whittaker functions, lattice models and …whittaker functions, lattice models and (non)symmetric...

Whittaker functions, lattice models and (non)symmetric polynomials Lecture 1

Whittaker functions, lattice models and(non)symmetric polynomials

Lecture notes for working seminar onsymmetric functions at Rutgers, fall 2019

Henrik P. A. Gustafsson

November 1, 2019

Aims and objectives

The aim of these lecture is to:

— Understand the connections between (non)symmetric polynomials, Whittaker functionsand lattice models.

— Gain familiarity with different techniques to compute and manipulate these objects.

After reading these notes the reader should be able to define, understand and relate thefollowing concepts:

— Roots, weights and characters.

— Schur polynomials and their connection to representation theory.

— Spherical and Iwahori Whittaker Whittaker functions on GLr(F ) for a non-archimedeanfield F , in particular F = Qp (the p-adic numbers).

— Compute standard p-adic integrals and in particular the spherical Whittaker functionfor GL2(Qp).

— Two-dimensional lattice models whose partition functions describe spherical andIwahori Whittaker functions.

— The Yang-Baxter equations for these lattice models and how it can be used to obtainfunctional relations for Whittaker functions.

1


Lecture 1

In this lecture we will first recall some facts about algebraic groups, weights and characters.We will then define the Schur polynomials and connect them to characters of GLr. Inthe last part we will show that local Whittaker functions for GLr(F ) where F is a non-archimedean field also can be expressed in terms of Whittaker functions. Recommendedreference literature: Bump, Lie groups; Hall Lie groups, Lie algebras, and representations:An elementary introduction; and Milne Algebraic groups. I can also recommend the noteshttp://sporadic.stanford.edu/bump/whittaker/whittaker.html by Bump.

1.1 Weights and characters

Let G be a linear algebraic group over Q, that is, a subgroup of GLr defined by polynomialequations in the matrix elements of g ∈ GLr and det(g)−1 with rational coefficients. For afield F , the group G(F ) is then the solutions to these equations in F . In this lecture serieswe will take for example F = C, or a local non-archimedean field like the p-adic numbers Qp.

Let T be a maximal torus of G. A torus is a subgroup T of G such that T (C) ∼= (C×)r forsome r ≥ 1 and it is maximal if it is not contained in any other torus.

The weight lattice Λ = X∗(T ) is defined as the group of rational characters on T , that is thehomomorphisms T → Gm where Gm takes a field F to the multiplicative group of invertibleelements F×. Since T (C) ∼= (C×)r and Gm(C) ∼= C×, we may identify Λ ∼= Zr using maps

t = diag(t1, . . . , tr) 7→ tλ =∏ri=1 t

λii λ = (λ1, . . . , λr) ∈ Zr . (1.1)

The group X∗(T ) of co-characters is similarly defined as the homomorphisms Gm → T .

Let (π, V ) be an irreducible representation of G, then the restriction of π to T is notirreducible unless V is one-dimensional. The restriction decomposes into one-dimensionalcharacters which are the weights in Λ associated to this particular representation. Themodule V decomposes into eigenspaces called weight spaces V (µ) = {v ∈ V : π(t)v = tµv}.In particular, if π is the adjoint representation: π(g)h = ghg−1, then the associated weightsare called the set of roots Φ of G.

Example 1.1

Let g = ( a bc d ) ∈ G = GL2 and t =(t1 00 t2

)∈ T . Then,

tgt−1 =

(a bt1/t2

ct2/t1 d

)= t(1,−1)

(0 b0 0

)+ t(−1,1)

(0 0c 0

)+ t(0,0)

(a 00 d

). (1.2)

The three different terms correspond to α, −α and 0.

In general for GLr the roots can be expressed in Λ ∼= Zr as εi − εj where εi is the standardbasis vector for Zr, and the simple roots can be expressed as αi = εi − εi+1. That is,α1 = (1,−1, 0, . . . , 0), α2 = (0, 1,−1, 0, . . . , 0) etc. Furthermore, Λ admits a Weyl invariantinner product which we will denote by (·, ·) for which {εi} is an orthonormal basis. We willdenote the Weyl group by W .

2

http://sporadic.stanford.edu/bump/whittaker/whittaker.html


It is often useful to embed the weight lattice of SLr into the weight lattice of GLr as thehyperplane satisfying

∑λi = 0 in the coordinates of the above basis keeping the GLr-

description of the roots. Note that the maximal torus of SLr(C) is isomorphic to (C×)r−1.

The co-roots α∨ ∈ X∗(T ) of G are defined by the dual pairing 〈α∨, β〉 = 2(α, β)/(α, α) for aroot β, extended by linearity.

