Kcalvertthesis

Variants of Schur-Weyl duality and Dirac cohomology

Kieran Calvert Balliol College University of Oxford

A thesis submitted for the degree of Doctor of Philosophy

Hilary 2019 To the memory of my mum and the strength, support and love of all three of my parents. Acknowledgements

I would like to express great appreciation for my supervisor Prof. Dan Ciubotaru, whose guidance has been invaluable. His expert knowledge, patience, and perseverance cannot be understated. I am incredibly thankful for all of his help. Special thanks go to all of my oﬃce mates, especially Christoph, Craig, Emily, Marcelo, and Richard. Without Kit, Sam and Georgia I would not have made it to the start of my PhD, and without Thomas and SJ I certainly would not have made it to the end. I wish to thank my family: my parents, and my siblings for their seemingly unbounded support. All the phone conversations and encouraging messages were surprisingly necessary. And ﬁnally to the start of my new family, Lisa, who contains all of my limit points. Her unwavering encouragement, support, and enthusiasm have bolstered me when I have needed it the most. Abstract

This thesis is divided into the following three parts.

Chapter 1: Realising the projective representations of Sn We derive an explicit description of the genuine projective representations of the symmetric group Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. In [15] Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of Sn are related to the characters of elliptic graded modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties Be of g. We use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan’s morphism for Hecke algebras in type A using spectrum data of the Jucys-Murphy elements.

Chapter 2: Dirac cohomology of the Dunkl-Opdam subalgebra We define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This presentation uncovers the Dunkl-Opdam subalgebra as a Drinfeld algebra. The Dunkl-Opdam subalgebra is the first natural occurrence of a non-faithful Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We formalise generalised graded Hecke algebras and extend a Langlands classification to generalised graded Hecke algebras. We then describe an equivalence between the irreducibles of the Dunkl-Opdam subalgebra and direct sums of graded Hecke algebras of type A. This equivalence commutes with taking Dirac cohomology. Chapter 3: Functors relating nonspherical principal series

We define an extension of the affine Brauer algebra called the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group, ⊗k and it naturally acts on EndK (X ⊗ V ). We study functors Fµ,k from the category of admissible O(p, q) or Sp2n(R) modules to representations of the type B/C affine Brauer algebra. Furthermore, these functors take non-spherical principal series modules to principal series modules for the graded Hecke algebra of type Dk, Cn−k or Bn−k.

5 Contents

Introduction 1 Statement of main results ...... 5

Chapter 1: Realising the projective representations of Sn ...... 5 Chapter 2: Dirac cohomology of the Dunkl-Opdam subalgebra . . . .6 Chapter 3: Functors relating non-spherical principal series modules .8

0 Preliminaries 11 0.1 Root Systems, Weyl Groups and Coxeter groups ...... 11 0.2 Graded Hecke Algebras and Drinfeld algebras ...... 16 0.2.1 Rational Cherednik Algebras and the Dunkl-Opdam subalgebra 19 0.3 Superalgebras ...... 21 0.4 Dirac cohomology for Drinfeld algebras ...... 22 0.4.1 The Dirac operator ...... 24 0.4.2 Vogan’s Dirac morphism ...... 25 0.4.3 Dirac cohomology ...... 27 0.5 Closed Linear Groups ...... 28 0.6 Lie groups and Lie algebras ...... 29 0.6.1 Field extensions and Lie algebras ...... 30 0.6.2 Compact Lie groups ...... 32 0.6.3 Split, Compact forms and Cartan involutions ...... 32 0.6.4 Iwasawa Decomposition ...... 34 0.7 Real reductive linear groups ...... 35 0.8 Schur-Weyl Duality ...... 39 0.8.1 Schur’s double centraliser result...... 39 0.8.2 Brauer Algebras ...... 40

i 1 Realising the projective representations of Sn 44 1.1 Introduction ...... 46 1.2 Deﬁnitions ...... 47 1.3 Extended Dirac cohomology ...... 53 ˜ 1.4 Branching graph for Sn ...... 58 ˜ 1.5 Spectrum data for Sn ...... 62 1.6 Explicit representation from spectrum data ...... 67 1.7 Description of Vogan’s morphism ...... 71 1.7.1 Proof of Theorem 1.7.7 ...... 74

