-sphere is a homogeneous space for SO(n + 1). The groupSO(n + 1) acts on Rn+1 by multiplication of a matrix in SO(n + 1) by a columnvector. The stabilizer of (0, . . . , 0, 1)t is

H =(

g′ 00 1

): g′ ∈ SO(n)

.

An H-invariant complement of h in g is

m =

x1

x2

...xn

−x1 −x2 · · · −xn 0

.

As an SO(n)-representation, m is equivalent to the standard representation on Rn.If X1, . . . , Xn is an orthonormal basis of m (with respect to the negative of thetrace form), then S(m)H ' span(

∑X2

j )k : k = 0, 1, 2, . . . . The correspondinginvariant differential operators are the powers of the Laplacian =

∑R(Yj)2.

Continuing with the hypothesis that G/H is a reductive homogeneous spaceand m is an invariant complement, many first order differential operators maybe constructed. The first we consider is exterior differentiation (the d-operator).By (5.3) the tangent space of G/H at the identity may be identified (as an H-representation) with m. For X∗ ∈ m∗, let e(X∗) : ∧pm∗ → ∧p+1m∗ be exteriormultiplication, i.e., e(X∗)ω = X∗ ∧ ω. Choose a basis Xj of m and let X∗

j bethe dual basis.Claim: δ =

∑Xj ⊗ e(X∗

j ) is an H-invariant in S(m)⊗HomC(∧pm∗,∧p+1m∗).First we check that δ is independent of the basis (and dual basis) used. Let Yj be

38

another basis of m with dual basis Y ∗j . Write Xj =

∑akjYk and X∗

j =∑

bljY∗l .

Then ∑j

Xj⊗e(X∗j ) =

∑j,k,l

akjbljYk ⊗ e(Y ∗l )

=∑k,l

(∑

j

akjblj)Yk ⊗ e(Y ∗l ).

(7.3)

However, the fact that Xj and X∗j are dual bases, gives us δkl = X∗

k(Xl) =∑p bpkY ∗

p (∑

q aqlYq) =∑

p bpkapl. That is, the matrices A = (aij) and B = (bij)satisfy At = B−1. Now it follows that the last term in (7.3) is

∑Yk ⊗ e(Y ∗

k ). Nowwe check that h ·

∑Xj ⊗ e(X∗

j ) =∑

Xj ⊗ e(X∗j ). Note that h ·

∑Xj ⊗ e(X∗

j ) =∑j(Ad(h)Xj)⊗ e(Ad(h)X∗

j ). Since (Ad(h)X∗j )(Ad(h)Xk) = X∗

j (Xk) = δjk we seethat Ad(h)Xj and Ad(h)X∗

j are a pair of dual bases. The claim now followsfrom independence of basis.

It follows that a G-invariant operator d corresponds to δ. The formula is d =∑j R(Xj)⊗ e(Xj). This is exterior differentiation.

Many G-invariant differential operators of order one on a symmetric space maybe easily constructed. A symmetric space is of the form G/K, where K is a maximalcompact subgroup. As discussed in Lecture 1, the Cartan decomposition g = k⊕ s

satisfies [k, s] ⊂ s. Therefore s is a K-invariant complement to k in g. Let E bea vector bundle over G/K. Define t(X) : E → s ⊗ E by t(X)v = X ⊗ v. Fixan orthonormal basis Xj of s. Then a calculation similar to the calculationabove shows that

∑Xj ⊗ t(Xj) is a K-invariant in S(s) ⊗ HomC(E, s ⊗ E). The

corresponding differential operator is∑

R(Xj)⊗ t(Xj).

One may say more. Suppose the E and F are homogeneous bundles over G/K.Then HomK(s⊗ E,F ) is isomorphic to a K-subrepresentation of DG(E ,F). (Thisfollows from the fact that s∗ ' s and s ⊗ HomC(E,F )K ' s ⊗ E∗ ⊗ FK 'HomK(s⊗E,F ).) The differential operator corresponding to p ∈ HomK(s⊗E,F )is p (

∑Xj ⊗ t(Xj)). In particular, each irreducible constituent F of s⊗ E gives

a first order differential operator. Each first order differential operator arises thisway.

Now we consider the homogeneous space G/P , where P is a parabolic subgroupof a reductive Lie group G. Then G/P is not a reductive homogeneous space. Thiscase will be of particular interest to us in the remaining lectures.

As usual, write P = LN with N the nilradical and L a reductive subgroup. Asabove, let (σ,E) and (%, F ) be finite dimensional representations of P .

By Prop. 6.11,

DG(E ,F) ' U(g) ⊗U(p)

HomC(E,F )P ' U(g) ⊗U(p)

(E∗ ⊗ F )P .

In the right hand side E∗ ⊗ F is a U(p)-module by the dual representation σ∗ of p

on E∗ and trivial action on F .39

The map

U(g) ⊗U(p)

(E∗ ⊗ F ) → (U(g) ⊗U(p)

E∗)⊗ F

u⊗ (e∗ ⊗ f) 7→ (u⊗ e∗)⊗ f

is an isomorphism as P -representations. Now, for very general reasons, we have

(U(g) ⊗U(p)

E∗)⊗ FP

' HomP (F ∗,U(g) ⊗U(p)

E∗)

' HomL,n(F ∗,U(g) ⊗U(p)

E∗).

(7.4)

It should be noted that N is connected, so in the second isomorphism above, equiv-ariance with respect to L and n is equivalent to equivariance with respect to P . IfN acts trivially on F then

(7.5) (U(g) ⊗U(p)

E∗)⊗ FP ' HomL(F ∗, U(g) ⊗U(p)

E∗n).

The following characterization of the invariant differential operators has nowbeen proved.

Proposition 7.6. For a parabolic subgroup P of a reductive group G, and homo-geneous vector bundles E and F defined by representations of P on E and F ,

DG(E ,F) ' Homg,L

(U(g)⊗U(p) F ∗,U(g)⊗U(p) E∗).

Proof. By the above discussion

DG(E ,F) ' HomL,n(F ∗,U(g) ⊗U(p)

E∗).

Now

HomL,n(F ∗,U(g)⊗U(p) E∗) ' Homp,L(F ∗,U(g)⊗U(p) E∗)

' Homg,L(U(g)⊗U(p) F ∗,U(g)⊗U(p) E∗).

The last isomorphism is the general fact for modules over rings (with identity) that

HomR(R⊗S

M,N) ' HomS(M,N).

Corollary 7.7. If, under the hypothesis of the proposition, N is trivial on F , then

DG(E ,F) ' HomL

(F ∗, U(g) ⊗

U(p)E∗n

).

As we will see in Lecture 11, this proposition and corollary reduce the search forinvariant differential operators to an algebraic question.

40

EXERCISES

(7.1) What are the invariant differential operators on SL(2,R)/SO(2)?

(7.2) Show that the formula for exterior differentiation found above coincides withthe usual one (defined for example in Example 6.2).

(7.3) Determine all G-invariant differential operators from the bundle Lk to Lm onSL(2,R)/P . Here Lm denotes the homogeneous bundle for the character

χm

((a 0c d

))= am

of P .

(7.4) Verify the isomorphisms (7.4) and (7.5).

41

LECTURE 8.Conformally invariant systems, I

Suppose that g is a real or complex Lie algebra and M is a smooth manifold.Assume that there is a map π : g → D(M) satisfying the following two conditions.

(A1) π is a Lie algebra homomorphism, i.e., π([X, Y ]) = [π(X), π(Y )] (thebracket on the righthand side being the commutator of differential operators).

(A2) For each X ∈ g, π(X) is a first order differential operator.By condition (2) each π(X) may be decomposed as π(X) = π0(X) + π1(X), withπ0(X) multiplication by a smooth function on M and π1(X) a vector field on M .

Given a vector bundle E → M , we say that E → M is a g-bundle if there is alinear map πE : g → D(E) satisfying

(1) πE([X, Y ]) = [πE(X), πE(Y )], and(2) [πE(X), f ] = π1(X)f ,.

for all X, Y ∈ g and all f ∈ C∞(M).

Definition 8.1. A conformally invariant system on E (with respect to π and πE)is a finite set of differential operators D1, . . . , Dn ∈ D(E) so that

(1) D1, . . . , Dn is linearly independent at each point of M , and(2) for each X ∈ g there is an n× n matrix C(X) of smooth functions on M so

that in D(E)

[πE(X), Dj ] =∑

i

Cij(X)Di.

The map C : g → Mn×n(C∞(M)) is called the structure operator. Two conformallyinvariant systems D1, . . . , Dn and D′

1, . . . , D′n are said to be equivalent if there is

an invertible matrix in A ∈ Mn×n(C∞(M)) so that

D′j =

∑i

AijDi.

A conformally invariant system D1, . . . , Dn is reducible if there is an equivalentconformally invariant system D′

1, . . . , D′n for which D′

1, . . . , D′m, m < n, is a con-

formally invariant system.

Observe that the common solution space of Dj = 0 is a g-invariant subspaceof C∞(M, E). Furthermore, two equivalent conformally invariant systems have thesame common solution spaces.

Lemma 8.2. Let C be the structure operator for a conformally invariant system.Then for X, Y ∈ g,

C([X, Y ]) = π1(X)C(Y )− π1(Y )C(X) + [C(X), C(Y )].

Proof. Let D1, . . . , Dn be a conformally invariant system with structure operatorC. Then

42

[πE([X, Y ]), Dj ] =∑

i

Cij([X, Y ])Di.

On the other hand

[πE([X, Y ]), Dj ] = [[πE(X), πE(Y )], Dj ],

by condition (1) of the definition of g-bundle,

= [πE(X), [πE(Y ), Dj ]]− [πE(Y ), [πE(X), Dj ]],

by the Jacobi identity,

= [πE(X),∑

k

Ckj(Y )Dk]− [πE(Y ),∑

k

Ckj(X)Dk]

=∑

k

([πE(X), Ckj(Y )]Dk + Ckj(Y )[πE(X), Dk]

)−∑

k

([πE(Y ), Ckj(X)]Dk − Ckj(X)[πE(Y ), Dk]

)=∑

k

(π1(X)Ckj(Y ))Dk +

∑i,k

Cik(X)Ckj(Y )Di

−∑

k

(π1(Y )Ckj(X))Dk −

∑i,k

Cik(Y )Ckj(X)Di,

by condition (2) of the definition of g-bundle,

=∑

i

(π1(X)Cij(Y )− π1(Y )Cij(X)

+∑

k

(Cik(X)Ckj(Y )− Cik(Y )Ckj(X)))Di.

The lemma now follows by comparing the two expressions for [πE([X, Y ]), Dj ].

The theory of conformally invariant systems is developed in great generality in[1]. However, at this point we will consider a special case of the above formalismwhich is of particular interest.

The situation we will focus on is related to the so-called principal series repre-sentations of a real reductive Lie group. Therefore, we let G be a real reductive Liegroup and P a parabolic subgroup of G. Then P has Levi decomposition P = LN

and we let P = LN be as in (3.7). We have no need to carefully define the prin-cipal series representations. The representations of interest to us are constructedby ‘inducing’ an irreducible finite dimensional representation (σ,E) of P . The irre-ducibility of E guarantees that σ(n) = IE for all n ∈ N . There is a representation ofG on the space C∞(G/P , E) of smooth sections of the homogeneous vector bundleE → G/P given by

(Π(g)ϕ)(x) = ϕ(g−1x)), ϕ ∈ C∞(G/P , E).

We have not said what a representation of a Lie group on an infinite dimensionalspace is - there are some continuity conditions which are required to hold - however

43

it suffices here to simply note two points. First, Π(g1)Π(g2) = Π(g1g2), and secondΠ may be differentiated in the sense that π(X)f = d

dtΠ(exp(tX))f exists and givesa function in C∞(G/P , E). It follows that π([X, Y ]) = [π(X), π(Y )].

Instead of studying C∞(G/P , E) we will use an alternative realization of therepresentation, which is often referred to as the ‘N -picture’. For this, we applythe Bruhat decomposition (3.11). Thus NP is a dense open subset of G. Thenrestricting sections ϕ ∈ C∞(G/P ) to N gives an injection C∞(G/P ) → C∞(N,E).

Two spaces of functions will play a role:

C∞(N,E) = F : N → E : F is smooth and

C∞σ (N) = F : N → E : F = ϕ|N for some ϕ ∈ C∞(G/P , E).

The differential of the action of G on C∞(G/P , E) may be transported to an actionof g on C∞

σ (N); this is done in the proof of Prop. 8.4.

Write the Bruhat decomposition of g ∈ NLN as

g = n(g)p(g) ∈ NP

with the further decomposition

g = n(g)l(g)n(g) ∈ NLN.

At the Lie algebra level write

X = Xn + Xp ∈ n⊕ p.

X = Xn + Xl + Xn ∈ n⊕ l⊕ n.

There is useful observation on the relationship between the above decompositions.

X =d

dtexp(tX)|t=0

=d

dtn(exp(tX))l(exp(tX)n(exp(tX))|t=0

=d

dtn(exp(tX))|t=0 +

d

dtl(exp(tX))|t=0 +

d

dtn(exp(tX))|t=0

is the decomposition of X with respect to g = n⊕ l⊕ n. Therefore,

Xn =d

dtn(exp(tX))|t=0,

Xl =d

dtl(exp(tX))|t=0,

Xn =d

dtn(exp(tX))|t=0,

Xp =d

dtp(exp(tX))|t=0.

(8.3)

44

Our immediate goal is to explicitly determine the action of the Lie algebra g onC∞

σ (N). For this we define right action of n on C∞(N,E). When Y ∈ n set

(R(Y )f)(n) =d

dtf(n exp(tY ))|t=0, n ∈ N

for f ∈ C∞(N,E). The notation R(u), u ∈ U(n) will be used for the naturalextension of R to the enveloping algebra.

Proposition 8.4. Let Y ∈ g. Then the action of g on C∞σ (N) arising from the

left translation on C∞(G/P , E) is given by

(8.5) (πσ(Y )f)(n) = σ((Ad(n−1)Y )p)f(n)− (R((Ad(n−1)Y )n)f)(n),

for n ∈ N . Two special cases are:

for Y ∈ l, πσ(Y )f = σ(Y )f −R(Ad((·)−1)Y − Y )f

for Y ∈ n, πσ(Y )f = −R(Ad((·)−1)Y )f.