A weight λ is called dominant if (λ, αi) ≥ 0 for all simple roots αi. In the basis {εi}i we getthat λi ≥ λi+1 which means that (λ1, . . . , λn) is an integer partition if λr ≥ 0.

Under some basic assumptions on the group G, there is a bijection λ←→ (πλ, Vλ) betweendominant weights and isomorphism classes of irreducible representations. There is anordering on the weights: µ � µ′ if µ− µ′ is a positive linear combination of the simple roots,and λ is the unique highest weight for the representation (πλ, Vλ).

The character χπ of a representation (π, V ) is χπ(g) = trV (π(g)). We restrict to the maximaltorus T , and recall that V decomposes into weight spaces V (µ). Let mµ = dim(V (µ)).Then, since π(t)v = tµv for v ∈ V (µ),

χπ(t) =∑µ∈Λ

trV (µ)(π(t)) =∑µ∈Λ

trV (µ)(1V (µ))tµ =

∑µ∈Λ

mµtµ . (1.3)

For a highest weight representation (πλ, Vλ), the character χλ := χπλ can be computed bythe Weyl character formula:

χλ(t) =

∑w∈W (−1)`(w)tw(λ+ρ)∑w∈W (−1)`(w)tw(ρ)

, (1.4)

where ρ = 12

∑α>0 α is the Weyl vector. For G = GLr we will shift this by a multiple of

(1, . . . , 1) to ρ = (r− 1, r− 2, . . . , 1, 0) to make all entries non-negative. This does not affectthe Weyl character formula. We also note for G = GLr that χλ+(k,...,k)(t) = det(t)kχλ(t)which means that we may assume that λ is a partition for the computation of χλ(t).

1.2 Schur polynomials

There are several ways of defining Schur polynomials. One which will be convenient for us isthe following determinant formula. Let λ be a partition of length r (which may be paddedby zeroes to obtain this length). Then we define the Schur polynomial sλ : Cr → C by

sλ(z1, . . . , zr) :=det(z

(λ+ρ)ji )

det(zρji )

. (1.5)

For brevity we will denote z = (z1, . . . , zr).

Both the numerator and the denominator are alternating polynomials, changing sign underpermutations of the variables, because of the properties of the determinant. The denominatoris called the Vandermonde determinant and can, by Gauss elimination, be shown to equal∏i<j(zi − zj). Since the numerator is alternating it also has zeroes at zi = zj , i < j

and therefore contains the Vandermonde determinant as a factor. Thus sλ(z1, . . . , zr) is asymmetric polynomial.

3


By writing out the determinants we may compare (1.5) with the Weyl character formula (1.4)for G = GLr which has the Weyl group W ∼= Sr. We get that

sλ(z) =

∑σ∈Sr sgn(σ)

∏i z

(λ+ρ)σii∑

σ∈Sr sgn(σ)∏i zρσii

= χλ(z) . (1.6)

Noting that sλ is polynomial and comparing with (1.3) we conclude that mµ = 0 unless µ isdominent, i.e. a partition.

Example 1.2

Consider r = 2 with λ = (1, 0) and S2 = {1, σ}. We have that sλ(z1, z2) =sgn(1)z21+sgn(σ)z22sgn(1)z1+sgn(σ)z2

= z1 + z2.

1.3 Whittaker functions and the Langlands dual group

To the group G we may associate the Langlands dual group G(C) as follows. Its maximaltorus T is dual to T in the sense that X∗(T ) is identified with X∗(T ) := Λ and the co-rootsof G are the roots of G. The Langlands dual associated to GLr is GLr(C).

We will now define the local Whittaker functions. Let G = GLr and N be the maximalunipotent subgroup of G consisting of lower triangular matrices with unit diagonal. Let Fbe a local non-archimedean field with ring of integers o. Let p be the maximal ideal of owith generator $ ∈ p and residue cardinality q = |o/p|. We fix a unitary character ψ onN(F ) such that ψ restricted to any single matrix element ni,i+1 is a character on F trivialon o but no larger fractional ideal.

For λ = (λ1, . . . , λr) ∈ Zr let $λ = diag($λ1 , . . . , $λr). These form a full set ofrepresentatives for the coset space T (F )/T (o). We will consider an unramified character τzon T (F ) trivial on T (o) and defined by τz($λ) = zλ =

∏i zλii . For F = Qp we have that

τz(t) =∏i z

ordp(ti)i .

We trivially extend τz to a character on the upper triangular matrices B(F ) and then as afunction on GLr(F ). It is a spherical function since τz(gk) = τz(g) for k in the maximalcompact subgroup GLr(o) of GLr(F ). Let f◦z (ntk) = δ1/2(t)τz(t) for t ∈ T (F ), n ∈ N(F )and k ∈ GLr(o) where δ1/2 is the modular quasicharacter which may be evaluated asδ1/2(t) = tρ where ρ = 1

2

∑α>0 α without a shift unlike ρ. We define the spherical Whittaker

function as

W ◦z (g) =

∫N(F )

f◦z (w0ng)ψ−1(n) dn (1.7)

which is determined by its values on T (F )/T (o).