2 Dirac cohomology of the Dunkl-Opdam subalgebra 77 2.1 Introduction ...... 79 2.2 Drinfeld algebras ...... 81 2.2.1 Non-faithful Drinfeld algebras ...... 82 2.2.2 The Dirac operator for (non-faithful) Drinfeld algebras . . . . 83 2.3 Generalised graded Hecke algebras ...... 86 2.3.1 Preliminaries for the Langlands classiﬁcation ...... 87 2.3.2 The Langlands classiﬁcation for generalized graded Hecke algebras 89 2.4 Dunkl-Opdam subalgebra ...... 92 2.4.1 The rational Cherednik algebra ...... 92 2.4.2 Dunkl-Opdam quadratic operators ...... 93 2.4.3 Dunkl-Opdam subalgebra admits a non-faithful Drinfeld presentation ...... 94 2.4.4 Dunkl-Opdam subalgebra is a generalised graded Hecke algebra 100

2.5 Constructing the representations of H(G(m, 1, n)) from H(Sn).... 100 2.6 Dirac cohomology of the Dunkl-Opdam subalgebra ...... 107

3 Functors relating non-spherical principal series 112 3.1 Introduction ...... 114 3.2 Preliminaries ...... 117 3.3 Brauer Algebras ...... 121 θ 3.4 Quotients of the aﬃne type B/C Brauer algebra Bk[m]...... 126 θ 3.5 Functors from HC(G)-mod to Bk-mod...... 129 3.6 Restricting functors to principal series modules ...... 130 3.7 Images of principal series modules ...... 134

ii 3.8 Principal series modules map to principal series modules ...... 140 3.9 Hermitian forms ...... 148

Bibliography 159

iii Introduction

Root systems (Definition 0.1.5) were introduced by Killing [34] to classify semisimple complex Lie algebras. The Weyl group (Definition 0.1.20) is the reflection group acting on a crystallographic root system. Weyl groups are examples of finite real reflection groups which can be classified by, not necessarily crystallographic, root systems. The class of complex reflection groups contains all real reflection groups. These groups were classified by Shephard and Todd [52]. The underlying theme of this thesis is the study of several collections of algebras that are extensions of (complex) reflection groups. Examples include graded Hecke algebras, extended graded Hecke algebras, generalised graded Hecke algebra, Drinfeld algebras (both faithful and non-faithful), symplectic algebras, the rational Cherednik algebra, the Dunkl-Opdam subalgebra, and (affine) Brauer algebras. The results of this thesis split into three chapters, each one characterised by a distinct question. This introduction highlights the defining questions.

The secondary theme of this thesis is Dirac cohomology, and it is a tool that we will continually use. The ﬁrst applications of Dirac cohomology were developed for Lie algebras [47],[2]. Barbasch, Ciubotaru, and Trapa [4] deﬁned Dirac cohomology for graded Hecke algebras. An integral component of Dirac cohomology is the Vogan morphism (Theorem 0.4.24), which for graded Hecke algebras links the irreducible projective representations of the Weyl group with central characters of the graded Hecke algebra. Set H(W ) to be the graded Hecke algebra associated to a root system R with Weyl group W . Let X be an H(W )-module, then the Dirac cohomology

HD(X) is naturally a projective representation of W , that is, a representation of the double cover Wf of W . This leads us to our ﬁrst question:

1 (Q1) Vershik and Sergeev [56] described the projective representations of the

symmetric group Sn using Jucys-Murphy elements. Are we able to rediscover this exposition with Dirac cohomology?

This question forms the basis for Chapter 1. It is much easier to describe the branching graph of the projective modules if we utilise the extended graded Hecke algebra of the symmetric group and the Dirac index [15] of this algebra. One can endow the extended graded Hecke algebra with a superalgebra structure (Deﬁnition 0.3.2). In this case, the Dirac index is an interpretation of Dirac cohomology for Hecke algebras in supertheory. We, therefore, study projective supermodules of the symmetric group. One can easily recover the analogous results on the ungraded projective modules. Using the Dirac index and combinatorics in [26], we reconstruct the branching graph of the genuine projective representations of the symmetric group, thus, creating a model for the isomorphism classes of projective representations of the symmetric group. As a corollary to studying the connection between Dirac cohomology and the projective representations of the symmetric group, we can explicitly calculate the Vogan morphism in the case of the symmetric group.

Let G be a finite group with finite complex representation V. Drinfeld [21] introduced the class of Drinfeld algebras (Definition 0.2.8) for a complex representation V of a finite group G and particular skew symmetric forms. Lusztig [39] introduced graded Hecke algebras as the associated graded algebra of a filtration of an affine Hecke algebra. They are of interest since through this association the graded Hecke algebra plays the equivalent role of a Lie algebra to a Lie group for the affine Hecke algebra. Graded Hecke algebras (Definition 0.2.3) are associated to a Weyl group and a parameter function. By employing a different presentation of a graded Hecke algebra, one can show that graded Hecke algebras are examples of Drinfeld algebras. Other notable examples of Drinfeld algebras are symplectic algebras [24] associated to a finite symplectic group (Definition 0.1.31) and within these the rational Chered- nik algebras [22] (Definition 0.2.11) associated to a complex reflection group with parameter function. We note that graded Hecke algebras are defined for Weyl groups and are examples of Drinfeld algebras with orthogonal representations. One could ask, can we extend the

2 class of graded Hecke algebras to complex reﬂection groups? Chapter 2 is dedicated to answering the following question.