Proof. Suppose that f = ϕ|N . Then as long as g−1n ∈ NP , we have

(πσ(g)f)(n) = ϕ(g−1n) = σ(p(g−1n)−1)f(n(g−1n)).

For g close enough to the identity, by the openness of NP , g−1n ∈ NP . Now,taking g = exp(tY ) and differentiating at t = 0 we get:

(πσ(Y )f)(n)

=d

dtσ(p(exp(−tY )n)−1)|t=0f(n) +

d

dtf(n(exp(−tY )n))|t=0

=d

dtσ(p(exp(−tAd(n−1)Y )−1))|t=0f(n) +

d

dtf(nn(exp(−tAd(n−1)Y ))|t=0

= σ((Ad(n−1)Y )p)f(n)− (R((Ad(n−1)Y )n)f)(n), by (8.3).

This gives (8.5).

For the two special formulas, write n = exp(X) ∈ N . Then Ad(n−1)Y =Y − [X, Y ] + 1

2 [X, [X, Y ]]− · · · . So

(Ad(n−1)Y )p = Y and (Ad(n−1)Y )n = Ad(n−1)Y − Y, when Y ∈ l,

(Ad(n−1)Y )p = 0 and (Ad(n−1)Y )n = Ad(n−1)Y, when Y ∈ n.

Now the two formulas follow.

We will often use the formulas of the Proposition when they are evaluated atn = e. Therefore we state this case separately as a corollary.

Corollary 8.6. In the action of g on C∞(N,E)

(πσ(Y )f)(e) = σ(Yp)f(e)− (R(Yn)f)(e),

(πσ(Y )f)(e) =

−(R(Y )f)(e), if Y ∈ n

σ(Y )f(e) if Y ∈ p.

45

Remark 8.7. It is an important observation that the formula of the propositionimplies that πσ extends to a representation of g on C∞(N,E).

Applying the proposition to the trivial one dimensional representation of P showsthat there is a map π : N → D(N) satisfying (A1) and (A2):

(π(X)f)(n) = −(R((Ad(n−1)X)n)f

)(n), n ∈ N,X ∈ g.

The homogeneous bundle E → G/P restricted to N is the trivial bundle N×E → N .We will, by slight abuse of notation, refer to this trivial bundle as E . Then the spaceof smooth functions C∞(N,E) is the space of smooth sections of E → N . Thisbundle is a g-bundle under the action of Prop. 8.4, extended to all of C∞(N,E).The second property of the definition of a g-bundle holds since the first term onthe right-hand side of (8.5) commutes with smooth function (in D(E)).

Considering g-bundles on E → N defined by πσ we make the following definition.

Definition 8.8. A conformally invariant system D1. . . . , Dn on E → N is calledstraight if and only if [πσ(X), Dj ] = 0 for all X ∈ n. Therefore, being straight isequivalent to the structure operator vanishing on n.

Theorem 8.9. ([1]) Any conformally invariant system on E → N is equivalent toa straight conformally invariant system.

Several comments are in order. If a differential operator D ∈ D(E) commuteswith all X ∈ n, then D is an N -invariant differential operator. In this case weknow from 6.11 (applied to G = N and H = e) that D is a linear combinationof (u⊗ T ) , for some u ∈ U(n) and T ∈ HomC(E,E).

The following proposition will be used later. It shows how certain computationsmay be reduced to computations at the identity.

Proposition 8.10. Suppose that D1, . . . , Dn ∈ D(E) and each Dj commutes withπσ(X), X ∈ n. Assume that D1, . . . , Dn are linearly independent at e and there isa map b : g → gl(n,C) so that

([πσ(X), Dj ]f)(e) =n∑

i=1

(b(X))ij(Dif)(e),

for all X ∈ g and all f ∈ C∞(N,E). Then D1, . . . , Dn is a straight conformallyinvariant system. The structure operator is given by C(X)(n) = b(Ad(n−1)X), forn ∈ N and X ∈ g.

Proof. As noted above, since Dj commutes with πσ(X), X ∈ n, Dj is invariantunder left translation `n−1 . Now the independence at an arbitrary n follows fromN -invariance.

46

We will use the fact that `n−1πσ(X)`n = πσ(Ad(n−1)X), which follows easilyfrom the last formula in (8.4).

([πσ(X),Dj ]f)(n) = (`n−1([πσ(X), Dj ]f))(e)

= ([πσ(Ad(n−1)X), Dj ](`n−1f))(e)

=∑

i

b(Ad(n−1X))(Dj(`n−1f))(e)

=∑

i

b(Ad(n−1X))(Djf)(n).

The conformal invariance, along with the formula for the structure operator, nowfollows.

EXERCISES

(8.1) If D1, . . . , Dm is a straight conformally invariant system, show that the struc-ture operator C(X) is a matrix of constants when X ∈ l. (Hint: Consider Lemma8.2.)

47

LECTURE 9.Prehomogeneous vector spaces and invariant theory

Let G be a complex Lie group and ρ : G → gl(V ) a holomorphic representation(that is, the differential is complex linear). Then (ρ, V ) is a prehomogeneous vectorspace if there exists an open G-orbit in V . Over the next several lectures showhow to use the invariant theory of a prehomogeneous vector space to constructconformally invariant systems of differential equations.

Suppose (ρ, V ) is a prehomogeneous vector space. A fact is that there is preciselyone open orbit and this open orbit is dense in V . If V = ρ(G)v then we call v ageneric element of V . The complement of the dense orbit is called the singularset. The singular set consists of the elements v ∈ V so that dim(StabG(v)) >

dim(G)−dim(V ), and v is generic if and only if dim(StabG(v)) = dim(G)−dim(V ).

Here are a few simple examples. Let G = GL(n,C) and V = Sym(n,C).Then ρ(g)X = gXgt defines a prehomogeneous vector space. The dense orbit isggt : g ∈ G. This is the orbit of I. The stabilizer of I is g ∈ G : ggt =I = O(n,C), therefore dim(StabG(v)) = dim(G) − dim(V ). It follows that B ∈Sym(n,C) is generic if and only if the corresponding bilinear form is nondegenerate,i.e., det(B) 6= 0. There are a finite number of orbits of G on V . These orbits are

Om ≡ B ∈ Sym(n,C) : rank(B) = m, m = 0, 1, . . . , n.

Another example is the action of G = (GL(1))n ' (C×)n on Cn−1 given by

(α1, . . . , αn) · (z1, . . . , zn−1) = (α1

α2z1,

α2

α3z2, . . . ,

αn−1

αnzn−1).

The generic elements are precisely those elements of Cn−1 having no coordinatesequal to zero.

An important family of examples arises as follows. Suppose P = LN is a para-bolic subgroup of a complex reductive group and suppose that N is abelian. Thenthe adjoint representation of L on n is a prehomogeneous vector space. In fact, L

has only a finite number of orbits on n. Examples of this form are called preho-mogeneous vector spaces of parabolic type. See [10, Ch. X] for a proof of a slightlymore general fact due to Vinberg.

Often a prehomogeneous vector space has a relatively invariant polynomial, thatis, a polynomial f(x) on V satisfying f(ρ(g)x) = χ(g)f(x), for some character χ ofG and all g ∈ G.

The first two examples given above have relatively invariant polynomials. In thefirst example det(ρ(g)B) = det(gBgt) = det(g)2 det(B), so f(X) = det(X) is arelatively invariant polynomial and χ(g) = det(g)2 is the corresponding characterof G. Note that the singular set is B ∈ V : f(B) = 0. Now consider thesecond example. The characters of G are χ(α1, . . . , αn) = αm1

1 · · ·αmnn , for all

(m1, . . . ,mn) ∈ Zn. Each monomial pk(Z) = zk11 · · · zkn−1

n−1 is a relative invariant.48

Then

pk((α1, . . . , αn) · Z) = αk11 αk2−k1

2 . . . αkn−1−kn−2n−1 α−kn−1

n pk(Z).

The semigroup of relatively invariant polynomials is generated by zj : j =1, . . . , n. Here the singular set is the union of the irreducible hypersurface zj = 0.

A theorem of Bernstein [3] states that for any polynomial f(X) on a vector spaceV there exists a polynomial differential operator P (s, xj , ∂xj

) on V , polynomial ins, and a polynomial b(s) so that

P (s, xj , ∂xj)f(x)s = b(s)f(x)s−1.

Such polynomials form an ideal in C[s]. One refers to the monic generator as theb-function of f(x). In general, the b-function (and the P (s, xj , ∂xj )) are difficult tofind.

One important situation for which P (s, xi, ∂xi) can be calculated is when f(x) is

the relative invariant polynomial for a prehomogeneous vector space for a reductivegroup G. Suppose that G is a complex reductive group and (ρ, V ) is a prehomo-geneous vector space for G. Assume that f(x) is a relatively invariant polynomial;f(ρ(g)x) = χ(g)f(x).

Proposition 9.1. (See [9, Prop. 2.21]) The dual (ρ∗, V ∗) is a prehomogeneousvector space with a relatively invariant polynomial. Denoting this relative invariantby f∗(x), we have f∗(ρ∗(g)y) = χ−1(g)f∗(y), g ∈ G, y ∈ V ∗ and deg(f∗(y)) =deg(f(x)).

To find P (s, xi, ∂xi) we use the following construction. For a finite dimensionalvector space W , given a polynomial p(y) ∈ P (W ∗) there is a unique differentialoperator p(∂x) on W with constant coefficients satisfying p(∂x)e〈y ,x〉 = p(x)e〈y ,x〉.To describe this differential operator, choose a basis ej of W and a dual basise∗j of W ∗ (i.e., a basis of W ∗ so that 〈e∗j , ek〉 = δjk). Write x =

∑xiei and

y =∑

yie∗i . Then p(∂x) is obtained from p(y) by replacing each yj by ∂xj = ∂

∂xj.

For (ρ, V ) and G as above we now may state the following proposition.

Proposition 9.2. (See [9, Prop. 2.22]) There is a polynomial b(s) so thatf∗(∂x)f(x)s+1 = b(s)f(x)s.

In fact, the polynomial b(s) is the Bernstein polynomial.

Example 9.3. The action of GL(p,C)×GL(q,C) on V = Mp×q(C) by ρ((l1, l2))X =l1Xl−1

2 defines a prehomogeneous vector space of parabolic type. Let’s assume thatp ≤ q. Then the generic elements are the matrices of rank p. There is a relativeinvariant if and only if p = q. Let us assume this. Then f(X) = det(X) is arelative invariant since f(l1Xl−1

2 ) = χ(l1, l2)f(X), with χ(l1, l2) = det(l1l−12 ). The

dual representation is equivalent to ρ1 : GL(p,C)×GL(q,C) → gl(Mp×p), defined49

by ρ1(l1, l2) = ρ(l2, l1). (Check: If 〈 , 〉 is the trace form, then X → 〈X , 〉 is a vec-tor space isomorphism V → V ∗. Also, (ρ∗(l1, l2)〈X , 〉)(Z) = 〈X , ρ(l−1

1 , l−12 )Z〉 =

Trace(Xl−11 Zl2) = Trace(l2Xl−1

1 Z) = 〈ρ1(l2, l1)X ,Z〉.) For the dual prehomoge-neous vector space the relative invariant is also det(Y ). The proposition states thatthere is a polynomial b(s) so that

(9.4) det(∂xij ) det(X)s+1 = b(s) det(X)s.

This may be seen as follows. We claim that

Fs(X) = det(∂xij ) det(X)s+1/ det(X)s

is a GL(p,C)×GL(p,C) invariant rational function on V . This holds because

Fs(l1Xl−12 ) = det(∂l2xij l−1

1) det(l1Xl−1

2 )s+1/ det(l1Xl−12 )s = fs(X).

By Exercise 9.2, Fs(X) must be a constant (depending on s). One may checkthat this constant is polynomial in s. We arrive at (9.4). A (somewhat involved)calculation shows that det(∂xij ) det(X)s+1 = (s + 1)(s + 2) . . . (s + p) det(X)s.

Now we turn to invariant theory. By this we simply mean the decomposition ofthe symmetric algebra S(V ) into the direct sum of irreducible G representationsfor a representation (ρ, V ) of G.

Let b = h + n be a Borel subalgebra of g. Therefore a set of positive roots ∆+

is determined by requiring α ∈ ∆+ if and only if g(α) ⊂ n. Then, as discussedin Lecture 4, the highest weight vectors are the vectors in V annihilated be n. Inparticular, the highest weight vectors of the irreducible constituents of S(V ) span

S(V )n ≡ u ∈ S(V ) : X · u = 0. for all X ∈ n.

Note that since X · u1u2 = (X · u1)u2 + u1(X · u2), S(V )n is a subalgebra of S(V ).

Since S(V ) ' P (V ∗), as G-representations, it is equivalent to consider the ques-tion of decomposing P (V ∗).

When G = SL(2,C) and V = C2 is the ‘standard’ representation, then

S(V )n = vm+ : m = 0, 1, . . . ' C[v+].

Here we are taking v+ to be a highest weight vector in V . The correspondingdecomposition of S(V ) is

S(V ) '∞∑

m=0

Vm,

with Vm the irreducible representation of SL(2,C) of dimension m + 1.

In later lectures we will use the following theorem which describes the decomposi-tion of S(V ) for a family of prehomogeneous vector spaces. Suppose that P = LN

is a parabolic subgroup of a complex reductive group G and N is abelian. Thedecomposition of S(n) may be described in a very nice way. To do this we mustintroduce the notion of strongly orthogonal roots. Two roots α and β are called

50

strongly orthogonal if and only if α ± β are not roots. (Note that this means that[g(±α), g(±β)] = 0.)