The values W ◦z ($λ) can be computed using the Casselman-Shalika formula which can bereinterpreted as the Weyl character formula (1.4) for the Langlands dual group GLr(C)which we showed in (1.6) was a Schur polynomial

W ◦z ($λ) =∏α>0

(1− q−1zα) ·

δ1/2($λ)sλ(z) λ ∈ Λ is dominant

0 otherwise .(1.8)

4


Lecture 2

Last time we defined Schur polynomials using the determinant formula and showed thatthey are indeed symmetric polynomials. We recalled some facts about representationtheory, especially for GLr, noting that characters of irreducible representations are Schurpolynomials. We also defined spherical Whittaker functions for GLr(F ) where F is a localnon-archimedean field and mentioned they are also related to Schur polynomials.

Today we will, as requested, compute the spherical Whittaker functions for GL2(Qp) bydirectly computing the p-adic integrals. We will therefore start by recalling some usefulfacts for p-adic numbers and computing some integrals that will be used later. While goingthrough the proof in the next section I advice to take a look at the summary of the steps atthe end of the section.

2.1 p-adic numbers and integrals

The ring of integers for F = Qp is o = Zp := {x ∈ Qp : |x|p ≤ 1} and p = pZp for which wemay take the generator $ = p and we have that q := |o/p| = p. We will also use the facts

(i) Z×p := {x ∈ Zp : |x|p = 1} = Zp\pZp.

(ii) Zp =⊔∞k=0 p

kZ×p .

(iii) Qp\Zp =⊔∞k=1 p

−kZ×p .

These can all be deduced from the formal Laurent series presentation of x ∈ Qp:

x = xkpk + xk+1p

k+1 + . . . with xi ∈ Z/pZ such that xk 6= 0 and |x|p = p−k. (2.1)

Note that terms further to the right have smaller and smaller p-adic norm.

We have an additive measure dx on Qp that is invariant under translations d(x+ a) = dxand scales as d(ax) = |a|p dx for a ∈ Qp. It is normalized such that

∫Zp dx = 1.

Let e : Qp → C× be a unitary additive character on Qp trivial on Zp but no larger fractionalideal mn Zp. Being additive it satisfies e(x+ y) = e(x) e(y). Note that e(1

p) is a p-th root ofunity since 1 = e(1) = e(1

p + · · ·+ 1p) = e(1

p)p where the sum is over p terms. We have that∑p−1n=0 e(np ) = (1− e(1

p)p)/(1− e(1p)) = 0. We will make use of the following integral.

Proposition 2.1. Let k, l ∈ Z. Then,∫p−kZp

e(plx) dx = pk

{1 l − k ≥ 0

0 l − k < 0(2.2)

Proof. By the variable substitution x = p−kx′ we can reduce to the case k = 0, that is,the integral

∫Zp e(plx) dx. For l ≥ 0 the argument plx ∈ Zp on which e is trivial, which

leaves us with∫Zp dx = 1. For l < 0 we split the integration domain into a sum over p parts:

0 ≤ n ≤ p − 1 where the elements of part n has p−l−1-coefficient equal to n. We get anoverall factor that is the sum over all p-roots of unity and hence the integral is zero.

5


2.2 Detailed computation of Whittaker functions on GL2(Qp)

We start by recalling the spherical function f◦z we integrate over to obtain the Whittakerfunction on GLr(Qp). By definition f◦z satisfies f◦z (ntg) = f◦z (nt)f◦z (g) for n ∈ N(F ) andt ∈ T (F ) and f◦z (gk) = f◦z (g) for k ∈ G(o). It is given by f◦z (ntk) =

∣∣tρ∣∣p

∏i z

ordp(ti)i .

Let µz = (µ1, . . . , µr) ∈ Cr where µi = − log zi/ log p, such that zi = p−µi and therefore

zordp(ti)i = p− ordp(ti)µi = |ti|µip . (2.3)

For the remainder of this section we will consider the group GL2(Qp). The Whittakerfunction defined in (1.7) can then be expressed as

W ◦z (t) =

∫Qp

fz

(w0

(1 x

1

)t)e(x) dx (2.4)

which is determined by its values at t = diag(t1, t2) = diag(pλ1 , pλ2) and where

w0 =

(0 11 0

)∈ GL2(Zp) . (2.5)

Theorem 2.2. Let λ be a partition of length two, that is, λ = (λ1, λ2) with λ1 ≥ λ2 ≥ 0.Then, the spherical Whittaker function on GL2(Qp) evaluated at t = diag(pλ1 , p

λ2) equals

W ◦z (t) = p12 (λ2−λ1)

(1− p−1 z1

z2

)sλ(z1, z2) . (2.6)

This agrees with (1.8) where δ1/2(t) = p12 (λ2−λ1) and zα = z1/z2 for the single simple root

α. The different steps enumerated in the margin will be summarized after the proof.