(Q2) Are there graded Hecke algebras associated to the classical complex reﬂection groups G(m, p, n) (Deﬁnition 0.1.25)?

Faithful Drinfeld algebras are the class of Drinfeld algebras restricted to faithful representations of a finite group G ⊂ GL(V ). Ram and Shepler [49] considered this question for faithful Drinfeld algebras. They prove that there is no Drinfeld algebra for G(m, p, n) with a faithful representation V of dimension n for m, n ≥ 3. We notice that Drinfeld’s original definition allows non-faithful representations. This nuance has largely been forgotten, perhaps because in practice there were no naturally occurring non-faithful Drinfeld algebras. We extend the search for Drinfeld algebras related to G(m, p, n) to include non-faithful representations. Ram and Shepler give a candidate for a ’Drinfeld-esque’ algebra associated to G(m, p, n). This algebra is isomorphic to the Dunkl-Opdam subalgebra HDO (Definition 0.2.15) of the rational Cherednik algebra associated to G(m, p, n). We construct new generators for the Dunkl-Opdam subalgebra which expose HDO as a non-faithful Drinfeld algebra. Therefore, we simultaneously support Ram and Shepler’s intuition and, consequently, discover a naturally occurring non-faithful Drinfeld algebra. Continuing on our sub-theme of Dirac cohomology, we extend Ciubotaru’s [13] Dirac cohomology for faithful Drinfeld algebras to non-faithful Drinfeld algebras. As a product, we define Dirac cohomology for HDO. Inspired by the construction of irreducible representations of the hyperoctahedral group (Example 0.1.4) which utilises induction from symmetric subgroups, we construct an equivalence of irreducibles between the Dunkl-Opdam subalgebra and direct sums of graded Hecke algebras of parabolic symmetric groups. Furthermore, this equivalence behaves well with respect to Dirac cohomology. We can transport the Dirac cohomology of an HDO-module X to the Dirac cohomology of the corresponding module of the graded Hecke algebra of parabolic symmetric group H(Sa0 )⊗...⊗H(Sam−1 ) (Definition 0.2.5). Dezelée[19] states that the Dunkl-Opdam subalgebra is a generalised graded Hecke algebra. We formalise this idea of generalised graded Hecke algebras into a formal

3 class of algebras. We extend Evens’ [25] Langlands classiﬁcation for graded Hecke algebras to generalised graded Hecke algebras.

Drinfeld algebras were originally introduced to deﬁne a Schur-Weyl duality with the

Yangian associated to sln. Schur-Weyl duality, [51, 60], was first discovered for gln and the symmetric group. This duality pairs gln and C[Sk] as double commutator algebras ⊗k acting on V for the matrix module V of gln. This motif was continued; Brauer [8] constructed the Brauer algebra Bk[m] (Definition 0.8.6) to be the commutant of the ⊗k special orthogonal group SOn(R), or the symplectic group Sp2n(R), on V . Arakawa and Suzuki [1] defined functors from category O of gln to modules of the graded Hecke algebra of the symmetric group H(Sk). This utilises a type of affine Schur-Weyl duality ⊗k on the space EndC(X ⊗ V ), where X ∈ O and V is the matrix module. These ⊗k functors rely on a map ρ : H(Sk) → Endgln (X ⊗ V ). The next step was to study, for ⊗k G = SOn(R) or Sp2n(R), the algebra EndG(X ⊗ V ), the affine Schur-Weyl duality for SOn(R) and Sp2n(R). The analogue of the graded Hecke algebra, in this case, is the affine Brauer algebra, Bk (Definition 3.3.2). The Lie groups SO2n+1(R) and Sp2n(R) are associated to root systems of type Bn and Cn respectively, with Weyl group W (Bn) the hyperoctahedral group. Disappointingly, the affine Brauer algebra,

Bk, is an extension of the symmetric group and is not of type B or C. This leads us to our ﬁnal question; the topic of ongoing work and the basis for Chapter 3.

(Q3) Is there a Schur-Weyl duality related to the real orthogonal and symplectic groups which contains the hyperoctahedral group in its commutator? That is, can we

ﬁnd an endomorphism space for O(p, q) and Sp2n(R) which exhibits a commutator algebra that is an extension of the hyperoctahedral group?