Form a family of strongly orthogonal roots as follows. Let ∆+ be a positivesystem of roots defining a Borel subalgebra containing p. Then as described earlier,p is determined by specifying one simple root β1. Then ∆(l) = span∆ \ β1∩∆and β1 is the unique simple root in ∆(n). Then let us write the simple roots asΠ = β1, . . . , βn. Let H1, . . . ,Hn be the basis of h dual to the basis Π of h∗.The lexicographic order gives the positive system ∆+. We will use this order tochoose a family of strongly orthogonal roots. Let γ1 be the greatest root in ∆(n).Then choose γ2 to be the greatest root in ∆(n) which is strongly orthogonal to γ1.Continue by choosing γj+1 to be the greatest root in ∆(n) strongly orthogonal toγ1, . . . , γj . Continue this until there are no roots in ∆(n) strongly orthogonal toγ1, . . . , γr.

Fact: mγ ≡r∑

j=1

mjγj is dominant with respect to ∆+(l) when m1 ≥ · · · ≥ mr.

Theorem 9.5. ([14]) For L, n and the choice of positive systems of roots as above,

S(n) '∑

m1≥···≥mr≥0

Emγ .

Stated slightly differently, this theorem says that there are ‘fundamental’ invari-ants u1, . . . , ur in S(n)n which freely generate the algebra. So S(n) ' C[u1, . . . , ur],as algebras. So, um1

1 · · ·umrr are the highest weight vectors of the irreducible con-

stituents of S(n).

Example 9.6. Consider the parabolic subgroup P = LN in GL(n,C), n = p + q,

defined by the simple root εp − εp+1 (for the positive system of roots as in (2.8)).Assume p ≤ q. In block form this is the subgroup

P =(

a b0 d

): a ∈ GL(p,C), d ∈ GL(q,C) and b ∈ Mp×q(C)

Then L ' GL(p,C)×GL(q,C), and the adjoint action is equivalent to the action(`1, `2)X = `1X`−1

2 , X ∈ Mp×q(C). The roots in n are εi − εj for 1 ≤ i ≤ p < j ≤p + q. The family of strongly orthogonal roots described above is

γ1 = ε1 − εn,

γ2 = ε1 − εn−1,

...

γp = εp − εn−p+1

Then S(n) '∑

m1≥···≥mr≥0

E(m1,...,mp,0,...,0,−mp,...,−m1).

51

Example 9.7. Consider the parabolic subgroup

P =(

a b0 (at)−1

): a ∈ GL(n,C) and b ∈ Sym(n,C)

in Sp(2n,C). Then L ' GL(n,C) and the action on n ' Sym(n,C) is `·X = `X`t,X ∈ Sym(n,C). The set of strongly orthogonal roots is γi = 2εi, i = 1, . . . , n.Therefore Schmid’s Theorem says that S(n) =

∑E(2m1,2m2,...,2mn),m1 ≥ · · · ≥

mn ≥ 0. It is usually useful to know explicitly the decomposition. For this we willuse the L-isomorphism S(n) ' P (n∗). One may easily check that n∗ is Sym(n,C)with the action ` ·X = (`t)−1X`−1. Let µj = γ1 + · · · + γj and let fj(X) be thedeterminant of the principal (upper left) j × j minor of X. Then

(a1

. . .an

· fj

)(X) = fj(

a1

. . .an

X

a1

. . .an

)

= (a1 · · · aj)2fj(X),

so fj has weight µj . In Exercise (9.3) you will show that fj(X) is annihilated by n, sois a highest weight vector. You will also show that the irreducible subrepresentationisomorphic to Eµj is the span of the determinants of the j × j minors.

EXERCISES

(9.1) Determine all parabolic subalgebras of the complex classical groups for whichn is abelian.

(9.2) Prove that the only G-invariant rational functions on a prehomogeneous vectorspace are the constants.

(9.3) In the setting of Example 9.7, describe the Lie algebra action of l on poly-nomials. For j = 0, 1 . . . n, show that fj(X) is annihilated by n and the span ofthe determinants of the j × j minors is an irreducible L-representation of highestweight µj .

52

LECTURE 10. An example of a conformally invariant system

In this lecture we will explicitly write down a conformally invariant system ofdifferential operators. Consider the homogeneous space G0/P 0, P0 = L0N0 andP 0 = θ(P0) with

G0 = GL(n,R)

L0 = (

`1 00 `2

): `1 ∈ GL(p,R), `2 ∈ GL(q,R)

N0 = (

I X0 I

): X ∈ Mp×q(R).

(10.1)

In this example N0 is abelian, a property which simplifies matters. We willassume that p ≤ q.

We will write g (resp. n, etc.) for the complexification of g0 (resp. n0, etc.).Also, we take G = GL(n,C) and P = LN the (complex) subgroup with Lie algebrap.

For the positive system of roots of Example 2.8, p is defined by S = εp − εp+1(see 3.5). We will use the notation n for the span of the root spaces g(α) withg(α) ⊂ l and α ∈ ∆+. Note that n is not to be confused with n, and h + n + n isthe Borel subalgebra defined by ∆+.

As noted in Lecture 9, n is a prehomogeneous vector space under the adjointaction of L. The L-action can be written as follows: L = GL(q,C) × GL(q,C)acting on Mp×q(C) by (`1, `2) ·X = `1X`−1

2 . The orbits are given by the rank ofX and the closures of the orbits are given by

Om = X ∈ Mp×q(R) : determinant of all (m + 1)× (m + 1) minors vanish.

Our first goal is to decompose S(n) ≡ P(n) as a L-representation. Theorem (9.5)tell us that the fundamental n-invariants have weights

γ1 = ε1 − εn = (1, 0, . . . , 0,−1)

γ1 + γ2 = (ε1 − εn) + (ε2 − εn−1) = (1, 1, 0, . . . , 0,−1,−1)

. . .

γ1 + . . . + γp =p∑1

(εi − εn−i+1) = (1, 1, . . . , 1, 0, . . . , 0,−1, . . .− 1).

The (unique) L-subrepresentations of P(n) having these highest weights are

Vm = spandet(m×m minors).

It will be important to introduce a little notation for the minors. Let Y = (yij) ∈Mp×q(C). For some m = 1, . . . p let

(10.2) R ⊂ 1, . . . q, S ⊂ 1, . . . p with #R = #S = m,53

and let ∆mR,S(Y ) be the determinant of the matrix obtained by deleting the i-th

rows for i /∈ R and the j-th columns for j /∈ S. An easy calculation shows that theweight of ∆m

R,S(Y ) is

(10.3)∑i∈S

εi −∑j∈R

εp+j .

Lemma 10.4. Vm = span∆mR,S(Y ) : R,S satisfy (10.2) is an irreducible L-

representation of highest weight γ1 + . . . + γm.

Proof. For L -invariance of Vm, suppose that ` = (e, `2) with `2 ∈ GL(q,C). Then`2Y has rows which are linear combinations of the rows of Y . Therefore the de-terminant of an m × m minor of ` · Y is a linear combination of determinants ofother m ×m minors. A similar argument applies for (`1, e) · Y = Y `−1

1 . A glanceat (10.3) shows that the only dominant weight which occurs is γ1 + . . . + γm. Weconclude that Vm is irreducible.

Our conformally invariant system will act on C∞(N0) = C∞(N0,C) with theaction of g coming from the principal series representation

C∞(G0/P 0,Cs) = f : G0 → C : f(g`n) = χs(`−1)f(g),

where χs(`1, `2) = |det(`1)det(`2)

|s/2.

Note that dχs(X, Y ) = s2 (Tr(X) − Tr(Y )). As described in Prop. (8.4), the

action of Z ∈ g on C∞(N0) is given by

(10.5) (πs(Z)f)(n) = dχs((Ad(n−1)Z)p)f(n)− (R((Ad(n−1)Z)n)f)(n), n ∈ N0.

Recall that (·)p denotes the projection to p in the decomposition g = n + (l + n).Similarly for (·)n.

Let us make this action explicit for our example. We will use the notation

nX = exp(

0 X0 0

)=(

I X0 I

), X ∈ Mp×q(R).

The matrix with 1 in the ij-place and 0 elsewhere is denoted by Ei,j . We writeX = (xi,j) (i.e, X =

∑xijEi,j).

Proposition 10.6. For G = GL(n,R), n = p + q and P = LN as in (10.1)

(1) (π(`)f)(n) = χs(`)f(`n`−1)

(2) π((

0 Eji

0 0

)= − ∂

∂xji

(3) π((

0 0Eij 0

)) = −s xji −

∑k,l xki xjl

∂∂xkl

.

54

Proof. The statement in (1) follows from the definition of the representation π ofG, realized on C∞(N). To prove (2) apply (10.5):

(π(

0 Eji

0 0

)f)(nX) = −(R(Ad

(I −X0 I

)(0 Eji

0 0

))f)(nX)

= −(R((

0 Eji

0 0

))f)(nX)

= −df

dt(nX+tEji

)

= −(∂f

∂xji)(nX).

The last identity in the Lemma also follows from (10.5). We need to compute

the n⊕ l⊕ n decomposition of Ad(n−X)(

0 0Eij 0

):

Ad(n−X)(

0 0Eij 0

)=(

0 −XEijX0 0

)+(−XEij 0

Eij EijX

).

Therefore, dχs

((Ad(n−X)

(0 0

Eij 0

))l+n

)= s

2 (−Tr(XEij)−Tr(EijX)) = −s xji.

We also have,

r(Ad(n−X)

(0 0

Eij 0

)n

)= −r

(−XEij 0

Eij EijX

)= −

∑k,l

xki xjl r

(0 Ekl

0 0

)= −

∑k,l

xki xjl∂

∂xkl.

Now (c) follows.

The construction of the conformally invariant system from Vm is as follows.There is a natural vector space isomorphism between P(n) and the constant co-efficient differential operators on n. Given P (Y ) ∈ P(n), P (∂x) is the constantcoefficient differential operator satisfying

(10.7) P (∂x) e〈Y,X〉 = P (Y ) e〈Y,X〉,

where 〈Y, X〉 = Tr((

0 0Y 0

)(0 X0 0

)) = Tr(Y X). Explicitly, given a monomial

P (Y ) = Πi,jymij

ij , P (∂X) = Πi,j( ∂∂xji

)mij .

Theorem 10.8. For m = 1, 2, . . . , p and s = −(m− 1)

∆mRS(∂x) : R,S as in (10.2)

is a conformally invariant system on C∞s (n).

The proof of this Theorem will occupy the remainder of this lecture. Our strategy

is to write a formula for the commutator[πs

(0 0

Eij 0

),∆m

RS(∂x)]

for arbitrary s.

55

We will obtain a sum of two terms (see (10.19)). The first term is of the formneeded for a conformally invariant system, the second term is the product of alinear function of s and sum of derivatives. The value s = −(m − 1) makes thesecond term vanish.

Note that [πσ(X),∆mRS(∂x)] = 0, for X ∈ n, as is clear by (2) of Prop. 10.6. The

computation of [πσ(X),∆mRS(∂x)] = 0, for X ∈ l is left as an exercise. One sees

that this commutator lies in spanC∆mRS(∂x), regardless of the value of s.

The computation of the commutator takes place in the Weyl algebra C[xij ,∂

∂xij].

Before beginning the computation we write down a few formulas which will be used.

In the Weyl algebra we have:

(10.9) [∂

∂xij, xij ] = 1, [

∂

∂xij, xkl] = 0 when (i, j) 6= (k, l).

(10.10) [u, v1 v2] = [u, v1] v2 + v1 [u, v2].

(10.11) [u, v1 v2 v3] = [u, v1] v2 v3 + v1 [u, v2] v3 + v1 v2 [u, v3].

The next formula simply uses the expansion of a determinant along a row orcolumn. We must take some care in writing this. Write R = r1, . . . , rm, S =s1, . . . , sm. When j ∈ R (resp. l ∈ S), let j = rj′ (resp. l = sl′) define j′ (resp.l′). Then for j ∈ R,

∑l∈S

(−1)j′+l′ xk,l ∆mR,S,j,l =

∆m

R,S if k = j

0 if k ∈ R \ j±∆m

R−j∪k,S if k /∈ R.

Here ∆mR,S,j,k = ∆m−1

R−j,S−k, the determinant of the matrix obtained from ∆mR,S

by omitting the j′-th row and the k′-th column. This, along with the similarexpansion along the i-th column, gives

(10.12)∑l∈S

(−1)j′+l′∆mR,S,j,l(∂X) ∂kl =

∆m

R,S(∂X) if k = j

0 if k ∈ R \ j±∆m

R−j∪k,S(∂X) if k /∈ R.

(10.13)∑k∈R

(−1)k′+i′ ∆mR,S,k,i(∂X) ∂kl =

∆m

R,S(∂X) if l = i

0 if k ∈ S \ i±∆m

R,S−i∪l(∂X) if l /∈ S.

Our final notation is that

δR(j) =

1 if j ∈ R

0 if j /∈ R.

We similarly define δS .56

Lemma 10.14.

[∆mR,S(∂X), xji] = δR(j) δS(i) (−1)i′+j′ ∆m

R,S,j,i(∂X).

Proof. This is clearly zero unless j ∈ R and i ∈ S. When j ∈ R and i ∈ S, (10.12)applies with k = j to give

[∆mR,S(∂X), xji] =

∑l∈S

(−1)l′+j′[∆m

R,S,j,l(∂X) ∂jl, xji

]= (−1)j′+l′ ∆m

R,S,j,l(∂X), by (10.9).

Lemma 10.15.∑k,l

[∆m

R,S(∂X), xki xjl∂kl

]= δR(j)δS(i)(1−m)∆m

R,S,j,i(∂X)

+ δR(j)

(xji∆m

R,S(∂X)) +∑k/∈R

xki

(±∆m

R−j∪k,S(∂X)))

+ δS(i)

(xji∆m

R,S(∂X) +∑l/∈S

xjl

(±∆m

R,S−i∪l(∂X)))

.

Proof. By Lemma(10.14) and (10.11),∑k,l

[∆m

R,S(∂X), xki xjl∂kl

]=∑k,l

(xki[∆m

R,S(∂X), xjl] ∂kl + [∆mR,S(∂X), xki]xjl ∂kl

).

(10.16)

The right hand side of (10.16) is∑kl

(δR(j)δS(l) xki(−1)j′+l′∆m

R,S,j,i(∂X) δkl

+ δR(k)δS(i) (−1)k′+i′ ∆mR,S,k,i(∂X)xjl ∂kl

)= δR(j)

∑k

xki

(∑l∈S

(−1)j′+l′ ∆mR,S,j,l(∂X) ∂kl

)

+∑

l

(∑k∈R

(−1)k′+i′∆mR,S,k,i(∂X) xjl∂kl

).