Proof. We have that(A)

w0

(1 x

1

)t = w0tw0

(1

xt2/t1 1

)w0 (2.7)

and f◦z(w0tw0

(1

xt2/t1 1

)w0

)= f◦z (w0tw0)f◦z

(( 1xt2/t1 1

))where f◦z (w0tw0) = |t2|

µ1+12

p |t1|µ2−1

2p .

Thus, W ◦z (t) equals

|t2|µ1+

12

p |t1|µ2−1

2p

∫Qp

fz

(( 1xt2/t1 1

))e(x) dx = |t2|

µ1−12

p |t1|µ2+

12

p

∫Qp

fz

(( 1x′ 1

))e(x′t1/t2) dx′

(2.8)where we have made the substitution x′ = xt2/t1.

If x′ ∈ Zp then the argument of f◦z is in GL2(Zp) on which f◦z is trivial. Otherwise, x′−1 ∈ Zpand we make the following, so called, p-adic Iwasawa decomposition with factors in N(F ),(B)

6


T (F ) and GL2(Zp) respectively.(1x′ 1

)=

(1 x′−1

1

)(x′−1

x′

)(1

1 x′−1

)(2.9)

Since ρ = (1, 0) we have that f◦z ((x′−1

x′

)) =

∣∣x′−1∣∣µ1+

12

p|x′|

µ2−12

p = |x′|µ2−µ1−1p and thus

f◦z

(( 1x′ 1

))=

{1 if x′ ∈ Zp|x′|µ2−µ1−1

p otherwise(2.10)

We can thus split up the integral as follows∫Qp

f◦z

(( 1x′ 1

))e(x′t1/t2) dx′ =

∫Zp

e(x′t1/t2) dx′ +

∫Qp\Zp

∣∣x′∣∣µ2−µ1−1

pe(x′t1/t2) dx′ (2.11)

We have that t1/t2 = pλ1−λ2 where λ1 − λ2 ≥ 0. Using Proposition 2.1, the first integralin (2.8) becomes ∫

Zp

e(x′pλ1−λ2) dx′ = 1 (2.12)

Since Qp\Zp =⊔∞k=1 p

−kZ×p we get that the second integral in (2.8) becomes(C) ∫Qp\Zp

|x|µ2−µ1−1p e(xpλ1−λ2) dx =

∞∑k=1

pk(µ2−µ1−1)

∫p−kZ×p

e(xpλ1−λ2) dx (2.13)

Note that since µi = − log zi/ log p we have that pkµi = p−k log zi/ log p = e−k log zi = z−ki .Thus, we get the prefactor pk(µ2−µ1−1) = z−k2 zk1p

−k

∞∑k=1

( z1

pz2

)k ∫p−kZ×p

e(xpλ1−λ2) dx (2.14)

We have that Z×p = Zp\pZp. Thus, with l = λ1 − λ2,(D) ∫p−kZ×p

e(plx) dx =

∫p−kZp

e(plx) dx−∫

p−k+1Zp

e(plx) dx (2.15)

Using Proposition 2.1 we get that (2.14) equals

∞∑k=1

( z1

pz2

)k pk(1− 1

p) if λ1 − λ2 − k ≥ 0

−pk−1 if λ1 − λ2 − k = −1

0 if λ1 − λ2 − k < −1

= −1

p

(z1

z2

)λ1−λ2+1+ (1− 1

p)

λ1−λ2∑k=1

(z1

z2

)k(2.16)

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In summary, by reorganizing the sums we get that,(E)

W ◦z (t) = p12 (λ2−λ1)zλ21 zλ12

(1− 1

p

(z1

z2

)λ1−λ2+1+ (1− 1

p)

λ1−λ2∑k=1

(z1

z2

)k)

= p12 (λ2−λ1)

(λ1−λ2∑k=0

zλ2+k1 zλ1−k2 − 1

p

(z1

z2

) λ1−λ2∑k=0

zλ2+k1 zλ1−k2

)

= p12 (λ2−λ1)

(1− p−1 z1

z2

) λ1−λ2∑k=0

zλ2+k1 zλ1−k2

(2.17)

which agrees with (2.6) where indeed s(λ1,λ2)(z1, z2) =∑λ1−λ2

k=0 zλ2+k1 zλ1−k2 .