Ciubotaru and Trapa [16] deﬁned functors from spherical principal series modules

(Definition 3.6.1) of O(p, q) and Sp2n(R) to representations of the graded Hecke algebra of type Bn, Cn and Dn. Let G = O(p, q) or Sp2n(R), with maximal compact subgroup ⊗k K. In [16], there is implicit use of the algebra EndK (X ⊗ V ). We make this explicit and construct an extension of the affine Brauer algebra, using the Cartan involution θ. θ This extension is called the type B/C affine Brauer algebra Bk (Definition 3.3.10). ⊗k This new algebra maps to EndK (X ⊗ V ). The affine type B/C Brauer algebra is an extension of the hyperoctahedral group. This algebra’s representations are the natural image of the functors defined in [16]. Furthermore, with this new viewpoint, we can

4 θ uniformly construct functors from G-representations to Bk-modules. We extend the results in [16] to non-spherical representations. Non-spherical principal series modules for O(p, q) or Sp2n(R) map to principal series modules of graded Hecke algebras of θ type Dk, Bn−k or Cn−k. We also deﬁne a star operation on Bk which allows us to study unitary module of the type B/C aﬃne Brauer algebra. Importantly, this star operation descends to a star operation on the graded Hecke algebra via a quotient of θ Bk. We then go on to show that the functors preserve Hermitian forms and unitarity.

Statement of main results

We give an overview of the main results of this thesis.

Chapter 1: Realising the projective representations of Sn

In Chapter 1, we reconstruct the branching graph of the genuine projective super representation of the symmetric group. With this data, we then create an explicit model for every genuine projective representation of Sn.

A partition λ = (λ1, ..., λr) is strict if λi > λi+1 for all i = 1, ...., r − 1. The isomorphism classes of genuine irreducible projective supermodules of the symmetric group are parametrised by strict partitions. We utilise a transversal of these supermodules (Proposition 1.3.8): 1 {τ˜λ = X−1(λ) ⊗ S : λ is a strict partition}. aλ We reconstruct the branching graph for the irreducible genuine projective representation of Sn.

˜ Theorem 1.4.9. The branching rules of the genuine projective irreducible Sn-supermodules are:

 2˜τλ(1) ⊕ 2˜τλ(2) ⊕ · · · ⊕ 2˜τλ(r−1) ⊕ τ˜λ(r) if n − r is odd and λr = 1, n  Resn−1τ˜λ = 2˜τλ(1) ⊕ 2˜τλ(2) ⊕ · · · ⊕ 2˜τλ(r−1) ⊕ 2˜τλ(r) if n − r is odd and λr > 1,  τ˜λ(1) ⊕ τ˜λ(2) ⊕ · · · ⊕ τ˜λ(r−1) ⊕ τ˜λ(r) if n − r is even.

Here λ is always a strict partition, λ(i) depends on a reduction rule, deﬁned in [26], which gives partitions λ(i) from λ.

5 Following [56] we deﬁne two subsets of Nn, one related to the action of the Jucys- Murphy elements: Sspec(n, λ)(Deﬁnition 1.5.5).

The other is related to combinatorial content data on standard shifted tableaux:

Scont(n, λ)(Deﬁnition 1.5.7).

The main result of Chapter 1 is the following theorem, originally proved in [56]; we have reconstructed this theorem using Dirac cohomology. We use a new result (Lemma 1.5.12) relating the squares of the Jucys-Murphy elements and the square of the Dirac operator.

Theorem 1.5.13. The set Sspec(n) ⊂ Nn, descending from eigenvalues of Jucys- Murphy elements, is equal to the set Scont(n), a combinatorial construction from contents of shifted Young tableaux. Furthermore,

Sspec(n, λ) = Scont(n, λ).

In section 1.6 we describe how one gets from the spectrum data to an explicit ˜ ˜ description of the genuine projective supermodules of Sn and the action of Sn in matrix form. In Section 1.7, we give an explicit description of the Vogan morphism for graded Hecke algebras of type A introduced in [4].

Chapter 2: Dirac cohomology of the Dunkl-Opdam subalgebra

In this chapter we focus on a potential candidate for a Drinfeld algebra associated to the complex reﬂection group G(m, 1, n), the Dunkl-Opdam subalgebra HDO (Deﬁnition 0.2.15). We give a new presentation of the Dunkl-Opdam subalgebra which exhibits it as a non-faithful Drinfeld algebra.