In the first term, by (10.12) only the k = j and k /∈ R terms survive. They give

(10.17) δR(j)

(xji∆m

R,S(∂X) +∑k/∈R

xki

(±∆m

R−j∪k,S(∂X)))

.

The second term requires a little more work. By (10.9)

xjl ∂kl = ∂klxjl − δjk.57

Combining (10.13) with this observation, we can show that the second term equals

δS(i)(∆m

R,S(∂X)xji+∑l/∈S

(±∆m

R,S−i∪l(∂X)xjl − qδR(j)(−1)i′+j′∆mR,S,j,l(∂X)

)).

Now we must move the xjl to the left of the expression using Lemma(10.14).The second term now is

δS(i)(xji∆m

R,S(∂X) + δR(j)(−1)j′+i′∆mR,S,j,i(∂X)

)+∑l/∈S

(±xjl∆m

R,(S−i∪l)(∂X)± δR(j)∆mR,S−i∪l,j,l(∂X)

)− qδR(j)(−1)i′+j′∆m

R,S,j,i(∂X).

Since ±∆R,S−i∪l,j,l(∂X) = ∆R,S,j,i(∂X), this equals

δS(i)

(xji∆m

R,S(∂X) +∑l/∈S

xjl(±)∆mR,S−i∪l(∂X)

)+ δR(j)δS(i)(−1)i′+j′(1−m)∆m

R,S,j,i(∂X).

(10.18)

Combining (10.17) and (10.18) we obtain the formula in the statement of theLemma.

We now conclude that[π(Eij),∆m

R,S ]

= δR(j) δS(i)(2s− 1 + m)(−1)i′+j′ ∆mR,S,j,i(∂X)

+ δR(j)

(xji∆m

R,S(∂X)) +∑k/∈R

xki

(±∆m

R−j∪k,S(∂X)))

+ δS(i)

(xji∆m

R,S(∂X) +∑l/∈S

xjl

(±∆m

R,S−i∪l(∂X)))

.

(10.19)

This completes the proof of the Theorem.

EXERCISES

(10.1) Compute [πσ(X),∆mRS(∂x)], for X ∈ l.

58

LECTURE 11. Conformally invariant systems, II

In this lecture the connection between conformally invariant systems and Vermamodules will be investigated. It turns out that each conformally invariant sys-tem acting on C∞(N, E) determines a finite dimensional p-subrepresentation ofU(g) ⊗U(p) E∗. Simple conditions on the conformally invariant system guaranteethat (a) F ⊂ U(g)⊗U(p) E∗n, (b) the module U(g)⊗U(p) E∗ is reducible (c) thereis a (nonzero) G-invariant differential intertwining operator between bundles E andF∗ on G/P .

Modules of the form U(g) ⊗U(p) V for finite dimensional irreducible represen-tations of p are called generalized Verma modules. Therefore, by the discussionof Lecture 7, the existence of conformally invariant systems acting on C∞(N,V)gives information about Homg(U(g)⊗U(p) F,U(g)⊗U(p) E∗), and in particular, thereducibility of generalized Verma modules.

Our first goal is to show how a conformally invariant system on C∞(N, E) gives ap-submodule of U(g)⊗U(p) E∗. We will keep in place our assumptions from Lecture8 that (σ,E) is a representation of P , E is the trivial bundle on N , E is a g-bundlevia πσ and all conformally invariant systems are straight. In addition we assumethat N acts on E by the identity.

We will begin by proving a well-known lemma which states that a Verma moduleis isomorphic to a space of distributions on N supported at the identity. The E∗-valued distributions on N supported at the identity may be defined as

D′e(N, E∗) ≡ Λ :C∞(N, E) → C : Λ is continuous

and Λ(f) = 0 if e /∈ supp(f).

The continuity of Λ is with respect to the C∞ topology on C∞(N, E). The Liealgebra g acts on D′

e(N, E∗) by

(X · Λ)(f) = −Λ(πσ(X)f), X ∈ g, f ∈ C∞(N, E).

Note that this makes D′e(N, E∗) a U(g)-module by

(u · Λ)(f) = Λ(πσ(u)f), u ∈ U(g),

where u → u is the involution defined in Lecture 6.

Note that N is diffeomorphic to Rn (via the exponential map). A standard factabout distributions on Rn is that those supported at 0 are precisely the distributionsof the form

Λ(f) =∑α

aα∂αf

∂xα

∣∣∣x=0

for some finite set of aα ∈ C. See, for example, [13, Thm. 6.25]. It follows thateach distribution in D′

e(N, E∗) is of the form

(11.1) Λ(f) = 〈e∗ , (R(u)f)(e)〉, for some u ∈ U(n).59

Lemma 11.2. The linear map

φ : U(g)⊗U(p) E∗ → D′e(N, E∗)

determined by

φ(u⊗ e∗) : f 7→ 〈e∗ , (πσ(u)f)(e)〉

is a U(g)-module isomorphism.

Proof. It should first be checked that φ is well-defined. For this it is enough tocheck that for Y ∈ p

φ(uY ⊗ e∗) = φ(u⊗ σ∗(Y )e∗).

This is a calculation:

φ(uY ⊗ e∗) = 〈e∗ , (πσ((uY ))f)(e)〉= 〈e∗ , (πσ(Y )πσ(u)f)(e)〉= −〈e∗ , σ(Y )(πσ(u)f)(e)〉, by (8.5), with n = e, Y ∈ p

= 〈σ∗(Y )e∗ , (πσ(u)f)(e)〉= φ(u⊗ σ∗(Y )e∗)(f).

The surjectivity of φ follows from (11.1) along with the fact that U(g)⊗U(p)E∗ '

U(n)⊗ E∗ as vector spaces and the fact of Exercise (11.1).

For injectivity, suppose uj ⊗ e∗i is a basis of U(n)⊗E∗ and φ(∑

aijuj ⊗ e∗j ) = 0.Then, by Exercise 11.1,

∑ij aij〈e∗i , (R(uj)f)(e)〉 = 0. Therefore, by independence

of e∗i ,∑

j aij(R(uj)f)(e) = 0, for all i and all f ∈ C∞(N, E). By N -invariance ofR(uj),

∑j aijR(uj)f = 0, for all i and f . By (5.13) this implies

∑j aijuj = 0, for

all i. By the independence of uj , all aij = 0.

The final item to check is that φ is a U(g)-homomorphism. For u1 ∈ U(g),

φ(u1u⊗ e∗) = 〈e∗ , (πσ((u1u))f)(e)〉= 〈e∗ , (πσ(u)πσ(u1)f)(e)〉= φ(u⊗ e∗)(π(u1)f)

=(u1 · φ(u⊗ e∗)

)(f).

Remark 11.3. This lemma was nearly proved earlier. In Lemma 6.7 we may takeF = C, the trivial one dimensional representation. Then

DG(E , F ) ' (U(g)⊗U(p) E∗)⊗C = U(g)⊗U(p) E∗.

However DG(E , F ) is the space of linear maps

D : C∞(G/P , E∗) → C∞(G)60

satisfying (6.6). Following such a map by evaluation at e gives a distribution sup-ported at e. Each such distribution is of this form by the standard result aboutdistributions mentioned earlier.

Now suppose that D1, . . . , Dm is a conformally invariant system of differentialoperators acting on C∞(N, E). Write

[πσ(X), Dj ] =∑

i

Cij(X)Di, Cij(X) ∈ C∞(N).

For each e∗ ∈ E∗ define the order zero distribution Λe∗ by

Λe∗(f) = 〈e∗ , f(e)〉, f ∈ C∞(N, E).

For D ∈ D(E) define(DΛe∗)(f) = 〈e∗ , (Df)(e)〉.

If Y ∈ p we use this definition and (8.6) to compute(Y ·(DjΛe∗)

)(f) = −DjΛe∗(πσ(Y )f)

= −〈e∗ , (Djπσ(Y )f)(e)〉= 〈e∗ ,

([πσ(Y ), Dj ]f

)(e)〉 − 〈e∗ ,

(πσ(Y )Djf

)(e)〉

=∑

i

Cij(Y )(e)〈e∗ , (Dif)(e)〉 − 〈e∗ , σ(Y )(Djf)(e)〉

=∑

i

Cij(Y )(e)(DiΛe∗)(f)− (DjΛσ∗(Y )e∗)(f).

Therefore,

(11.4) Y · (DjΛe∗) =∑

i

Cij(Y )(e)DiΛe∗ −DjΛσ∗(Y )e∗ .

This proves the following proposition.

Proposition 11.5. If D1, . . . , Dm is a conformally invariant system acting onC∞(N, E), then

(11.6) F ≡ spanDjΛe∗ : e∗ ∈ E∗, j = 1, . . . ,m

is a p-submodule of D′e(N, E∗).

By applying Lemma 11.2 we get the following corollary.

Corollary 11.7. For the finite dimensional p-module F defined by (11.6),Homp(F,U(g)⊗U(p) E∗) 6= 0.

Note that there is no reason for n to act trivially on F .

A simple condition on D1, . . . , Dm guarantees that the n-action on F is trivial.Recall from Exercise (3.3) that for each parabolic subalgebra p there is an H0 ∈ h

so that (a) α(H0) ∈ Z, (b) g(α) ⊂ l if and only if α(H0) = 0, and (c) g(α) ⊂ n ifand only if α(H0) > 0. This allows us to define the notion of homogeneity for aconformally invariant system. A conformally invariant system is called homogeneous

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when there is a constant d ∈ C so that [πσ(H0), Dj ] = dDj for all j = 1, . . . ,m (or,equivalently, Cij(H0) = dδij).

Lemma 11.8. If D1, . . . , Dm is a homogeneous conformally invariant system withstructure operator C, then C(Y )(e) = 0 for all Y ∈ n.

Proof. Let Y ∈ n. By Lemma 8.2

C([H0, Y ])(e) = (π1(H0)C(Y ))(e)− (π1(Y )C(H0))(e) + [C(H0), C(Y )](e).

Since H0 ∈ h ⊂ l, (8.5) gives π1(H0) = 0 at e, so the first term above vanishes. Theother two terms vanish since C(H0) is a constant multiple of the identity matrix.Now for Y ∈ g(−α) ⊂ n,

0 = C([H0, Y ]) = −α(H0)C(Y )(e),

But α(H0) 6= 0, so C(Y )(e) = 0. By linearity of C, C(Y ) = 0 for all Y ∈ n.

Theorem 11.9. If D1, . . . , Dm is a homogeneous conformally invariant systemacting on C∞(N, E), then for the p-representation F = spanCDjΛe∗ : j =1, . . . ,m, e∗ ∈ E∗,

Homl(F, U(g)⊗U(p) E∗n) 6= 0.

Proof. By Corollary 11.7 it is enough to verify that n acts by zero on F . However,this is clear from (11.4) by applying Lemma 11.8 and the fact that the n action onE (hence on E∗) is trivial.

In order to deduce reducibility of U(g)⊗U(p)E∗ we need a condition for F to con-

tain a constituent not equivalent to E∗. Assume that D1, . . . , Dm is a homogeneousconformally invariant system with [πσ(H0), Dj ] = dDj for all j. By irreducibilityof E, σ(H0) acts by a scalar c on E (by Schur’s Lemma). Hence the action onE∗ is by −c. Now (11.4) tells us that πσ(H0)DjΛe∗ = (d + c)DjΛe∗ . Therefore ifd 6= 0, F and E∗ have different H0-weights. We may conclude that if d 6= 0, thenU(g)⊗U(p) E∗ is reducible.

In order to apply 7.7 to obtain the existence of G-invariant differential operatorsbetween homogeneous bundles on G/P , the group P must be considered. Recallthat P is not in general connected, so the existence of a p homomorphism F →U(g) ⊗U(p) E∗n is not sufficient for the existence of an invariant intertwiningoperator.

The adjoint representation of P on U(g) and the dual representation on E∗ givea representation of P on U(g) ⊗ E∗. It is easily checked that this gives a well-defined representation of P on U(g) ⊗U(p) E∗. Furthermore, P acts on D′

e(N, E∗)by (p · Λ)(f) = Λ(`p−1f). Then it is easily seen that the isomorphism of Lemma11.2 is a P -intertwining map.

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Theorem 11.10. Under the hypothesis of Theorem 11.9, if F is P -stable, thenDG(E ,F∗) 6= 0.

Proof. The statement follows from the above discussion.

Given a homogeneous conformally invariant system a formula for the invariantdifferential operator is easily given. To do this, choose a basis fj of the image ofF in U(g) ⊗U(p) E∗n. Let f∗j be the dual basis. Also fix a basis e∗k of E∗.Write fj =

∑k ujk⊗e∗k, ujk ∈ U(g). Then

∑j fj⊗f∗j is in (U(g)⊗U(p)E

∗)⊗F ∗P .For ϕ ∈ C∞(G/P , E)

(11.11) (Dϕ)(g) =∑j,k

〈ek , (R(ujk)ϕ)(g)〉f∗j .

Example 11.12. Let G be GL(p + q,R) as in Lecture 10. Then in the nota-tion of that Lecture, for each m = 1, . . . , p consider the conformally invariant sys-tem ∆m

R,S(∂X). This is a homogeneous system for E = Cs, s = −(m − 1).The p-module F is irreducible of highest weight γ1 + · · · + γm and is dual tospanC∆m

R,S(Y ). We may conclude from this that

Homg(U(g)⊗U(p) F,U(g)⊗U(p) C∗s) 6= 0,

in particular, U(g) ⊗U(p) C∗s is reducible. In this example dim(E) = 1, so we may

omit the basis e∗k from the discussion. Let Λ1 be the distribution Λ1(f) = f(e).Then

F = span∆mR,S(∂X)Λ1.

Therefore, F ∗ is equivalent to span∆mR,S as P representation.

