UNIPOTENT HECKE ALGEBRAS: the Structure, Representation Theory, and Combinatorics

UNIPOTENT HECKE ALGEBRAS: the structure, representation theory, and combinatorics

By F. Nathaniel Edgar Thiem

A            

D  P (M)

at the UNIVERSITY OF WISCONSIN – MADISON 2005 i Abstract

Unipotent Hecke algebras The Iwahori-Hecke algebra and Gelfand-Graev Hecke algebras have been critical in developing the representation theory of finite groups and its implications in algebraic combinatorics. This thesis lays the foundations for a theory of unipotent Hecke algebras, a family of Hecke algebras that includes both the classical Gelfand-Graev Hecke algebra and a generalization of the Iwahori-Hecke algebra (the Yokonuma Hecke algebra). In particular, this includes a combinatorial analysis of their structure and representation theory. My main results are (a) a braid-like multiplication algorithm for a natural basis; (b) a canonical commutative subalgebra that may be used for weight space decompositions of irreducible modules. When the underlying group is the general linear group GLn(Fq) over a finite field Fq with q elements, I also supply (c) an explicit construction of the natural basis; (d) a generalization of a Robinson-Schensted-Knuth correspondence (a bijective proof of representation theoretic identities); and (e) an explicit construction of the irreducible Yokonuma algebra modules using the known irreducible module structure of the Iwahori-Hecke algebra. ii Acknowledgements

It has been a joy to develop these results while participating in Lie theory group at the University of Wisconsin-Madison. Georgia Benkart and Paul Terwilliger have been particularly helpful with advice and mathematical ideas, and I enjoyed many discussions with Stephen Griffeth, Michael Lau, and Matt Ondrus (among others). I am especially grateful to my advisor Arun Ram, whose insights, encouragement and patient help guided me into the world of representation theory, and taught me how to talk, write, and research in the world of mathematics. Some grants also supported this research, offering extra time for research (VIGRE DMS-9819788) and funds to travel and interact with other communities (NSF DMS- 0097977, NSA MDA904-01-1-0032). Lastly, throughout all the difficulties and stresses of a doctoral program my wife Claudia has been (and continues to be) an invaluable companion. iii Contents

Abstract i

Acknowledgements ii

1 Introduction 1

2 Preliminaries 5 2.1 Heckealgebras ...... 5 2.1.1 Modulesandcharacters...... 6 2.1.2 Weightspacedecompositions ...... 9 2.1.3 Charactersinagroupalgebra ...... 10 2.1.4 Heckealgebras ...... 11 2.2 ChevalleyGroups...... 12 2.2.1 Liealgebraset-up...... 13 2.2.2 From a Chevalley basis of gs toaChevalleygroup ...... 15 2.3 Somecombinatoricsofthesymmetricgroup ...... 18 2.3.1 A pictorial version of the symmetric group S n ...... 18 2.3.2 Compositions,partitions,andtableaux ...... 19 2.3.3 Symmetricfunctions ...... 21 2.3.4 RSKcorrespondence ...... 23

3 Unipotent Hecke algebras 25 3.1 Thealgebras...... 25 3.1.1 ImportantsubgroupsofaChevalleygroup ...... 25 3.1.2 UnipotentHeckealgebras ...... 27 3.1.3 The choices µ and ψ ...... 28 3.1.4 Anaturalbasis ...... 31 3.2 Parabolic subalgebras of Hµ ...... 33 3.2.1 Weight space decompositions for Hµ-modules ...... 36 3.3 Multiplicationofbasiselements ...... 37 3.3.1 Chevalleygrouprelations ...... 37 3.3.2 LocalHeckealgebrarelations ...... 40 3.3.3 GlobalHeckealgebrarelations...... 46

4 A basis with multiplication in the G = GLn(Fq) case 50 4.1 UnipotentHeckealgebras...... 50 4.1.1 Thegroup GLn(Fq)...... 50 iv

4.1.2 A pictorial version of GLn(Fq)...... 52 4.1.3 The unipotent Hecke algebra Hµ ...... 53 4.2 An indexing for the standard basis of Hµ ...... 55 4.3 Multiplication in Hµ ...... 61 4.3.1 Pictorial versions of eµveµ ...... 61 4.3.2 Relationsformultiplyingbasiselements...... 66 4.3.3 Computing ϕk viapainting,pathsandsinks...... 70 4.3.4 Amultiplicationalgorithm ...... 74

5 Representation theory in the G = GLn(Fq) case 79 5.1 The representation theory of Hµ ...... 80 5.2 AgeneralizationoftheRSKcorrespondence...... 82 G 5.3 Zelevinsky’s decomposition of IndU (ψµ)...... 84 5.3.1 Preliminariestotheproof(1)...... 85 G 5.3.2 The decomposition of IndU(ψ(n))(2)...... 88 G 5.3.3 Decomposition of IndU(ψµ)(3)...... 92 5.4 A weight space decomposition of Hµ-modules...... 93

6 The representation theory of the Yokonuma algebra 97 6.1 Generaltype...... 97 6.1.1 Areductiontheorem ...... 99 6.1.2 The algebras Tγ ...... 102 6.2 The G = GLn(Fq)case ...... 103 6.2.1 The Yokonuma algebra and the Iwahori-Hecke algebra . . . . . 103 6.2.2 The representation theory of Tν ...... 105 6.2.3 The irreducible modules of H11 ...... 108

A Commutation Relations 112

Bibliography 114 1 Chapter 1

Introduction

Combinatorial representation theory takes abstract algebraic structures and attempts to give explicit tools for working with them. While the central questions may take an abstract form, the answers from this perspective use known combinatorial objects in computations and constructive descriptions. This thesis examines a family of Hecke algebras, giving algorithms that elucidate some of the structural and representation theoretic results classically developed from an algebraic and geometric point of view. Thus, a classical description of a natural basis becomes an actual construction of this basis, and an efficient algorithm for computing products in this basis replaces a classical formula. A large body of literature in combinatorial representation theory is dedicated to the study of the symmetric group S n, giving rise to a wealth of combinatorial ideas. Some of these ideas have been generalized to Weyl groups, a larger class of groups whose structure is essential in the study of Lie groups and Lie algebras. Even more generally, the Iwahori-Hecke algebra gives a “quantum” version of the Weyl group situation. Iwahori [Iwa64] and Iwahori-Matsumoto [IM65] introduced the Iwahori-Hecke algebra as a first step in classifying the irreducible representations of finite Chevalley groups and reductive p-adic Lie groups. Subsequent work (e.g. [CS99] [KL79] [LV83]) has established Hecke algebras as fundamental tools in the representation theory of Lie groups and Lie algebras, and advances on subfactors and quantum groups by Jones [Jon83], Jimbo [Jim86], and Drinfeld [Dri87] gave Hecke algebras a central role in knot theory [Jon87], statistical mechanics [Jon89], mathematical physics, and operator algebras. Unipotent Hecke algebras are a family of algebras that generalize the Iwahori-Hecke algebra. Another classical example of a unipotent Hecke algebra is the Gelfand-Graev Hecke algebra, which has connections with unipotent orbits [Kaw75], Kloosterman sums [CS99], and in the p-adic case, Hecke operators (see [Bum98, 4.6,4.8]). Inter- polating between these two classical examples will be fundamental to my analysis. There are three ingredients in the construction of a Hecke algebra: a group G, a subgroup U and a character χ : U → C. The Hecke algebra associated with the triple (G, U, χ) is the centralizer algebra

G H(G, U, χ) = EndCG(IndU (χ)).

It has a natural basis indexed by a subset Nχ of U-double coset representatives in G. 2

A unipotent Hecke algebra is

G Hµ = H(G, U, ψµ) = EndCG(IndU(ψµ)), where U is a maximal unipotent subgroup of a ﬁnite Chevalley G (such as GLn(Fq)), and ψµ is an arbitrary linear character. In this context, the classical algebras are

the Yokonuma Hecke algebra [Yok69b] ←→ ψµ trivial, the Gelfand-Graev Hecke algebra [GG62] ←→ ψµ in general position.

After reviewing some of the underlying mathematics in Chapter 2, Chapter 3 ana- lyzes unipotent Hecke algebras of general type. The main results are

(a) a generalization to unipotent Hecke algebras of the braid calculus used in Iwahori- Hecke algebra multiplication. I use two approaches: the ﬁrst gives “local” relations similar to braid relations, and the second gives a more global algorithm for computing products. In particular, this provides another solution to the diﬃcult problem of multiplication in the Gelfand-Graev Hecke algebra (see [Cur88] for a geometric approach to this problem).

(b) the construction of a large commutative subalgebra Lµ in Hµ, which may be used as a Cartan subalgebra to analyze the representations of Hµ via weight space decompositions;

Chapters 4, 5 and part of 6 consider the case when G = GLn(Fq), the general linear group over a ﬁnite ﬁeld with q elements. These chapters can all be read independently of Chapter 3, and I would recommend that readers unfamiliar with Chevalley groups begin with Chapter 4. Chapter 4 describes the structure of the unipotent Hecke algebras when G = GLn(Fq); in this case, unipotent Hecke algebras are indexed by compositions µ of n. The main results are

(c) an explicit basis. Suppose Hµ corresponds to the composition µ, and Nµ indexes the basis of Hµ. There is an explicit bijection from Nµ to the set of polynomial matrices over Fq with row-degree sums and column-degree sums equal to µ (see Chapter 4 for details). In particular, this gives an explicit construction of the matrices in Nµ . (d) a pictorial approach to multiplying the natural basis elements, inspired by (a) above.

Chapter 5 proceeds to develop the representation theory of unipotent Hecke algebras G in the case G = GLn(F). This chapter relies heavily on a decomposition of IndU(ψµ) given by Zelevinsky in [Zel81]. The main results are 3

(e) a generalization of the RSK correspondence which provides a combinatorial proof of the identity 2 dim(Hµ) = dim(V) ; IrreducibleX Hµ-modules V

(f) a computation of the weight space decomposition of an Hµ-module with respect to a commutative subalgebra Lµ (see (b) above). Chapter 6 concludes by studying the representation theory of Yokonuma Hecke algebras (the special case when ψµ is trivial). The main results are (g) a reduction in general type from the study of irreducible Yokonuma Hecke algebra modules to the study of irreducible modules for a family of “sub”algebras;

(h) an explicit construction of the irreducible modules of the Yokonuma algebra in the case G = GLn(Fq). 4 Chapter 2

Preliminaries

2.1 Hecke algebras

This section gives some of the main representation theoretic results used in this thesis, and deﬁnes Hecke algebras. Three roughly equivalent structures are commonly used to describe the representation theory of an algebra A: (1) Modules. Modules are vector spaces on which A acts by linear transformations.

(2) Representations. Every choice of basis in an n-dimensional vector space on which A acts gives a representation, or an algebra homomorphism from A to the algebra of n × n matrices.

(3) Characters. The trace of a matrix is the sum of its diagonal entries. A character is the composition of the algebra homomorphism of (2) with the trace map. The following discussion will focus on modules and characters.

2.1.1 Modules and characters Let A be a ﬁnite dimensional algebra over the complex numbers C. An A-module V is a ﬁnite dimensional vector space over C with a map

A × V −→ V (a, v) 7→ av

such that

(c1)v = cv, c ∈ C, (ab)v = a(bv), a, b ∈ A, v ∈ V,

(a + b)(c1v1 + c2v2) = c1av1 + c2av2 + c1bv1 + c2bv2, c1, c2 ∈ C.

An A-module V is irreducible if it contains no nontrivial, proper subspace V′ such that av′ ∈ V′ for all a ∈ A and v′ ∈ V′. An A-module homomorphism θ : V → V′ is a C-linear transformation such that

aθ(v) = θ(av), for a ∈ A, v ∈ V, 5

and an A-module isomorphism is a bijective A-module homomorphism. Write

′ ′ HomA(V, V ) = {A-module homomorphisms V → V }

EndA(V) = HomA(V, V).

Let Aˆ be an indexing set for the irreducible modules of A (up to isomorphism), and ﬁx a set γ {A }γ∈Aˆ of isomorphism class representatives. An algebra A is semisimple if every A-module V decomposes

γ γ γ γ V mγ(V)A , where mγ(V) ∈ Z≥0, mγ(V)A = A ⊕···⊕ A .

Mγ∈Aˆ mγ(V) terms | {z } In this thesis, all algebras (though not necessarily all Lie algebras) are semisimple.

Lemma 2.1 (Schur’s Lemma). Suppose γ, µ ∈ Aˆ with corresponding irreducible modules Aγ and Aµ. Then

γ µ 1, if γ = µ, dim(HomA(A , A )) = δγµ = ( 0, otherwise.

Suppose A ⊆ B is a subalgebra of B and let V be an A-module. Then B ⊗A V is the vector space over C presented by generators {b ⊗ v | b ∈ B, v ∈ V} with relations

ba ⊗ v = b ⊗ av, a ∈ A, b ∈ B, v ∈ V,

(b1 + b2) ⊗ (v1 + v2) = b1 ⊗ v1 + b1 ⊗ v2 + b2 ⊗ v1 + b2 ⊗ v2, b1, b2 ∈ B, v1, v2 ∈ V.

Note that the map B × (B ⊗A V) −→ B ⊗A V (b′, b ⊗ v) 7→ (b′b) ⊗ v makes B ⊗A V a B-module. Deﬁne induction and restriction (respectively) as the maps

B IndA : {A-modules} −→ {B-modules} V 7→ B ⊗A V ResB : {B-modules} −→ {A-modules} A . V 7→ V

Suppose V is an A-module. Every choice of basis {v1, v2,..., vn} of V gives rise to an algebra homomorphism

ρ : A → Mn(C), where Mn(C) = {n × n matrices with entries in C}. 6

The character χV : A → C associated to V is

χV (a) = tr(ρ(a)).

′ The character χV is independent of the choice of basis, and if V V are isomorphic A-modules, then χV = χV′ . A character χV is irreducible if V is irreducible. The degree of a character χV is χV (1) = dim(V), and if χ(1) = 1, then a character is linear; note that linear characters are both irreducible characters and algebra homomorphisms. Deﬁne γ γ R[A] = C-span{χ | γ ∈ Aˆ}, where χ = χAγ , with a C-bilinear inner product given by

γ µ hχ , χ i = δγµ. (2.1) Note that the semisimplicity of A implies that any character is contained in the subspace

γ Z≥0-span{χ | γ ∈ Aˆ}.

If χV is a character of a subalgebra A of B, then let

B IndA(χV ) = χB⊗AV B and if χV′ is a character of B, then ResA(χV′ ) is the restriction of the map χV′ to A. Proposition 2.2 (Frobenius Reciprocity). Let A be a subalgebra of B. Suppose χ is a character of A and χ′ is a character of B. Then

B ′ B ′ hIndA(χ), χ i = hχ, ResA(χ )i.

2.1.2 Weight space decompositions Suppose A is a commutative algebra. Then dim(Aγ) = 1, for all γ ∈ Aˆ. The corresponding character χγ : A → C is an algebra homomorphism, and we may identify the label γ with the character χγ, so that av = χγ(a)v = γ(a)v, for a ∈ A, v ∈ Aγ. If A is a commutative subalgebra of B and V is an B-module, then as a A-module

V = Vγ, where Vγ = {v ∈ V | av = γ(a)v, a ∈ A}. Mγ∈Aˆ

The subspace Vγ is the γ-weight space of V, and if Vγ , 0 then V has a weight γ. For large subalgebras A, such a decomposition can help construct the module V. For example, 7

, 1. If0 vγ ∈ Vγ, then {vγ}Vγ,0 is a linearly independent set of vectors. In particular, if dim(Vγ) = 1 for all Vγ , 0, then {vγ} is a basis for V.

2. If V is irreducible, then BVγ = V. Chapter 6 uses this idea to construct irreducible modules of the Yokonuma algebra.

2.1.3 Characters in a group algebra Let G be a ﬁnite group. The group algebra CG is the algebra

CG = C-span{g ∈ G} with basis G and multiplication determined by the group multiplication in G. A G- module V is a vector space over C with a map

G × V −→ V (g, v) 7→ gv

such that

(gg′)v = g(g′v), g(cv + c′v′) = cgv + c′gv′, for c, c′ ∈ C, g, g′ ∈ G, v, v′ ∈ V.

Note that there is a natural bijection

{CG-modules} ←→ {G-modules} , V 7→ V

so I will use G-modules and CG-modules interchangeably. Let Gˆ index the irreducible CG-modules. The inner product (2.1) on R[G] has an explicit expression

h·, ·i : R[G] × R[G] −→ C 1 (χ, χ′) 7→ χ(g)χ′(g−1), |G| Xg∈G and if U is a subgroup of G, then induction is given by

G IndU (χ): G −→ C 1 g 7→ χ(xgx−1). |U| Xx∈G xgx−1∈U 8

2.1.4 Hecke algebras An idempotent e ∈ CG is a nonzero element that satisﬁes e2 = e. For γ ∈ Gˆ, let χγ(1) e = χγ(g−1)g ∈ CG. γ |G| Xg∈G These are the minimal central idempotents of CG, and they satisfy

2 γ γ 1 = eγ, eγ = eγ, eγeµ = δγµeγ, CGeγ dim(G )G . Xγ∈Gˆ In particular, as a G-module,

CG CGeγ. Mγ∈Gˆ Let U be a subgroup of G with an irreducible U-module M. The Hecke algebra H(G, U, M) is the centralizer algebra

G H(G, U, M) = EndCG(IndU (M)).

If eM ∈ CU is an idempotent such that M CUeM , then

G IndU(M) = CG ⊗CU CUeM = CG ⊗CU eM CGeM and the map

∼ θ :e CGe −→ H(G, U, M) φ : CGe −→ CGe M M M where g M M (2.2) eMgeM 7→ φg keM 7→ keMgeM, is an algebra anti-isomorphism (i.e. θM(ab) = θM(b)θM(a)). The following theorem connects the representation theory of G to the representation theory of H(G, U, M). Theorem 2.3 and the Corollary are in [CR81, Theorem 11.25]. Theorem 2.3 (Double Centralizer Theorem). Let V be a G-module and let H = EndCG(V). Then, as a G-module,

V dim(H γ)Gγ Mγ∈Gˆ

if and only if, as an H-module,

V dim(Gγ)H γ, Mγ∈Gˆ

where {H γ | γ ∈ Gˆ, H γ , 0} is the set of irreducible H-modules. 9

Corollary 2.4. Suppose U is a subgroup of G. Let M CUeM be an irreducible U- module. Write H = H(G, U, M). Then the map

{G-submodules of CGeM} −→ {H-modules} V 7→ eMV is a bijection that sends Gγ 7→ H γ for every irreducible Gγ that is isomorphic to a G γ G .((submodule of IndU(M) (G ֒→ IndU (M

2.2 Chevalley Groups

There are two common approaches used to define finite Chevalley groups. One strategy considers the subgroup of elements in an algebraic group fixed under a Frobenius map (see [Car85]). The approach here, however, begins with a pair (g, V) of a Lie algebra g and a g-module V, and constructs the Chevalley group from a variant of the exponential map. This perspective gives an explicit construction of the elements, which will prove useful in the computations that follow (see also [Ste67]).

2.2.1 Lie algebra set-up A ﬁnite dimensional Lie algebra g is a ﬁnite dimensional vector space over C with a C-bilinear map [·, ·]: g × g −→ g (X, Y) 7→ [X, Y] such that

(a) [X, X] = 0, for all X ∈ g,

(b) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0, for X, Y, Z ∈ g.

Note that if A is any algebra, then the bracket [a, b] = ab − ba for a, b ∈ A gives a Lie algebra structure to A. A g-module is a vector space V with a map

g × V −→ V (X, v) 7→ Xv

such that

(aX + bY)v = aXv + bYv, for all a, b ∈ C, X, Y ∈ g, v ∈ V,

X(av1 + bv2) = aXv1 + bXv2, for all a, b ∈ C, X ∈ g, v1, v2 ∈ V, [X, Y]v = X(Yv) − Y(Xv), for all X, Y ∈ g, v ∈ V. 10

A Lie algebra g acts diagonally on V if there exists a basis {v1,..., vr} of V such that

Xvi ∈ Cvi, for all X ∈ g and i = 1, 2,..., r.

The center of g is

Z(g) = {X ∈ g | [X, Y] = 0 for all Y ∈ g}.

Let γ ∈ gˆ index the irreducible g-modules gγ. A Lie algebra g is reductive if for every ﬁnite-dimensional module V on which Z(g) acts diagonally,

γ V = mγ(V)g , where mγ(V) ∈ Z≥0. Mγ∈gˆ

If g is reductive, then

g = Z(g) ⊕ gs where gs = [g, g] is semisimple.

Let g be a reductive Lie algebra. A Cartan subalgebra h of g is a subalgebra satisfying

(a) hk = [hk−1, h] = 0 for some k ≥ 1 (h is nilpotent),

(b) h = {X ∈ g | [X, H] ∈ h, H ∈ h} (h is its own normalizer).

If hs is a Cartan subalgebra of gs, then h = Z(g) ⊕ hs is a Cartan subalgebra of g. Let

∗ ∗ h = HomC(h, C) and hs = HomC(hs, C).

As an hs-module, gs decomposes

gs hs ⊕ (gs)α, where (gs)α = hX ∈ gs | [H, X] = α(H)X, H ∈ hsi. ∗ 0M,α∈hs

∗ The set of roots of gs is R = {α ∈ h | α , 0, (gs)α , 0}. Asetof simple roots is a subset {α1, α2,...,αℓ} ⊆ R of the roots such that

∗ (a) Q ⊗C hs = Q-span{α1, α2,...,αℓ} with {α1,...,αℓ} linearly independent,

(b) Every root β ∈ R can be written as β = c1α1 + c2α2 + ··· + cℓαℓ with either {c1,..., cℓ} ⊆ Z≥0 or {c1,..., cℓ} ⊆ Z≤0. Every choice of simple roots splits the set of roots R into positive roots

+ R = {β ∈ R | β = c1α1 + c2α2 + ··· + cℓαℓ, ci ∈ Z≥0} 11

and negative roots

− R = {β ∈ R | β = c1α1 + c2α2 + ··· + cℓαℓ, ci ∈ Z≤0}.