Theorem 2.4.13. There exists a presentation of HDO given by elements {z˜i : i =

1, ..., n} and generators {si, gj : i ≤ i ≤ n − 1, 1 ≤ j ≤ n} in G(m, 1, n) such that:

−1 siz˜jsi = si(˜zj),

giz˜j =z ˜jgi ∀i, j = 1, ..., n,

[˜zi, z˜j] ∈ CG(m, 1, n), ∼ n where G(m, 1, n) = Sn o (Zm) . The group Sn is generated by transpositions si, th i = 1, ..., n − 1 and gj generates the j copy of Zm.

6 We deﬁne a class of algebras, the generalised graded Hecke algebras (Deﬁnition

2.3.1), which includes the algebra HDO and the generalised graded Hecke algebras of type B and D deﬁned in [19]. We extend the Langlands classiﬁcation, proved by Evens [25] for graded Hecke algebras, to generalised graded Hecke algebras.

Theorem 2.3.8. Let GH(W o T ) denote a generalised graded Hecke algebra. A parabolic subalgebra is denoted by GHP , with GHPS denoting the semisimple part of GHP (Deﬁnition 2.3.2). (i) Every irreducible GH-module V can be realised as a quotient of ˆ ˆ GH(W o T ) ⊗GHP U, where U = U ⊗ Cν is such that U is an irreducible tempered ∗+ GHPs -module and Cν is a character of S(a) deﬁned by ν ∈ a .

(ii) If U is as in (i) then GH(W o T ) ⊗GHP U has a unique irreducible quotient to be denoted J(P,U). 0 0 0 0 ˆ ∼ ˆ 0 ˆ ∼ ˆ (iii) If J(P, U ⊗ Cν) = J(P , U ⊗ Cν ) then P = P , U = U as GHPs -modules and ν = ν0.

We show that the semisimpliﬁcation of the module cateory of the Dunkl-Opdam subalgebra HDO(G(m, 1, n)) is equivalent to the semisimpliﬁcation of the module category of a direct sum of graded Hecke algebras for parabolic symmetric groups.

m Pm−1 Theorem 2.5.6. Let A = {a = (a0, .., am−1) ∈ N : i=0 ai = n}, note that N includes zero. The irreducible representations of H(G(m, 1, n)) split into blocks which are induced from products of H(Sa)-representations:

∼ G irr(H(G(m, 1, n))) = irr(H(Sa0 )) ⊗ ... ⊗ irr(H(Sam−1 )). a∈A

This bijection of irreducible modules deﬁnes an equivalence of the semisimpliﬁcation of the module categories, the Grothndieck groups, G ( (G(m, 1, n)) and G (L ⊗ 0 H 0 a∈A HSa0 ... ⊗ ). Hsam−1

The maps F and F −1 (Lemma 2.5.4) are functors that exhibit this equivalence. We then show that this equivalence behaves well with respect to Dirac cohomology; we can transport an HDO-module to its corresponding module under the equivalence and then take its Dirac cohomology. Dirac cohomology of a H module X is the kernel of a dirac operator D ∈ H ⊗ C(V ) modulo Ker D ∩ Im D.

7 Theorem 2.6.7. Given an irreducible -representation V then F (V ) as an ⊗ HDO HSa0 ... ⊗ -module is isomorphic to F (V ) ∼ X ⊗ ... ⊗ X . The Dirac cohomology HSam−1 = a0 am−1 of V is M c HD(Xa0 ) ⊗ ... ⊗ HD(Xam−1 ) , c∈C where HD(X) is the type A Dirac cohomology of the HSk -module X, P is the parabolic subset associated to a and C is a set of coset representatives of Sn/SP . Let HD(•) denote the functor taking the relevant module to its Dirac cohomology. We have the following commutative diagram:

G ( (G(m, 1, n))) F G (L ⊗ ... ⊗ ) 0 H 0 a∈A HSa0 HSam−1

HD(•) HD(•) L CG(^m, 1, n)-mod a∈A CSfa0 ⊗ ... ⊗ CS^am−1 -mod. Fg−1

Here G0(H(G(m, 1, n)) is th Grothendieck group of the isomorphism classes of representations of H(G(m, 1, n)).

Chapter 3: Functors relating non-spherical principal series modules

In this chapter we study Schur-Weyl duality for symplectic and orthogonal real groups. ⊗k Following on from [16], we study the algebra EndK (X ⊗V ) for (g,K) module X and the reflection representation V . This inspires the formulation of a new algebra, the θ type B/C affine Brauer algebra Bk. From the viewpoint of this algebra, the functors θ defined in [16] have a natural image category, Bk-mod. This new perspective allows us to highlight ideas used in [16] and extend these results. Fix G to be either O(p, q) or Sp2n(R). Let X be a G-representation, K the maximal compact subgroup of G and let µ be an irreducible K type. We study the functors defined by Ciubotaru and Trapa [16], ⊗k Fµ,k(X) = HomK (µ, X ⊗ V ).