We now summarize the steps:

(A) We move the torus element to the front conjugated by w0, and the remaining w0 tothe back so that we may use the transformation properties of f◦z .

(B) The remaining argument of f◦z is lower triangular. If the argument is not in GL2(Zp)on which f◦z is trivial we have to factorize it further and pick up the remaining toruselement, again using the transformation properties of f◦z . The integral is then split upinto two pieces depending on whether the argument is already in GL2(Zp) or if weneed to make the factorization.

(C) The first piece was evaluated by Proposition 2.1, but for the second piece we need tosplit up the integration domain further into “shells of equal p-adic norm”.

(D) Each such shell is the difference of two integrals that can be computed byProposition 2.1. Because of the condition for this integral to be non-vanishing,we get a bound for which shells can contribute.

(E) Putting it all together and reorganizing the sums the different contributing shellsbecome exactly the terms in the Schur polynomials.

8


Lecture 3

Last time we computed the spherical Whittaker functions on GL2(Qp) by directlycomputing the p-adic integrals.

Today we will return to the general case GLr(F ) where F is a non-archimedean field anddescribe how the spherical Whittaker coefficients can be expressed as the partition functionof a two dimensional solvable lattice model.

3.1 The spherical lattice model

We will now construct the lattice model whose partition function gives the spherical Whittakervectors. The lattice model is also sometimes called the Tokuyama model or the sphericalmodel.

Construct a grid as in Figure 1 where we label the rows from 1 to r in increasing order fromtop to bottom and the columns from 0 to some sufficiently large N increasing from right toleft. Our results will not depend on N .

1

2

3

01234

Figure 1: Lattice with numbering of rows and columns and vertex marked as dots.

Each crossing is called a vertex and each of the four lines bordering a vertex is called anedge. To each edge we assign one of two edge states: + or − . But we may only makeassignments according to the following rules. Let ρ = (r − 1, r − 2, . . . , 0).

1. The edges around a vertex must match one of the allowed configurations shown inTable 1.

2. The boundary edges must be assigned edge states according to the following rules:

(a) The left and bottom boundary edges must be + .

(b) The right boundary edges must be − .

(c) For a given partition λ of length r we assign − to the columns with labels λi + ρion the top boundary. The remaining top boundary edges are chosen to be + .

Note that, while the boundary is fixed by λ there are generally several ways to fill in theinterior while still satisfying rule 1. A configuration of allowed edges for a given boundary iscalled a lattice state, or state for short. The set of states for a given λ is called an ensembleand will be denoted by Sλ. Note also that if three edge states around a vertex is given,

9


Table 1: Allowed vertex configurations and their Boltzmann weights.

+

+

+

+

−

−

−

−

+

−

+

−

−

+

−

+

−

+

+

−

+

−

−

+

1 z −v z (1− v)z 1

there is at most one allowed vertex configuration for that vertex (or, equivalently, at mostone allowed edge state for the remaining edge). Finally, note that the number of − at thenorth and west edges of a vertex equal the number of − at the south and east edges — wehave a conservation of − . A similar conservation law follows for whole rows of vertices aswell.

We show an example of all possible states given λ = (1, 1) in Figure 2. Exercise: show thatwe cannot have a state where the vertical edges between the horizontal rows 1 and 2 are+ + − from left to right.

z1

z2

012− − +

− + +

+ + +

+ + − −

+ − − −

z1

z2

012− − +

+ − +

+ + +

+ − − −

+ + − −

[−v 1 z1

1 z2 z2

] [1 z1 z1

1 1 z2

]Figure 2: The possible states for λ = (1, 1), i.e. λ+ ρ = (2, 1) together with the Boltzmann weight foreach vertex underneath in a similar grid pattern.

Each row of horizontal edges, labeled by the row number i is assigned a complex variablezi and each vertex x in this row is given a weight β(x) ∈ C according to the second rowin Table 1 with z = zi and v a parameter which we will eventually set as v−1 = q = |o/p|(which was equal to p for F = Qp in Lecture 2). These weights are called Boltzmann weights.A lattice state s is then given a total weight which is the product of its vertex weights

β(s) =∏

vertex x∈sβ(x) (3.1)

which is a function in z1, . . . , zr. The weight for each state in Figure 2 is shown beneath thestate.

Given a partition λ, which specifies an ensemble Sλ we can define the associated partitionfunction Z◦λ as

Zλ(z1, . . . , zr) =∑s∈Sλ

β(s) (3.2)

We note that increasing N only adds extra factors of 1 and thus does not change the weightsof the states nor the partition function.

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For the example shown in Figure 2 we get that

Z◦(1,1)(z1, z2) = −vz1z22 + z2

1z2 = z1

(1− v z2

z1

)s(1,1)(z1, z2) (3.3)

where s(1,1)(z1, z2) = z1z2.