In fact, R− = −R+. Example. Consider the Lie algebra gl2 = gl2(C) = {2 × 2 matrices with entries in C}, and bracket [·, ·] given by [X, Y] = XY − YX. Write

gl2 = Z(gl2) ⊕ sl2,

c 0 where Z(gl2) = { 0 c | c ∈ C} and sl2 = {X ∈ gl2 | tr(X) = 0}. Also,

a 0 0 0 0 c sl2 = { 0 −a | a ∈ C} ⊕ c 0 | c ∈ C ⊕ 0 0 | c ∈ C .

∗ a 0 ∗ a 0 Let α ∈ h = { 0 b | a, b ∈ C} be the simple root given by α 0 b = a − b, so that + − R = {α} and R = {−α}.

2.2.2 From a Chevalley basis of gs to a Chevalley group

For every pair of roots α, −α, there exists a Lie algebra isomorphism φα : sl2 → hgα, g−αi. Under this isomorphism, let

0 1 1 0 0 0 Xα = φα 0 0 ∈ (gs)α, Hα = φα 0 −1 ∈ hs, X−α = φα 1 0 ∈ (gs)−α. Note that [Xα, X−α] = Hα. The following classical result can be found in [Hum72, Theorem 25.2].

Theorem 2.5 (Chevalley Basis). There is a choice of the φα such that the set {Xα, Hαi | α ∈ R, 1 ≤ i ≤ ℓ} is a basis of gs satisfying

(a) Hα is a Z-linear combination of Hα1 ,Hα2 ,...,Hαℓ .

(b) Let α, β ∈ R such that β , ±α, and suppose l, r ∈ Z≥0 are maximal such

{β − lα,...,β − α,β,β + α,...,β + rα} ⊆ R.

Then ±(l + 1)X , if r ≥ 1, [X , X ] = α+β α β ( 0 otherwise.

A basis as in Theorem 2.5 is a Chevalley basis of gs (For an analysis of the choices involved see [Sam69] and [Tit66]). Let V be a ﬁnite dimensional g-module such that V has a C-basis {v1, v2,..., vr} that satisﬁes

(a) There exists a C-basis {H1,..., Hn} of h such that 12

(1) Hαi ∈ Z≥0-span{H1,..., Hn},

(2) Hiv j ∈ Zv j for all i = 1, 2,..., n and j = 1, 2,..., r.

(3) dimZ(Z-span{H1, H2,..., Hn}) ≤ dimC(h). Xn (b) α v ∈ Z-span{v , v ,..., v } for α ∈ R, n ∈ Z and i = 1, 2,..., r. n! i 1 2 r ≥0

(c) dimZ(Z-span{v1, v2,..., vr}) ≤ dimC(V). (Condition (a) guarantees that Z(g) acts diagonally. If Z(g) = 0, then the existence of such a basis is guaranteed by a theorem of Kostant [Hum72, Theorem 27.1]). Let V be a ﬁnite dimensional g-module that satisﬁes (a)-(c) above. For H ∈ h, let

H (H − 1)(H − 2) ··· (H − (n + 1)) = ∈ End(V). n ! n!

As a transformation of the basis v1, v2,..., vr, the element Hi = diag(h1, h2,..., hr) ∈ ∗ End(V) with h j ∈ Z. Let y ∈ C . Temporarily abandon precision (i.e. allow inﬁnite sums) to obtain

H (y − 1)n i = diag (y − 1)n h1 , (y − 1)n h2 ,..., (y − 1)n hr n ! n n n Xn≥0 Xn≥0 Xn≥0 Xn≥0 = diag(yh1 , yh2 ,..., yhr ) ∈ End(V) (by the binomial theorem).

This computation motivates some of the definitions below. Let hZ = Z-span{H1, H2,..., Hn}. (2.3) ∗ The finite field Fq with q elements has a multiplicative group Fq and an additive group + Fq . Let Vq = Fq-span{v1, v2,..., vr}. (2.4)

The ﬁnite reductive Chevalley group GV ⊆ GL(Vq) is

∗ GV = hxα(a), hH(b) | α ∈ R, H ∈ hZ, a ∈ Fq, b ∈ Fqi,

where Xn x (a) = an α , and (2.5) α n! Xn≥0 λ1(H) λ2(H) λr(H) hH(b) = diag(b , b ,..., b ), where Hvi = λi(H)vi. (2.6)

Remarks. 13

1. If we “allow” inﬁnite sums, then

n Hi hH (b) = (b − 1) . i n ! Xn≥0

2. If g = gs, then GV = hxα(t)i.

Example. Suppose (as before) g = gl2 and let 1 0 V = C-span , ( 0! 1!) be the natural g-module given by matrix multiplication. Then h has a basis

a 0 1 0 0 0 h = | a, b ∈ C = C-span , . ( 0 b ! ) ( 0 0 ! 0 1 !) By direct computation,

1 t ta 0 xα(t) = and h a 0 (t) = b for a, b ∈ Z, 0 1 ! 0 b 0 t ! and GV = GL2(Fq) (the general linear group).

2.3 Some combinatorics of the symmetric group

This section describes some of the pertinent combinatorial objects and techniques.

2.3.1 A pictorial version of the symmetric group S n

Let si ∈ S n be the simple reﬂection that switches i and i + 1, and ﬁxes everything else. The group S n can be presented by generators s1, s2,..., sn−1 and relations

2 si = 1, si si+1 si = si+1 si si+1, si s j = s j si, for |i − j| > 1.

Using two rows of n vertices and strands between the top and bottom vertices, we may pictorially describe permutations in the following way. View

ith vertex ?? ? • • •?? • • • ?? si as ··· ?? ··· . ?? •••••• ? 14

Multiplication in S n corresponds to concatenation of diagrams, so for example,

•?? • • • • • s1 ?? ?? • • • •OOO • • = = OO = s s . OOO 2 1 • •? • ••• O ?? s2 ?? ••• ? •••

Therefore, S n is generated by s1, s2,..., sn−1 with relations • • • • • • • • • • • • • • • • • • = , = , = . •• •• ••• ••• •••• •••• 2.3.2 Compositions, partitions, and tableaux

A composition µ = (µ1, µ2,...,µr) is a sequence of positive integers. The size of µ is |µ| = µ1 + µ2 + ··· + µr, the length of µ is ℓ(µ) = r and

µ≤ = {µ≤1, µ≤2,...,µ≤r}, where µ≤i = µ1 + µ2 + ··· + µi. (2.7)

If |µ| = n, then µ is a composition of n and we write µ |= n. View µ as a collection of boxes aligned to the left. If a box x is in the ith row and jth column of µ, then the content of x is c(x) = i − j. For example, if

µ = (2, 5, 3, 4) = ,

then |µ| = 14, ℓ(µ) = 4, µ≤ = (2, 7, 10, 14), and the contents of the boxes are

0 1 -1 0 1 2 3 . -2 -1 0 -3 -2 -1 0

Alternatively, µ≤ coincides with the numbers in the boxes at the end of the rows in the diagram 1 2 3 4 5 6 7 . 8 9 10 11 12 13 14

A partition ν = (ν1,ν2,...,νr) is a composition where ν1 ≥ ν2 ≥ ··· ≥ νr > 0. If |ν| = n, then ν is a partition of n and we write ν ⊢ n. Let

P = {partitions} and Pn = {ν ⊢ n}. (2.8) 15

′ ′ ′ ′ Suppose ν ∈P. The conjugate partition ν = (ν1,ν2,...,νℓ) is given by

′ νi = Card{ j | ν j ≥ i}.

In terms of diagrams, ν′ is the collection of boxes obtained by ﬂipping ν across its main diagonal. For example,

if ν = , then ν′ = .

Suppose ν, µ are partitions. If νi ≥ µi for all 1 ≤ i ≤ ℓ(µ), then the skew partition ν/µ is the collection of boxes obtained by removing the boxes in µ from the upper left-hand corner of the diagram λ. For example, if

ν = and µ = , then ν/µ = .

A horizontal strip ν/µ is a skew shape such that no column contains more than one box. Note that if µ = ∅, then ν/µ is a partition. A column strict tableau Q of shape ν/µ is a ﬁlling of the boxes of ν/µ by positive integers such that

(a) the entries strictly increase along columns,

(b) the entries weakly increase along rows.

The weight of Q is the composition wt(Q) = (wt(Q)1, wt(Q)2,...) given by

wt(Q)i = number of i in Q.

For example,

1 1 3

Q = 1 2 has wt(Q) = (3, 1, 3, 0, 1). 3 3 5 16

2.3.3 Symmetric functions

The symmetric group S n acts on the set of variables {x1, x2,..., xn} by permuting the indices. The ring of symmetric polynomials in the variables {x1, x2,..., xn} is

Λn(x) = { f ∈ Z[x1, x2,..., xn] | w( f ) = f, w ∈ S n}.

For r ∈ Z≥0, the rth elementary symmetric polynomial is

er(x : n) = xi1 xi2 ··· xir , where by convention er(x : n) = 0, if r > n, 1≤i1

r r r pr(x : n) = x1 + x2 + ··· + xn. For any partition ν ∈P, let

eν(x : n) = eν1 (x : n)eν2 (x : n) ··· eνℓ (x : n)

pν(x : n) = pν1 (x : n)pν2 (x : n) ··· pνℓ (x : n).

The Schur polynomial corresponding to ν is

s (x : n) = det(e ′ (x : n)), (2.9) ν νi −i+ j for which Pieri’s rule gives

sν(x : n)s(n)(x : n) = sγ(x : n) [Mac95, I.5.16]. (2.10) horizontalX strip γ/ν |γ/ν|=n

For each t ∈ C, the Hall-Littlewood symmetric function is

xi − tx j P (x : n; t) = w xν1 xν2 ··· xνn ν  1 2 n x − x  ν∈ ν  ν >ν i j  wXS n  Yi j    1  x −tx ν1 ν2 νn i j = w x1 x2 ··· xn , vν(t)  xi − x j  wX∈S n  Yi< j      where vν(t) ∈ C is a constant as in [Mac95, III.2]; and they satisfy

Pν(x : n; 0) = sν(x : n), P(1n)(x : n; t) = en(x : n).

For t ∈ C,

Λn(x) = Z-span{eν(x : n)} = Z-span{sν(x : n)} = Z-span{Pν(x : n; t)}, (2.11) 17

and if we extend coeﬃcients to Q, then

Q ⊗ Λn(x) = Q-span{pν(x : n)}.

Note that for n > m there is a map

Λn(x) −→ Λm(x) f (x1, x2,..., xn) 7→ f (x1,..., xm, 0,..., 0)

which sends sν(x : n) 7→ sν(x : m), eν(x : n) 7→ eν(x : m), etc. Let {x1, x2,...} be an inﬁnite set of variables. The Schur function sν(x) is the inﬁnite sequence

sν(x) = (sν(x : 1), sν(x : 2), sν(x : 3),...),

and one can deﬁne elementary symmetric functions er(x), power sum symmetric functions pr(x), and Hall-Littlewood symmetric functions Pν(x; t), analogously. The ring of symmetric functions in the variables {x1, x2,...} is

Λ(x) = Z-span{sν(x) | ν ∈ P},

and let ΛC(x) = C-span{sν(x) | ν ∈ P}.

2.3.4 RSK correspondence The classical RSK correspondence provides a combinatorial proof of the identity 1 = s (x)s (y) [Knu70] 1 − x y ν ν i, j>0 i j ν⊢n Y Xn≥0 by constructing for each ℓ ≥ 0 a bijection between the matrices b ∈ Mℓ(Z≥0) and the set of pairs (P(b), Q(b)) of column strict tableaux with the same shape. The bijection is as follows. If P is a column strict tableau and j ∈ Z>0, let P ← j be the column strict tableau given by the following algorithm

(a) Insert j into the the ﬁrst column of P by displacing the smallest number ≥ j. If all numbers are < j, then place j at the bottom of the ﬁrst column.

(b) Iterate this insertion by inserting the displaced entry into the next column.

i1 i2 ··· in A two-line array is a two-rowed array with i1 ≤ i2 ≤ ··· ≤ in j1 j2 ··· jn ! and jk ≥ jk+1 if ik = ik+1. If b ∈ Mℓ(Z≥0), then let ~b be the two-line array with bi j pairs i j . For b ∈ Mℓ(Z≥0), suppose

i i ··· i ~b = 1 2 n . j1 j2 ··· jn ! Then the pair (P(b), Q(b)) is the ﬁnal pair in the sequence

(∅, ∅) = (P0, Q0), (P1, Q1), (P2, Q2),..., (Pn, Qn) = (P(b), Q(b)),

where (Pk, Qk) is a pair of column strict tableaux with the same shape given by

Qk is deﬁned by sh(Qk) = sh(Pk) with Pk = Pk−1 ← jk and ik in the new box sh(Qk)/sh(Qk−1).

For example,

1 1 0 11223 b =  0 0 2  corresponds to ~b = 21332 !  0 1 0      and provides the sequence

1 2 1 1 1 2 1 1 1 2 3 1 1 3 (∅, ∅), ( 2 , 1 ), ( 1 2 , 1 1 ), , , , , , 3 2 3 3 2 2 2 3 2 2 so that (P(b), Q(b)) = 1 2 3 , 1 1 3 . 2 3 2 2 19 Chapter 3

Unipotent Hecke algebras

3.1 The algebras

3.1.1 Important subgroups of a Chevalley group

Let G = GV be a Chevalley group deﬁned with a g-module V as in Section 2.2.2. Recall that R+ is the set of positive roots according to some ﬁxed set of simple roots {α1, α2,...,αℓ} of g (Section 2.2.1). The group G contains a subgroup U given by

+ U = hxα(t) | α ∈ R , t ∈ Fqi, which decomposes as

U = Uα, where Uα = hxα(t) | t ∈ Fqi, αY∈R+ with uniqueness of expression for any ﬁxed ordering of the positive roots [Ste67, Lemma 18]. For each α ∈ R+, the map

∼ + Uα −→ Fq xα(t) 7→ t is a group homomorphism. For α ∈ R, deﬁne the maps

s : h∗ −→ h∗ α (3.1) γ 7→ γ − γ(Hα)α, s : h = Z(g) ⊕ h −→ h α s (3.2) H + Hβ 7→ H + Hβ − β(Hα)Hα, for β ∈ R.

The Weyl group of G is W = hsα | α ∈ Ri and has a presentation given by

2 mij , W = hs1, s2,..., sℓ | si = 1, (si s j) = 1, 1 ≤ i j ≤ ℓi, mi j ∈ Z>0, si = sαi .

If w = si1 si2 ··· sir with r minimal, then the length of w is ℓ(w) = r. Let hZ be as in (2.3). If q > 3, then the subgroup

∗ T = hhH(t) | H ∈ hZ, t ∈ Fqi 20 has its normalizer in G given by

∗ −1 N = hwα(t), h | α ∈ R, h ∈ T, t ∈ Fqi, where wα(t) = xα(t)x−α(−t )xα(t).

−1 If α ∈ R, then hHα (t) = wα(t)wα(1) . Write hα(t) = hHα (t) and hi(t) = hαi (t). There is a natural surjection from N onto the Weyl group W with kernel T given by

π : N −→ W ∗ wα(t) 7→ sα, for α ∈ R, t ∈ Fq, (3.3) h 7→ 1, for h ∈ T.

Suppose v ∈ N. Then for each minimal expression

π(v) = si1 si2 ... sir , with ℓ(π(v)) = r, there is a unique decomposition of v as

v = v1v2 ··· vrvT , where vk = wik (1) and vT ∈ T. (3.4)

Write ξi = wi(1). (3.5)

3.1.2 Unipotent Hecke algebras

+ ∗ Let G be a ﬁnite Chevalley group. Fix a nontrivial homomorphism ψ : Fq → C . If

+ µ : R → Fq satisﬁes µα = 0 for all α notsimple, (3.6) α 7→ µα then the map ψ : U −→ C∗ µ (3.7) xα(t) 7→ ψ(µαt) is a linear character of U. In fact, with the exception of a few degenerate special cases of G (which can be avoided if q > 3), all linear characters of U are of this form [Yok69a, Theorem 1]. The unipotent Hecke algebra H(G, U, ψµ) is

G Hµ = EndCG(IndU (ψµ)). (3.8)

Using the anti-isomorphism (2.2), view Hµ as the subset of CG 1 H = e CGe , where e = ψ (u−1)u. (3.9) µ µ µ µ |U| µ Xu∈U 21

3.1.3 The choices µ and ψ

This section assumes all linear characters of U are of the form ψµ for some nontrivial + ∗ homomorphism ψ : Fq → C and µ as in (3.6). The group T normalizes Uα, since

−1 + ∗ hxα(t)h = xα(α(h)t), for α ∈ R , t ∈ Fq, h ∈ T, α(h) ∈ Fq. Thus, if χ : U → C∗ is a linear character, then

hχ : U −→ C∗ −1 xα(t) 7→ χ(hxα(t)h ) is a linear character for every h ∈ T. Proposition 3.1. Suppose χ : U → C∗ is a linear character. For every h ∈ T,

G G h IndU (χ) = IndU ( χ). Proof. Recall that 1 1 IndG (hψ )(g) = hψ (xgx−1) = ψ (hxgx−1h−1). U µ |U| µ |U| µ Xx∈G Xx∈G xgx−1∈U xgx−1∈U

Since T normalizes U, the sum over all x ∈ G such that xgx−1 ∈ U is the same as the sum over hx ∈ G such that hxgx−1h−1 ∈ U. Thus, 1 IndG (hψ )(g) = ψ (x′gx′−1) = IndG (ψ )(g). U µ |U| µ U µ Xx′∈G x′gx′−1∈U

∗ The type of a linear character χ : U → C is the set J ⊆{α1, α2,...,αℓ} such that , χ(xαi (t)) 1 for all t ∈ Fq if and only if αi ∈ J. A unipotent Hecke algebra H(G, U, χ) has type J if χ has type J. Proposition 3.2.

+ ∗ ′ + ∗ (a) Fix a nontrivial homomorphism ψ : Fq → C . If ψ : Fq → C is a homomorphism, then there exists k ∈ Fq such that

′ + ψ (t) = ψ(kt), for all t ∈ Fq .

(b) The linear characters χ : U → C∗ and hχ : U → C∗ have the same type for any h ∈ T. 22

(c) Let J ⊆ {α1, α2,...,αℓ} and SJ = {type J linear characters of U}. Then the number of T-orbits of SJ is |Z |(q − 1)|J| J , where Z = {h ∈ T | α(h) = 1, for α ∈ J}. |T| J

(d) The number of distinct (nonisomorphic) type J unipotent Hecke algebras is at most |Z |(q − 1)|J| J . |T| Proof. (a) The map + + ∗ Fq −→ Hom(Fq , C ) ψ :F+ → C∗ k 7−→ k q t 7→ ψ(kt) is a group homomorphism, and the kernel is trivial because ψ is nontrivial. Since + |Hom(Fq , C)| = q, the map is also surjective. + + (b) Since Uα Fq , the restriction of χ to Uα gives a linear character of Fq . Part (a) + implies that for every α ∈ R there exists kα ∈ Fq such that

χ(xα(t)) = ψ(kαt)

Furthermore, if χ has type J, then kα , 0 if and only if α ∈ J. Write

χ = ψ(k1,...,kℓ) where ki = kαi .

−1 Since hxα(t)h = xα(α(h)t) for h ∈ T,

h χ = ψ(α1(h)k1,...,αℓ(h)kℓ),

and α(h)kα , 0 if and only if kα , 0 if and only if α ∈ J. Thus, the action of T preserves the type of χ. (c) Suppose a group K acts on a set S. Then by [Isa94, Theorem 4.18] the number of orbits of the K action is 1 Card{s ∈S | g(s) = s}. |K| Xg∈K In this case, K = T acts on the set

S = SJ = {ψ(k1,k2,...,kℓ) | ki = 0, unless αi ∈ J}. Furthermore,

h ψ(k1,k2,...,kℓ) = ψ(k1,k2,...,kℓ) if and only if αi(h) = 1 for all αi ∈ J.

|J| Finally, |SJ| = (q − 1) . 23

Examples.

1. If J = ∅, then |J| = 0 and |ZJ| = |T|, so there is a unique unipotent Hecke algebra of type J; in this case, ψµ is trivial and Hµ is the Yokonuma algebra.

2. A character is in general position if it has type J = {α1, α2,...,αℓ}. In this case, ZJ is the center of G, and |J| = ℓ, so the number of type J unipotent Hecke algebras is at most |Z(G)|(q − 1)ℓ . |T|

3.1.4 A natural basis The group G has a double-coset decomposition G = UvU, [Ste67, Theorem 4 and B = UT ] (3.10) Gv∈N and if

Nµ = {v ∈ N | eµveµ , 0} −1 −1 (3.11) = {v ∈ N | u, vuv ∈ U implies ψµ(u) = ψµ(vuv )}

then the set {eµveµ | v ∈ Nµ} is a basis for Hµ [CR81, Prop. 11.30]. Theorem 3.7 gives a set of relations similar to those of the Yokonuma algebra (Ex- ample 1, below) for evaluating the product

(eµueµ)(eµveµ), with u, v ∈ Nµ.

in any unipotent Hecke algebra Hµ. Examples.

1. The Yokonuma Hecke algebra. If µα = 0 for all positive roots α, then ψµ = 11is the

trivial character and N11 = N. Let Tv = e11ve11 for v ∈ N, with Ti = Tξi (ξi as in (3.5)) and

TH(t) = ThH (t). If v ∈ N has a decomposition v = v1v2 ··· vrvT (as in (3.4)), then

Tv = Ti1 Ti2 ··· Tir TvT (See Chapter 6).