Realizing C∞(G/P ,Cs) as C∞s (N) (as in (11.11)) the intertwining operator

C∞s (N) → C∞

σµ(N,F∗

µ) may be written explicitly as follows. Let fR,S be thebasis corresponding to ∆m

R,SΛ1, then the dual basis is f∗R,S = ∆mR,S . Then

(Df)(n) =∑R,S

(∆mR,Sf(n))f∗R,S .

Now let’s turn things around. Suppose that we have F ⊂ U(g)⊗U(p) E∗n. (SoHomg(U(g) ⊗U(p) F,U(g) ⊗U(p) E∗) 6= 0.) It will be shown in the remainder ofthis lecture that F determines a conformally invariant system.

As noted in Lecture 4 there are several ways to define a representation of p onEnd(E). We choose to use the action defined by Y · T = −T σ(Y ), for Y ∈ p,T ∈ End(E). Therefore, End(E) is a U(p)-module under the action u · T =T σ(u) as in the paragraph preceding (6.6). Note that the p-representationEnd(E) is equivalent to E∗ ⊗ E when the action on the tensor product is on E∗

only; Y · e∗ ⊗ e = (σ∗(Y )e∗) ⊗ e. In what follows the module U(g) ⊗U(p) End(E)is defined with the U(p)-module structure on End(E) described above. It followseasily that U(g)⊗U(p) End(E) ' (U(g)⊗U(p) E∗)⊗ E.

63

Lemma 11.13. Defining

u⊗ T 7→ Du⊗T

with (Du⊗T f

)(n) = T

((πσ(u)(`n−1f))(e)

)(11.14)

gives, by linear extension, an isomorphism

U(g)⊗U(p) End(E) → DN (E).

Moreover, for Y ∈ p

(11.15)([πσ(Y ), Du⊗T ]f

)(e) =

(DY u⊗T f

)(e) + σ(Y )

(Du⊗T f

)(e).

Proof. First it needs to be checked that the map in (11.14) is well-defined onU(g)⊗U(p) End(E). For Y ∈ p,(

DuY⊗T f)(e) = T

((πσ((uY ))f)(e)

)= −T

((πσ(Y )πs(u)f)(e)

)= −T

(σ(Y )(πs(u)f)(e)

), by Lemma 8.6

=(Du⊗(Tσ(Y ))f

)(e).

In Lecture 8 we have shown that DN (E) ' U(n) ⊗ End(E) (by taking G = N

in Prop. 6.11). We have also seen (Prop. 6.11) that U(g)⊗U(p) End(E) ' U(n)⊗End(E) as vector spaces. Thus, U(g)⊗U(p)E

∗ ' DN (E). However the isomorphismof Prop. 6.11 is not given in the form of the statement of the lemma. We must seethat ((u⊗ T ) f)(e) = (Du⊗T f)(e), for u ∈ U(n). This is left for Exercise (11.1).

The proof of (11.15) is a calculation. Let Y ∈ p.([πσ(Y ), Du⊗T ]f

)(e)

=(πσ(Y )Du⊗T f

)(e)−

(Du⊗T πσ(Y )f

)(e)

= σ(Y )(Du⊗T f)(e)− T (πσ(u)πσ(Y )f)(e)

= σ(Y )(Du⊗T f)(e) + T((πσ((Y u))f)(e)

)= σ(Y )(Du⊗T f)(e) +

(DY u⊗T f

)(e).

Suppose F ⊂ U(g)⊗U(p) E∗ is any p-submodule. Fix a basis f1, . . . , fn of F andlet the representation of p be written in terms of matrix coefficients:

Y · fj =∑

i

aij(Y )fi, bij(Y ) ∈ C.

Also fix a basis e1, . . . , em of E and write

σ(Y )ek =∑

l

blk(Y )fl, blk(Y ) ∈ C.

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Consider fj ⊗ ek ∈ (U(g) ⊗U(p) E∗) ⊗ E ' U(g) ⊗U(p) End(E) and set Djk ≡Dfj⊗ek

∈ DN (E). Then the lemma tells us that for Y ∈ p,([πσ(Y ), Djk]f

)(e)

=(DY ·fj⊗ek

f)(e) + σ(Y )

((Djkf)(e)

)=∑

i

aij(Y )(Dikf)(e) +∑

l

blk(Y )(Djlf)(e).(11.16)

Theorem 11.17. Suppose F ⊂ U(g)⊗U(p) E∗ is a p-submodule and let fj , ek andDjk be as above. Then Djk is a conformally invariant system.

Proof. By the N -invariance of each Djk it suffices to prove independence at e.Suppose

∑cjk(Djkf)(e) = 0, for all f ∈ C∞(N, E). Then by the lemma,

∑fj ⊗

ek = 0. But fj⊗ek is independent, so cjk = 0, for all j, k. Prop. 8.10 and (11.16)imply Djk is a conformally invariant system.

EXERCISES

(11.1) Prove that πσ(u) = R(u) as operators on C∞(N, E), when u ∈ U(n).

(11.2) Prove that if D1, . . . , Dm is a (not necessarily homogeneous) conformallyinvariant system with the property that each Dj annihilates the constants, thenthe generalized Verma module U(g)⊗U(p) E∗ is reducible.

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LECTURE 12.Conformally Invariant Systems: The case of Abelian nilradicals.

Assume that g0 is a simple real Lie algebra and that g0 contains a parabolicsubalgebra p0 = l0⊕n0 with n0 abelian. Let p0 = l0 +n0 be the opposite parabolic.Denote by g , p, p, l, n and n, the complexifications of g0, p0, p0, l0, n0 and n0.

Following the setup for Theorem 9.5, there is a lexicographic order on h∗ de-fined by a set H1, . . . ,Hr with the following properties. (a) Letting ∆+ be thecorresponding positive system,

h +∑

α∈∆+

g(α) ⊂ p.

(b) β > α for all β ∈ ∆(n) and α ∈ ∆(l). (c) There is a unique simple root β1

in ∆(n). (d) The maximal strongly orthogonal set of roots appearing in Theorem9.5 is constructed by taking γ1 to be the largest root in ∆(n) (with respect to thelexicographic order). Then inductively, γj+1 is the largest root in ∆(n) stronglyorthogonal to γ1, . . . , γj .

Recall that if κ is the killing form and if λ ∈ h∗, an element Hλ is defined byκ(Hλ,H) = λ(H),H ∈ h. Define

h− =∑

CHγi .

Then we have the orthogonal decomposition

h = h− ⊕ h+, where h+ = H ∈ h : γi(H) = 0 for all i.

Example 12.1. Let g0 = sl(p + q,R) (p ≤ q, n = p + q), g = sl(p + q,C) and let

h =

a1

a2

. . .an

: ai ∈ C and Σai = 0

.

The roots ∆+(g, h) are εi − εj , 1 ≤ i < j ≤ p + q. Then the parabolic subalgebraswith abelian n are of the form(

A B0 D

): A ∈ gl(p,C), D ∈ gl(q,C), B ∈ Mp×q(C), T r(A) + Tr(D) = 0

with p + q = n. Then ∆(n) = εi − εj : i ≤ p < j and the unique simple root in∆(n) is β1 = εp−εp+1. The lexicographic order may be defined by H1, . . . ,Hn−1,

66

with

H1 =p∑

i=1

Eii

Hj =j−1∑i=1

Eii, for 1 < j ≤ p

Hj =j∑

i=1

Eii, for p < j < n.

Then it is easily checked that the maximal set of strongly orthogonal roots is γ1 =ε1 − εn, γ2 = ε2 − εn−1, . . . , γp = εp − εn−p+1. It is also easily checked from thedefinitions that h− consists of diagonal matrices:

h− = diag(t1, t2, . . . , tp, 0, . . . , 0,−tp, . . . ,−t2,−t1) .

The following lemma has a somewhat long proof and we will not give it. However,it is very easily verified for the above example.

Lemma 12.2. ([12]) Let ρ denote restriction of roots from h to h−. There are twocases:

(1) ρ(∆) ∪ 0 = ± 12 (γi ± γj)|h− : 1 ≤ i, j ≤ p. In this case the non-zero

ρ-image of some subsets of ∆ are given by

ρ(∆+(l)) = 12(γi − γj)|h− : 1 ≤ i < j ≤ p,

ρ(∆(n)) = 12(γi + γj |h− : 1 ≤ i, j ≤ p.

(2) ρ(∆) ∪ 0 = ± 12 (γi ± γj)|h− ,± 1

2γi|h− : 1 ≤ i, j ≤ p. In this case thenon-zero ρ-image of some subsets of ∆ are given by

ρ(∆+(l)) = 12(γi − γj)|h− : 1 ≤ i < j ≤ p ∪ 1

2γi|h−,

ρ(∆(n)) = 12(γi + γj)|h− : 1 ≤ i, j ≤ p ∪ 1

2γi|h−.

Example 12.3. We return to the case g = sl(p+q,C). Assume that H ∈ h−. Thenγi(H) = 2ti, and for α ∈ ∆+(l), α(H) is either ti−tj(i < j), ti or 0. The possibilityof α(h) = 0 occurs only when p < q. If α ∈ ∆(n), then α(H) is ti + tj , i < j or ti.

It will be important for us to define some subalgebras of g in a systematic fashion.This is done as follows.

Definition 12.4. Let p be the number of strongly orthogonal roots γi.(1) ∆p = α ∈ ∆ : α|h− 6= ± 1

2γi, for any i.(2) gp = (h +

∑α∈∆p g(α))ss, where the subscript means the semisimple part2.

2Recall that a reductive Lie algebra can be written as g = gss ⊕ z. We refer to gss as thesemisimple part of g. It may be easily checked that gss = [g, g]. In the case at hand h+

Pα∈∆p g(α)

is reductive; taking the semisimple part is simply removing the part of h which commutes withthe root vectors for roots in ∆p.

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(3) ∆m = α ∈ ∆p : α|h− ∈ spanRρ(γi) : i ≤ m(4) gm = (h +

∑α∈∆m g(α))ss.

Lemma 12.5. For m = 1, . . . , p the following hold.

(1) ∆m is a root subsystem of ∆.(2) gm is a simple Lie algebra and pm = gm ∩ p is a parabolic subalgebra

with abelian nilradical. The Levi decomposition of pm is lm + nm wherelm = gm ∩ l and nm = gm ∩ n.

We will not give a proof of this lemma, however, in our example in sl(p + q,C)the lemma clearly holds since the subalgebras gm are as follows.

gm =

A 0 B

0 0 0C 0 D

; A,D ∈ gl(m,C), B, C ∈ Mm×m(C), T r(A + D) = 0

.

Note that each gm ' sl(2m,C) and the parabolic subalgebra pm is the ‘middleparabolic’ subalgebra.

In what follows Eα will be a root vector for α. Also, Hγj = [Eγj , E−γj ] if theroot vectors are normalized properly. We will assume such a normalization.

The goal of this lecture is to build conformally invariant systems on appropriateline bundles over G/P . If λ is the fundamental weight for the unique simple rootβ1 in ∆(n), then there are one dimensional representations of l with weight −sλ.Let us denote these representations by C−s. Theorem 11.17 will be used as follows.The invariant theory (in particular, Theorem 9.5) will suggest certain candidatesfor irreducible l-representations F which embed3 into U(g)⊗U(p) Csn. However,as we will see, this can happen only for certain values of s. The remainder of thissection contains a computation of these values of s. We give a sketch of a veryclever manipulation of root vectors due to Wallach. for more details see [16]. Itshould be pointed out that the special case of abelian n which we are consideringis crucial to the computation.

We begin the computation by writing down the candidates for the irreduciblel-modules we hope to embed in U(g)⊗U(p) Csn.

As in Lecture 9, set n =∑

α∈∆+(l) g(α). Similarly, we set nj = n∩lj . Then Theo-rem 9.5 says that S(n)n ' C[u1, . . . , up], where each um is the highest weight vectorfor an irreducible subrepresentation of S(n) of highest weight γ1 + · · · + γm. Wewill denote this subrepresentation by Fm. The candidates for l-subrepresentationsof U(g)⊗U(p) Csn are Fm ⊗Cs.

Proposition 12.6. ([16]) um ∈ U(nm) and is the highest weight vector of a one-dimensional lm-representation with highest weight

∑m1 γi.

3Defining n to act trivially on F gives a p-representation on F . Therefore, such an embeddingof F gives a p-homomorphism. Thus Theorem 11.17 applies.

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Note that since n is abelian U(n) = S(n). Therefore, Fj ⊗ Cs ⊂ U(n) ⊗ Cs =U(g) ⊗U(p) Cs. We will need to find the ‘special values’ s for which Fj ⊗ Cs ⊂U(g) ⊗U(p) Csn. Thus, one needs to compute Y v ⊗ 1 = [Y, v] ⊗ 1 ∈ U(g) ⊗U(p)

Cs, for all Y ∈ n and all v ∈ Fj . The following lemma shows that this can beaccomplished by computing just one bracket.

Lemma 12.7. Let E−γmbe a nonzero root vector for −γm. If, for some particular

value of s, [E−γm , um]⊗1 = 0 in U(g)⊗U(p) Cs, then Fm⊗Cs ⊂ U(g)⊗U(p) Csn.

Proof. Any representation of a g on a vector space V defines a U(g)-module struc-ture on V . (See Exercise (5.3).) Therefore the adjoint representation of g on g

makes g a U(g)-module. We will temporarily use the notation u · Z for the actionof u ∈ U(g) on Z ∈ g in this module.

Recall that um spans a one-dimensional representation of lm, therefore the de-rived algebra [lm, lm] acts by zero on um. In other words, we have the identity[Y, um] = 0 in U(g) for all Y ∈ [lm, lm]. Similarly, [lm, lm] annihilates Cs. We willuse these two facts a number of times in the following calculations.

We claim that for any W ∈ g and u ∈ U([lm, lm]), [u ·W,um]⊗ 1 = u[W,um]⊗ 1is an identity in U(g) ⊗U(p) Cs. We prove this for all u = Y1Y2 . . . Yk, Yi ∈ [lm, lm]by induction on k. First k = 1. So, let Y ∈ [lm, lm].