In general we have the following equality

Z◦λ(z) = zρ∏i<j

(1− q−1 zj

zi

)sλ(z) = zρδ−1/2($λ)W ◦w0z($λ) (3.4)

We will postpone the proof/motivation of this statement to Lecture 5.

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Lecture 4

Last time we introduced a lattice model whose partition function we claimed computedthe spherical Whittaker functions of GLr(F ) which are given by Schur polynomials. Wewill prove this statement in the next lecture.

Today we will introduce an important tool that will be used in that proof called theYang-Baxter equation; a tool used in statistical mechanics to ‘solve’ lattice models. If thereis time we will also discuss another important step of the proof which is to split up thespherical Whittaker function into smaller pieces called Iwahori Whittaker functions.

4.1 The Yang-Baxter equation

In the Yang-Baxter equation we introduce another type of vertices shown in Table 2. Theseare called R-matrix configurations and should be interpreted as mixing two horizontal rowsof edges in the lattice model.

Table 2: R-matrix configurations. The location of the parameters zi and zj follow the first entry.

+zi

+zj + zi

+ zj −

− −

− +

− +

− −

+ −

+ −

+ +

− +

− −

+

zj − vzi zi − vzj v(zi − zj) zi − zj (1− v)zi (1− v)zj

The Yang-Baxter equation for this system can then be expressed as

Z

d zi∗

azi e zj∗

bzj

f

∗

c

= Z

bzj ∗

e zjazi ∗

d zi

f

∗

c

(4.1)

where the interior states ∗ are summed over. It is customary to drop the function Z andjust write equality of the systems. The validity of this equation can easily be checked eitherby hand (tedious) or writing a simple computer program.

Note that the Yang-Baxter equation also holds as part of a bigger system, and that we mayuse it repeatedly to move the R-matrix further and further to the left.

As a warmup exercise for what we will do in the next lecture let us now apply the Yang-Baxterequation to the partition function Zλ(z) defined in the previous lecture. A schematicalillustration of the procedure is shown in Figure 3.

We start by attaching an R-matrix between rows k and k + 1 to the left boundary (whichconsists of + ). By this we mean that we impose a = b = + in (4.1) and assign the remainingedge states according to the rules of the previous lecture, with the following exception. We

12


+z2

+z1

+

+

+ + + + +

− z1

− z2

−−

=

+ + + + +

− z1+z2

− z2+z1

−+

−+

=

+z2

+z1

+

+

+ + + + +

− z1

− z2

−−

Figure 3: Repeated use of the Yang-Baxter equation on the partition function. The boxes correspond toareas of interior edge states that are summed over.

should now sum over the two edge states that used to belong to the boundary, but are nowattached to the R-matrix; in a way, they have become internal edge states.

Looking at Table 2 we see that there is only one R-matrix element (the first entry)contributing in the left-hand side of (4.1), with weight zk+1 − vzk. The old boundaryfor Zλ thus remain unchanged.

With repeated use of (4.1) in the interior of Zλ(z) we end up with a right-hand side wherethe R-matrix is attached to the right boundary (which consists of − ). By the right-hand sideof the Yang-Baxter equation we need to impose d = e = − . This only gives one contribution(from the second entry in Table 2) of weight zk − vzk+1, and yet again the old boundaryfor Zλ(z) is unchanged. By this manipulation we note from (4.1) that zk and zk+1 haveswitched places.

We get that(zk+1 − vzk)Zλ(z) = (zk − vzk+1)Zλ(wkz) (4.2)

where wk swaps zk and zk+1.

We have that

wkzρ =

zk+1

zkzρ wk

[∏i<j

(1− v zj

zi

)]=

(1− v zk

zk+1

)(

1− v zk+1

zk

)∏i<j

(1− v zj

zi

)(4.3)

Thus, inserting (3.4) into (4.2) we get that

zρ∏i<j

(1− v zj

zi

)sλ(z) = Zλ(z) =

zk − vzk+1

zk+1 − vzkZλ(wkz)

=zk − vzk+1

zk+1 − vzkzk+1

zk

(1− v zk

zk+1

)(

1− v zk+1

zk

)︸︷︷︸

=1

zρ∏i<j

(1− v zj

zi

)sλ(wkz)

(4.4)

Thus, we have used the Yang-Baxter equation to show that sλ(wkz) = sλ(z), that is, thatsλ is symmetric.

13


4.2 Iwahori Whittaker functions

For simplicity we will restrict to F = Qp, but the general case follows analogously.