Thus, the Yokonuma algebra H11 has generators Ti, Th for 1 ≤ i ≤ ℓ, h ∈ T (see [Yok69b]) with relations, T 2 = q−1T (−1) + q−1 T (t−1)T , 1 ≤ i ≤ ℓ, i Hαi Hαi i ∗ Xt∈Fq

mij T i T jTi ··· = T j T iT j ··· , (si s j) = 1,

mij terms mij terms | {z } | {z } TiTh = TsihTi, h ∈ T,

ThTk = Thk, h, k ∈ T. 24

These relations give an “eﬃcient” way to compute arbitrary products (e11ue11)(e11ve11) in H11. There is a surjective map from the Yokonuma algebra onto the Iwahori-Hecke algebra that sends Th 7→ 1 for all h ∈ T. “Setting Th = 1” in the Yokonuma algebra relations recovers relations for the Iwahori-Hecke algebra,

2 = −1 + −1 − ··· = ··· Ti q q (q 1)Ti, T i T j T j Ti .

mij terms mij terms | {z } | {z } Furthermore, there is a surjective map from the Iwahori Hecke algebra onto the Weyl group given by mapping Ti 7→ si and q 7→ 1. Thus, by “setting Ti = si and q = 1” we retrieve the Coxeter relations of W,

2 = ··· = ··· si 1, si s j si s j s i s j .

mij terms mij terms | {z } | {z } For a more detailed discussion of the Yokonuma algebra, see Chapter 6.

2. The Gelfand-Graev Hecke algebra. By deﬁnition, if µα , 0 for all simple roots G α, then ψµ is in general position. In this case, the Gelfand-Graev character IndU(ψµ) is multiplicity free as a G-module ([Yok68],[Ste67, Theorem 49]). The corresponding Hecke algebra Hµ is therefore commutative. However, decomposing the product (eµueµ)(eµveµ) into basis elements is more challenging than in the Yokonuma case [Cha76, Cur88, Rai02].

3.2 Parabolic subalgebras of Hµ

Let ψµ : U → G be as in (3.7). Fix a subset J ⊆{α1, α2,...,αℓ} such that

, if µαi 0, then αi ∈ J.

For example, if ψµ is in general position, then J = {α1, α2,...,αℓ}, but if ψµ is trivial, then J could be any subset. Let

WJ = hsi ∈ W | αi ∈ Ji, PJ = hU, T, WJ i and RJ = Z-span{αi ∈ J} ∩ R.

Then PJ has subgroups

+ LJ = hT, WJ, Uα | α ∈ RJi and UJ = hUα | α ∈ R − RJi (3.12)

(a Levi subgroup and the unipotent radical of PJ, respectively). Note that

UJ LJ = PJ , UJ ∩ LJ = 1, and, in fact, PJ = UJ ⋊ LJ . 25

Deﬁne the idempotents of CU,

1 −1 ′ 1 eµJ = ψµ(u )u and eJ = u, (3.13) |LJ ∩ U| |UJ| u∈XLJ∩U uX∈UJ

′ so that eµ = eµJeJ . The group homomorphisms

P −→ L P −→ G J J and J for l ∈ L , u ∈ U , lu 7→ l lu 7→ lu J J

induce maps

InfPJ :{L -modules} −→ {P -modules} LJ J J ′ , M 7→ eJ M G { } −→ { } IndPJ : PJ-modules G-modules ′ ′ M 7→ CG ⊗CPJ M

G C whose composition is the map IndfLJ . Note that in the special case when LJ e is an irreducible LJ -module with corresponding idempotent e, then G { } −→ { } IndfLJ : LJ-modules G-modules ′ CLJ e 7→ CGeeJ. The map ψ : U → C∗ restricts to a linear character ResU (ψ ): L ∩ U → C∗. To µ U∩LJ µ J make the notation less heavy-handed, write ψ : L ∩ U → C∗, for ResU (ψ ). µ J U∩LJ µ

Lemma 3.3. Let ψµ be as in (3.7). Then

IndG (ψ ) IndfG (IndLJ (ψ )). U µ LJ U∩LJ µ

G Proof. Recall IndU (ψµ) CGeµ. On the other hand,

IndLJ (ψ ) CL e implies IndfG (IndLJ (ψ )) CGe e′ , U∩LJ µ J µJ LJ U∩LJ µ µJ J

′ where eµJ is as in (3.13). But eµJeJ = eµ, so IndG (ψ ) CGe Ce e′ IndfG (IndLJ (ψ )). U µ µ µJ J LJ U∩LJ µ Theorem 3.4. The map

θ : End (IndLJ (ψ )) −→ H CLJ U∩LJ µ µ eµJveµJ 7→ eµveµ, for v ∈ LJ ∩ Nµ, is an injective algebra homomorphism. 26

′ Proof. Let v ∈ N ∩ LJ . Since eµJveµJ ∈ CLJ , eJ ∈ CUJ and LJ ∩ UJ = 1,

′ eµJveµJ = 0 ifandonlyif eJeµJveµJ = 0.

′ Because LJ normalizes UJ, both eµJ and v commute with eJ. Therefore,

′ ′ eµJveµJ = 0 ifandonlyif eJ eµJveJ eµJ = eµveµ = 0, and {eµJveµJ , 0} if and only if v ∈ Nµ ∩ LJ . Consequently, θ is both well-deﬁned and injective. Consider

′ ′ ′ θ(eµJ ueµJ)θ(eµJ veµJ) = eµueµeµveµ = eµJeJueµJeJveJ eµJ.

′ Since u commutes with eJ,

′ ′ θ(eµJueµJ)θ(eµJveµJ) = eµJeJueµJveJ eµJ = eµueµJveµ

= θ(eµJueµJveµJ), and so θ is an algebra homomorphism. Write L = θ(End (IndLJ (ψ ))) ⊆ H (3.14) J CLJ U∩LJ µ µ

The LJ are “parabolic” subalgebras of Hµ in that they have a similar relationship to the representation theory of Hµ as parabolic subgroups PJ have with the representation theory of G.

3.2.1 Weight space decompositions for Hµ-modules An important special case of Theorem 3.4 is when

, J = Jµ = {αi ∈{α1, α2,...,αℓ} | µαi 0},

so that ψµ has type Jµ. Write Lµ = LJµ , Wµ = WJµ , etc.

Corollary 3.5. The algebra Lµ is a nonzero commutative subalgebra of Hµ.

Proof. As a character of U ∩ L , ψ is in general position, so IndLµ (ψ ) is a Gelfand- µ µ Lµ∩U µ Graev module and Lµ is a Gelfand-Graev Hecke algebra (and therefore commutative). 27

Since Lµ is commutative, all the irreducible modules are one-dimensional; let Lˆµ be an indexing set for the irreducible modules of Lµ. Suppose V is an Hµ-module. As an Lµ-module,

V Vγ where Vγ = {v ∈ V | xv = γ(x)v, x ∈Lµ}.

Mγ∈Lˆµ

If γ ∈ Lˆµ, then Vγ is the γ-weight space of V, and V has a weight γ if Vγ , 0. See Chapter 6 for an application of this decomposition. Examples

1. In the Yokonuma algebra ψµ = 11, J11 = ∅ and L11 = e11CTe11.

2. In the Gelfand-Graev Hecke algebra case, Jµ = {α1, α2,...,αℓ} and Lµ = Hµ (conﬁrming, but not proving, that Hµ is commutative).

3.3 Multiplication of basis elements

This section examines the decomposition of products in terms of the natural basis

v′ ′ (eµueµ)(eµveµ) = cuv (eµv eµ). ′ vX∈Nµ In particular, Theorem 3.7, below, gives a set of braid-like relations (similar to those of the Yokonuma algebra) for manipulating the products, and Corollary 3.9 gives a recur- sive formula for computing these products.

3.3.1 Chevalley group relations The relations governing the interaction between the subgroups N, U, and T will be critical in describing the Hecke algebra multiplication in the following section. They can all be found in [Ste67]. The subgroup + U = hxα(t) | α ∈ R , t ∈ Fqi + has generators {xα(t) | α ∈ R , t ∈ Fq}, with relations

−1 −1 i j xα(a)xβ(b)xα(a) xβ(b) = xγ(zγa b ), (U1) γ=iYα+ jβ∈R+ i, j∈Z>0

xα(a)xα(b) = xα(a + b), (U2)

where zγ ∈ Z depends on i, j,α,β, but not on a, b ∈ Fq [Ste67]. The zγ have been explicitly computed for various types in [Dem65, Ste67] (See also Appendix A). 28

∗ The subgroup N has generators {ξi, hH(t) | i = 1, 2,...,ℓ, H ∈ hZ, t ∈ Fq}, with relations 2 ξi = hi(−1), (N1) mij ξi ξ j ξiξ j ··· = ξ j ξ i ξ jξ i ··· , where (si s j) = 1 in W, (N2)

mij terms mij terms | {z } | {z } ξihH(t) = hsi(H)(t)ξi, (N3)

hH(a)hH(b) = hH(ab), (N4) ′ hH(a)hH′ (b) = hH′ (b)hH(a), for H, H ∈ h, (N5) ′ hH(a)hH′ (a) = hH+H′ (a), for H, H ∈ h, (N6) = λ j(H1) λ j(Hk) = hH1 (t1)hH2 (t2) ... hHk (tk) 1, if t1 ··· tk 1 for all 1 ≤ j ≤ n, (N7)

where λ j : h → C depends on V as in (2.6). The double-coset decomposition of G (3.10) implies G = hU, Ni. Thus, G is gener- + ∗ ated by {xα(a),ξi, hH(b) | α ∈ R , a ∈ Fq, i = 1, 2,...,ℓ, H ∈ hZ, b ∈ Fq} with relations (U1)-(N7) and −1 ξi xα(t)ξi = xsi(α)(ciαt), where ciα = ±1 (ciαi = −1) (UN1) −1 hxα(b)h = xα(α(h)b), for h ∈ T, (UN2) −1 −1 −1 , ξi xi(t)ξi = xi(t )hi(t )ξi xi(t ), where xi(t) = xαi (t) and t 0. (UN3)

where for α ∈ R and hH(t) ∈ T, α(H) α(hH(t)) = t . (3.15)

∗ Fix a ψµ : U → C as in (3.7). For k ∈ Fq, let 1 e (k) = ψ(−µ kt)x (t) with the convention e = e (1). (3.16) α q α α α α Xt∈Fq Note that for any given ordering of the positive roots, the decomposition

U = Uα implies eµ = eα. (3.17) αY∈R+ αY∈R+ In particular, given any α ∈ R+, we may choose the ordering of the positive roots to have eα appear either ﬁrst or last. Therefore, since eα is an idempotent,

eµeα = eµ = eαeµ. (3.18)

If w = si1 si2 ··· sir ∈ W with r minimal, then let + − Rw = {α ∈ R | w(α) ∈ R } (3.19) = {αir , sir (αir−1 ),..., sir sir−1 ··· si2 (αi1 )}, where the second equality is in [Bou02, VI.1, Corollary 2 of Proposition 17]. 29

Lemma 3.6. Let v ∈ N, and let w = π(v) (with π : N → W as in (3.3)).

−1 + ξieα(k)ξi = esiα(ciαk), for α ∈ R , 1 ≤ i ≤ n − 1, (E1) −1 veαv = ewα, for α < Rw, v ∈ Nµ, (E2) −1 −1 heα(k)h = eα(kα(h) ), for h ∈ T, (E3)

eµ xα(t) = ψ(µαt)eµ = xα(t)eµ. (E4)

Proof. (E1) Using relation (UN1), we have 1 1 ξ e (k)ξ−1 = ψ(−µ kt)ξ x (t)ξ−1 = ψ(−µ kt)x (c t) i α i q α i α i q α siα iα Xt∈Fq Xt∈Fq

1 ′ ′ = ψ(−µαciαkt )xsiα(t ) = esiα(ciαk). q ′ tX∈Fq

(E2) Suppose α < Rw. Since v ∈ Nµ,

−1 ψ(µαt) = ψµ(xα(t)) = ψµ(vxα(t)v ) = ψµ(xwα(kt)) (by (UN1))

= ψ(µwαkt), for some k =∈ Z,0.

In particular, µα = kµwα and we may conclude

−1 1 1 −1 ′ ′ veαv = ψ(−µαt)xwα(kt) = ψ(−µαk t )xwα(t ) = ewα. q q ′ Xt∈Fq tX∈Fq

−1 (E3) Since hxα(t)h = xα(α(h)t), we have 1 he (k)h−1 = ψ(−kt)x (α(h)t) = ψ(−ktα(h)−1)x (t) = e (kα(h)−1). α q α α α Xt∈Fq Xt∈Fq

(E4) Note that 1 1 e x (t) = ψ(−µ a)x (a)x (t) = ψ(−µ a)x (a + t) α α q α α α q α α aX∈Fq aX∈Fq

1 ′ ′ 1 ′ ′ = ψ(−µα(a − t))xα(a ) = ψ(−µαa )ψ(µαt)xα(a ) q ′ q ′ aX∈Fq aX∈Fq

= ψ(µαt)eα

Therefore, by (3.18), eµ xα(t) = eµeα xα(t) = ψ(µαt)eµ. 30

3.3.2 Local Hecke algebra relations

Let u = u1u2 ··· uruT ∈ N decompose according to si1 si2 ... sir ∈ W. For1 ≤ k ≤ r deﬁne + constants ck = ±1 and roots βk ∈ R by the equation

−1 x (c t) = (u + ··· u ) x (t)(u + ··· u ). (3.20) βk k k 1 r αik k 1 r

Note that Rπ(u) = {β1,β2,...,βr} (see (3.19)). Deﬁne fu ∈ Fq[y1, y2,..., yr] by

µβ1 c1 µβ2 c2 µβr cr fu = − y1 − y2 −···− yr. (3.21) β1(uT ) β2(uT ) βr(uT ) and for k = 1, 2,..., r, let

uk(t) = ξik (t), where ξi(t) = ξi xi(t). (3.22)

Theorem 3.7. Let u = u1u2 ··· uruT , v = v1v2 ··· vsvT ∈ Nµ decompose according to

si1 si2 ··· sir ∈ W and s j1 s j2 ··· s js ∈ W, respectively, as in (3.4). Then (a) 1 = ◦ ··· ··· (eµueµ)(eµveµ) r (ψ fu)(t) eµ u1(t1)u2(t2) ur(tr) v1v2 vs heµ, q r Xt∈Fq −1 where h = vT v uT v ∈ T.

(b) The following local relations suﬃce to compute the product (eµueµ)(eµveµ).

2 −1 −1 (ψ ◦ f )(t)ξi(t)ξi = (ψ ◦ f )(0)ξi + (ψ ◦ f )(t)xi(t )ξihi(t)xi(t ), (H1) ∗ Xt∈Fq Xt∈Fq

ξi xα(t) = xsi(α)(ciαt)ξi, (H2) −1 xα(t)h = hxα(α(h) t), (H3)

eµ xα(t) = ψ(µαt)eµ = xα(t)eµ, (H4) ±1 ±1 r (ψ ◦ f )(t)(ψ ◦ g)(t) = (ψ ◦ ( f + g))(t), for f, g ∈ Fq[y1 ,..., yr ],t ∈ Fq, (H5)

hα(t)ξi = ξihsi(α)(t), (H6) m n , ξi(a)xα(b) = xsiγ(cisi(γ)zγa b )ξi(a), where α αi, (H7) + γ=mYαi+nα∈R m,n≥0

Φ(a)ξi(a)xi(b) = Φ(a − b)ξi(a), for some map Φ : Fq → CG, (H8)

aX∈Fq aX∈Fq

hα(a)hα(b) = hα(ab), (H9) ξi ξ j ξiξ j ··· = ξ j ξ i ξ jξ i ··· , where mi j is the order of si s j in W. (H10)

mij terms mij terms | {z } | {z } 31

Proof. (a) Order the positive roots so that by (3.17)

eµueµveµ = eµu eα eβ1 eβ2 ··· eβr veµ (deﬁnition βk) αY

= eµ ewα ueβ1 eβ2 ··· eβr veµ (Lemma 3.6,E2) αY

= eµueβ1 eβ2 ··· eβr veµ (Lemma 3.6, E4)

= eµu1u2 ··· uruT eβ1 eβ2 ··· eβr veµ µ µ µ = e u u ··· u e ( β1 )e ( β2 ) ··· e ( βr )u ve (Lemma 3.6,E3) µ 1 2 r β1 β1(uT ) β2 β2(uT ) βr βr(uT ) T µ µ c µ c µ c = e u e ( β1 1 )u e ( β2 2 ) ··· u e ( βr r )u ve (Lemma 3.6,E1) µ 1 αi1 β1(uT ) 2 αi2 β2(uT ) r αir βr(uT ) T µ µ c µ c µ cr − = e u e ( β1 1 )u e ( β2 2 ) ··· u e ( βr )v ··· v v v 1u ve µ 1 αi1 β1(uT ) 2 αi2 β2(uT ) r αir βr(uT ) 1 s T T µ eµ µβ c1t1 µβ crtr = ψ(− 1 )u1(t1) ··· ψ(− r )ur(tr)v1 ··· vsheµ (deﬁnition eα) qr β1(uT ) βr(uT ) t1,...,Xtr∈Fq 1 = ◦ ··· ··· H r (ψ fu)(t)eµu1(t1) ur(tr)v1 vsheµ, (by ( 5)) q r Xt∈Fq

−1 where h = vT v uT v ∈ T, as desired. (b) First, note that these relations are in fact correct (though not necessarily suﬃcient): (H1) comes from (UN3); (H2) comes from (UN1); (H3) comes from (UN2); (H4) comes from (E4); (H5) comes from the multiplicativity of ψ; (H6) comes from (N3); (H7) comes from (U1) and (UN1); (H8) comes from (U2); (H9) is (N4); and (H10) is (N2). It therefore remains to show suﬃciency. By (a) we may write 1 = ψ ◦ ··· ··· (eµueµ)(eµveµ) r ( f )(t)eµu1(t1) ur(tr)v1 vsheµ q r Xt∈Fq

for some f ∈ Fq[y1,..., yr] and h ∈ T. Say tk is resolved if the only part of the sum depending on tk is (ψ ◦ f ). The product is reduced when all the tk are resolved. I will show how to resolve tr and the result will follow by induction. Use relation (H2) to deﬁne the constant d and the root γ ∈ R by

··· −1 ··· = = (v1v2 vs) xαir (t)(v1v2 vs) xγ(dt) (where ℓ(π(v)) s). (3.23) There are two possible situations:

Case 1. γ ∈ R+,

Case 2. γ ∈ R−. 32

In Case 1, 1 = ψ ◦ ··· ··· (eµueµ)(eµveµ) r ( f )(t)eµu1(t1) ur xir (tr)v1 vsheµ (by (a)) q r −−−−→ Xt∈Fq 1 = ψ ◦ ··· ··· H r ( f )(t)eµu1(t1) ur−1(tr−1)urv1 vs xγ(dtr)heµ (by ( 2)) q r −−−−−→ Xt∈Fq 1 = ψ ◦ ··· ··· γ −1 H r ( f )(t)eµu1(t1) ur−1(tr−1)urv1 vshxγ(d (h) tr)eµ (by ( 3)) q r Xt∈Fq 1 = ψ ◦ ··· ··· ψ µ γ −1 H r ( f )(t)eµu1(t1) ur−1(tr−1)urv1 vsh ( γd (h) tr)eµ (by ( 4)) q r Xt∈Fq 1 = ψ ◦ ··· ··· ψ ◦ µ γ −1 r ( f )(t)eµu1(t1) ur−1(tr−1)urv1 vsh( γd (h) yr)(tr)eµ q r ←−−−−−−−−−−−−−−−−−− Xt∈Fq 1 = ψ ◦ ··· ··· , H r ( g)(t)eµu1(t1) ur−1(tr−1)urv1 vsheµ (by ( 5)) q r Xt∈Fq

−1 where g = f + µγdγ(h) yr. We have resolved tr in Case 1. − In Case 2, γ ∈ R , so we can no longer move xir (tr) past the v j. Instead,

(eµueµ)(eµveµ) e = µ ◦ ··· −1 ··· r (ψ f )(t)u1(t1) ur−1(tr−1)ur xir (tr)urur v1 vsheµ q r Xt∈Fq e = µ u (t ) ··· u (t ) (ψ ◦ f )(t′, t )u x (t )u u−1v ··· v he qr 1 1 r−1 r−1 r r ir r r r 1 s µ ′ r−1 ∈ t X∈Fq tXr Fq e = µ ··· ψ ◦ ′, 2 −1 ··· H r u1(t1) ur−1(tr−1)( f )(t 0)ur ur v1 vsheµ (by ( 1)) q ′ r−1 t X∈Fq e + µ ··· ψ ◦ ′, −1 −1 −1 −1 ··· r u1(t1) ur−1(tr−1) ( f )(t tr)xir (tr )hir (tr )ur xir (tr )ur v1 vsheµ q ′ r−1 ∗ −−−−−→ t X∈Fq tXr∈Fq e = µ ··· ψ ◦ ′, ··· r u1(t1) ur−1(tr−1)( f )(t 0)urv1 vsheµ q ′ r−1 ←−−−−−−−−−− t X∈Fq e + µ ··· ψ ◦ ′, −1 − ··· , r u1(t1) ur−1(tr−1) ( g)(t tr)xir (tr )hir ( tr)v1 vsheµ q ′ r−1 ∗ −−−−−→ t X∈Fq tXr∈Fq (by (H2,H3,H4)) 33

−1 −1 where g = f − µγdγ(h) yr (same as in the analogous steps in Case 1). e = µ ◦ ′ ··· ··· r (ψ f )(t , 0)u1(t1) ur−1(tr−1)urv1 vsheµ q ′ r−1 t X∈Fq e + µ ··· ψ ◦ ′, −1 ··· − , H r u1(t1) ur−1(tr−1) ( g)(t tr)xir (tr )v1 vshγ( tr)heµ (by ( 6)) q ′ r−1 ∗ ←−−−−−−−−−−−−−−−−− t X∈Fq tXr∈Fq e = µ ψ ◦ ′, ··· ··· r ( f )(t 0)u1(t1) ur−1(tr−1)urv1 vsheµ q ′ r−1 t X∈Fq e + µ ψ ◦ ϕ ′, , ··· ··· ′ r ( (g))(t tr) xβ(aβ(t tr)) u1(t1) ur−1(tr−1)v1 vsh eµ q ′ r−1 + ←−−−−−−−−− tX∈Fq βY∈R ∗ tr∈Fq (by (H7,H8))

where ϕ : Fq[y1,..., yr] → Fq[y1,..., yr] catalogues the substitutions to g due to (H8), −1 the aβ(y1, y2,..., yr ) ∈ Fq[y1,..., yr−1, yr] are determined by repeated applications of ′ (H7) and (H8), and h = hγ(−tr)h ∈ T.