[Y ·W,um]⊗ 1 = [[Y, W ], um]⊗ 1

= ([Y, [W,um]]− [W, [Y, um]])⊗ 1

= [Y, [W,um]]⊗ 1

= Y [W,um]⊗ 1− [W,um]Y ⊗ 1

= Y [W,um]⊗ 1

(12.8)

For u = Y1uY2 . . . Yk = Y1u′,

[u ·W,um]⊗ 1 = [Y1 · (u′W ), um]⊗ 1

= Y1[u′ ·W,um]⊗ 1, by (12.8),

= Y1u′[W,um]⊗ 1, by induction,

= u[W,um]⊗ 1.

Since nm is irreducible for [lm, lm], for any Y ∈ nm there is some uY ∈ U([lm, lm])so that Y = uY · E−γm . It now follows from the claim that if [E−γm , um] ⊗ 1 =0, then [Y, um] ⊗ 1 = [uY · E−γm ] ⊗ 1 = uY [E−γm , um] ⊗ 1 = 0. In particular[E−γ1 , um]⊗ 1 = 0.

Now we will use the fact that um is a highest weight vector to say that [X, um] =0, for X ∈ n. We now claim that for any u ∈ U(n), u[W,um]⊗ 1 = [u ·W,um]⊗ 1,

69

for any W ∈ n. To see this suppose u = X ∈ n. Then

[X ·W,um]⊗ 1 = [[X, W ], um]⊗ 1

= [X, [W,um]]⊗ 1− [W, [X, um]]⊗ 1

= [X, [W,um]]⊗ 1

= X[W,um]⊗ 1 + [W,um]X ⊗ 1

= X[W,um]⊗ 1.

This, along with induction (as in the proof of the first claim) establishes the claim.

Now it follows that since any Y ∈ n is some u · E−γ1 , u ∈ U(n) (since E−γ1 is alowest weight vector in n, [Y, um]⊗ 1 = 0, for all Y ∈ n. For any ` ∈ L and Y ∈ n,[Y, Ad(`)um] ⊗ 1 = Ad(`)[Ad(`−1)Y, um] ⊗ 1 = 0. Finally, by irreducibility of Fm,we have that [Y, v]⊗ 1 = 0 for all Y ∈ n and v ∈ Fm.

Hence we need to compute [E−γm , um]. Because of the symmetric nature of ourdefinitions, it is enough to compute [E−γp , up]. This computation that can be doneentirely in the smaller Lie algebra gp. This computation is in [16].

In the computation, the only properties of up that will be used are

(1) up is the highest weight of a lp-module,(2) the weight of up is

∑p1 γi,

(3) n · up = 0,(4) the multiplicity of Fm in S(n) is one.

In particular, explicit formulas for up are not needed.

Nonetheless it is interesting to see what um is for the sl(p+q,C) example. Theseare given by minors in the following sense. Let X = (xij) be in ∈ Mn×n(C) ' n.Let ∆m be the determinant of the minor of a matrix X which is obtained by deletingthe last p−m rows and the first q−m columns. Then replace each xij by the rootvector Ei,p+j . Since all root vectors in n commute, this defines an element in S(n).More explicitly,

um =∑

σ∈Sm

(−1)σE1,p+q−m+σ(1)E2,p+q−m+σ(2) · · ·Em,p+q−m+σ(m).

In particular, for sl(2p,C),

up =∑

σ∈Sp

(−1)σE1,p+σ(1)E2,p+σ(2) · · ·Em,p+σ(m).

We need to introduce one more definition and a preliminary Lemma.

Definition 12.9. Cp = α ∈ ∆+(l) : ρ(α) = 12 (γi − γp)|h− : i < p.

Lemma 12.10. (1) [E−γp , v] = 0 for all v ∈ np−1.(2) If α ∈ Cp and β ∈ ∆(np−1), then α + β is not a root.(3) If Z ∈ np, then [Z,E−γp ] = 0.

70

Proof. To prove (1), observe that if α ∈ ∆(np−1), then ρ(α) = 12 (γi + γj) with

i, j ≤ p − 1. On the other hand, if i, j ≤ p − 1, then 12 (γi + γj) − γp is not one of

the possible restrictions of roots to h−. The argument that proves (2) is similar:ρ(α) = 1

2 (γi − γp) for some i < p while ρ(β) = 12 (γk + γl) for some k, l ≤ p − 1.

There is no root in ∆ with restriction ρ(α) + ρ(β). The proof of (3) is similar.

We now sketch the computation of

(12.11) [E−γp, up]⊗ 1

and find the special value of s making (12.11) zero in U(g)⊗U(p) Cs.

In the remainder of the lecture we briefly describe this computation and find sj .It will be useful to keep in mind our example and to think of up as the determinantof an appropriate minor. Then, the next Proposition can be interpreted as anexpansion of a determinant by the last row and the furthest most left column ofthe appropriate minor, similar to the computations we did in Lecture 10.

Lemma 12.12. ([16])

(12.13) up = Eγpup−1 +

∑α,β∈Cp,α≤β

[Eα, Eγp] [Eβ , Eγp

]Zα,β

where Zα,β ∈ U(np−1)

Proof. (Sketch) Write up =∑

aα1,...,αpEα1 . . . Eαp with the sum running overα1, . . . , αp so that

∑αi =

∑p1 γi. Then

up = Eγpv +∑

αi 6=γp

aα1,...,αpEα1 . . . Eαp .

By Theorem 12.2 v is in U(np−1).

We want to argue that v is a multiple of up−1. It is clear that v has weight∑p−11 γi. We need to show that [Z, v] = 0 for all Z ∈ np. This characterizes up−1

(up to a scalar multiple) by the multiplicity one of Theorem 9.5. Therefore wecompute [E−γp , up]. By Lemma(12.10), we know that [E−γp , v] = 0. Hence,

[E−γp , up] = vH−γp + w1 +∑

α∈∆l

wαEα, w1, wα ∈ U(np).

Assume that Z ∈ np. Since up is the highest weight vector of a L-module, wehave [Z, up] = 0. We also know, by Lemma (12.10), that [Z,E−γp ] = 0. Thus,applying ad(Z) to the previous formula gives

0 = [Z, [E−γp, up]] = [Z, v]H−γp

+ v[Z,H−γp] + [Z,w1]

+∑

α∈∆(l)

[Z,wα]Eα +∑

α∈∆lC

wα[Z,Eα]

= [Z, v]H−γp + [Z,w1] +∑

α∈∆(l

vαEα.

(12.14)

71

In equation (12.14), [Z, v]H−γp ∈ U(n)h, [Z,w1] ∈ U(n) and the last term is inU(n)(n ⊕ n). We conclude from the PBW Theorem that each of the three termsin the last line of (12.14) is zero, therefore [Z, v] = 0 for all Z ∈ np. Since theL-irreducibles in U(n) have multiplicity one, we have v = c up−1.

It is a little more delicate to show that c 6= 0 and we will omit the proof.

For the second term in (12.13), note that since the weight must be γ1 + · · ·+ γp

and no Eγp occurs, there must be a pair Eα+γp , Eβ+γp which does occur. (Thisfollows from the Lemma 12.2 and the fact that γp is simple.) For α+γp and β +γp

to be roots, α and β must be in Cp. Requiring α ≤ β is simply to ensure that thesame term is not written twice.

Lemma 12.15. ([16])

[E−γp , up] = up−1H−γp +14〈γp, γp〉cpup−1 +

∑α∈∆+(l)

w′αEα

where w′α ∈ U(n) and cp is the cardinality of Cp = α ∈ ∆+(l) : ρ(α) = γi−γp

2 , i <

p

Proof. (Sketch) Set wp =∑

α≤β∈Cp[Eα, Eγp ] [Eβ , Eγp ]Zα,β . By Proposition (12.12),

we need to show that

[E−γp, wp] =

14〈γp, γp〉cpup−1 +

∑α∈∆(l)

w′αEα

Direct computations lead to the following formula:

(12.16) [E−γp, wp] = −1

2〈γp, γp〉

∑α,β∈Cp,α≤β

[Eβ , [Eα, Eγp ]]Zα,β +∑

α∈∆(l)

w′αEα.

We need to identify∑

α,β∈Cp, α≤β [Eβ , [Eα, Eγp ]]Zα,β with −cp

2 up−1. To do so,we need to establish a relation between wp and up−1. Once again we use the factthat up is a highest weight vector. For δ ∈ Cp, [Eδ, up] = 0. Lemma 12.10 impliesthat [Eα, Zα,β ] = [Eδ, up−1] = 0. Hence, by Lemma 12.12, for δ ∈ Cp

0 = [Eδ, up] = up−1[Eδ, Eγp ] + [Eδ, wp].

This implies that for δ ∈ Cp,

−up−1[Eδ, Eγp ] =∑

α≤β∈Cp

[Eδ, [Eα, Eγp ]] [Eβ , Eγp ]Zα,β

+∑

α≤β∈Cp

[Eα, Eγp ] [Eδ, [Eβ , Eγp ]]Zα,β .

Since [Eδ, [Eα, Eγp]]Zα,β does not contain [Eδ, Eγp

] (by Lemma 12.2), the PBWTheorem (applied to U(n) and a basis of root vectors) tells us that for each δ ∈ Cp,

(12.17)∑

α≤δ∈Cp

[Eδ, [Eα, Eγp ]]Zα,δ +∑

δ≤β∈Cp

[Eδ, [Eβ , Eγp ]]Zα,δ = −up−1.

72

Adding equation (12.17) over δ ∈ Cp gives

(12.18)∑

α,δ∈Cp,α≤δ

Zα,δ[Eδ, [Eα, Eγp ]] +∑

α,δ∈Cp,δ≤β

Zδ,β [Eδ, [Eβ , Eγp ]] = −cpup−1

Now we observe that sum of roots in Cp is not a root, and therefore the Jacobiidentity gives [Eα[Eβ , Eγp ]] = [Eβ [Eα, Eγp ]]. Equation (12.18) now implies

(12.19) −cp

2up−1 =

∑α≤δ∈Cp

Zα,δ[Eδ, [Eα, Eγp ]].

The lemma follows.

Theorem 12.20. Let m = 1, . . . , p and s = −cm. Also let Em = Fm ⊗Cs. Then

Homl(Em, U(g)⊗U(p) Csn) 6= 0.

Furthermore, Homg(U(g) ⊗U(p) Em,U(g) ⊗U(p) Cs) 6= 0 and for the bundles Em

and C−s on G/P , D(C−s, Em) 6= 0.

Proof. Observe that this Lemma implies that

[Eγp, up]⊗ 1 = up−1H−γp

⊗ 1 +14〈γp, γp〉cpup−1 ⊗ 1 = 0

if and only if sλ(Hγp) = 〈γp,γp〉4 cp. By Lemma(12.2),

λ(Hγp) = λ(Hα) =

12〈β1, β1〉 =

12〈γp, γp〉.

The last inequality follows from the fact that the simple root β1 in ∆(n) has thesame length as each of the strongly orthogonal roots γi.

Corollary 12.21. For our sl(p + q,C) example the special values are s = m− 1.

EXERCISES

(12.1) Do the computation to show that the special value of s corresponding toup in the sl(p,C) example is −(p − 1) by using the formula for up in terms of thedeterminant.

For the sl(p + q,C) example write down the corresponding conformally invariantsystem. (See Theorem 11.17.) Compare with Lecture 10.

73

LECTURE 13. Conformally Invariant Systems for Heisenberg Type

parabolic subgroups, I

Consider a complex simple Lie algebra g other than sl(2,C). It is a fact thateach such g contains a parabolic subalgebra p = l ⊕ n with n a Heisenberg Liealgebra. Up to conjugacy there is a unique such parabolic subalgebra.

Definition 13.1. Let V be a complex vector space with a nondegenerate alternat-ing bilinear form ω (called a symplectic form). It follows that V has even dimension,say 2n. Then define a 2n + 1-dimensional Lie algebra by h2n+1 ≡ V ⊕C with Liebracket defined by [(v, z), (v′, z′)] = (0, ω(v, v′)). A Lie algebra is a Heisenberg Liealgebra if it is isomorphic to h2n+1 (n ≥ 1).

Exercise: Verify that this bracket operation defines a Lie algebra. Show that thesubalgebra

n =

0 x1 . . . xn z

y1

...yn

0

of gl(n + 1,C) is isomorphic to h2n+1.

A Lie algebra n is said to be 2-step nilpotent if n is nonabelian and [n[n, n]] = 0.It may be shown that h2n+1 is the unique (up to isomorphism) Lie algebra whichis 2-step nilpotent and has one-dimensional center.

We say that a parabolic subalgebra p = l ⊕ n is of Heisenberg type if n is aHeisenberg Lie algebra.

In earlier lectures we were successful in using invariant theory to construct con-formally invariant systems and gain information on reducibility of Verma moduleswhen the parabolic subalgebra is of abelian type, that is when n is abelian. The par-abolic subalgebras of Heisenberg type may be considered ‘close’ to those of abeliantype since [X[Y, Z]] is always zero. On the other hand it may be considered as theopposite extreme since the center is as small as possible. We will see how methodssimilar to those used for parabolics of abelian type may be used for the Heisenbergtype parabolics.

Now suppose G0 is a real simple Lie group with Lie algebra g0. A parabolicsubalgebra p0 of g0 is of Heisenberg type when p = p0 ⊗ C is a subalgebra ofg = g0 ⊗ C of Heisenberg type. Not all simple real Lie algebras have parabolicsubalgebras of Heisenberg type. For example so(n, 1) does not. Each split4 simpleLie group does have one. Let us fix a parabolic subalgebra p0 of Heisenberg typeand let P0 be the corresponding parabolic subgroup of G0.

4A real Lie algebra is split means amin is a Cartan subalgebra. A group with such a Lie algebrais called split.

74

Our goal in the final two lectures is to show how to construct conformally in-variant systems on C∞(G0/P0,Cs), for appropriate homogeneous line bundles Cs.As in the abelian case, this will be possible for certain ‘special values’ of s.

In order to study the relevant invariant theory, let G be a complex Lie groupwith Lie algebra g and let P = LN be the parabolic subgroup with Lie algebrap = p0⊗C. Then, by Vinberg’s theorem ([10, Ch. X]), n/[n, n] is a prehomogeneousvector space under the adjoint action of L. By the complete reducibility of theadjoint action of L on n we may write the L-decomposition as n = V + ⊕ z, wherez is the one-dimensional center of n. Then V + ' n/[n, n] is the prehomogeneousvector space which will play a key role in our construction of conformally invariantsystems.