We recall from Lectures 1 and 2 that the spherical Whittaker function was defined as

W ◦z (g) =

∫N(Qp)

f◦z (w0ng)ψ−1(n) dn f◦z (ntk) =∣∣tρ∣∣

p

∏i

zordp(ti)i (4.5)

where t is a torus element, n upper triangular and k ∈ K = GLr(Zp).

Instead of considering functions right-invariant under K, we will relax and only requireinvariance under a subgroup of K called the Iwahori subgroup. The Iwahori subgroup J isthe elements of GLr(Zp) which are upper triangular mod p.

One can show that (see for example arXiv:1002.2996)

GLr(Qp) =⊔w∈W

B(Qp)wJ K = GLr(Zp) =⊔w∈W

B(Zp)wJ (4.6)

Using this decomposition we define the so called Iwahori standard basis elements for g = bw′jwith b ∈ B(Qp) and j ∈ J as

fwz (bw′j) =

{f◦z (b) if w = w′

0 otherwise.(4.7)

Let f(g) =∑

w∈W fwz (g). We can decompose g = bk and k = bw′j where b ∈ B(Zp) ⊂ K.Then f(bk) = f(bbw′j) = fw

′z (bb) = f◦z (bb) = f◦z (b) which shows that

∑w∈W fwz (g) is a

spherical function and indeed equal to f◦z .

We also define the corresponding Iwahori Whittaker function as

Wwz (g) =

∫N(Qp)

fwz (w0ng)ψ−1(n) dn . (4.8)

Now the spherical Whittaker function can now, similar to above, be expressed as

W ◦z (g) =∑w∈W

Wwz (g) . (4.9)

Instead of being related to Schur polynomials like W ◦z , the Iwahori Whittaker functionsWw

z are related to non-symmetric variants of Hall-Littlewood polynomials (a limit of non-symmetric Macdonald polynomials).

14

https://arxiv.org/abs/1002.2996


Lecture 5

In Lecture 3 we claimed that the partition function of the spherical lattice model is aspherical Whittaker function as detailed in (3.4). In Lecture 4 we described the Yang-Baxterequation for the spherical lattice model and described a generalization of spherical Whittakerfunctions which are called Iwahori Whittaker functions.

Today we will give (a sketch of) a proof of (3.4) following [BBBG19]. To match the notationtherein we will slightly rewrite the right-hand side using s−w0λ(z−1) = sλ(w0z) = sλ(z)which can be shown from the determinant formula. For simplicity we will still work withF = Qp. We thus want to show that

Z◦λ(z) = zρδ−1/2(p−w0λ)W ◦z−1(p−w0λ) . (5.1)

where pλ = diag(pλ1 , . . . , pλr). Throughout this lecture we will let v = p−1.

The strategy is to construct a new Iwahori lattice model and use its associated Yang-Baxterequations to show that this new partition function Zwλ (z) equalsWw

λ (z) (up to normalization).The Iwahori lattice model will be constructed in such a way that

∑w Z

wλ (z) = Z◦λ(z), thus

proving (5.1).

5.1 Some facts about Iwahori Whittaker functions

Before we introduce the Iwahori lattice model and prove (5.1) we will need the followingfacts which are proven in [BBBG19].

Let us denoteφwλ (z) = δ−1/2(p−w0λ)Ww

z−1(p−w0λ) . (5.2)

Fact 1: φ1λ(z) = zλ

LetTkf(z) =

f(z)− f(skz)

zαk − 1− vf(z)− z−αkf(skz)

zαk − 1. (5.3)

Fact 2:

φskwλ (z) =

{Tkφ

wλ (z) if `(skw) > `(w),

T−1k φwλ (z) if `(skw) < `(w).

(5.4)

A special case of the this fact was first proven in [BBL15].

Fact 3: We recall from Lecture 4 that

W ◦z (g) =∑w∈W

Wwz (g) (5.5)

15


5.2 Iwahori lattice model

The Iwahori lattice model is very similar to the spherical model we introduced in Lecture 3but with a few important distinctions: Each − will be replaced by a color from an orderedpalette of r colors (for GLr). This will introduce some extra possible vertex configurationsand states. Recall that the − in the spherical model traces out different paths in the lattice(going from north-west to south-east). We construct the boundaries and admissible states insuch a way that each path is now assigned its unique color starting with a certain order ofcolors on the top boundary. For weights where two paths cross we ensure that the sum overpossibilities (with fixed north and west input edges) matches the weight for the sphericalmodel:

−

−

−

−

= −

−

−

−

+ − −

−

−

(5.6)

The actual weights can be found in [BBBG19], but we will not need to display all of themhere.

For the Iwahori model we will specify an order of the colors on the right boundary (readfrom top to bottom) as a permutation w of the order of colors on the top boundary (readfrom left to right). We then define Zwλ (z) to be the partition function with boundary givenby λ for the top column positions and w for the order of the colors on the right boundary.