1 = ψ ◦ ′, ··· ··· r ( f )(t 0)eµu1(t1) ur−1(tr−1)urv1 vsheµ q ′ r−1 t X∈Fq 1 + ◦ ′ ··· ··· ′ r (ψ ϕ(g))(t , tr)eµ ψ(µβaβ(t, tr)) u1(t1) ur−1(tr−1)v1 vsh eµ q ′ r−1 + ←−−−−−−−−−−− tX∈Fq βY∈R ∗ tr∈Fq (by (H4)) 1 = ◦ ′ ··· ··· r (ψ f )(t , 0)eµu1(t1) ur−1(tr−1)urv1 vsheµ q ′ r−1 t X∈Fq 1 + ψ ◦ ′, ··· ··· ′ , H r ( g2)(t tr)eµu1(t1) ur−1(tr−1)v1 vsh eµ (by ( 5)) q ′ r−1 tX∈Fq ∗ tr∈Fq

−1 where g2 = ϕ(g)+ β∈R+ µβaβ(y1,..., yr−1, yr ). Now tr is resolved for Case 2, as desired. P

Lemma 3.8 (Resolving tk). Let u = u1u2 ··· uk ∈ N decompose according to si1 si2 ··· sik (with uT = 1). Suppose v ∈ N and f ∈ Fq[y1, y2,..., yk]. deﬁne γ ∈ R and d ∈ C by the −1 equation v xik (t)v = xγ(dt). Then 34

Case 1 If ℓ(π(ukv)) >ℓ(π(v)), then

(ψ ◦ f )(t)eµu1(t1) ···uk(tk)veµ k Xt∈Fq

= (ψ ◦ ( f + µγdcγyk))(t)eµu1(t1) ··· uk−1(tk−1)ukveµ. k Xt∈Fq

Case 2 If ℓ(π(ukv)) <ℓ(π(v)), then

(ψ ◦ f )(t)eµu1(t1) ··· uk(tk)veµ = (ψ ◦ f )(t)eµu1(t1) ··· uk−1(tk−1)ukveµ k k Xt∈Fq Xt∈Fq, tk=0 −1 −1 + (ψ ◦ (ϕk( f ) − µγdyk ))(t)eµu1(t1) ··· uk−1(tk−1)hik (tk )veµ, k Xt∈Fq, ∗ tk∈Fq

±1 ±1 ±1 ±1 where ϕk : Fq[y1 ,..., yk ] → Fq[y1 ,..., yk ] is given by

−1 (ψ ◦ f )(t)eµu1(t1) ··· uk−1(tk−1)xik (tk ) = (ψ ◦ ϕk( f ))(t)eµu1(t1) ··· uk−1(tk−1). k ←−−−−− k Xt∈Fq Xt∈Fq ∗ ∗ tk∈Fq tk∈Fq

Proof. This Lemma puts v in the place of uT v in the proof of Theorem 3.7, (b), and summarizes the steps taken in Case 1 and Case 2.

3.3.3 Global Hecke algebra relations

Fix u = u1u2 ··· uruT ∈ Nµ, decomposed according si1 si2 ··· sir ∈ W (see (3.4)). Suppose ′ ′ v ∈ Nµ and let v = uT v . For 0 ≤ k ≤ r, let τ = (τ1, τ2,...,τr−k) be such that τi ∈ {+0, −0, 1}, where +0, −0, and 1 are symbols. If τ has r − k elements, then the colength of τ is ℓ∨(τ) = k. For example, if r = 10 and τ = (−0, 1, +0, +0, 1, 1), then ℓ∨(τ) = 4. For i ∈{+0, −0, 1}, let

(i, τ) = (i, τ1, τ2, ··· , τr−k).

By convention, if ℓ∨(τ) = r, then τ = ∅. Suppose ℓ∨(τ) = k. Deﬁne 1 Ξτ , = ψ ◦ τ ··· τ , (u v) r ( f )(t)eµu1(t1) uk(tk)v (t)eµ (3.24) q τ Xt∈Fq 35

where

if τi−k = +0, then ti ∈ Fq, τ r Fq = t ∈ Fq | for k < i ≤ r, if τi−k = −0, then ti = 0,  ; (3.25)  ∗   if τi−k = 1, then ti ∈ Fq.   −  τ =  τ1 1 τ1 τr−k 1−τr−k  v (t) hik+1 (tk+1) uk+1 ··· hir (tr) ur v,  (3.26) with +0 = −0 = 0 ∈ Z,1 = 1 ∈ Z in (3.26); and f τ is deﬁned recursively by

∅ µβ1 c1 µβ2 c2 µβr cr f = fu = − y1 − y2 −···− yr, (as in (3.21)), (3.27) β1(uT ) β2(uT ) βr(uT ) τ (i,τ) f + µγτ dτyk, if i = ±0, f = τ −1 (3.28) ( ϕk( f ) − µγτ dτyk , if i = 1, where (vτ)−1 x (t)vτ = x (d t) and the map ϕ is as in Lemma 3.8, Case 2. αik γτ τ k Remarks.

∅ ′ ′ 1. By (3.24) and Theorem 3.7 (a), Ξ (u, v) = (eµueµ)(eµv eµ) (recall, v = uT v ). 2. If ℓ∨(τ) = 0 so that τ is a string of length r, then 1 Ξτ , = ψ ◦ τ τ (a) (u v) r ( f )(t)eµv eµ has no remaining factors of the form uk(tk), q τ Xt∈Fq τ τ (b) Ξ (u, v) = 0 unless v (t) ∈ Nµ for some t ∈ Fq. The following corollary gives relations for expanding Ξτ(u, v) (beginning with Ξ∅(u, v)) as a sum of terms of the form Ξτ′ with ℓ∨(τ′) = ℓ∨(τ) − 1. When each term has colength ′ 0 (or length r), then we will have decomposed the product (eµueµ)(eµv eµ) in terms of the basis elements of Hµ. In summary, while we compute f τ recursively by removing elements from τ, we ′ compute the product (eµueµ)(eµv eµ) by progressively adding elements to τ.

′ Corollary 3.9 (The Global Alternative). Let u, v ∈ Nµ such that u = u1u2 ··· uruT ′ decomposes according to a minimal expression in W. Let v = uT v . Then

′ ∅ (a) (eµueµ)(eµv eµ) = Ξ (u, v) (b) If ℓ∨(τ) = k, then

(+0,τ) τ τ τ Ξ (u, v), if ℓ(π(ukv )) >ℓ(π(v )), Ξ (u, v) = (−0,τ) (1,τ) τ τ ( Ξ (u, v) + Ξ (u, v), if ℓ(π(ukv )) <ℓ(π(v )). 36

Proof. (a) follows from Remark 1. (b) Suppose ℓ∨(τ) = k. Note that 1 Ξτ , = ψ ◦ τ ··· τ (u v) r ( f )(t)eµu1(t1) uk(tk)v eµ q τ Xt∈Fq 1 = (ψ ◦ f τ)(t′, t′′)e u (t ) ··· u (t )vτe qr µ 1 1 k k µ ′′ r−k τ ′ k t ∈X(Fq ) tX∈Fq

r−k τ r−k where (Fq ) = {(tk+1,..., tr) ∈ Fq | restrictions according to τ} (as in (3.25)). Apply Lemma 3.8 to the inside sum with f := f τ, v := vτ. Note that the Lemma relations imply

′ k {t ∈ Fq}, if in Case 1, ′ k ′ k {t ∈ Fq} becomes  {t ∈ Fq | tk = 0}, if in Case 2, first sum,  ′ k ∗  {t ∈ Fq | tk ∈ Fq}, if in Case 2, second sum,   f (+0,τ), if in Case 1, τ (−0,τ) f becomes  f , if in Case 2, first sum,  (1,τ)  f , if in Case 2, second sum.  +  v( 0,τ), if in Case 1, τ (−0,τ) v becomes  v , if in Case 2, first sum,  (1,τ)  v , if in Case 2, second sum.  Thus,  Ξ(+0,τ)(u, v), if Case 1, Ξτ(u, v) = ( Ξ(−0,τ)(u, v) + Ξ(1,τ)(u, v), if Case 2, as desired. 37 Chapter 4

A basis with multiplication in the G = GLn(Fq) case

4.1 Unipotent Hecke algebras

4.1.1 The group GLn(Fq)

Let G = GLn(Fq) be the general linear group over the finite field Fq with q elements. Define the subgroups

diagonal monomial T = , N = , ( matrices ) ( matrices ) (4.1) permutation 1 ∗ W = , and U = .. , matrices  .  ( )  0 1      where a monomial matrix is a matrix with exactly one nonzero entry in each row and column (N = WT). If necessary, specify the size of the matrices by a subscript such as Gn, Un, Wn, etc. If a ∈ Gm and b ∈ Gn are matrices, then let a ⊕ b ∈ Gm+n be the block diagonal matrix a 0 a ⊕ b = . 0 b !

Let xi j(t) ∈ U be the matrix with t in position (i, j), ones on the diagonal and zeroes elsewhere; write xi(t) = xi,i+1(t). Let hεi (t) ∈ T denote the diagonal matrix with t in the ith slot and ones elsewhere, and let si ∈ W ⊆ N be the identity matrix with the ith and (i + 1)st columns interchanged. That is,

1 t xi(t) = Idi−1 ⊕ 0 1 ⊕ Idn−i−1, hεi (t) = Idi−1 ⊕ (t) ⊕ Idn−i, 0 1 (4.2) si = Idi−1⊕ 1 0 ⊕ Idn−i−1, where Idk is the k × k identity matrix. Then

∗ W = hs1,s2,... sn−1i, T = hhε (t) | 1 ≤ i ≤ n, t ∈ F i, N = WT, i q (4.3) U = hxi j(t) | 1 ≤ i < j ≤ n, t ∈ Fqi, G = hU, W, Ti. 38

The Chevalley group relations for G are (see also Section 3.3.1)

xi j(a)xrs(b) = xi j(b)xrs(a)xis(δ jrab)xr j(−δisab), (i, j) , (r, s), (U1)

xi j(a)xi j(b) = xi j(a + b), (U2) 2 si = 1, (N1)

si si+1 si = si+1 si si+1 and si s j = s j si, |i − j| > 1, (N2)

hεi (b)hεi (a) = hεi (ab), (N4)

hε j (b)hεi (a) = hε j (b)hεi (a), (N5)

sr xi j(t) = xsr (i)sr( j)(t)sr, (UN1) −δri δrj xi j(a)hεr (t) = hεr (t)xi j(t t a), (UN2) −1 −1 , si xi(t)si = xi(t )si xi(−t)hεi (t)hεi+1 (−t ), t 0, (UN3)

where δi j is the Kronecker delta.

4.1.2 A pictorial version of GLn(Fq) For the results that follow, it will be useful to view these elements of CG as braid-like diagrams instead of matrices. Consider the following depictions of elements by diagrams with vertices, strands between the vertices, and various objects that slide around on the strands. View

ith vertex? ?? • • •?? • • • ?? si as ··· ?? ··· , (4.4) ?? •••••• ? ith vertex 99 • • 9• • • ··· t ··· hεi (t) as 1 1 1 1 , (4.5) ••••• ith vertex jth vertex ?? • • ?• • • • • • −→ ◦ ··· a ··· b ··· xi j(ab) as ◦ ←− , (4.6) •••••••• where each diagram has two rows of n vertices. Multiplication in G corresponds to the concatenation of two diagrams; for example, s2 s1 is

•?? • • • • • s1 ?? ?? • • • •OOO • • s s = = = OO . 2 1 OOO • •? • ••• O ?? s2 ?? ••• ? ••• 39

In the following Chevalley relations, curved strands indicate longer strands, so for ex- ◦ −→a b ample (UN1) indicates that ◦ and ←− slide along the strands they are on (no matter how long). The Chevalley relations are • • • • • • −→ ◦ −→ ◦ a 1 • • • • 1 ◦ ←− ◦ b −→ ◦ ←− 1 a ◦ ◦ ◦ −→a ←− −→ ◦ = ◦ b (U1) = 1 a+b (U2) −→a ←− −→ ◦ ◦ ←−− ◦ 1 1 b ←− −→ ◦ ◦ ←− 1 b ••• •••◦ ←− •• •• • • • • = (N1) •• •• • • • • • • • • • • • • • • = and = (N2) ••• ••• •••• •••• • • • • • • a a b = ab (N4) = (N5) a b b • • •• •• • • r s • • •−→ • ◦ • • a b ◦ ←− −−−−→ ◦ −→ ◦ ar−1 bs = (UN1) a b = ◦ ←− (UN2) −→ ◦ ◦ ←− a b ◦ ←− •• •• •• •r •s −1 ab• −(ab•) −→ ◦ • • b −a ◦ ←− −→ ◦ a b , ◦ ←− = ab 0. (UN3) −−−→ ◦ −1 b a−1 •• ◦ ←−−− • •

4.1.3 The unipotent Hecke algebra Hµ + ∗ Fix a nontrivial group homomorphism ψ : Fq → C , and a map

µ :{(i, j) | 1 ≤ i < j ≤ n} −→ Fq with µi j = 0 for j , i + 1. (4.7) (i, j) 7→ µi j Then ψ : U −→ C∗ µ (4.8) xi j(t) 7→ ψ(µi jt)

is a group homomorphism. Since µi j = 0 for all j , i + 1, write

µ = (µ(1), µ(2),...,µ(n−1), µ(n)), where µ(i) = µi,i+1 and µ(n) = 0. (4.7) 40

The unipotent Hecke algebra Hµ of the triple (G, U, ψµ) is 1 H = End IndG (ψ ) e CGe , where e = ψ (u−1)u. (4.9) µ G U µ µ µ µ |U| µ Xu∈U ∗ The type of a linear character ψµ : U → C is the composition ν such that

µ(i) = 0 ifandonlyif i ∈ ν≤ (with ν≤ as in Section 2.3.2). Proposition 4.1. Let ν be a composition. Suppose χ and χ′ are two type ν linear characters of U. Then H(G, U, χ) H(G, U, χ′).

Proof. Note that for h = diag(h1, h2,..., hn) ∈ T,

−1 −1 hxi j(t)h = xi j(hih j t) for1 ≤ i < j ≤ n, t ∈ Fq.

Thus, T normalizes Ui j = hxi j(t) | t ∈ Fqi and acts on linear characters of U by hχ(u) = χ(huh−1), for h ∈ T, u ∈ U. This action preserves the type of χ. The map

G ∼ G ′ Ind (χ) CGe −→ CGe ′ Ind (χ ) U χ χ U where χ′ = hχ for h ∈ T, geχ 7→ gheχ′ is a G-module isomorphism, so H(G, U, χ) H(G, U, hχ). It therefore suﬃces to prove that T acts transitively on the linear characters of type µ. By Proposition 3.2 (c), the number of T-orbits of type µ linear characters is |Z |(q − 1)n−ℓ(µ) |{h ∈ T | hx (t)h−1 = x (t), µ , 0}|(q − 1)n−ℓ(µ) µ = i i (i) |T| (q − 1)n (q − 1)ℓ(µ)(q − 1)n−ℓ(µ) = (q − 1)n = 1. Therefore T acts transitively on the type µ linear characters of U.

+ ∗ Proposition 4.1 implies that given any ﬁxed character ψ : Fq → C , it suﬃces to specify the type of the linear character ψµ. Note that the map (given by example) Compositions µ = (µ , µ ,...,µ ) ←→ (1) (2) (n) ( µ = (µ1, µ2,...,µℓ) |= n ) ( µ(i) ∈{0, 1} and µ(n) = 0 ) 1 0 (4.10) µ = 1 1 1 1 0 ↔ (1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0) 1 1 0 1 1 1 0 41

is a bijection. In the following sections identify the compositions µ = (µ1,...,µℓ) with the map µ = (µ(1),...,µ(n)) using this bijection. The classical examples of unipotent Hecke algebras are the Yokonuma algebra H(1n) [Yok69b] and the Gelfand-Graev Hecke algebra H(n) [Ste67].

4.2 An indexing for the standard basis of Hµ

The group G has a double-coset decomposition G = v∈N UvU, so if F Nµ = {v ∈ N | eµveµ , 0} −1 −1 (4.11) = {v ∈ N | u, vuv ∈ U implies ψµ(u) = ψµ(vuv )}

then the isomorphism in (4.9) implies that the set {eµveµ | v ∈ Nµ} is a basis for Hµ [CR81, Prop. 11.30]. Suppose a ∈ Mℓ(Fq[X]) is an ℓ × ℓ matrix with polynomial entries. Let d(ai j) be the degree of the polynomial ai j. Deﬁne the degree row sums and the degree column sums of a to be the compositions

→ → → → ↓ ↓ ↓ ↓ d (a) = (d (a)1, d (a)2,..., d (a)ℓ) and d (a) = (d (a)1, d (a)2,..., d (a)ℓ),

where ℓ ℓ → ↓ d (a)i = d(ai j) and d (a) j = d(ai j). Xj=1 Xi=1 Let

→ ↓ Mµ = {a ∈ Mℓ(µ)(Fq[X]) | d (a) = d (a) = µ, ai j monic, ai j(0) , 0}. (4.12)

For example,

X + 11 1 X + 2 (1+0+0+1=2) 3  X + 3 X + 2X + 3 1 X + 2  (1+3+0+1=5)  1 X2 + 4X + 2 X + 2 1  (0+2+1+0=3) ∈ M . (∗)   (2,5,3,4)  1 1 X2 + 3X + 1 X2 + 2  (0+0+2+2=4)    (1+1+0+0=2) (0+3+2+0=5) (0+0+1+2=3) (1+1+0+2=4) 

i1 i2 ir n Suppose ( f ) = (a0 + a1X + a2X + ··· + arX + X ) ∈ M(n) isa1 × 1 matrix with a0, a1,..., ar , 0. Let

( f ) •

v( f ) = = w(n)(w(i1)a0 ⊕ w(i2−i1)a1 ⊕···⊕ w(n−ir )ar) ∈ N, (4.13) • 42 where •T •M •2 ··· • • TTTTMM 2 qqjjjj TTMTMTM22 qjqjqjj w(k) = jqjTqM2jTMT ∈ Wk, jjjqjq 2 MTMTTT ••jjjjqqq··· •••2 MM TTT and by convention v(1) is the empty strand (not a strand). For example, v(a+bX3+cX4+X6) is

a a a c c • • • •b • • OOO oo ?? OOO ooo ?? oOoO ?? + 3+ 4+ 6 oo OO ? a a a c c (a bX •cX X ) • ooo • OOO• • • ??• • • • •b • • o TTTT TTTT TTTT ggg ggg TTT TTT TgTgTggg ggggg = = TTTT gTgTgTgT gTgTgTgT . ggggTgTTggggTgTT TTT gggg ggggTTTT TTTT TTTT • •YYY •TT •? • j• ee• ••••••ggg ggg T T T YYYYYTYTTT ?? jjjejeeeee YYYTYTYT??jejejeee eejejejYT?jTeYTYTYY eeeeejejj ?? TTYTYYYY ••••••eeeee jjjj ? TTT YYYYY

For each a ∈ Mµ deﬁne a matrix va ∈ N pictorially by

(aℓ1) (a21) (a11) (aℓ2) (a22) (a12) (aℓℓ) (a2ℓ) (a1ℓ) •ZZ ··· •Q • •WW ··· • • ··· • ··· g• d• ZZZZZZZQQQ WWWW mmm ggdgdgddddd ZZZQZQZZ WWmWmW ddgdggdgdd QQZQZZZZZ mmm WWWW ddddgdgdgg Q QQ ZZmZmZmZZZ WWW Wdddddddgggg va = QQmm ZZZZZdZdddddWWWgWggg , (4.14) mmmQQQ dddddd ZZZg ZgZgZg WWWW mmm dddddQdQdQ gggg ZZZZZZZWWWW dmdmdd dd QQgggg ZZ ZZWZWZWW ddddmdmdm ggggQQQ ZZWZWWZWZZ •ddd···dddd •••mmm ggg···g ••QQ ··· • ···WWZWZ••ZZZZ where ai j is the (i, j)th entry of a, and the top vertex associated with ai j goes to the bottom vertex associated with a ji. Note that if ai j = 1, then both its strand and corresponding vertices vanish. The fact that a ∈ Mµ guarantees that we are left with two rows of exactly n vertices. For example, if a isasin(∗), then va is

(3+X) (1+X) (2+4X+X2) (3+2X+X3) (1+3X+X2) (2+X) (2+X2) (2+X) (2+X) •V • •V • •T • • • • VVVV VVVV TTT ddddcdcdcdccccccc VVVV VVVV TTT cdcdcddcdcdcdcccc VVVV VVVV cTcTTcTcdcdcdcdcdcdcd VVVV VVVcV ccccccdcdcdcdddd TTT VVVV ccccccccccdddVdVdVdVd TTT ccccccccVVcVcV ddddddd VVVV TTT cccccc ccc dddddVdVVV VVVV TTTT •••••••••cccccccccc ddddddd VVV VVV TT [3]• [1]• [2• 4]• [3• •2 2]• [1• 3]• [2]• [2• 2]• [2]• [2]• TTTT VVVV RRR // TTTT OOO eeeeedddddd TTT VVVV RRR / TTT OO eeedededdddd TTT T VVVVRR/R TTTTOOO e dededededdd TTT VV/VRVRR dedeTdeTdeTdOedO = TTT / VVRVR ddededededede TT OTO TTT // VRVRVRV ddddedededee TOTOTO . TTTT / ddddddReVdReVReVeVe TOTOTT ddTdTdTddd/eeeeee RRVRVVV OOTTT ddddddd eeeTeTeT/T RRRVVVV OOOTTTT ddddddd eeeee / TTT RRR VVVV OO TTT ••••••••••••••ddddddd eeeeee / TTT RR VVV OO TTT 43

Viewed as a matrix,

µ 1 columns µ 2 columns µ 3 columns ··· ··· z }| { z }| { z }|0 { _ _ _ _ _ v µ  0 (a13) 0  1  0 ______ rows  v(a )   12 0 0 0   ______  v(a )   11 0 0 0 00     0 ···   0 0 ______  v  µ  (a )  2  0 0 23 0 0 

rows  0 ______  v  v =  (a22) 0 0 00 0  a    ______  v(a ) 0 0 00 0 00 0   21     0 0 0 ···   0 0 0 ______  v  µ  (a )  3  0 0 33 0 0 0    rows  0 ______  v(a )   32 0 0 0000 0   ______  v(a )   31 0 0 00000000 0       . . .   . 0 0 0 . 0 0 0 . 0 0 0 .. .  . . . . .       Theorem 4.2. Let Nµ be as in (4.11) and Mµ be as in (4.12). The map

Mµ −→ Nµ a 7→ va, given by (4.14) is a bijection.