θ The extension of the aﬃne Brauer algebra, Bk, is deﬁned using operators related to θ ⊗k the Cartan involution θ of G. This new algebra Bk acts on X ⊗ V and commutes with the action of K.

8 We show that the functors Fµ,k take the category of admissible G-modules to θ Bk-modules. The algebras Hk(c) are graded Hecke algebras associated to the hyperoctahedral group of rank k, with parameter function related to c ∈ C. ν Let Xδ (Deﬁnition 3.6.1) be a non-spherical principal series module for G. Given a ν principal series representation Xδ we associate to it characters µ, µ (Table 3.6.1 ) with ν scalars cµ and cµ (Table 3.7.1). We describe the image of Xδ under the functors Fµ,k and ∼ ∼ Fµ,n−k. As vector spaces Hk(c) = S(ak)⊗W (Bk) and Hn−k(cµ) = S(an−k)⊗W (Bn−k).

The character νk is the restriction of ν to ak and νn−k is the restriction to an−k.

ν Theorem 3.8.14. For G = Sp2n(R) or O(p, q) p + q = 2n + 1, the module Fµ,k(Xδ ) is isomorphic to the Hk(cµ) principal series module

Hk(cµ) X(νk) = Ind νk. S(ak)

ν The module Fµ,n−k(Xδ ) is isomorphic to the Hn−k(cµ) principal series module

Hk(cµ) X(¯νn−k) = Ind ν¯n−k. S(¯an−k)

For G = Sp2n(R) and O(n + 1, n), every principal series module is induced from a character. Hence we can entirely describe the Hecke algebra modules resulting from functors Fµ,k and Fµ,n−k of any principal series module. Casselman’s theorem [12] states that every irreducible representation in HC(G) is a subrepresentation of a principal series module.

Theorem 3.8.15. Let G be O(n + 1, n) or Sp2n(R), then G is split. Let X be an ν irreducible G-module. Let Xδ be a principal series representation that contains X, θ θ then the Bk and Bn−k-modules

Fµ,k(X) and Fµ,n−k(X) are submodules of the Hk(cµ) and Hn−k(cµ)-modules

X(νk) and X(¯νn−k).

θ We deﬁne two star-operations on Bk[m] which descend to the two star-operations on the graded Hecke algebra. Furthermore, we show that if X is a Hermitian (respectively unitary) module of G then the image of X under the functor Fµ,k is a Hermitian (resp. θ unitary) module for Bk[m]. We also show that the Langlands quotient is preserved.

9 ν Theorem 3.9.26. Let Xδ be a principal series module for G = O(p, q) or Sp2n(R). ν ν The Langlands quotient X = X / radh, i ν is mapped by F , to the Langlands δ δ Xδ µ,k quotient of the (c )-module, X(ν ) = X(ν )/ radh, i . Similarly, Xν is mapped Hk µ k k Xνk δ by Fµ,n−k, to the Hn−k(cµ)-module X(¯νn−k).

We then give a non-unitary test for non-spherical principal series modules.

Theorem 3.9.29. [Non-unitary test for principal series modules] If either X(νk) or X(¯νn−k) are not unitary, as Hk(cµ) and Hn−k(cµ)-modules, then the Langlands ν quotient of the minimal principal series module Xδk , for G = O(p, q) or Sp2n(R) is not unitary.

This result gives a functorial proof of the non-unitarity criterion proved with petite K-types by Barbasch, Pantano, Paul and, Salamanca-Riba [3, 46]. We also obtain a non-unitary test for any Harish Chandra module; in the split case one could check unitarity of Hecke algebra modules, but in general, one has to work with Brauer algebras.

Theorem 3.9.30. [Non-unitary test for Harish-Chandra modules] Let X be a Harish

Chandra module. For G = Sp2n(R) or O(p, q) p+q = 2n+1, if for any character µ and θ k = 1, .., n the Bk-module Fµ,k(X) is not unitary, then the Langlands quotient X of X is not a unitary G-module. In the case when G is split then X is a subrepresentation ν of Xδ and Fµ,k(X), Fµ,n−k(X) are Hecke algebra modules. In this case, if either

Fµ,k(X), Fµ,n−k(X) is not unitary as a Hecke algebra module then X is not unitary as a G-module.

10 Chapter 0

Preliminaries

This chapter is entirely preparatory. In Section 0.1 we introduce root systems, Weyl groups, real and complex reflection groups. Section 0.2 defines graded Hecke algebras and Drinfeld algebras. Section 0.3 is a very brief discussion on superalgebras and supermodules. We define Dirac cohomology for Drinfeld algebras in Section 0.4. Sections 0.5, 0.6 and 0.7 introduce the very basics of Lie theory. In Section 0.8 we discuss Schur-Weyl duality and introduce the Brauer algebra.