In addition to the decomposition n = V + ⊕ z write n = V − ⊕ z′.

13.1. The invariant theory. A covariant is an irreducible representation (ρ,W )of L along with a nonzero L-equivariant polynomial map F : V + → W . By apolynomial map we of course mean a map for which each coordinate is polynomialin V +. So a covariant comes from an element of P (V +) ⊗ WL. But this isHomL(W ∗, P (V +)). There are four natural covariants τ1, . . . , τ4 which will play arole in our construction.

In order to define these covariants we need to set up a little notation. Fix aCartan subalgebra h of g and a positive system of roots ∆+ = ∆+(h, g). Let Π bethe corresponding system of simple roots. Denote by γ the highest root. This is, bydefinition, the highest weight of the adjoint representation. Choose a root vectorEγ ∈ g(γ) normalized so that −κ(Eγ , θ(Eγ)) = 1. By scaling κ if necessary weassume that 〈γ , γ〉 = 2. It is a fact that the Heisenberg parabolic is the parabolicsubalgebra p defined (as in Lecture 3) by S = Π \ α ∈ Π : 〈α , γ〉 = 0. Thismeans that

p = l⊕ n = (h +∑

α∈∆,〈α ,γ〉=0

g(α)) + (∑

α∈∆,〈α ,γ〉>0

g(α)).

One can also easily see that z = CEγ . Let Hγ (as defined earlier) be givenby γ(H) = 〈H , Hγ〉. Note that it follows that α(Hγ) = 〈α , γ〉. In particular,ad(Hγ)Eγ = 2Eγ . When α ∈ ∆(V +), 〈α , γ〉 = 1. Therefore, there is a grading ofg by eigenvalues of ad(Hγ). Set g(k) = X ∈ g : ad(Hγ)(X) = kX. Then

g = g(−2) ⊕ g(−1) ⊕ g(0) ⊕ g(1) ⊕ g(2),

and [g(j), g(k)] ⊂ g(j+k). Since g(0) = l, g(±1) = V ± and g(±2) = g(±γ), this gradingis

g = g(−γ) ⊕ V − ⊕ l⊕ V + ⊕ g(γ).

Since g(γ) is one dimensional there is a character χ : L → C× so that Ad(`)Eγ =χ(`)Eγ . This character will play an important role in all that follows. Since the

75

Killing form pairs gγ and g−γ it follows that (g(γ))∗ ' g(−γ) and the representationof L on g(−γ) is by χ(`−1).

Example 13.2. Consider g = sl(r + 1,C), r ≥ 2. It is convenient to form the so-called extended Dynkin diagram by attaching −γ by the same rules as were usedto form the Dynkin diagram.

α1 α2 . . .αr−1 αr

−γ

PPPPPPPPPPPPPP

nnnnnnnnnnnnnn

Then γ = α1 + . . . + αr = ε1 − εr+1 and ∆(h, l) = εi − εj : 2 ≤ i < j ≤ r.The prehomogeneous vector space (Ad, V +) is the same as GL(1,C)×GL(r−1,C)acting on Cr−1 ×Cr−1 by

(λ, g)(v1, v2) = (λ(gt)−1v1, λ det(g)gv2).

Here we are thinking of Cr−1 as column vectors. The character χ is χ(λ, g) =λ2 det(g).

Example 13.3. Let g = so(2r,C). The extended Dynkin diagram isαr−1

α1 α2 . . . αr−2

>>>>

>>>>

−γ

αr

Here γ = α1+2α2+. . .+2αr−2+αr−1+αr. A representation equivalent to (Ad, V +)is GL(2) × SO(2r − 4) acting naturally on the tensor product C2 ⊗ C2r−4. Ourcharacter χ is χ(g1, g2) = det(g1).

Now we are ready to define the four natural covariants. For 1 ≤ k ≤ 4 andX ∈ V +, we define τk(X) ∈ g by

τk(X) =1k!

ad(X)k(E−γ).

Since ad(X) increases the Hγ-weight by 1 we have polynomial maps

τ1 : V + → V −

τ2 : V + → l

τ3 : V + → V +

τ4 : V + → g(γ)

76

Lemma 13.4. For ` ∈ L, X ∈ V + and 1 ≤ k ≤ 4, we have

τk(Ad(`)X) = χ(`)Ad(`) τk(X).

Proof. Note that

τk(X) =1k

ad(X)(τk−1(X)

).

Assume that the Lemma holds for k − 1, then

τk(Ad(`)X) =1k

ad(Ad(`)X)(τk−1(Ad(`)X)

)=

1k

ad(Ad(`)X)(χ(`)Ad(`)τk−1(X)big)

=1k

χ(`)ad(Ad(`)X)(Ad(`)τk−1(X)

)=

1k

χ(`)Ad(`)ad(X)(τk−1(X)

)= χ(`)Ad(`)τk(X).

To complete the induction argument we check when k = 1.

τ1(Ad(`)X) = [Ad(`)X, E−γ ]

= Ad(`)[X, Ad(`−1)E−γ ]

= Ad(`)[X, χ(`−1)−1E−γ ]

= χ(`)Ad(`)[X, E−γ ]

= χ(`)Ad(`)τ1(X).

Since gγ is one-dimensional, there is a quartic polynomial ∆ on V + such thatτ4(X) = ∆(X)Eγ . The lemma implies that, ∆(Ad(`)X) = χ2(`)∆(X). Therefore,if ∆ is nonzero then it is a relative invariant for the prehomogeneous vector spaceV +, associated to the character χ2. Nothing done so far guarantees that the mapsintroduced are non-zero. Indeed, in type Cr one can check that τ3 = τ4 = 0 whileτ1 and τ2 are non-zero. For each of the other Lie algebras τ1, . . . , τ4 are all nonzero.

We assume that g is not of type Cr and prove that τk 6= 0 for 1 ≤ k ≤ 4. A keyobservation is that under this assumption there exists a simple root δ ∈ ∆(V +, h)having the property that δ′ = γ−δ ∈ ∆(V +, h) and γ−2δ is not a root. A little tech-nical lemma shows that it is possible to normalize root vectors E±δ, E±(γ−δ), E±γ

and vectors Hδ,Hγ−δ in such a way that the linear map satisfying

Hδ → E11 − E22, Eδ → E12, E−δ → E21

Hδ′ → E22 − E33, Eδ′ → E23, E−δ′ → E32

Hγ → E11 − E33, Eα → E13, E−γ → E31

is an isomorphism between the subalgebra of g generated by E±δ, E±δ′ and sl(3).

Lemma 13.5. For scalars x and y we have77

(1) τ1(xEδ + yEδ′) = yEδ − xEδ′ ,(2) τ2(xEδ + yEδ′) = xy

2

(Hδ −Hδ′

),

(3) τ3(xEδ + yEδ′) = −x2y2 Eδ + xy2

2 Eδ′ ,(4) τ4(xEδ + yEδ′) = x2y2

4 Eγ .

Proof. In view of the previous observation, it is enough to do the calculations onsl(3).

Proposition 13.6. The point Xo =√

2(Eδ + Eδ′) ∈ V + is a generic point of the

prehomogeneous vector space (ad, V +).

Proof. In Type A2 the Proposition is verified directly. Otherwise, one needs toargue that ∆ generates the semigroup of relative invariant polynomials of (Ad, V +).Since ∆(Xo) 6= 0, the Proposition follows.

EXERCISES

(13.1) Verify the statements made in Examples 13.2 and 13.3.

(13.2) For the simple Lie algebras so(2r + 1,C) and sp(2r,C) give the same infor-mation as was given in Example 13.2.

(13.3) Show that for sp(2r,C) both τ3 and τ4 are zero.

(13.4) For each of the classical simple Lie algebras, other than sp(2r,C), find theroots δ and δ′

78

LECTURE 14. Conformally Invariant Systems for Heisenberg Type

parabolic subgroups, II

This lecture is a continuation of the previous lecture. We will only describe someof the computations. Details may be found in [2]

Since n = V + + z is a Heisenberg Lie algebra, the space V + carries a non-degenerate alternating form ω given by

[X1, X2] = ω(X1, X2)Eγ , for X1, X2 ∈ V +.

The form is indeed non-degenerate since for each Eα ∈ V +, Eγ−α ∈ V + and[Eγ−α, Eα] 6= 0. By applying Ad(`) to the above formula we have

(14.1) ω(Ad(`)X1,Ad(`)X2) = χ(`)ω(X1, X2).

Note that there is an involution of the set of roots in V + given by α 7→ γ − α. Setα′ = γ−α, for each α ∈ ∆(V +), so α 7→ α′ is the involution of ∆(V +). In order tocarry out the necessary calculations we fix a basis Eα : α ∈ ∆(V +) consisting ofroot vectors normalized in a special way5. The basis Eα : α ∈ ∆(V +) satisfiesω(Eα, Eβ′) = 0 if and only if α 6= β. When α = β, we set ω(Eα, Eα′) = ωα.Therefore,

ω(Eα, Eβ′) = ωαδα,β .

It follows that any vector Y ∈ V + may be expanded as

(14.2) Y =∑

α∈∆(V +)

ω−1α ω(Y, Eα′)Eα.

Check: Write Y =∑

yβEβ . Then∑α

ω−1α ω(

∑β

yβEβ , Eα′)Eα =∑α,β

ω−1α yβω(Eβ , Eα′)Eα =

∑α

yαEα = Y.

14.1. Embedding of covariants into the enveloping algebra. The covari-ants τ1, . . . , τ4 will be used to find certain l-subrepresentations of U(n). Thesel-subrepresentations of U(n) will later be used to construct the systems of the dif-ferential operators which will turn out to be conformally invariant systems.

Recall that

τk : V + → Wk, k = 1, . . . , 4

are L-equivariant polynomial maps. That is

τk ∈ P (V +)⊗WkL ' HomL(W ∗k , P (V +)).

5One must be precise about the normalization of root vectors in order to construct the con-formally invariant systems. The details of this normalization is given in [2]. However, for thecomputations presented in this lecture, these details are not needed

79

From Lecture 13 we know that

W1 = V − ⊗Cχ

W2 = l⊗Cχ

W3 = V + ⊗Cχ

W4 = z⊗Cχ ' Cχ2 .

Our goal is to get an element of U(n) for each polynomial τk. This is accomplishedas follows. The symplectic form ω gives an isomorphism

(V +)∗ ' V + ⊗Cχ−1

by formula (14.1). Now P (V +) ' S((V +)∗) ' S(V + ⊗Cχ−1). Since Wk occurs ashomogeneous polynomials of degree k, we have

W ∗k → Sk(V +)⊗Cχ−k ,

thereforeW ∗

k ⊗Cχk → Sk(V +) → U(n).

The following table records the subrepresentations of U(n) determined by the fourcovariants. Here, the Killing form is used to identify certain dual spaces withsubspaces of g.

k W ∗k ⊗Cχk

1 V +

2 [l, l]⊗Cχ

3 V − ⊗Cχ2

3 Cχ2

In the second entry we have removed the center of l. This is because the centermaps to 0, thus does not contribute.

In the k = 1 case, the embedding of V + into S(V +) is clear. Up to a multiplica-tive constant this happens in only one way.

When k = 2 it is not so clear how [l, l] ⊗ Cχ embeds into S(V +). In order toget an explicit embedding, we must trace through the isomorphisms which lead usfrom τ2 to [l, l] ⊗Cχ ⊂ S(V +). There are three points which must be clear. Thefirst is how τ2 gives [l, l]⊗Cχ−1 ⊂ P (V +). Since the Killing form identifies l withitself, Z maps to the polynomial (in X)

κ(τ2(X), Z).

The second point is that P (V +) ' S((V +)∗). This is a general fact for any(finite dimensional) vector space V . Given

u =1n!

∑σεSm

v∗σ(1)v∗σ(2) . . . v∗σ(m), v∗i ∈ V ∗,

there is a polynomial

Pu(X) = v∗1(X)v∗2(X) . . . v∗m(X).80

The linear extension of the map u 7→ Pu is an isomorphism S(V ∗) → P (V ).

The third point to make explicit how (V +)∗ ' V +⊗Cχ−1 . This is accomplishedby Y 7→ ω(Y, · ).

Now, combining the last two isomorphisms gives and isomorphism

(14.3) S(V + ⊗Cχ−1) → P (V +).

Explicitly, if

u =1n!

∑σεSm

Y ∗σ(1)Y

∗σ(2) . . . Y ∗

σ(m), Y ∗i ∈ (V +)∗,

the corresponding polynomial is

(14.4) Pu(X) = ω(Y1, X)ω(Y2, X) . . . ω(Ym, X).

Back to τ2. For Z ∈ [l, l] and α ∈ ∆(V +) write

ad(Z)(Eα) =∑

β∈∆(V +)

Mβ,α(Z)Eβ .

The functions Mβ,α(Z) are called matrix coefficients for the representation of [l, l]on V +. We claim that the map [l, l]⊗Cχ−1 → S(V +) determined by τ2 is

(14.5) Z 7→ 12

∑α,β∈∆(V +)

ω−1β Mα,β′(Z)(EαEβ + EβEα)

Proof. This is a computation using the remarks above. We show that the polyno-mial defined by the 1

2

∑α,β∈∆(V +) ω−1

β Mα,β′(Z)(EαEβ + EβEα) via formula (14.4)81

is κ(τ2(X), Z).∑α,β

ω−1β Mα,β′(Z)ω(Eα, X)ω(Eβ , X)

=∑

β

ω−1β

(∑α

Mα,β′(Z)ω(Eα, X))ω(Eβ , X)

=∑

β

ω−1β ω(

∑α

Mα,β′(Z)Eα, X)ω(Eβ , X)

=∑

β

ω−1β ω([Z,Eβ′ ], X)ω(Eβ , X)

=∑

β

ω−1β ω(Eβ′ , [X, Z])ω(Eβ , X),

by differentiating (14.1) and using the fact that Z ∈ [l, l],

= ω(∑

β

ω−1β ω([Z,X], Eβ′)Eβ , X

)= ω([Z,X], X), by (14.2)

= ω(X, [X, Z])

=12κ(E−γ , ω(X, [X, Z])Eγ), assuming the normalization κ(E−γ , Eγ) = 2,

=12κ(E−γ , [X, [X, Z]])

=12κ([X, [X, E−γ ]], Z)

= κ(τ2(X), Z).