Remark. When µ = (n) this theorem says that the map ( f ) 7→ v( f ) of (4.13) is a bijection between M(n) and N(n). Proof. Using the remark following the theorem, it is straightforward to reconstruct a from va. Therefore the map is invertible, and it suﬃces to show

(a) the map is well-deﬁned (va ∈ Nµ), (b) the map is surjective.

To show (a) and (b), we investigate the diagrams of elements in Nµ. Suppose v ∈ Nµ. Let

vi be the entry above the ith vertex of v, v(i) be the number of the bottom vertex connected to the ith top vertex, 44

so that {v1, v2,..., vn} are the labels above the vertices of v and (v(1), v(2),..., v(n)) is the permutation determined the diagram (and ignoring the labels vi). By (4.8),

ψ(t), if j = i + 1 and µ(i) = 1, ψµ(xi j(t)) = (A) ( 1, otherwise.

−1 −1 Recall that v ∈ Nµ if and only if u, vuv ∈ U implies ψµ(u) = ψµ(vuv ). That is, v ∈ Nµ if and only if for all 1 ≤ i < j ≤ n such that v(i) < v( j),

−1 ψµ(xi j(t)) = ψµ(vxi j(t)v ) −1 = ψµ(xv(i)v( j)(vitv j )) ψ(v tv−1), if v( j) = v(i) + 1 and µ = 1, = i j (v(i)) (B) ( 1, otherwise.

Compare (A) and (B) to obtain that v ∈ Nµ if and only if for all 1 ≤ i < j ≤ n such that v(i) < v( j),

(i) If µ(i) = 1 and µ(v(i)) = 0, then j , i + 1,

(ii) If µ(i) = 0 and µ(v(i)) = 1, then v( j) , v(i) + 1,

(iii) If µ(i) = µ(v(i)) = 1, then j = i + 1 if and only if v( j) = v(i) + 1,

′ (iii) If µ(i) = µ(v(i)) = 1 and v( j) = v(i) + 1, then vi = vi+1. We can visualize the implications of the conditions (i)−(iii)′ in the following way. Par- tition the vertices of v ∈ Nµ by µ. For example, µ = (2, 3, 1) partitions v according to v1 v2 v3 v4 v5 v6 • • • • • • . •••••• Suppose the ith top vertex is not next to a dotted line and the v(i)th bottom vertex is immediately to the left of a dotted line. Then condition (i) implies that v(i + 1) < v(i), so

vi vi+1 • • (I) • Similarly, condition (ii) implies

vi • ? (II) • • 45

and conditions (iii) and (iii)′ imply v v v v •i •i+1 •i •i ? or (vi+1 = vi in the second). (III) • • ••

In the case µ = (n) condition (III) implies that every v ∈ N(n) is of the form

a1 a1 a1 a2 a2 a2 ar ar ar •T •T··· •T •, •, ··· •, ··· • • ··· • TTTTTTTTTTTT, , , jjjjjjjjjjjj TTTTTTTT,,TTTT,, jj,,jjjjjjjjjj TTTT,TTjTjTj,TjTjTjTj,jjjjj = ⊕ ⊕···⊕ , jj,jTTjTj,jTTjTj,TjTT w(n)(a1w(i1) a2w(i2) arw(ir )) jjj jj,,j jTjT,j,T TT,,T TTT jjjjjjjjjjj,j , TTT,TTTTTTTTT ••jjjjjj···jj•jjj···j ••, , ··· •••, TTT TTT T···TT• ∗ where (i1, i2,..., ir) |= n, ai ∈ Fq, and w(k) ∈ Wk is as in (4.13). In fact, this observation proves that the map ( f ) 7→ v( f ) is a bijection between M(n) and N(n) (mentioned in the remark). Note that since the diagrams va satisfy (I), (II) and (III), va ∈ Nµ, proving (a). On the other hand, (I), (II) and (III) imply that each v ∈ Nµ must be of the form v = va for some a ∈ Mµ, proving (b). Remark: This bijection will prove useful in developing a generalization of the RSK correspondence in Chapter 5. Let mµ = {a ∈ Mℓ(Z≥0) | row-sums and column-sums are µ}

and for a ∈ mµ, let ℓ(a) = Card{ai j , 0 | 1 ≤ i, j ≤ ℓ} Corollary 4.3. Let µ |= n. Then

ℓ(a) n−ℓ(a) dim(Hµ) = (q − 1) q

aX∈mµ Proof. Theorem 4.2 gives dim(Hµ) = |Mµ|.

We can obtain a matrixa ¯ ∈ Mµ from a matrix a ∈ mµ by selecting for each 1 ≤ i, j ≤ ℓ(µ), a monic polynomial fi j with nonzero constant term and degree ai j. Conversely, everya ¯ ∈ Mµ arises uniquely out of such a construction. For a ﬁxed a ∈ mµ, the total number of ways to choose appropriate polynomials is

ℓ(a) n−ℓ(a) Card{ f ∈ Fq[X] | f monic, f (0) = 0 and d( f ) = ai j} = (q − 1) q . 1Y≤i, j≤ℓ aij,0

The result follows by summing over all a ∈ mµ. 46

4.3 Multiplication in Hµ

This section examines the implications of the unipotent Hecke algebra multiplication relations from Chapter 3 in the case G = GLn(Fq). The ﬁnal goal is the algorithm given in Theorem 4.5.

4.3.1 Pictorial versions of eµveµ Note that the map

π : N = WT−→ W (4.15) wh 7→ w, for w ∈ W, h ∈ T, is a surjective group homomorphism. Since W ⊆ N, adjust the decomposition of (3.4) as follows. Let u ∈ N with π(u) = si1 ··· sir for r minimal. Then u has a unique decomposition

u = u1u2 ... uruT , where uk = sik , uT ∈ T. (4.16)

For t ∈ Fq, write uk(t) = sik xik (t). Fix a composition µ. The decomposition

U = Ui j where Ui j = hxi j(t) | t ∈ Fqi, 1≤Yi< j≤n implies 1 e = e (1), where e (k) = ψ(−µ kt)x (t). (4.17) µ i j i j q i j i j 1≤Yi< j≤n Xt∈Fq Pictorially, let

ikth vertex ?? • • • ? • • • ?? ? uk as ··· k? ··· (4.18) ?? •••••• ? i th vertex k ? • • ?• • • • ??? ? uk(tk) as ··· k ··· (4.19) ??? •••••• ? ith vertex jth vertex ?? • • ?• ··· • • • /o /o /o /o /o /o ei j(k) as ··· • k • ··· , (4.20) ••• ··· ••• • • • • • µ µ µ eµ as • (1) • (2) • ··· • (n−1) •. (4.21) ••••• 47

Examples: 1. If u = u1u2 ··· u8uT ∈ N decomposes according to s3 s1 s2 s3 s1 s4 s2 s3 ∈ W with uT = diag(a, b, c, d, e), then

a b c d e a b c d e • •?? •AA Ð• • • •?? •AA Ð• • ?? A ÐÐ ? AA Ð ? 8 ?? 8 Ð ?? LL ?? LL ? Ñ LLL ? ÑÑ LLL 7 Ñ 6 7 Ñ 6 pp == rr pp == rr u = ppp rrr , u (t )u (t ) ··· u (t )u = ppp = rrr 5 4 1 1 2 2 8 8 T 5 = 4 == ppp == ppp 3 pp 3 pp pp NNN pp NNN ppp NN ppp NN 2 1 2 A 1 Ð AA } >> Ð AA } >> •••••ÐÐ A }} > •••••ÐÐ A }}} > and • µ(1) • µ(2) • µ(3) • µ(4) • a b c d e • •? •A • • ?? AA ÐÐ ?? 8 Ð ?? LL ? Ñ LLL 7 Ñ 6 pp == rr eµueµ = ppp rrr . 5 = p 4 = ppp 3 Np ppp NNN 2 pp N 1 Ð AA } >> •ÐÐµ(1) A• µ(2) •}}µ(3) >• µ(4) • 2. If n = 5, then (4.17) implies

• /o1 /o • • • • • /o /o /o 1 /o /o /o • • /o /o/o /o/o 1 /o /o/o /o/o • • /o /o/o /o/o/o /o 1 /o /o/o /o/o/o /o • • /o1 /o • • µ(1) • µ(2) • µ(3) • µ(4) • = • /o /o /o 1 /o /o /o • • /o /o/o /o/o 1 /o /o/o /o/o • • /o1 /o • • /o /o /o 1 /o /o /o • •••• /o1 /o •

(eµ = e45(1)e35(1)e34(1)e25(1)e24(1)e23(1)e15(1)e14(1)e13(1)e12(1)). The elements ei j(k) also interact with U and N as follows (see also Section 3.3.1)

srei j(k)sr = esr(i)sr( j)(k), (E1)

eµvei j(1) = eµv, v ∈ Nµ, (πv)(i) < (πv)( j), (E2)

δri −δrj ei j(k)hεr (t) = hεr (t)ei j(t t k), (E3)

eµ xi j(t) = ψ(µi jt)eµ = xi j(t)eµ, (E4) or pictorially, • • • • r• s• • • • /o /o k /o /o • • /o−1 /o • = (E1) • /o /o k /o /o • = krs (E3) • /o /o k /o /o • •• •• •• •r •s 48

• • • • • k • • k •

−→ ◦ −→ ◦ a b = ψ(kab) and a b = ψ(kab) (E4) ◦ ←− ◦ ←− • k • • k • •• ••

and for v ∈ Nµ, • • ··· • • • • ··· • • ··· • /o /o /o 1 /o /o /o • ··· ··· ··· • ••• • ••• v = v (E2)

• ··· • µ(i) ··· µ( j−1) • ··· • • ··· • µ(i) ··· µ( j−1) • ··· •

Suppose u = u1u2 ··· uruT ∈ Nµ with uT = diag(h1, h2,..., hn). Then using (4.17), (E3), (E1) and (E2), we can rewrite eµueµ as

µ(1) µ(2) µ(3) ···µ(n−1) • • • • h h h ··· h •1 •2 •3 •n h1 h2 h3 ··· hn • •• • 1 = ◦ r (ψ fu)(t) u1(t1)u2(t2) ··· ur(tr) (4.22) ··· q r u1u2 ur Xt∈Fq • µ(1) • µ(2) • µ(3) ··· µ(n−1) • • µ(1) • µ(2) • µ(3) ··· µ(n−1) • where fu ∈ Fq[y1, y2,..., yr] is given by

, ,..., = −µ −1 − µ −1 −···− µ −1 , fu(y1 y2 yr) i1 j1 hi1 h j1 y1 i2 j2 hi2 h j2 y2 ir jr hir h jr yr (4.23)

for (ik, jk) = (a, b), if the kth crossing in u crosses the strands coming from the ath and bth top vertices. Example (continued). Suppose u = u1u2 ··· u8uT ∈ N decomposes according to s3 s1 s2 s3 s1 s4 s2 s3 ∈ W and uT = diag(a, b, c, d, e) ∈ T (as in Example 1 above). Consider the ordering • • • /o1 /o • • • /o /o /o 1 /o /o /o • • /o /o /o 1 /o /o /o • •/o /o/o /o/o /o 1 /o /o/o /o/o /o• • /o /o/o /o/o 1 /o /o/o /o/o • e = , µ •/o /o/o /o/o/o /o/o 1 /o /o/o /o/o/o /o/o • • /o1 /o • • /o1 /o • • /o1 /o • • /o /o/o /o 1 /o /o/o /o ••• 49

or eµ = e13(1)e23(1)e12(1)e45(1)e15(1)e25(1)e14(1)e35(1)e24(1)e34(1). Then

e ue = e ue (1)e (1)e (1)e (1)e (1)e (1)e (1)e (1)e (1)e (1) µ µ µ ←−−−−−−−−−−13 23 12 45 15 25 14 35 24 34 = eµue12(1)e45(1)e15(1)e25(1)e14(1)e35(1)e24(1)e34(1) (by (E2)) = e u u ... u u e (1)e (1)e (1)e (1)e (1)e (1)e (1)e (1) µ 1 2 8 T ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−12 45 15 25 14 35 24 34 a d a b a c b c = e u u ... u e ( )e ( )e ( )e ( )e ( )e ( )e ( )e ( )u (by (E3)) µ 1 2 8 12 b 45 e 15 e 25 e 14 d 35 e 24 d 34 d T ←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− a d a b a c b c = e u e ( )u e ( )u e ( )u e ( )u e ( )u e ( )u e ( )u e ( )u (by (E1)) µ 1 34 b 2 12 e 3 23 e 4 34 e 5 12 d 6 45 e 7 23 d 8 34 d T e a d a c = µ ◦ ··· 8 (ψ f )(t)u1 x34( t1)u2 x12( t2)u3 x23( t3) u8 x34( t8)uT , q 8 b e e d Xt∈Fq where by (4.17) , f = −y1 − y2 − y8. Therefore, by renormalizing e = µ ◦ ′ ··· eµueµ 8 (ψ fu)(t )u1 x34(t1)u2 x12(t2)u3 x23(t3) u8 x34(t8)uT q ′ 8 tX∈Fq e = µ ◦ ′ ··· 8 (ψ fu)(t )u1(t1)u2(t2)u3(t3) u8(t8)uT , q ′ 8 tX∈Fq

−1 −1 −1 where fu = −ba y1 − ed y2 − dc y8. Pictorially, by sliding the ei j(1) down along the strands until they get stuck, this computation gives

• µ(1) • µ(2) • µ(3) • µ(4) • a b c d e a b c d e • •? •A • • • •?? •AA Ð• • ?? AA Ð ? AA Ð ? ÐÐ ?? 8 Ð ?? 8 L ?? LL ?? LL ? ÑÑ LLL ÑÑ LL 7 Ñ 6 7 6 1 p == r pp == rr ppp = rrr eµueµ = ppp rrr = (ψ ◦ fu)(t) 5 p 4 r (4.24) 5 4 q8 == pp == pp Xt∈F8 = pp ppp q 3 p 3 N pp NNN pp NNN pp NN ppp N 2 p 1 2 A 1 > Ð AAA } >> Ð A } > Ðµ(1) A µ(2) }}µ(3) > µ(4) •ÐÐµ(1) A• µ(2) •}}µ(3) >• µ(4) • •Ð • •} • •

−1 −1 −1 where fu = −ba y1 −ed y2 −dc y8 (as in (4.23)), since (i1, j1) = (1, 2), (i2, j2) = (4, 5), (i3, j3) = (1, 5), etc.

4.3.2 Relations for multiplying basis elements.

Let u = u1u2 ··· uruT ∈ Nµ decompose according to a minimal expression in W as in (4.16). Let v ∈ Nµ and use (N3) and (N4) to write uT v = w·diag(a1, a2, ··· , an) for some 50

w = π(v) ∈ W (see (4.7)). Then use (4.22) to write

• µ(1) • µ(2) • µ(3) ···µ(n−1) • a•1 a•••2 a3 an w 1 = ◦ ______(eµueµ)(eµveµ) r (ψ fu)(t) (4.25) q r Xt∈Fq u1(t1)u2(t2) ··· ur(tr)

• µ(1) • µ(2) • µ(3) ···µ(n−1) •

∅ (This form corresponds to Ξ (u, uT v) of Corollary 3.9).

Example (continued). If u is as in (4.24) and v = s2 s3 s2 s1 s2 · diag( f, g, h, i, j) ∈ N, then

• µ(1) • µ(2) • µ(3) • µ(4) • f d gc ah bi e j •UUU •BB n• n• • UUUU Bnnn nnn nnUnUBBUU nnn nnn nnBBnUUUU •nnn • UUU• • < >> Ó << > ÓÓ 1 < 8 Ó = ◦ << Ô JJJ (eµueµveµ) 8 (ψ fu)(t) < Ô JJ . q 7 Ô 6 X∈ 8 r : t Fq rr :: ttt 5 rr 4 tt :: rr : rr 3 rL rr LLL rrr L 2 > 1 ; ÓÓ >> ÐÐ ;; •ÓÓ µ(1) >• µ(2) •ÐÐµ(3) ;• µ(4) •

Consider the crossing in (4.25) corresponding to ur(tr). There are two possibilities. Case 1 the strands that cross at r do not cross again as they go up to the top of the diagram (ℓ(urw) >ℓ(w)), Case 2 the strands that cross at r cross once on the way up to the top of the diagram (ℓ(urw) <ℓ(w)). In the ﬁrst case,

• ··· • µ(i) ··· µ( j−1) • ··· • a a•1 a•i •j a•n w 1 ______= ψ ◦ , eµueµveµ r ( fu)(t) q r r Xt∈Fq ______u1(t1) ··· ur−1(tr−1)

• µ(1) • µ(2) • µ(3) ···µ(n−1) • 51

so (UN1), (UN2) and (E4) imply

• µ(1) • µ(2) • µ(3) ···µ(n−1) • a•1 a•••2 a3 an 1 urw e ue ve = (ψ ◦ f (+0))(t) (4.26) µ µ µ r ______q r Xt∈Fq u1(t1) ··· ur−1(tr−1)

• µ(1) • µ(2) • µ(3) ···µ(n−1) •

(+0) −1 where f = fu + µi ja jai yr. In the second case,

• ··· • µ(i) ··· µ( j−1) • ··· • a a•1 a•i •j a•n w 1 ______= ◦ eµueµveµ r (ψ fu)(t) . q r r Xt∈Fq ______u1(t1) ··· ur−1(tr−1)

• µ(1) • µ(2) • µ(3) ···µ(n−1) •

Use (UN3) and (N1) to split the sum into two parts corresponding to tr = 0 and tr , 0,

• ··· • µ(i) ··· µ( j−1) • ··· • a a•1 a•i •j a•n

urw 1 = ψ ◦ (−0) ______eµueµveµ r ( f )(t) q r Xt∈Fq tr=0 ______u1(t1) ··· ur−1(tr−1)

• µ(1) • µ(2) • µ(3) ···µ(n−1) • • ··· • µ(i) ··· µ( j−1) • ··· • a a t −1 a •1 •i r −a•jtr •n w

1 ______−→_ _ _◦ _ _ −1 tr + ◦ ◦ ←− r (ψ fu)(t) q r ◦ t∈Fr −→ Xq 1 t−1 ∗ ◦ r tr∈Fq ______←−−_ _ u1(t1) ··· ur−1(tr−1)

• µ(1) • µ(2) • µ(3) ···µ(n−1) • 52

(−0) where f = fu. Now use (UN1), (UN2), (U2), (U1) and (E4) on the second sum to get

• µ(1) • µ(2) • µ(3) ···µ(n−1) • a•1 a•••2 a3 an u w 1 r = ψ ◦ (−0) eµueµveµ r ( f )(t) ______q r Xt∈Fq tr=0 u1(t1) ··· ur−1(tr−1) • µ(1) • µ(2) • µ(3) ···µ(n−1) • (4.27) • ··· • µ(i) ··· µ( j−1) • ··· • −1 a1 a t −a t an • •••i r j r w 1 + ψ ◦ (1) ______r ( f )(t) q r Xt∈Fq u1(t1) ··· ur−1(tr−1) ∗ tr∈Fq • µ(1) • µ(2) • µ(3) ···µ(n−1) •

(1) −1 −1 where f = ϕr( fu) + µi ja jai yr , and ϕr( f ) is deﬁned by

r ◦ r (ψ ◦ f )(t) −→ −1 = (ψ ◦ ϕr( f ))(t) . (∗) 1 tr r ◦ ←−− r Xt∈Fq ______Xt∈Fq ______∗ ∗ tr∈Fq tr∈Fq u1(t1) ··· ur−1(tr−1) u1(t1) ··· ur−1(tr−1)

• µ(1) • µ(2) • µ(3) ···µ(n−1) • • µ(1) • µ(2) • µ(3) ···µ(n−1) • Remarks:

(a) We could have applied these steps for any f , u, and v, so we can iterate the process with each sum.

(b) The most complex step in these computations is determining ϕr. The following section will develop an eﬃcient algorithm for computing the right-hand side of (∗).