0.1 Root Systems, Weyl Groups and Coxeter groups

Definition 0.1.1. Let V be a real Euclidean vector space, i.e. V has a positive definite symmetric bilinear form (, ). A reflection s is a linear operator on V which sends a non-zero vector α ∈ V to its negative and fixes the hyperplane Hα orthogonal to α.

We denote this reﬂection by sα, (v, α) s (v) = v − 2 α. α (α, α)

The reﬂection sα is an orthogonal operator on V .

Definition 0.1.2. A finite reflection group is a finite subgroup of the orthogonal group,

O(V ), generated by reflections sα. We will denote a finite reflection group by W .

n Example 0.1.3. The symmetric group. Set n ≥ 2, let V = R with basis e1, ..., en, define the orthogonal transformation, sij, by the permutation which exchanges ei and ej and fixes all other basis vectors. The subgroup of O(V ) generated by the permutation matrices sij is the symmetric group Sn. Set αij = ei − ej. The transposition sij is the reflection sαi,j . The symmetric group is a finite reflection group.

11 n Example 0.1.4. Again set V = R with basis e1, ..., en. The hyperoctahedral group

W (Bn) is the subgroup of O(V ) generated by sαjk and sei for i = 1, .., n and 1 ≤ i < j ≤ n. The group W (Bn) is isomorphic to the semidirect product:

∼ n W (Bn) = Sn o (Z2) ,

n where Sn acts on (Z2) by permuting the entries.

Let Φ be a ﬁnite subset of V , such that

Φ ∩ Rα = {α, −α} for all α ∈ Φ,

sα(Φ) = Φ for all α ∈ Φ.

Deﬁnition 0.1.5. Let Φ be as above and deﬁne W to be the subgroup O(V ) generated by sα for all α ∈ Φ. Then the pair (Φ,W ) is a root system and Φ is the set of roots of W .

Note that we do not require a root system to be integral, this will be included in the deﬁnition of crystalographic root system. Fix a total ordering < on V . Set the positive roots Φ+ to be {α ∈ Φ : 0 < α}.

Definition 0.1.6. Given a finite reflection group W with roots Φ we say ∆ is a set of simple roots if the vectors in ∆ are linearly independent and

+ Φ = spanN(∆).

Deﬁnition 0.1.7. For a root system (W, Φ, ∆) deﬁne

1 X ρ = α. 2 α∈Φ+

Theorem 0.1.8. [31, Theorem 1.5] The finite reflection group W , with a simple system ∆ ⊂ Φ is generated by the simple reflections

{sα : α ∈ ∆}.

Deﬁnition 0.1.9. Let ∆P be a subset of ∆, we deﬁne the parabolic subgroup WP of

W to be the group generated by sα, α ∈ ∆P . Let ΦP = spanN ∆P ∩ Φ then WP , ∆P , ΦP is a root system.

12 Definition 0.1.10. Let W be a finite reflection group, Φ roots for W and simple roots ∆. For a group element w ∈ W we define the length of w ∈ W to be:

l(w) = min{j : w = sα1 sα2 ...sαj for αi ∈ ∆}.

Deﬁnition 0.1.11. Let w0 be an element of longest length in W .

The longest element w0 is unique for any ﬁxed ∆, it is an involution. Deﬁne m(α, β) to be the order of sαsβ in W .

Theorem 0.1.12. [31, Theorem 1.9] Fix W and ∆ ⊂ Φ, then W is isomorphic to the abstract group generated by sα, α ∈ ∆ subject to the relations

m(α,β) (sαsβ) = e for α, β ∈ ∆.

By Theorem 0.1.12 a given finite reflection group W is parametrised by ∆ and m(α, β). We associate to this data a diagram. First we define a general Coxeter graph.

Deﬁnition 0.1.13. A Coxeter graph Γ is a set of vertices S with edges between s and s0 labelled by integers m(s, s0) ≥ 3. If s and s0 ∈ S are not connected by an edge we set m(s, s0) = 2.

Definition 0.1.14. Let W be a finite reflection group with roots Φ and simple roots ∆. Define the Coxeter graph of W to have vertices α ∈ ∆ and edges between α 6= β when m(α, β) ≥ 3. Label such an edge by m(α, β).

Proposition 0.1.15. [31, Proposition 2.2] Let W1 and W2 be ﬁnite reﬂection groups with the same Coxeter graphs, then there is isometry of spaces V1 and V2 inducing an isomorphism between W1 and W2.