Similar computations may be done for τ3 and τ4.

14.2. The conformally invariant systems. Consider the one dimensional rep-resentations χs, s ∈ R, of L. Extend these representations to all of P by making N

act trivially. Denote by Cs, or Cχs , the space of this representation. Note that thedifferential of χs has weight sγ. We will use the notation Csγ for the representationspace for the Lie algebra representation of weight sγ. Then consider C∞(N,Cs)and the action of g by πs as in Prop. 8.4.

Each embedding W ∗k ⊗ Cχ−k → S(V +) → U(n) gives a family of differential

operators by the right action. For certain special values of s these families ofoperators give conformally invariant systems. We will show in detail how this goesfor k = 1. In fact we will do this two ways. Then we will give a short explanationof the situation of k = 2, 3 and 4.

Consider the case k = 1. The embedding V + → U(n) gives a family of differentialoperators R(Y ) : Y ∈ V +. We let Ω1 be the system R(Eα) : α ∈ ∆(V +).

Proposition 14.6. When s = 0, Ω1 is a conformally invariant systems.82

Proof. By Theorem 11.17 it suffices to show that V +⊗1−sγ ⊂U(g)⊗U(p) C−sγ

n,when s = 0. This is a simple computation using the facts that n acts by 0 on 1−sγ

(by definition) and [l, l] acts by 0 on 1−sγ (since χ is one dimensional). Let Y ∈ V +.

Since [E−γ , Y ] ∈ n we have

E−γY ⊗ 1−sγ = [E−γ , Y ]⊗ 1−sγ + Y E−γ ⊗ 1−sγ = 0.

For β ∈ ∆(V +),

E−βY⊗1−sγ = [E−β , Y ]⊗ 1−sγ + Y E−β ⊗ 1−sγ

=∑α

yα[E−β , Eα]⊗ 1−sγ

=∑α

yα ⊗ [E−β , Eα]1−sγ

= s yβ〈γ , β〉1⊗ 1−sγ .

The last equality holds because [E−β , Eα] ∈ [l, l] unless α = β, [E−β , Eα] = H−α

and −sγ(H−α) = s〈γ , α〉. Now recall that the parabolic subalgebra p is defined byγ, thus 〈γ , α〉 > 0 for all α ∈ ∆(n). We may conclude that Y ⊗ 1−sγ is annihilatedby n if and only if s = 0.

A corollary to the proof is that Homg(U(g)⊗U(p) V +,U(g)⊗U(p) C−sγ) 6= 0.

A proof of the Proposition may also be given using commutators. Formula(11.15) along with Prop. 8.10 will be applied. In using Formula (11.15) we mustcalculate WY ⊗ 1, in U(g)⊗U(p) Hom(Cs,Cs) ' (U(g)⊗U(p) C−s)⊗Cs, for w ∈ p

and X ∈ V +.

WY⊗1−sγ = [W,Y ]⊗ 1−sγ + Y W ⊗ 1−sγ

= [W,Y ]⊗ 1−sγ

= ([W,Y ]p + [W,Y ]n)⊗ 1−sγ

= −sdχ([W,Y ]l)⊗ 1−sγ + ([W,Y ]n)⊗ 1−sγ

Formula (11.15) now gives, for W ∈ p, Y ∈ V +,([πs(W ), R(Y )]f

)(e) =

(R([W,Y ]n)f

)(e)− sdχ([W,Y ]l)f(e) + sdχ(X)(R(Y )f)(e).

When s = 0, Prop. 8.10 may be applied to conclude that Ω1 is a conformallyinvariant system.

This completes the case of k = 1. We will briefly discuss the other cases. Forτ2, [l, l] ⊗ Cχ → U(n). One observes, by looking at each simple Lie algebra, thatoften [l, l] decomposes into the direct sum of two simple ideals. In these casesone simple ideal is isomorphic to sl(2,C). We will write [l, l] = lsmall ⊕ lbig, withlsmall ' sl(2,C). Then (14.5) gives two systems of differential operators, one bytaking Z ∈ lsmall and the other by taking Z ∈ lbig. Call these two systems Ωsmall

2

and Ωbig2 .

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For k = 3, computations similar to those which gave Formula (14.5) give a systemΩ′

3. It turns out that this does not give a conformally invariant systems (for anyvalue of s). The point is that V − ⊗ Cχ2 occurs in U(n) in two different ways. Asecond term must be added to Ω′

3 to get a system Ω3. Similarly τ4 gives a systemΩ4 (with one operator). The following table gives the special values of s for whichthe system is conformally invariant.

Type Ω1 Ωbig2 Ωsmall

2 Ω3 Ω4

Ar(r ≥ 2) 0 0 (r − 1)/2 none (r − 2)/2Br(r ≥ 3) 0 r − 5/2 1 none r − 2Cr(r ≥ 2) 0 −1/2 none none noneDr(r ≥ 5) 0 r − 3 1 none r − 5/2E6 0 2 none 3 9/2E7 0 3 none 5 15/2E8 0 5 none 9 27/2F4 0 3/2 none 2 3G2 0 2/3 none 1/3 1/2

EXERCISES

(14.1) Let p be a parabolic subalgebra of Heisenberg type. Use Prop. 8.10 tofind a formula for

([πs(E−γ), R(Y )]f

)(n), for W ∈ p, Y ∈ V + and arbitrary n ∈

N . (Hint: Write n = exp(X + zEγ), X ∈ V +. Then you will need to compute[Ad(n−1)E−γ , Y ]. Do this by using the fact that Ad(exp(U)) = exp(ad(U)) for anyU ∈ g. Since N is nilpotent, the series expansion of exp(ad(U))(Y ) is finite forU ∈ n.)

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LECTURE 15. Gyoja’s conjecture

There is a conjecture of Gyoja which relates the reducibility of Verma modulesto the roots of the b-function of a certain naturally defined polynomial. A versionof this conjecture will be stated at the end of this lecture. First, an example isgiven which motivates the conjecture.

Consider the example of sl(2p,C) and the ‘middle’ parabolic, i.e., the maximalparabolic subalgebra for which εp − εp+1 is the unique simple root in ∆(n). Thenthe prehomogeneous vector space n is Mp×p(C) with action of L = S(GL(p,C) ×GL(p,C)) given by ρ(`1, `2)X = `1X`−1

2 . ∆(X) = det(X) is a relatively invariantpolynomial. As mentioned in Lecture 9,

∆(∂X)∆(X)s+1 = (s + 1)(s + 2) . . . (s + p)∆(X)s.

Our computations in Lecture 10 (and again in Lecture 12) show that the Vermamodule U(g) ⊗U(p) Csλ, with λ equal to the fundamental weight for εp − εp+1, isreducible for s = 0,−1,−2, . . . ,−(p− 1). These values of s are precisely the rootsof the b-function shifted by one. It is a fact that the Verma module is reducible ifand only if s ∈ −p + Z>0. This may be stated as the Verma module U(g)⊗U(p) Cs

is reducible if and only if b(s+ k) = 0 for some k ∈ Z>0. This illustrates the strongconnection between reducibility of Verma modules and roots of certain b-functions

Assume for the rest of this lecture that p = l + n us a maximal parabolic sub-algebra. Assume that p ⊃ h +

∑α∈∆+

g(α). Let β be the unique simple root in ∆(n),

and let λβ be the corresponding fundamental weight.

In order to state Gyoja’s conjecture we need a polynomial to fill the role playedby det(X) in the above example.

Denote by V = V β the irreducible representation of lowest weight −λβ and letv− be the lowest6 weight vector. Then the dual representation V ∗ = (V β)∗ hashighest weight λβ . Let v∗− be the lowest weight vector in V ∗. Define

∆β(X) = 〈v∗− , exp(X)v−〉, for X ∈ n.

Lemma 15.1. ∆β(X) is polynomial in X ∈ n.

Proof. Since X ∈ n, X is nilpotent. Therefore, for any representation ρ, there is an

integer k so that ρ(X)k = 0. Therefore, exp(X)v− =k∑

i=0

Xi

i!v−. The lemma now

follows.

Here are a few examples of ∆β(X).

6A lowest weight vector in a representation ρ is a weight vector v− with the property that

ρ(X)v− = 0 for all X ∈ g(−α), for α ∈ ∆+.

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Example 15.2. Let g = sl(p + q,C), and assume p ≤ q. Set n = p + q. Takeβ = εp−εp+1, so the parabolic subalgebra p is the same as in earlier examples. Thenλ = λβ =

∑ni=q+1 εi and V ' ∧pCn with lowest weight vector v− = eq+1 ∧ · · · ∧ en.

The dual is V ∗ '∧q Cn. (This fact is easily checked by giving a nondegenerate

pairing∧p Cn ×

∧q Cn → C. This is accomplished as follows. Since∧n Cn =

C e1 ∧ · · · ∧ en, there is a scalar Cv,w so that

v1 ∧ · · · ∧ vp ∧ w1 ∧ · · · ∧ wq = Cv,we1 ∧ · · · ∧ en.

Then a nondegenerate pairing is defined by 〈v1 ∧ · · · ∧ vp , w1 ∧ · · · ∧ wq〉 = Cv,w.)The lowest weight vector in V ∗ is therefore v∗− = ep+1 ∧ · · · ∧ en. Now let(

0 X0 0

), X ∈ Mp×q(C)

so that

exp(

0 X0 0

)=(

Ip X0 Iq

)Write X = (xij), so

∆β(X) = 〈ep+1 ∧ · · · ∧ en ,

(I X0 I

)eq+1 ∧ · · · ∧ en〉

= 〈ep+1 ∧ · · · ∧ en , (eq + Xeq−p+1) ∧ · · · ∧ (ep+q + Xeq)〉

= 〈ep+1 ∧ · · · ∧ en , (p∑

j=1

xj,q−p+1ej) ∧ · · · ∧ (p∑

j=1

xj,qej)〉

= (−1)pq det(X)

where X is the furthest p× p minor to the right in X.

Example 15.3. Suppose that g is a simple complex Lie algebra, other thansl(n,C), and p is a parabolic subalgebra of Heisenberg type. The the fundamentalweight λ = γ is the highest root. So V = g, the adjoint representation. Writing anelement of n as X + zEγ (with X ∈ V + and z ∈ C), we have

∆β(X + zEγ) = κ(E−γ ,Ad(exp(X + zEγ))E−γ)

= κ(E−γ ,

4∑k=0

ad(X + zEγ)k

k!E−γ)

= κ(E−γ , (ad(X)4 + z2ad(Eγ))E−γ)

= ∆(X) + z2.

(The last equality assumes some specific normalization of the root vectors E±γ .)

In [5] several conjectures are stated. The following is a version of one of them.

Let p be a maximal parabolic subalgebra of a simple complex Lie algebra g. Letβ be the unique simple root in ∆(n) and g the corresponding fundamental weight.

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Then λ extends to a one-dimensional representation of p. Let b(s) be the b-functionfor ∆β(X).

Conjecture 15.4. The generalized Verma module U(g) ⊗U(p) Csλ is reducible ifand only if b(s + m) = 0, for some m = 1, 2, 3 . . . .

This conjecture is known to hold for p of abelian type. Considerable progress ismade in the case of parabolics of Heisenberg type in [2]. See Section 7 of [2] for adiscussion of the b-function.

EXERCISES

(15.1) Consider so(2n,C) defined by the symmetric bilinear form b on C2n withmatrix (

0 II 0

).

Let p be the parabolic subalgebra for which ∆(n) = ε1 ± εj : j = 2, . . . , n.Compute the polynomial ∆β(X) for the parabolic subalgebra p.

(12.2) For sp(2n,C) and β any simple root, let pβ be the parabolic subalgebraobtained by omitting β. Compute ∆β(X).

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References

[1] L. Barchini, A. C. Kable, and R. Zierau, Conformally invariant systems of differential oper-ators, Preprint.

[2] , Conformally invariant systems of differential operators and prehomogeneous vectorspaces of heisenberg parabolic type, Preprint. Available athttp://www.math.okstate.edu/ zierau/papers.html.

[3] I. N. Bernstein, The analytic continuation of generalized functions with respect to a param-eter, Funct. Anal. and Appl. 6 (1972), 26–40.

[4] C. W. Curtis, Linear Algebra, an Introductory Approach, Undergraduate Texts in Mathe-matics, Springer-Verlag, New York, 1984.

[5] A. Gyoja, Highest weight modules and b-functions of semi-invariants, Publ. RIMS, KyotoUniv. 30 (1994).

[6] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press,New York, 1978.

[7] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Textsin Mathematics, vol. 9, Springer-Verlag, Berlin, 1972.

[8] N. Jacobson, Basic Algebra. I., W. H. Freeman and Co., San Francisco, 1974.[9] T. Kimura, Introduction to Prehomogeneous Vector Spaces, Translations of Mathematical

Monographs, vol. 215, Amer. Math. Soc., Providence, RI, 1998.[10] A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140,

Birkhauser, New York, 2002.[11] A. Koranyi and H. M. Reimann, Equivariant first order differential operators on boundaries

of symmetric spaces, Inventiones Mathematicae 139 (2000), 371–390.[12] C. C. Moore, Compactifications of symmetric spaces ii, the cartan domains, Amer. J. of

Math.[13] W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.[14] W. Schmid, Die randwerte holomorpher Funktionen auf hermitesch symmetrischen Raumen,

Invent. Math. 9 (1969), 1–18.[15] V. S. Varadarajan, Lie Groups, lie Algebras, and Their Representations, Prentice-Hall, En-

gelewood Cliffs, NJ, 1974.[16] N. Wallach, Analytic continuation of the discrete series II, Transactions of the AMS 251

(1979), 19–37.

Oklahoma State University, Mathematics Department, Stillwater, Oklahoma 74078

E-mail address: leticia@math.okstate.edu

Oklahoma State University, Mathematics Department, Stillwater, Oklahoma 74078

E-mail address: zierau@math.okstate.edu

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