4.3.3 Computing ϕk via painting, paths and sinks.

k Painting algorithm (u ). Suppose u = u1u2 ··· ur ∈ N decomposes according to

si1 si2 ··· sik ∈ W (assume uT = 1). Paint ﬂows down strands (by gravity). Each step is il- lustrated with the example u = u1u2 ··· u8, decomposed according to s3 s1 s2 s3 s1 s4 s2 s3 ∈ W. 53

(1) Paint the left [respectively right] strand exiting k red [blue] all the way to the bottom of the diagram. • •> •@ • • >> @@ ÑÑ >> @ ÑÑ >> 8 >> ÒÒÒÒ 7 ÒÒ 6 qqq << ss qqqq < ss 5 q 4 s where red is , blue is , and k is 8 . << qq < qqq 3 qM qqq MMM qqq MM 2 @@ 1 = ÑÑ @@@@ ~~ == •••••ÑÑ @@ ~~ = (2) For each crossing that the red [blue] strand passes through, paint the right [left] strand (if possible) red [blue] until that strand either reaches the bottom or crosses the blue [red] strand of (1). • •> •@ • • • •> •@ • • • •> •@ • • >> @@ ÑÑ >> @@ ÑÑ >> @@ ÑÑ >> @ ÑÑ >> @ ÑÑ >> @ ÑÑ >> 8 >> 8 >> 8 >> ÒÒÒÒ >> ÒÒÒÒ >> ÒÒÒÒ 7 ÒÒ 6 7 ÒÒ 6 7 ÒÒ 6 qq <<< s qq <<< s qq <<< qqqq <<< sss qqqq <<< sss qqqq <<< 5 qq 4 s 5 qq 4 s 5 qq 4 << qq <<<< qq <<<< < qqq << qqq << 3 q 3 qM 3 M q MMM q MMM MMM qqq MM qqq MMMM MMMM 2 qq M 1 2 qq M 1 2 M 1 @@@ = @@@ === @@@ === ÑÑ @@@ ~~~~ == ÑÑ @@@ ~~~~ ==== ÑÑ @@@ ~~~~ ==== •••••ÑÑ @@ ~~~~ = •••••ÑÑ @@ ~~~~ = •••••ÑÑ @@ ~~~~ = (4.28) Let

k u = u1(t1)u2(t2) ··· uk(tk) paintedaccording totheabovealgorithm (4.29)

Sinks and paths. The diagram u k has a crossed sink at j if j is a crossing between a red strand and a blue one, or · ·

j Note that since u is decomposed according to a minimal expression in W, there will be no crossings of the form ?? · · · · j′ (since would imply for some j′ ≥ j.) j j j The diagram u k has a bottom sink at j if a red strand enters jth bottom vertex and a blue strand enters the ( j + 1)st bottom vertex, or · ·

•? •?_ ? ?? jth vertex ( j + 1)st vertex 54

A red [respectively blue] path p from a sink s (either crossed or bottom) in u k is an increasing sequence j1 < j2 < ··· < jl = k, such that in u k

(a) jm is directly connected (no intervening crossings) to jm+1 by a red [blue] strand,

(b) if s is a crossed sink, then j1 = s, (b′) if s is a bottom sink, then

• in a red path, the sth bottom vertex connects to the crossing j1 with a red strand.

• in a blue path, the (s + 1)st bottom vertex connects to the crossing j1 with a blue strand. Let

k red paths from k blue paths from P=(u , s) = and P (u , s) = (4.30) ( s in u k ) : ( s in u k ) The weight of a path p is

k yi, if p ∈ P=(u , s),  p Yswitches wt(p) =  strands at i (4.31)  k  (−yi), if p ∈ P:(u , s).   p Yswitches  strands at i  k  Each sink s in u (either crossed j or bottom j) has an associated polynomial gs ∈ −1 Fq[y1, y2,..., yk−1, yk ] given by

−1 ′ gs = wt(p)yk wt(p ). (4.32) k p∈PX=(u ,s) k p′∈P:(u ,s)

Example (continued). If u = u1u2 ··· u8 decomposes according to s3 s1 s2 s3 s1 s4 s2 s3 (as in (4.28)), then

8 8  ÑÑ Ñ  8 ÑÑÑ ÑÑÑ  7 Ñ 7 ÑÑ   p ==   6   pppp ===    8  ppp =  8   P (u , 4) =  5 p , 4  P (u , 4) =   =  ====  :    ==     3 NN    NNN  NN 1     1 >>>     >>>> >>>>  •   >>>> >•    •          55

with wt(1 < 3 < 5 < 7 < 8) = y5, wt(1 < 4 < 7 < 8) = y1y7, and wt(6 < 8) = 1. The corresponding polynomial is

−1 −1 g4 = y5y8 + y1y7y8 . (4.33)

r Lemma 4.4. Let u = u1u2 ··· ur and ϕr be as in (4.27) and (∗); suppose u is painted as above. Then

ϕr( f ) = f + µ( j)g j. {y j7→y j−g | j a crossed sink} j j aX bottom sink

Proof. In the painting, • • −→a marks a strand travelled by ◦ • • and • • ◦ a marks a strand travelled by ←− . • • Substitutions due to crossed sinks correspond to the normalizations in relation (U2), and the sum over bottom sinks comes from applications of relation (E4).

8 Example (continued) Recall u = s3 s1 s2 s3 s1 s4 s2 s3. Then u has crossed sinks at 2 , 3 , and 4 . The only bottom sink is at 4. Therefore,

−1 −1 y 7→y −g ϕ ( f ) = f 4 4 4 + µ g = f y 7→y +y y−1y + µ (y y + y y y ). 8 (4) 4 4 4 7 8 6 (4) 5 8 1 7 8 y 7→y −g 3 3 3 y 7→y +y y−1y 3 3 5 8 6 y27→y2−g 2 y 7→y +y−1y 2 2 8 6

(for example, g4 was computed in (4.33)).

4.3.4 A multiplication algorithm

Theorem 4.5 (The algorithm). Let G = GLn(Fq) and u, v ∈ Nµ. An algorithm for multiplying eµueµ and eµveµ is

(1) Decompose u = u1u2 ··· uruT according to some minimal expression in W (as in (4.16)).

(2) Put eµueµveµ into the form speciﬁed by (4.25), with uT v = w · diag(a1, a2,..., an) (w = π(v) ∈ W).

(3) Complete the following

(a) If ℓ(urw) >ℓ(w), then apply relation (4.26). 56

r (b) If ℓ(urw) <ℓ(w), then apply relation (4.27), using (u1u2 ··· ur) and Lemma 4.4 to compute ϕr. (4) If r > 1, then reapply (3) to each sum with r := r − 1 and with

(a) w := urw, after using (3a) or using (3b), in the ﬁrst sum, (b) w := w, after using (3b), in the second sum.

(5) Set all diagrams not in Nµ to zero.

Sample computation. Suppose n = 3 and µ(i) = 1 for all 1 ≤ i ≤ 3 (i.e.. the Gelfand- Graev case). Then a a a a a b a b b a b c • • • •2 •2 • •D • • •D • • 2 2 zz DD DD zz N =  , 22 z2z2 , D D , DDzz | a, b, c ∈ F∗ µ 2z2z 22 DD zzDD q  zz 2 2 DD zz DD  ••• •••z ••• •••z    Suppose   a b c d e e •DD • z• •DD • • DD zz DD u = zDzD and v = DD . zz DD DD •••zz D ••• D 1. Theorem 4.5 (1): Let u = u1u2u3uT ∈ Nµ decompose according to s2 s1 s2 ∈ W, with uT = diag(a, b, c). 2. Theorem 4.5 (2): By (4.25)

• 1 • 1 • cd• ae• be• a b c d e e •J • • •J • • JJ tt JJ × × JJ tt J×J× ×× 1 (eµ tJtJ eµ)(eµ × JJ × eµ) = (ψ ◦ fu)(t) tt JJ ×× ×J×J 3 3 tt JJ × × JJ q 3 ll •••t •••× × Xt∈Fq lll 2 ll RRR RRR R 1 vv • 1 •vv 1 • = = = b c with uT v s2 s1 · diag(cd, ae, be) (so w s2 s1), and fu − a y1 − b y3 (as in (4.23)).

3 3. Theorem 4.5 (3b): Since ℓ(u3w) < ℓ(w), paint u1(t1)u2(t2)u3(t3) to get (u1u2u3) (as in (4.29)),

• 1 • 1 • cd• ae• be• 1 = (ψ ◦ fu)(t) q3 3 3 lll Xt∈Fq llll 2 lll RRR RRRR RR 1 • 1 • 1 • 57

Now apply (4.27),

• 1 • 1 • • 1 • 1 • ae cdt ae −bet−1 cd• • be• •3 • •3 1 1 = ◦ (−0) + ◦ (1) 3 (ψ f )(t) 3 (ψ f )(t) q 3 q 3 Xt∈Fq 2 R Xt∈Fq 2 R RRR ∗ RRR t3=0 RR t3∈Fq RR R 1 R 1 vv vv • 1 •vv 1 • • 1 •vv 1 •

(−0) b c where f = − a y1 − b y3 and by Lemma 4.4, be b b c f (1) = ϕ ( f ) + µ y−1 = − y + y y−1 − y + y−1. 3 u 13 cd 3 a 1 a 2 3 b 3 3

4. Theorem 4.5 (4): Set r := 2 with w := urw = s1 in the ﬁrst sum and w := w in the second sum.

5. Theorem 4.5 (3a) (3b): In the ﬁrst sum, ℓ(u2 s1) < ℓ(s1), so paint u1(t1)u2(t2) to get 2 (u1u2) . In the second sum, ℓ(u2 s2 s1) >ℓ(s2 s1), so apply (4.26),

• 1 • 1 • • 1 • 1 • ae cdt ae −bet−1 cd• • be• •3 • •3 1 1 = ◦ (−0) + ◦ (+0,1) 3 (ψ f )(t) 3 (ψ f )(t) q 3 q 3 Xt∈Fq Xt∈Fq 2 1 t =0 t ∈F∗ C 3 3 q vv CCC 1 • 1 •vv 1 • • 1 • 1 •

(+0,1) = b + b −1 c + −1 b −1 = b c + −1 where f − a y1 a y2y3 − b y3 y3 − µ(3) a y2y3 − a y1 − b y3 y3 . Now apply (4.27) to the ﬁrst sum,

• 1 • 1 • • 1 •−1 1 • cd ae be cdt2 −aet2 be • • • • •Ù • ÙÙ 1 (−0,−0) 1 (1,−0) Ù = (ψ ◦ f )(t) + (ψ ◦ f )(t) ÙÙ q3 q3 ÙÙ 3 1 3 Ù 1 Xt∈Fq Xt∈Fq Ù t =t =0 vv t ∈F∗,t =0 Ù vv 2 3 • 1 •vv 1 • 2 q 3 •ÙÙ 1 •vv 1 • • 1 • 1 • cdt ae −bet−1 •3 • •3 1 + ◦ (+0,1) 3 (ψ f )(t) q 3 Xt∈Fq 1 C t ∈F∗ vv CC 3 q • 1 •vv 1 C•

(−0,−0) b c (1,−0) (−0) ae −1 b −1 ae −1 where f = − a y1 − b y3 and f = ϕ( f ) + µ(1) cd y2 = − a y1 − y2 y1 + cd y2 .

6. Theorem 4.5 (4): Set r = 1 with w := u2 s1 = 1 in the ﬁrst sum, w := s1 in the second sum, and w := s1 s2 s1 in the third sum. 58

7. Theorem 4.5 (3a) (3a) (3b): In the ﬁrst sum ℓ(s21) > ℓ(1), so apply (4.26); in the second sum ℓ(s2 s1) >ℓ(s1), so apply (4.26); in the third sum, ℓ(s2 s1 s2 s1) <ℓ(s1 s2 s1), so 1 paint u1(t1) to get u1 ,

• 1 • 1 • • 1 •−1 1 • cd ae be cdt2 −aet2 be • •99 Õ• •LL Ò• Õ• 1 + − − 9 Õ 1 + − LL Ò Õ = (ψ ◦ f ( 0, 0, 0))(t) 99 ÕÕ + (ψ ◦ f ( 0,1, 0))(t) LÒLÒL ÕÕ 3 ÕÕ99 3 ÒÒ LLL ÕÕ q 3 Õ 9 q 3 Ò LÕL Xt∈Fq ÕÕ 99 Xt∈Fq ÒÒ ÕÕ LL • 1 •Õ 1 • ∗ •Ò 1 •Õ 1 L• t2=t3=0 t2∈Fq,t3=0 • 1 • 1 • cdt ae −bet−1 •3 • •3 1 + ◦ (+0,1) 3 (ψ f )(t) q 3 Xt∈Fq ∗ 1 t3∈Fq vvvv • 1 •vvvv 1 •

(+0,−0,−0) = b c + b = c (+0,1,−0) = b + ae −1 −1 where f − a y1 − b y3 a y1 − b y3 and f − a y1 cd y2 − y2 y1. Now apply (4.27) to the third sum

• 1 • 1 • • 1 • 1 • ae cdt −aet−1 cd• • be• •2 •2 be• 99 Õ LLL Ò Õ 1 (+0,−0,−0) 9 Õ 1 (+0,1,−0) LL Ò Õ = (ψ ◦ f )(t) 99 ÕÕ + (ψ ◦ f )(t) ÒLÒL ÕÕ 3 ÕÕ99 3 ÒÒ LLL ÕÕ q 3 Õ 9 q 3 Ò LÕL Xt∈Fq ÕÕ 99 Xt∈Fq ÒÒ ÕÕ LL • 1 •Õ 1 • ∗ •Ò 1 •Õ 1 L• t2=t3=0 t2∈Fq,t3=0 • 1 • 1 • • 1 • 1 • cdt ae −bet−1 cdt t −aet−1 −bet−1 •<3 •9 •3 •L1 3 •1 •3 1 << 99 1 LLL + (ψ ◦ f (−0,+0,1))(t) << 99 + (ψ ◦ f (1,+0,1))(t) LLL 3 << 99 3 LLL q 3 < 9 q 3 LL Xt∈Fq < 99 Xt∈Fq LL ∗ • 1 <• 1 • ∗ • 1 • 1 L• t1=0,t3∈Fq t1,t3∈Fq

(−0,+0,1) c −1 where f = − b y3 + y3 and ae b c ae f (1,+0,1) = ϕ ( f (+0,1)) + µ y−1y−1 = − y − y + y−1 + y−1 + y−1y−1. 1 (1) cd 1 3 a 1 b 3 3 1 cd 1 3

8. Theorem 4.5 (5): The ﬁrst sum contains no elements of Nµ, so set it to zero. The = −1 = a second sum contains elements of Nµ when be −aet2 , so set t2 − b . The third sum ae contains elements of Nµ when cdt3 = ae, so set t3 = cd . All the terms in the fourth sum 59 are basis elements.

• 1 • 1 • −ab−1cd be be •J • • JJ Õ Ø 1 JJ Õ Ø (+0,1,−0) JÕÕJ ØØ = 0 + (ψ ◦ f )(t) ÕÕ JJ Ø q3 Õ JØJØ ∈F3 Õ Ø JJ Xt q ÕÕ Ø JJ t =− a ,t =0 •Õ 1 •Ø 1 • 2 b 3 • 1 • 1 • • 1 • 1 • − −1 −1 ae ae −a 1bcd cdt1t3 −aet −bet •9 •6 • •J •1 •3 9 6 JJ 1 99 6 1 JJ (−0,+0,1) 9 66 (1,+0,1) JJ + (ψ ◦ f )(t) 99 6 + (ψ ◦ f )(t) JJ q3 99 66 q3 JJ Xt∈F3 9 6 Xt∈F3 JJ q 9 6 q JJ t =0,t = ae • 1 • 1 • t ,t ∈F∗ • 1 • 1 • 1 3 cd 1 3 q • 1 • 1 • • 1 • 1 • −ab−1cd be be ae ae −a−1bcd •JJ • • •9 •6 • JJ ÕÕ ØØ 99 66 1 be JJÕ Ø 1 ae cd 9 6 = ψ(− ) ÕÕJJ Ø + ψ(− + ) 99 6 2 Õ JJ ØØ 2 9 66 q cd ÕÕ JØJ q bd ae 99 6 Õ Ø JJ 9 6 •ÕÕ 1 •ØØ 1 J• • 1 9• 1 •6 • 1 •−1 1 •−1 cdt1t3 −aet1 −bet3 •JJ • • JJ 1 b c −1 −1 ae −1 −1 JJ + ψ(− t − t + t + t + t t ) JJ . 2 1 3 3 1 1 3 JJ q ∗ a b cd JJ t1X,t3∈Fq JJ • 1 • 1 J• 60 Chapter 5

Representation theory in the G = GLn(Fq) case

This chapter examines the representation theory of unipotent Hecke algebras when G = GLn(Fq). The combinatorics associated with the representation theory of GLn(Fq) generalizes the tableaux combinatorics of the symmetric group, and one of the main results of this chapter is to also give a generalization of the RSK correspondence (see Section 2.3.4). + ∗ Fix a homomorphism ψ : Fq → C . Recall that for any composition µ |= n, Hµ is a unipotent Hecke algebra (see Chapter 4). A fundamental result concerning Gelfand- Graev Hecke algebras is

Theorem 5.1 ([GG62],[Yok68],[Ste67]). For all n > 0, H(n) is commutative. and it will follow from Theorem 5.7, via the representation theory of unipotent Hecke algebras.

5.1 The representation theory of Hµ

Let S be a set. An S-partition λ = (λ(s1),λ(s2),...) is a sequence of partitions indexed by the elements of S. Let PS = {S-partitions}. (5.1) The following discussion deﬁnes two sets Θ and Φ, so that Θ-partitions index the irreducible characters of G and Φ-partitions index the conjugacy classes of G. ∗ ∗ ∗ Let Ln = Hom(Fqn , C ) be the character group of Fqn . If γ ∈ Lm, then let

∗ ∗ γ(r) : Fqmr −→ C x 7→ γ(x1+qr +q2r+···+qm(r−1) )

Thus if n = mr, then we may view Lm ⊆ Ln by identifying γ ∈ Lm with γ(r) ∈ Ln. Deﬁne

L = Ln. [n≥0 61

The Frobenius maps are F : F¯ → F¯ F : L → L q q and , x 7→ xq γ 7→ γq where F¯ q is the algebraic closure of Fq. The map ¯ ∗ {F-orbits of Fq} −→ { f ∈ Fq[t] | f is monic, irreducible, and f (0) , 0} k−1 q q2 qk−1 qi qk ¯ ∗ {x, x , x ,..., x } 7→ fx = (t − x ), where x = x ∈ Fq Yi=1 is a bijection such that the size of the F-orbit of x equals the degree d( fx) of fx. Let f is monic, irreducible Φ = f ∈ C[t] | and Θ = {F-orbits in L}. (5.2) ( and f (0) , 0 ) If η is a Φ-partition and λ is a Θ-partition, then let |η| = d( f )|η( f )| and |λ| = |ϕ||λ(ϕ)| Xf ∈Φ Xϕ∈Θ be the size of η and λ, respectively. Let the sets PΦ and PΘ be as in (5.1) and let Φ Φ Θ Θ Pn = {η ∈P | |η| = n} and Pn = {λ ∈P | |λ| = n}. (5.3)

Theorem 5.2 (Green [Gre55]). Let Gn = GLn(Fq). Φ η (a) Pn indexes the conjugacy classes K of Gn, Θ λ (b) Pn indexes the irreducible Gn-modules Gn. Suppose λ ∈PΘ.A column strict tableau P = (P(ϕ1), P(ϕ2),...) of shape λ is a column strict ﬁlling of λ by positive integers. That is, P(ϕ) is a column strict tableau of shape (ϕ) λ . Write sh(P) = λ. The weight of P is the composition wt(P) = (wt(P)1, wt(P)2,...) given by number of wt(P)i = |ϕ| . i in P(ϕ) ! Xϕ∈Θ If λ ∈PΘ and µ is a composition, then let ˆ λ Hµ = {column strict tableaux P | sh(P) = λ, wt(P) = µ} (5.4) and ˆ Θ ˆ λ Hµ = {λ ∈P | Hµ is not empty}. (5.5) The following theorem is a consequence of Theorem 2.3 and a theorem proved by Zelevinsky [Zel81] (see Theorem 5.5). A proof of Zelevinsky’s theorem is in Section 5.3. ˆ λ Theorem 5.3. The set Hµ indexes the irreducible Hµ-modules Hµ and λ ˆ λ dim(Hµ ) = |Hµ |. 62

5.2 A generalization of the RSK correspondence

For a composition µ |= n, let Nµ be as in (4.11) and Mµ as in (4.12). The (Hµ, Hµ)-bimodule decomposition

λ λ λ 2 ˆ λ 2 Hµ Hµ ⊗ Hµ implies |Nµ| = dim(Hµ) = dim(Hµ ) = |Hµ | .

Mλ∈Hˆ µ λX∈Hˆ µ λX∈Hˆ µ

Theorem 5.4, below, gives a combinatorial proof of this identity. Encode each matrix a ∈ Mµ as a Φ-sequence

( f1) ( f2) (a , a ,...), fi ∈ Φ,

( f ) where a ∈ Mℓ(µ)(Z≥0) is given by

( f ) ai j = highest power of f dividing ai j. Note that this is an entry by entry “factorization” of a such that

a( f ) ai j = f ij . Yf ∈Φ

Recall from Section 2.3.4 the classical RSK correspondence

Pairs (P, Q) of column strict M (Z≥ ) −→ ℓ 0 ( tableaux of the same shape ) b 7→ (P(b), Q(b)).

Theorem 5.4. For a ∈ Mµ,let P(a) and Q(a) be the Φ-column strict tableaux given by

( f1) ( f2) ( f1) ( f2) P(a) = (P(a ), P(a ),...) and Q(a) = (Q(a ), Q(a ),...) for fi ∈ Φ.

Then the map

Pairs (P, Q) of Φ-column Nµ −→ Mµ −→  strict tableaux of the same     shape and weight µ    v 7→ av 7→ (P(av), Q(av)),  is a bijection, where the ﬁrst map is the inverse of the bijection in Theorem 4.2.

By the construction above, the map is well-deﬁned, and since all the steps are invertible, the map is a bijection. 63

Example. suppose µ = (7, 5, 3, 2) and f, g, h ∈ Φ are such that d( f ) = 1, d(g) = 2, and d(h) = 3. Then g f 2h 1 1  h 1 g 1  av = ∈ M  1 1 f f 2     g 1 1 1    corresponds to the sequence  

0200 ( f ) 1000 (g) 0100 (h) 0000 0010 1000 ( f1) ( f2) =        (av , av ,...)   ,   ,     0012   0000   0000           0000   1000   0000                 and  

( f ) (g) (h) ( f ) (g) (h) (P(a ), Q(a )) = 2 2 4 , 1 1 , 1 2 1 1 3 , 1 4 , 1 2 . v v 3 4 3 ! 3 3 2 !

G 5.3 Zelevinsky’s decomposition of IndU(ψµ) This section proves the theorem

Theorem 5.5 (Zelevinsky [Zel81]). Let U be the subgroup of unipotent upper-triangular matrices of G = GLn(Fq), µ |= n and ψµ be as in (4.8). Then

G ˆ λ λ IndU (ψµ) = Card(Hµ )G .

Mλ∈Hˆ µ

Theorem 5.3 follows from this theorem and Theorem 2.3. The proof of Theorem 5.5 is in 3 steps.

(1) Establish the necessary connection between symmetric functions and the representation theory of G.

(2) Prove Theorem 5.5 for the case when ℓ(µ) = 1.

(3) Generalize (2) to arbitrary µ.