The state on the right of Figure 2 splits up into two cases:

z1

z2

012− − +

+ − +

+ + +

+ − − −

+ + − −

−→z1

z2

012− − +

+ − +

+ + +

+ − − −

+ + − −

−− − −−

−− −

w = 1

+z1

z2

012− − +

+ − +

+ + +

+ − − −

+ + − −

−−−− −

−− −

w = s1

There is a new Yang-Baxter equation for this model which we will use to relate differentpartition functions. Similar to arguments in Lecture 4, we will only need the R-matrixvertices shown in Table 3. The rest can be found in [BBBG19, Fig 9]

Table 3: R-matrix configurations for the Iwahori model. The colors c and d satisfy c > d if the c comesbefore b in the order of colors on the top boundary read from left to right.

+zi

+zj + zi

+ zj

dd

c c

d

d

c

c · · ·

zj − vzi

{(1− v)zi if c < d

(1− v)zj if c > d

{zi − zj if c < d

v(zi − zj) if c > d· · ·

16


Theorem 5.1. The partition function for the Iwahori lattice model computes IwahoriWhittaker functions:

Zwλ (z) = zρφwλ (z) (5.7)

Proof. We will make an iteration over the length of w. The base case is when w = 1. Inthis case the partition function consists of only one term: the first path must go one stepdown and directly to the right, the second path two steps down and directly to the right etc.just as in the w = 1 case in the figure above.

With the full Boltzmann weights from [BBBG19] the partition function is easily computedas

Z1λ(z) = zλ+ρ = zρφ1

λ(z) (5.8)

where the last equality follows from Fact 1.

When (repeatedly) applying the Yang-Baxter equation on rows k and k + 1 of a generalsystem with partition function Zwλ (z), the condition c < d is equivalent to `(skw) > `(w).In the same way as in Lecture 4 we get the following functional relation where it is assumedthat a < b

(+

+ +

+

)Zwλ (z) =

(bb

a a

)Zwλ (skz) +

(b

b

a

a

)Zskwλ (skz) if `(skw) > `(w)

(aa

b b

)Zwλ (skz) +

(a

a

b

b

)Zskwλ (skz) if `(skw) < `(w)

(5.9)

Applying sk to this whole expression (acting on the z) and solving for Zskwλ (z) one findsthat

Zskwλ (z) =

{TkZ

wλ (z) if `(skw) > `(w),

T−1k Zwλ (z) if `(skw) < `(w).

(5.10)

where Tk = zρTkz−ρ.

With a reduced word w = si1 · · · sim we therefore get that

Zwλ (z) = Ti1 · · ·TimZ1λ(z) = Ti1 · · ·Timzρφ1

λ(z) = zρTi1 · · ·Timφ1λ(z) = zρφwλ (z) (5.11)

where we in the last step have used Fact 2.

Corollary 5.2. It follows from Fact 3 and Theorem 5.1 that

zρδ−1/2(p−w0λ)W ◦z−1(p−w0λ) =∑w∈W

Zwλ (z) (5.12)

To prove the relation between the spherical model and the spherical Whittaker function (5.1),it remains to show that Z◦λ(z) =

∑w∈W Zwλ (z). From how we created the Iwahori model

with the procedure pictured in (5.6) this seems plausible, but I have deliberately skippedover some details like the fact that the Yang-Baxter equation requires the existence of

17


extra vertex configurations with vertical edges having more than one color. We call thesenon-strict states and they do not have a counterpart in the spherical model. The fact thatthe contributions from these non-strict states cancel out when taking the sum over Weylwords was shown in [BBBG19, §6], but the proof actually used (5.1) to do this.

We believe there should also be a combinatorial proof the cancellation of these states withoutrelying on (5.1), which we would need give an independent proof thereof. The original proofof (5.1) does, of course, not rely on this cancellation and is proved in a completely differentmanner in [BBF11].

References

[BBBG19] B. Brubaker, V. Buciumas, D. Bump, and H. P. A. Gustafsson, “Colored five-vertexmodels and Demazure atoms,” arXiv:1902.01795 [math.CO].

[BBF11] B. Brubaker, D. Bump, and S. Friedberg, “Schur Polynomials and The Yang-BaxterEquation,” Communications in Mathematical Physics 308 no. 2, (Oct, 2011) 281,arXiv:0912.0911 [math.CO].

[BBL15] B. Brubaker, D. Bump, and A. Licata, “Whittaker functions and Demazure operators,”J. Number Theory 146 (2015) 41–68, arXiv:1111.4230 [math.RT].

18

http://arxiv.org/abs/1902.01795

http://dx.doi.org/10.1007/s00220-011-1345-3



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