The proof below uses the ideas of Zelevinsky’s proof, but explicitly uses symmetric functions to prove the results. Speciﬁcally, the following discussion through the proof of Theorem 5.7 corresponds to [Zel81, Sections 9-11] and Theorem 5.5 corresponds to [Zel81, Theorem 12.1]. 64

5.3.1 Preliminaries to the proof (1)

Let µ |= n and G = Gn. The group

g1 ∗  g2   Pµ =  .  | gi ∈ Gµi = GLµi (Fq) (5.6)  ..          0 gℓ         has subgroups  

Idµ1 ∗

 Idµ2  L = G ⊕ G ⊕···⊕ G and U =   , (5.7) µ µ1 µ2 µℓ µ  ..   .     0 Idµ   ℓ      where Idk is the k × k identity matrix. Note that Pµ = LµUµ and Pµ = NG(Uµ). The indﬂation map is a composition of the inﬂation map and the induction map,

G −→ −→ IndfLµ : R[Lµ] R[Pµ] R[G] χ 7→ InfPµ (χ) 7→ IndG (InfPµ (χ)), Lµ Pµ Lµ

Pµ InfL (χ): Pµ → C where µ for l ∈ Lµ and u ∈ Uµ. lu 7→ χ(l), Suppose λ ∈ PΘ and η ∈ PΦ (see (5.3)). Let χλ be the irreducible character corresponding to the irreducible G-module Gλ and let κη be the characteristic function corresponding to the conjugacy class Kη (see Theorem 5.2), given by

η η 1, if g ∈ K , κ (g) = for g ∈ G|η|. ( 0, otherwise, Deﬁne

λ Θ η Φ R = R[Gn] = C-span{χ | λ ∈P } = C-span{κ | η ∈P }. Mn≥0 The space R has an inner product deﬁned by

λ ν hχ , χ i = δλν,

and multiplication

χλ ◦ χν = IndfGr+s (χλ ⊗ χν), for λ ∈PΘ,ν ∈PΘ. (5.8) L(r,s) r s 65

(ϕ) (ϕ) For each ϕ ∈ Θ, let {Y1 , Y2 ,...} be an inﬁnite set of variables, and let

(ϕ) ΛC = ΛC(Y ), Oϕ∈Θ

(ϕ) (ϕ) (ϕ) where ΛC(Y ) is the ring of symmetric functions in {Y1 , Y2 ,...} (see Section 2.3.3). ( f ) ( f ) For each f ∈ Φ, deﬁne an additional set of variables {X1 , X2 ,...} such that the symmetric functions in the Y variables are related to the symmetric functions in the X variables by the transform

(ϕ) k|ϕ|−1 ( fx ) pk(Y ) = (−1) ξ(x)p k|ϕ| (X ), (5.9) d( fx) ∈XF∗ x qk|ϕ|

( f ) where ξ ∈ ϕ, f ∈ Φ is the irreducible polynomial that has x as a root, and p a (X ) = 0 x b a < if b Z≥0. Then ( f ) ΛC = ΛC(X ). Of ∈Φ (ϕ) ( f ) For ν ∈ P, let sν(Y ) be the Schur function and Pν(X ; t) be the Hall-Littlewood symmetric function (as in Section 2.3.3). Deﬁne

(ϕ) −n(η) ( f ) −d( f ) sλ = sλ(ϕ) (Y ) and Pη = q Pη( f ) (X ; q ), (5.10) Yϕ∈Θ Yf ∈Φ

( f ) ℓ(µ) where n(η) = f ∈Φ d( f )n(η ) and for a composition µ, n(µ) = i=1 (i − 1)µi. The ring P Θ P Φ ΛC = C-span{sλ | λ ∈P } = C-span{Pη | η ∈P } has an inner product given by hsλ, sνi = δλν. Theorem 5.6 (Green [Gre55],Macdonald [Mac95]). The linear map

ch : R −→ ΛC λ Θ χ 7→ sλ, for λ ∈P η Φ κ 7→ Pη, for η ∈P , is an algebra isomorphism that preserves the inner product. A unipotent conjugacy class Kη is a conjugacy class such that η( f ) = ∅ unless f = t − 1. Let U = C-span{κη | η( f ) = ∅, unless f = t − 1} ⊆ R be the subalgebra of unipotent class functions. Note that by (5.10) and Theorem 5.6,

(t−1) ch(U) =ΛC(X ). 66

Consider the projection π : R → U which is an algebra homomorphism given by

χλ(g), if g ∈ G is unipotent, (πχλ)(g) = λ ∈PΘ. ( 0, otherwise,

−1 (t−1) Thenπ ˜ = ch ◦ π ◦ ch : ΛC → ΛC(X ) is given by

(ϕ) k|ϕ|−1 ( fx ) π˜(pk(Y )) = π˜ (−1) ξ(x)p k|ϕ| (X ) (by (5.9))  d( fx)   X∗   x ∈ F k|ϕ|   q  (5.11)  k|ϕ|−1 (t−1)  = (−1) ξ(1)p k|ϕ| (X ) + 0  1 k|ϕ|−1 (t−1) = (−1) pk|ϕ|(X ).

G 5.3.2 The decomposition of IndU(ψ(n)) (2) G The representation IndU (ψ(n)) is the Gelfand-Graev module, and with Theorem 2.3, The- orem 5.7, below, proves that H(n) is commutative. Theorem 5.7. Let U be the subgroup of unipotent upper-triangular matrices of G = GLn(Fq). Then

G (ϕ) ch(IndU(ψ(n))) = sλ, where ht(λ) = max{ℓ(λ ) | ϕ ∈ Θ}. Θ Xλ∈Pn ht(λ)=1

Proof. Let

−1 Ψ : R −→ C ˜ ch Ψ λ λ G and Ψ : ΛC −→ R −→ C. (5.12) χ 7→ hχ , IndU (ψ(n))i For any ﬁnite group H and γ, χ ∈ R[H], let

1 : H → C 1 1 H , e = h, and hχ, γi = χ(h)γ(h−1). h 7→ 1 H |H| H |H| Xh∈H Xh∈H The proof is in six steps.

˜ (1) ∗ (a) Ψ(ek(Y )) = δk1, where 1 = 1Fq ,

λ λ Θ (b) Ψ(χ ) = dim(e(n)G ) for λ ∈P ,

˜ ˜ ˜ (1) ∗ (c) Ψ( fg) = Ψ( f )Ψ(g) for all f, g ∈ ΛC(Y ), where 1 = 1Fq . (d) Ψ ◦ π =Ψ, 67

(ϕ) (1) (ϕ) (e) Ψ˜ ( f (Y )) = Ψ˜ ( f (Y )) for all f ∈ ΛC(Y ),

(f) Ψ˜ (sλ) = δht(λ)1. (a) An argument similar to the argument in [Mac95, pgs. 285-286] shows that

−1 (1) ch (ek(Y )) = 1Gk

(see [HR99, Theorem 4.9 (a)] for details). Therefore, by Frobenius reciprocity and the orthogonality of characters,

˜ (1) G Ψ(ek(Y )) = h1Gk , IndU (ψ(n))i = h1Uk , ψ(n)iUk = δk1.

λ G (b) Since there exists an idempotent e such that G CGe and IndU (ψ(n)) CGe(n), the map λ e(n)CGe −→ HomG(G , CGe(n))

γg : CGe → CGe(n) e(n)ge 7→ xe 7→ xege(n) is a vector space isomorphism(using an argument similar to the proof of [CR81, (3.18)]). Thus,

λ λ G λ G Ψ(χ ) = hχ , IndU(ψ(n))i = dim(HomG(G , IndU (ψ(n)))) λ = dim(e(n)CGe) = dim(e(n)G ).

(1) (1) (c) By (a), Ψ˜ (er(Y ))Ψ˜ (es(Y )) = δr1δs1. It therefore suﬃces to show that

(1) (1) (1) (1) (1) Ψ˜ (er(Y )es(Y )) = δr1δs1, (since ΛC(Y ) = C[e1(Y ), e2(Y ),...]).

Suppose r + s = n and let P = P(r,s). Then

˜ (1) (1) Gn Ψ(er(Y )es(Y )) =Ψ(IndP (1P)) = dim(e(n)CGeP).

Since T ⊆ P, eP = e(1n)eP, G = v∈N UvU, and N = WT, F e(n)CGeP = e(n)CGe(1n)eP = C-span{e(n)we(1n)eP | w ∈ W}.

If there exists 1 ≤ i ≤ n such that w(i + 1) = w(i) + 1, then

e(n)we(1n) = e(n)wxi,i+1(t)e(1n) = e(n) xw(i),w(i)+1(t)we(1n) = ψ(t)e(n)we(1n).

Therefore, e(n)we(1n) = 0 unless w = w(n). If r > 1 of s > 1, then there exists 1 ≤ i ≤ n such that xi+1,i(t) ∈ P(r,s), so

e(n)w(n)eP = e(n)w(n) xi+1,i(t)eP = e(n) xn−i,n−i+1(t)w(n)eP = ψ(t)e(n)w(n)eP = 0. 68

In particular, dim(e(n)CGeP) = 0.

If r = s = 1, then P(1,1) is upper-triangular, so

e(2)w(2)eP , 0

(1) (1) and dim(e(2)CGeP) = 1, giving Ψ˜ (er(Y )es(Y )) = δr1δs1. (d) By Frobenius reciprocity, hχλ, IndGn (ψ )i = hResGn (χλ), ψ i Un (n) Un (n) Un = hResGn (π(χλ)), ψ i = hπ(χλ), IndGn (ψ )i, Un (n) Un Un (n) so Ψ=Ψ ◦ π. (e) Induct on n, using (c) and the identity

n−1 n−1 (1) (1) r−1 (1) (1) ′ (−1) pn(Y ) = nen(Y ) − (−1) pr(Y )en−r(Y ), [Mac95, I.2.11 ] Xr=1 (1) to obtain Ψ˜ (pn(Y )) = 1. Note that

(ϕ) (ϕ) |ϕ|n−1 (t−1) (1) Ψ˜ (pn(Y )) = Ψ˜ (π(pn(Y ))) = Ψ˜ ((−1) p|ϕ|n(X )) = Ψ˜ (π(p|ϕ|k(Y )))

(1) (1) = Ψ˜ (p|ϕ|k(Y )) = 1 = Ψ˜ (pk(Y )). (1) Since Ψ˜ is multiplicative on ΛC(Y ),

(ϕ) (1) Ψ˜ (pν(Y )) = 1 = Ψ˜ (pν(Y )), for all partitions ν.

(ϕ) (ϕ) In particular, since Ψ˜ is linear and ΛC(Y ) = C-span{pν(Y )},

(ϕ) (1) (ϕ) Ψ˜ ( f (Y )) = Ψ˜ ( f (Y )), for all f ∈ ΛC(Y ).

Note that (e) also implies that Ψ˜ is multiplicative on all of ΛC. (f) Note that

Ψ˜ = Ψ˜ (ϕ) = Ψ˜ (1) = Ψ˜ (1) (sλ)  sλ(ϕ)(Y )  sλ(ϕ)(Y ) (sλ(ϕ)(Y )), Yϕ∈Θ  Yϕ∈Θ  Yϕ∈Θ         (1) where the last two equalities follow from (e) and (c), respectively. By deﬁnition sν(Y ) = (1) det(e ′ (Y )), so νi −i+ j 1, if ℓ(ν) = 1, Ψ˜ (s (Y(1))) = ν ( 0, otherwise, implies G ˜ ch(IndU(ψ(n))) = Ψ(sλ)sλ = sλ. Θ Θ λX∈Pn Xλ∈Pn ht(λ)=1 69

G 5.3.3 Decomposition of IndU(ψµ) (3) Suppose λ,ν ∈PΘ.A column strict tableau P of shape λ and weight ν is a column strict ﬁlling of λ such that for each ϕ ∈ Θ,

sh(P(ϕ)) = λ(ϕ) and wt(P(ϕ)) = ν(ϕ).

We can now prove the theorem stated at the beginning of this section: Theorem 5.5 ([Zel81]) Let U be the subgroup of unipotent upper-triangular matrices of G = GLn(Fq), µ |= n and ψµ be as in (4.8). Then

G ˆ λ λ IndU (ψµ) = Card(Hµ )G .

Mλ∈Hˆ µ Proof. Note that Pµ ′ IndU (ψµ) CPµeµ = CPµe[µ]e[µ], where 1 −1 ′ 1 e[µ] = ψµ(u )u and e[µ] = u. (5.13) |U ∩ Lµ| |Uµ| u∈XU∩Lµ uX∈Uµ Thus,

IndPµ (ψ ) InfPµ IndLµ (ψ ) U µ Lµ U∩Lµ µ G G G Pµ µ1 µ2 µℓ InfL IndU (ψ(µ1)) ⊗ IndU (ψ(µ2)) ⊗···⊗ IndU (ψ(µℓ)) µ µ1 µ2 µℓ In particular, by the deﬁnition of multiplication in R (5.8),

G Γµ = ch(IndU(ψµ)) = Γµ1 Γµ2 ··· Γµℓ , where Γµi = sλ. λ∈PΘX, λ = µi ht( ) 1

Θ Θ Pieri’s rule (2.10) implies that for λ ∈Pr , ν ∈Ps and ht(ν) = 1,

sλ sν = sγ, so Γµ = Kλµ sλ, Θ Xˆ γ/λ λX∈PΘ γ∈Pr+s,|Hν |,0 where K = Card{∅ = γ ⊂ γ ⊂ γ ⊂···⊂ γ = λ | |Hˆ γi+1/γi | = 1} λµ 0 1 2 ℓ (µi+1) ˆ λ = Card{column strict tableaux of shape λ and weight µ} = |Hµ |. By Green’s Theorem (Theorem 5.6), ch is an isomorphism, so

G −1 ˆ λ −1 ˆ λ λ IndU(ψµ) = ch (Γµ) = |Hµ |ch (sλ) = Card(Hµ )G .

λX∈Hˆ µ Mλ∈Hˆ µ 70

5.4 A weight space decomposition of Hµ-modules

Let µ = (µ1, µ2,...,µℓ) |= n and let Pµ, Lµ and Uµ be as in (5.6) and (5.7). Recall that 1 e = ψ (u−1)u. µ |U| µ Xu∈U

Theorem 5.8. For a ∈ Mµ, let Ta = eµvaeµ with va as in (4.14). Then the map

H(µ1) ⊗ H(µ2) ⊗···⊗H(µℓ) −→ Hµ

T( f1) ⊗ T( f2) ⊗···⊗ T( fℓ) 7→ T( f1)⊕( f2)⊕···⊕( fℓ), for ( fi) ∈ M(µi) is an injective algebra homomorphism with image Lµ = eµPµeµ = eµLµeµ. Proof. Note that

1 ℓ ⊗···⊗ = −1 −1 ⊗···⊗ T( f1) T( fℓ) 2 ψµ(xi yi ) x1v( f1)y1 xℓv( fℓ)yℓ. |U ∩ Lµ|   xi,Xyi∈Uµ Yi=1 i    Since U = (Lµ ∩ U)(Uµ), Lµ ∩ U Uµ1 × Uµ2 ×···× Uµℓ , and ψµ is trivial on Uµ, 1 T = ψ (x−1y−1)x(v ⊕ v ⊕···⊕ v )y ( f1)⊕( f2)⊕···⊕( fℓ) |U|2 µ ( f1) ( f2) ( fℓ) xX,y∈U 1 = ψ −1 −1 ⊕···⊕ −1 −1 ′ ⊕···⊕ ′ , 2 µ(x1 y1 xℓ yℓ )e[µ] x1v( f1)y1 xℓv( fℓ)yℓe[µ] |U ∩ Lµ| xi,Xyi∈Uµi

′ ′ where e[µ] is as in (5.13). Since Lµ ⊆ NG(Uµ), the idempotent e[µ] commutes with

x1v( f1)y1 ⊕···⊕ xℓv( fℓ)yℓ and

′ ℓ e µ T = [ ] ψ (x−1y−1) x v y ⊕···⊕ x v y . ( f1)⊕( f2)⊕···⊕( fℓ) |L ∩ U|2  µ i i  1 ( f1) 1 ℓ ( fℓ) ℓ xi,Xyi∈Uµ Yi=1 i     ′  Consequently, the map multiplies by e[µ] and changes ⊗ to ⊕, so it is an algebra homomorphism. Since the map sends basis elements to basis elements, it is also injective.

Remark. This is the GLn(Fq) version of Corollary 3.5.

Let Lµ be as in Theorem 5.8. By Theorem 5.1 each H(µi) is commutative, so Lµ is γ commutative and all the irreducible Lµ-modules Lµ are one-dimensional. Theorem 5.3 implies that ˆ H(µi) = {Θ-partitions λ | |λ| = µi, ht(λ) = 1}

indexes the irreducible H(µi)-modules. Therefore, the set ˆ ˆ ˆ ˆ ˆ Lµ = H(µ1) × H(µ2) ×···× H(µℓ) = {γ = (γ1,γ2,...,γℓ) | γi ∈ H(µi)} (5.14) 71

indexes the irreducible Lµ-modules. Identify γ ∈ Lˆµ with the map γ : Lµ → C such that

γ yv = γ(y)v, for all y ∈Lµ, v ∈Lµ.

For γ ∈ Lˆµ, the γ-weight space Vγ of an Hµ-module V is

Vγ = {v ∈ V | yv = γ(y)v, for all y ∈Lµ}. Then V Vγ.

Mγ∈Lˆµ Θ Let λ ∈P and γ ∈ Lˆµ.A column strict tableau P of shape λ and weight γ is column strict ﬁlling of λ such that for each ϕ ∈ Θ,

(ϕ) (ϕ) (ϕ) (ϕ) (ϕ) (ϕ) sh(P ) = λ and wt(P ) = (|γ1 |, |γ2 |,..., |γℓ |),

(ϕ) (ϕ) where |γi | is the number of boxes in the partition γi (which has length 1). Let ˆ λ λ Hγ = {P ∈ Hµ | sh(P) = λ, wt(P) = γ}. For example, suppose λ = (ϕ1), (ϕ2), (ϕ3) and γ = (ϕ1), (ϕ2), (ϕ3) ⊗ (ϕ1), (ϕ3) ⊗ (ϕ1), (ϕ2) ⊗ (ϕ2) . Then 1 1 3 (ϕ1), 1 3 (ϕ2), 1 (ϕ3) , 1 1 3 (ϕ1), 1 4 (ϕ2), 1 (ϕ3) Hˆ λ = 2 2 4 2 2 2 3 2 . γ  1 1 2 (ϕ1), 1 3 (ϕ2), 1 (ϕ3) , 1 1 2 (ϕ1), 1 4 (ϕ2), 1 (ϕ3)   2 3 4 2 2 3 3 2    λ ˆ  Theorem 5.9. Let Hµ be an irreducible Hµ-module and γ ∈ Lµ. Then 

λ ˆ λ dim(Hµ )γ = Card(Hγ ). Proof. By Theorem 2.3 and Proposition 2.2,

dim((H λ) ) = hResHµ (H λ), Lγi = hResG (Gλ), Pγi = hGλ, IndG (Pγ)i, µ γ Lµ µ µ Pµ µ Pµ µ where Pγ = InfPµ (Lγ). Therefore, µ Lµ µ

λ λ λ dim((Hµ )γ) = cγ, where sγ1 sγ2 ··· sγℓ = cγ sλ. λX∈PΘ

λ ˆ λ Pieri’s rule (2.10) implies cγ = |Hγ |. 72 Chapter 6

The representation theory of the Yokonuma algebra

6.1 General type

Let 11: U → C∗ be the trivial character, and let 1 e = u ∈ CG (6.1) 11 |U| Xu∈U

G be the idempotent so that IndU (11) CGe11. Then the Yokonuma algebra

H11 = EndCG(CGe11) e11CGe11 has a basis

{e11ve11 | v ∈ N}, indexed by N = hξi, h | i = 1, 2,...,ℓ, h ∈ Ti,

where ξi = wi(1) = xi(1)x−αi (−1)xi(1), and T = hhH(t) | H ∈ hZi. Recall that hi(t) = h (t). Hαi The map CT −→ H11 h 7→ e11he11 is an injective algebra homomorphism (see Chapter 3 (E3)). For v ∈ N, write

Tv = e11ve11 ∈ H11, with Ti = Tξi , and h = Th, h ∈ T.

Theorem 6.1 (Yokonuma). H11 is generated by Ti,i = 1,...,ℓ,andh ∈ T with relations

Tih = si(h)Ti, 2 −1 −1 Ti = q hi(−1) + q hi(t)Ti, ∗ Xt∈Fq T i T jTi ··· = T j T iT j ··· , where mi j is the order of si s j in W,

mij mij

| {zThT}k = T|hk,{z h,}k ∈ T. 73

Proof. Consider the product ξiξ j ∈ N for i , j. Note that

e11ξiξ je11 = e11 eα ξiξ jeαi e11 (by (3.17)) αY,αi

= e11ξi eα eαi eξ je11 (by Chapter 3, (E1)) αY,αi

= e11ξie11ξ je11 = (e11ξie11)(e11ξ je11).

Thus, if v = v1v2 ··· vrvT ∈ N decomposes according to si1 si2 ··· sir ∈ W (see (3.4)), then

Tv = e11v1v2 ··· vrvT e11 = Ti1 Ti2 ··· Tir vT . (6.2)

The Yokonuma algebra is therefore generated by Ti, i = 1, 2,...,ℓ, and h ∈ T. The necessity of the relations is a direct consequence of (6.2), (N3), (UN3), and (N2) (the last three are from Chapter 3). For suﬃciency, use a similar argument to the one used in the proof of Theorem 3.5 in [IM65].

6.1.1 A reduction theorem Let Tˆ index the irreducible CT-modules. Since T is abelian, all the irreducible modules γ are one-dimensional, and we may identify the label γ ∈ Tˆ of V = C-span{vγ} with the homomorphism γ : T → C∗ given by

hvγ = γ(h)vγ, for h ∈ T.

Suppose V is an H11-module. As a T-module

V = Vγ, where Vγ = {v ∈ V | hv = γ(h)v, h ∈ T}. Mγ∈Tˆ

Note that 1 H = H τ , where τ = γ(h−1)h. (6.3) 11 11 γ γ |T| Mγ∈Tˆ Xh∈T

Recall the surjection π : N → W of (3.3). Use this map to identify

w = si1 si2 ... sir ∈ W ←→ w = ξi1 ξi2 ...ξir ∈ N, for r minimal. (6.4)

Thus, the W-action on Tˆ

(wγ)(h) = γ(w−1(h)), for w ∈ W, h ∈ T, and γ ∈ Tˆ, 74

implies that for γ ∈ Tˆ, vγ ∈ Vγ,

−1 −1 hTwvγ = Tww (h)vγ = γ(w h)Twvγ for all h ∈ T.

Thus, Tw(Vγ) ⊆ Vwγ. (6.5) Let Wγ = {w ∈ W | w(γ) = γ} and Tγ = τγH11τγ = C-span{τγTwτγ | w ∈ Wγ}. (6.6)

Remarks

1. Since the identity of H11 is e11 and the identity of Tγ is τγ, the algebra Tγ is not a H11 subalgebra of H11. In fact, Tγ EndH11 (IndCT (γ)).

2. If V is an H11-module, then

τγV = Vγ, and so TγV = TγVγ.

In particular, Vγ is an Tγ-module.

Theorem 6.2. Let V be an irreducible H11-module such that Vγ , 0. Then

(a) Vγ is an irreducible Tγ-module;

(b) if M is an irreducible Tγ-module, then H11 ⊗Tγ M is an irreducible H11-module; (c) the map ∼ H11 Ind (V ) = H ⊗T V −→ V Tγ γ 11 γ γ Tw ⊗ vγ 7→ Twvγ

is an H11-module isomorphism.

Proof. (a) Let 0 , vγ ∈ Vγ. The irreducibility of V implies that H11vγ = V, so for any v ∈ Vγ, there exist elements cw,ν ∈ C such that

v = cw,νTwτνvγ (by (6.3)) Xw∈W ν∈Tˆ

= cw,γTwτγvγ (by the orthogonality of characters of T) wX∈W

= cw,γτγTwτγvγ. (by (6.5) and since v ∈ Vγ) wX∈Wγ 75

Since an arbitrarily chosen v is contained in Tγvγ, we have Tγvγ = Vγ, making Vγ an irreducible Tγ-module. (b) Suppose M is an irreducible Tγ-module. Let

H11 V = Ind (M) = H ⊗T M Tγ 11 γ

= C-span{Twτν ⊗ v | w ∈ W,ν , γ ∈ Tˆ, v ∈ M}

= C-span{Tw ⊗ v | w ∈ W/Wγ, v ∈ M} where the last equality follows from

τν ⊗ v = τν ⊗ τγv = τντγ ⊗ v = 0 ⊗ v, for ν , γ.

Note that Vγ = C-span{e11 ⊗ v | v ∈ M} M. In particular, Vγ is an irreducible Tγ-module and H11Vγ = V. ′ Suppose V has a nontrivial H11-submodule V . Since V is induced from Vγ, there must by some Tw ∈ H11 such that

′ Tw(V ) ∩ Vγ , 0.

′ ′ ′ But V is an H11-module, so V ∩ Vγ , 0. As an irreducible Tγ-module Vγ ⊆ V , and by the construction of V, ′ H11V ⊇ H11Vγ = V.

Therefore, V contains no nontrivial, proper submodules, making V an irreducible H11- module. (c) Follows from the proof of (b). Let Tˆ/W betheset of W-orbits in Tˆ. Identify Tˆ/W with a set of orbit representatives, and for γ ∈ Tˆ/W, let Tˆγ index the irreducible modules of Tγ. Corollary 6.3. The map

Irreducible Pairs (γ,λ), ←→ ˆ ( H11-modules ) ( γ ∈ Tˆ/W,λ ∈ Tγ ) IndH11 (T λ) ↔ (γ,λ) Tγ γ is a bijection.

6.1.2 The algebras Tγ Let B = UT be a Borel subgroup of G. Since

B −→ T uh 7→ h 76

is a surjective homomorphism, γ : T → C∗ extends to a linear character γ of B given by γ(uh) = γ(h). Note that 1 e τ = γ(b−1)b (e as in (6.1)). 11 γ |B| 11 Xb∈B Thus, G Tγ = τγe11CGe11τγ EndCG(IndB (γ)) = H(G, B,γ), where if γ is the trivial character 1B, then H(G, B, 1B) is the Iwahori-Hecke algebra.

Proposition 6.4. Let γ ∈ Tˆ be such that Wγ is generated by simple reﬂections. Then Tγ is presented by generators {Ti | si ∈ Wγ} with relations

−1 −1 ∗ 2 q + q (q − 1)Ti, if γ(hαi (t)) = 1 for all t ∈ Fq, Ti = −1 ( γ(hαi (−1))q , otherwise, T i T jTi ··· = T j T iT j ··· , where mi j is the order of si s j in W.

mij mij | {z } | {z } Proof. This follows from Theorem 6.1 and

∗ (q − 1)τγ, if γ(hαi (t)) = 1 for all t ∈ Fq, hαi (t) τγ = γ(hαi (t))τγ = ∗ ∗ ( 0, otherwise. Xt∈Fq Xt∈Fq

6.2 The G = GLn(Fq) case 6.2.1 The Yokonuma algebra and the Iwahori-Hecke algebra

n If ψµ = 11is the trivial character of U, then µ = (1 ). Let Ti = e11 sie11 and recall that hε j (t) = Id j−1 ⊕ (t) ⊕ Idn− j. In the case G = GLn(Fq), H11 has generators Ti, hε j (t), for ∗ 1 ≤ i < n,1 ≤ j ≤ n and t ∈ Fq with relations

2 −1 −1 −1 Ti = q + q hεi (−1) hεi (t)hεi+1 (t )Ti, ∗ Xt∈Fq

TiTi+1Ti = Ti+1TiTi+1, TiT j = T jTi, |i − j| > 1, ′ hε j (t)Ti = Tihε j′ (t), where j = si( j), ∗ hεi (a)hεi (b) = hεi (ab), hεi (a)hε j (b) = hε j (b)hεi (a), a, b ∈ Fq. The Yokonuma algebra has a decomposition 1 H = H τ , where τ = γ(h−1)h. 11 11 γ γ |T| Mγ∈Tˆ Xh∈T 77

ˆ ∗ ∗ Fix a total ordering ≤ on the set Fq = {ϕ1,ϕ2,...,ϕq−1} of linear characters of Fq, ∗ ∗ ˆ ∗ such that ϕ1 : Fq → C is the trivial character. Suppose γ ∈ T. Since hhεi (t) | t ∈ Fqi ∗ Fq,

γ(h) = γ(hε1 (h1))γ(hε2 (h2)) ··· γ(hεn (hn)), where h = diag(h1, h2,..., hn) ∈ T, ˆ ∗ = γ1(h1)γ2(h2) ··· γn(hn), where γi ∈ Fq. ˆ Thus, every γ ∈ T can be written γ = ϕi1 ⊗ ϕi2 ⊗···⊗ ϕin . Let ν = (ν1,ν2,...,νn) |= n with νi ≥ 0. Write

= ⊗···⊗ ⊗ ⊗···⊗ ⊗···⊗ ⊗···⊗ = γν ϕ1 ϕ 1 ϕ2 ϕ 2 ϕn ϕ n and τν τγν . (6.7)

ν1 terms ν2 terms νn terms | {z } | {z } | {z } Note that the γν are orbit representatives of the W-action in Tˆ. Let

Wν = {w ∈ W | w(γν) = γν} = Wν1 ⊕ Wν2 ⊕···⊕ Wνn ,

Tν = τνH11τν = C-span{τνTwτν | w ∈ Wν}.

Note that Wν is generated by its simple reﬂections.

If si is in the kth factor of Wν = Wν1 ⊕ Wν2 ⊕···⊕ Wνn , then write

si ∈ Wνk .

ν Lemma 6.5. Let ν |= n and Tw = τνTwτν for w ∈ Wν. Then Tν is presented by generators ν { Ti | si ∈ Wν} with relations

ν 2 −1 −1 ν Ti = q + ϕk(−1)q (q − 1) Ti, for si ∈ Wνk ν ν ν ν ν ν Ti Ti+1 Ti = Ti+1 Ti Ti+1 ν ν ν ν Ti T j = T j Ti, for |i − j| > 1.

Proof. Note that if si ∈ Wν then γν(hεi (t)) = γν(hεi+1 (t)). Thus, the lemma follows from the Yokonuma algebra relations.

The Iwahori-Hecke algebra T ⊆ H11 is the algebra

Tn = H(n), (recall ϕ1 is the trivial character).

In Tn, write Ii = τ(n)Tiτ(n). 78

Corollary 6.6. Let ν = (ν1,...,νn) |= n. Index the generators Ii of Tνk by {i | si ∈ Wνk }. Then the map

∼ Tν −→ Tν1 ⊗Tν2 ⊗···⊗Tνn ν Ti 7→ ϕk(−1)1 ⊗···⊗ 1 ⊗Ii ⊗ 1 ⊗···⊗ 1, for si ∈ Wνk , k − 1 terms n − k terms | {z } | {z } is an algebra isomorphism.

Proof. Let (k) ν = (0,..., 0,νk, 0,..., 0) |= νk. k − 1 terms n − k terms Then the map | {z } | {z }

(k) Tν −→ Tνk ν(k) Ti 7→ ϕk(−1)Ii is an algebra isomorphism by Lemma 6.5. Corollary 6.6 follows by applying this map to tensor products.

6.2.2 The representation theory of Tν

By Corollary 6.6, understanding the representation theory of Tν is the same as under-

standing the representation theory of the Iwahori-Hecke algebras Tνi . Let µ ⊢ n be a partition. A standard tableau of shape µ is a column strict tableau of shape µ and weight (1n) (i.e. every number between from 1 to n appears exactly once). Suppose P is a standard tableau of shape µ. Let

cP(i) = the content of the box containing i, q − 1 (6.8) CP(i) = . 1 − qcP(i)−cP(i+1)

Theorem 6.7 ([Hoe74, Ram97, Wen88]). Let G = GLn(Fq). Then

µ (a) The irreducible Tn-modules Tn are indexed by partitions µ ⊢ n,

µ (b) dim(Tn ) = Card{standard tableau P | sh(P) = µ},

µ n (c) Let Tn = C-span{vP | sh(P) = µ, wt(P) = (1 )}. Then

−1 −1 IivP = q CP(i)vP + q (1 + CP(i))vsi P

where vsiP = 0 if siP is not a column strict tableau. 79

ˆ ∗ (ϕ1) (ϕ2) (ϕq−1) Recall from Chapter 5, an Fq-partition λ = (λ ,λ ,...,λ ) is a sequence of ˆ ∗ partitions indexed by Fq. Let |λ| = |λ(ϕ1)| + |λ(ϕ2)| + ··· + |λ(ϕq−1)| = total number of boxes,

Let ν = (ν1,ν2,...,νn) |= n with νi ≥ 0 and γν as in (6.7). A standard ν-tableau Q = (Q(ϕ1), Q(ϕ2),..., Q(ϕq−1)) of shape λ is a column strict ﬁlling of λ by the numbers 1, 2, 3,..., n such that (a) each number appears exactly once,

(ϕk) (b) j ∈ Q if s j ∈ Wνk . Let ˆ λ Tν = {standard ν-tableau Q | sh(Q) = λ}, (6.9) ˆ ˆ ∗ ˆ λ Tν = {Fq-partition λ | Tν , ∅} ˆ ∗ (ϕi) = {Fq-partition λ | |λ | = |νi|}. (6.10)

Example. If ν = (3, 0, 1, 0), then γν = ϕ1 ⊗ ϕ1 ⊗ ϕ1 ⊗ ϕ3. The set Tˆν is

(ϕ ) (ϕ1) (ϕ1), ∅(ϕ2), (ϕ3), ∅(ϕ4) , 1 , ∅(ϕ2), (ϕ3), ∅(ϕ4) , , ∅(ϕ2), (ϕ3), ∅(ϕ4) , and, for example,

(ϕ1) (ϕ3) ( , ) (ϕ1) ϕ ϕ ϕ (ϕ1) ϕ ϕ ϕ ˆ 1 2 ( 2) 4 ( 3) ( 4) 1 3 ( 2) 4 ( 3) ( 4) Tν = 3 , ∅ , , ∅ , 2 , ∅ , , ∅ . n o ˆ λ (ϕk) Let Q ∈ Tν . Suppose s j ∈ Wνk so that j ∈ Q . Write

Q j = ϕk, (6.11)

(ϕk) cQ( j) = the content of the box containing j in Q , (6.12) q − 1 CQ( j) = (6.13) 1 − qcQ( j)−cQ( j+1)

Corollary 6.8. Let ν = (ν1,ν2,...,νn) |= n with νi ≥ 0 and ℓ(ν) = n. Then λ ˆ (a) The irreducible Tν-modules Tν are indexed by λ ∈ Tν. λ ˆ λ (b) dim(Tν ) = Card(Tν ). λ ˆ λ (c) Let Tν = C-span{vQ | Q ∈ Tν }. Then ν −1 −1 TivQ = q Q j(−1)CQ(i)vQ + q Q j(−1)(1 + CQ(i))vsiQ,

where vsiQ = 0 if siQ is not a column strict tableau. Proof. Transfer the explicit action of Theorem 6.7 across the isomorphism Tν Tν1 ⊗ Tν2 ⊗···⊗Tνn of Corollary 6.6. 80

6.2.3 The irreducible modules of H11

The goal of this section is to construct the irreducible H11-modules. According to Corol- lary 6.3 and (6.7), the map Pairs (ν, λ), ν |= n, Irreducible ←→ ( νi ≥ 0,ℓ(ν) = n,λ ∈ Tˆν ) ( H11-modules ) (6.14) (ν, λ) ↔ IndH11 (T λ) Tν ν is a bijection. Let W/ν be the set of minimal length coset representatives of W/Wν. Then

H11 λ λ Ind (T ) = H ⊗T T Tν ν 11 ν ν

= C-span{Tw ⊗ vQ | w ∈ W/ν, sh(Q) = λ, wt(Q) = γν}. (6.15) Note that the map Pairs (w, Q), = = n ˆ λ ←→ {tableau Q | sh(Q) λ, wt(Q) (1 )} (w ∈ W/ν, Q ∈ Tν ) (6.16) (w, Q) 7→ wQ is a bijection (since w ∈ W/ν implies w preserves the relative magnitudes of the entries in Q(ϕ) for all ϕ). For example, under this bijection •2 •D •M •M • • h• 2 DD MM MMzz hhh 2 D MMz MMhhh 1 3 (ϕ1) 5 (ϕ2) (ϕ3) 2 6 (ϕ1) 3 (ϕ2) (ϕ3) 2 D hzMhMhh MM 7 ↔ 1 2 Dhhhz M M , 2 4 , 6 , 4 7 , 5 , 2 hhhh DzDz MM MM ! •••••••hhhh2 zz D MM MM (since 2 < 3 in Q(ϕ1) on the left side, 4 < 6 in Q(ϕ1) on the right side). Write

vwQ = Tw ⊗ vQ. If |λ| = n, then let ˆ λ n H11 = { tableau Q | sh(Q) = λ, wt(Q) = (1 )}, (6.17) ˆ ˆ ∗ H11 = {Fq-partitions λ | |λ| = n}. (6.18) Remarks. 1. In the notation of Chapter 5, ˆ ˆ λ H11 = {Θ-partition λ | H11 , ∅}.

2. The map Pairs (ν, λ), ν |= n, ˆ ←→ H11 ( νi ≥ 0,ℓ(ν) = n,λ ∈ Tˆν ) (ν, λ) ↔ λ ˆ is a bijection, so by (6.14), H11 indexes the irreducible H11-modules. 81

Suppose Q is a tableau of shape λ and weight (1n). As in (6.11)-( 6.13), if Q(ϕ) has the box containing j, then write

Q j = ϕ, (ϕ) cQ( j) = the content of the box containing j in Q , q − 1 CQ( j) = . 1 − qcQ( j)−cQ( j+1) Theorem 6.9.

λ ˆ (a) The irreducible H11-modules H11 are indexed by λ ∈ H11. λ ˆ λ (b) dim(H11 ) = Card(H11 ). λ ˆ λ (c) Let H11 = C-span{vQ | Q ∈ H11 } as in (6.15) and (6.17). Then

hvQ = Q1(h1)Q2(h2) ··· Qn(hn)vQ, for h = diag(h1, h2,..., hn) ∈ T, −1 q vsiQ, if Qi ≻ Qi+1, − −1 + − −1 + = TivQ =  Qi( 1)q CQ(i)vQ Qi( 1)q (1 CQ(i))vsi Q, if Qi Qi+1,  v , if Q ≺ Q ,  si Q i i+1   where vsiQ = 0 if siQ is not a column strict tableau. Proof. (a) follows from Remark 2, and (b) follows from (6.15) and (6.17). (c) Directly compute the action of H11 on

H λ = IndH11 (T λ) = C-span{T ⊗ v | w ∈ W/λ,˜ Q ∈ Tˆ λ} 11 Tν ν w Q ν

For 1 ≤ i < n, if ℓ(siw) <ℓ(w), then

2 TiTw ⊗ vQ = Ti Tsiw ⊗ vQ −1 −1 −1 = q Tsiw ⊗ vQ + q hεi (−1) hεi (t)hεi+1 (t )Tw ⊗ vQ ∗ Xt∈Fq −1 −1 −1 = q Tsiw ⊗ vQ + q (wQ)i(−1) (wQ)i(t)(wQ)i+1(t )Tw ⊗ vQ ∗ Xt∈Fq

Since ℓ(siw) < ℓ(w) can hold only if (wQ)i , (wQ)i+1, the orthogonality of characters implies

−1 TiTw ⊗ vQ = q Tsiw ⊗ vQ + 0. (6.19)

If ℓ(siw) >ℓ(w), then there are two cases: Case 1: ℓ(siws j) <ℓ(siw) for some s j ∈ Wγ˜ , Case 2: ℓ(siws j) >ℓ(siw) for all s j ∈ Wγ˜ . 82

In Case 1,

TiTw ⊗ vQ = Tsiw ⊗ vQ = Tsiws j T j ⊗ vQ = Tsiws j ⊗ T jvQ −1 −1 = Tsiws j ⊗ Q j(−1)q CQ( j)vQ + Q j(−1)q (1 + CQ( j))vs j Q −1 −1 = Q j(−1)q CQ( j)Tsiws j ⊗ vQ + Q j(−1)q (1 + CQ( j))Tsiws j ⊗ vs j Q. (6.20)

In Case 2,

TiTw ⊗ vQ = Tsiw ⊗ vQ. (6.21) Make the identiﬁcation of (6.16) in equations (6.19), (6.20), and (6.21) to obtain the λ H11-action on H11 .

Sample Computations. If n = 10 for H11, then

T1v (ϕ1) (ϕ2) (ϕ3) = v (ϕ1) (ϕ2) (ϕ3) , 3 5 10 , 1 6 , 2 3 5 10 , 2 6 , 1 4 7 8 9 ! 4 7 8 9 !

−1 T2v (ϕ1) (ϕ2) (ϕ3) = q v (ϕ1) (ϕ2) (ϕ3) , 3 5 10 , 1 6 , 2 2 5 10 , 1 6 , 3 4 7 8 9 ! 4 7 8 9 ! −1 T3v (ϕ1) (ϕ2) (ϕ3) = −ϕ1(−1)q v (ϕ1) (ϕ2) (ϕ3) + 0, 3 5 10 , 1 6 , 2 3 5 10 , 1 6 , 2 4 7 8 9 ! 4 7 8 9 !

ϕ1(−1)q T4v (ϕ1) (ϕ2) (ϕ3) = v (ϕ1) (ϕ2) (ϕ3) 3 5 10 , 1 6 , 2 q + 1 3 5 10 , 1 6 , 2 4 7 8 9 ! 4 7 8 9 !

−1 q + ϕ1(−1) q + v (ϕ1) (ϕ2) (ϕ3) . q + 1 3 4 10 , 1 6 , 2 ! 5 7 8 9 !

A Character Table H11 for n = 2:

a b a b H • • •> • 11 >Ð>Ð • • •ÐÐ > •

(ϕi) ( ) ϕi(ab) ϕi(−ab)

(ϕ ) i −1 ϕi(ab) −ϕi(−ab)q (ϕi) (ϕ j) ( , ) ϕi(a)ϕ j(b) + ϕi(b)ϕ j(a) 0 83 Appendix A

Commutation Relations

The following relations are lifted directly from [Dem65, Proposition 5.4.3]. Let G be a finite Chevalley group over a finite field Fq with q elements, defined as in Section 2.2.2. Let R = R+ ∩ R− be as in Section 2.2.1. Let α, β ∈ R such that

β , −α and |α(Hβ)|≤|β(Hα)|.

Let l, r ∈ Z≥0 be maximal such that {β − lα,...,β − α,β,β + α,...,β + rα} ⊆ R. Note that l + r ≤ 3 [Hum72, Section 9.4]. Thus, the following analysis includes all the possible l and r values for α and β. • r = 0 implies xβ(b)xα(a) = xα(a)xβ(b), • l = 0 and r = 1 implies

xβ(b)xα(a) = xα(a)xβ(b)xα+β(±ab),

• l = 0 and r = 2 implies 2 xβ(b)xα(a) = xα(a)xβ(b)xα+β(±ab)x2α+β(±a b),

• l = 0 and r = 3 implies 2 3 3 2 xβ(b)xα(a) = xα(a)xβ(b)xα+β(±ab)x2α+β(±a b)x3α+β(±a b)x3α+2β(±a b ),

• l = 1 and r = 1 implies

xβ(b)xα(a) = xα(a)xβ(b)xα+β(±2ab),

• l = 1 and r = 2 implies 2 2 xβ(b)xα(a) = xα(a)xβ(b)xα+β(±2ab)x2α+β(±3a b)xα+2β(±3ab ),

• l = 2 and r = 1 implies

xβ(b)xα(a) = xα(a)xβ(b)xα+β(±3ab),

where a, b ∈ Fq and ±1 depends in part on the original choice of Chevalley basis made in Section 2.2.2. 84 Bibliography

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