Deﬁnition 0.1.16. A Coxeter System (W, Φ, ∆) is irreducible if its associated Coxeter graph is connected.

If S has cardinality n we deﬁne a symmetric n × n matrix A such that π A(s, s0) = − cos for all s, s0 ∈ S. m(s, s0)

Deﬁnition 0.1.17. The Coxeter Graph Γ is positive if the associated matrix A is positive deﬁnite, that is xtAx > 0 for x ∈ Rn. The Coxeter graph is semi positive if xtAx ≥ 0.

13 Theorem 0.1.18. [31, Theorem 2.7] The following graphs are the only connected Coxeter graphs of positive type.

4 Bn

E8 4 F4 n Gn 5 H3 5 H4 m I2(m)

Deﬁnition 0.1.19. A lattice in a vector space V is the Z span of a basis. A group G ⊂ O(V ) is said to be crystallographic if it stabilises a lattice in V .

Deﬁnition 0.1.20. A Weyl group is a reﬂection group that is also crystallographic.

Proposition 0.1.21. [31, Proposition 2.8] If W is crystallographic, then for every α 6= β, m(α, β) is either, 2, 3, 4 or 6.

A Root system Φ is crystallographic if

2(α, β) ∈ for all α, β ∈ Φ. (α, α) Z

For Weyl groups there are at most two diﬀerent lengths of roots.

Deﬁnition 0.1.22. A Dynkin diagram is a Coxeter graph which is directed.

To the associated Coxeter graph Γ of W , a Weyl group, we mandate that the edge m(α, β) is directed from β to α if (α, α) < (β, β). This endows the Coxeter graph of a Weyl group as a Dynkin Diagram.

14 Theorem 0.1.23. [31, Theorem 2.8] The following graphs are the only connected Dynkin diagrams. An Bn Cn

E8 F4 G2 The diagrams are in one-to-one correspondence with crystallographic root systems.

Definition 0.1.24. Let V be a complex Euclidean vector space i.e. V has a positive definite Hermitian form (, ). A complex reflection s is a linear operator on V which sends a non-zero vector α ∈ V to λα for a root of unity λ ∈ C \ 1 and fixes the hyperplane Hα orthogonal to α. We may denote this complex reflection by sα,λ, on V : (v, α) s (v) = v − (1 − λ) α. α (α, α) Definition 0.1.25. A complex reflection group is a finite subgroup of GL(V ) which is generated by complex reflections.

Deﬁnition 0.1.26. Let m, p, n ∈ N such that p|m. Let η be a primitive mth root of unity. Deﬁne the abelian group:

n a1 a2 an X Ap,m = {(η , η , ..., η ): ai = 0 mod p}. i=1

n The symmetric group naturally acts on (Zm) and Ap,m by permuting the coordinates. Deﬁne the group G(m, p, n)

G(m, p, n) = Sn o Ap,m.

The group G(m, 1, n) is a complex reﬂection group acting on V = Cn, generated by Sn and sei,η for i = 1, ..., n.

Theorem 0.1.27. [52, Table VII] All irreducible complex reﬂection groups are of the form G(m, p, n) or one of 34 exceptional types, labelled by Shephard-Todd numbers 4 − 37.

15 ∼ Remark 0.1.28. The group G(1, 1, n) = Sn and G(2, 1, n), G(2, 2, n) are isomorphic ∼ to W (Bn) and W (Dn) respectively. The cyclic group Zm = G(m, 1, 1).

Deﬁnition 0.1.29. A C-vector space V is symplectic if it has a bilinear form ω : V × V → C such that:

ω(v0, v) = −ω(v, v0) for all v, v0, ∈ V,

ω(u, v) = 0 for all u ∈ U implies v = 0.

The symplectic group on V is the group

Sp(V ) = {A ∈ GL(V ): ω(Av, Av0) = ω(v, v0) for all v, v0 ∈ V }.

Deﬁnition 0.1.30. Let V be a symplectic C-vector space with symplectic form ω.

A symplectic reﬂection is an element s ∈ Sp(V ) such that rk(IdV −s) = 2. For a symplectic reﬂection we can decompose V = ker(IdV −s) ⊕ Im(IdV −s) and this decomposition is orthogonal with respect to ω.

Definition 0.1.31. Let V be a symplectic vector space, a finite subgroup G ⊂ Sp(V ) is a symplectic reflection group if it can be generated by symplectic reflections.

Remark 0.1.32. Let G = G(m, p, n) be a classical complex reﬂection group acting on the vector space h = Cn, let h∗ denote its dual. Set V = h ⊕ h∗ and endow V with a symplectic form deﬁned by

0 Id ω(x, y) = xt n y for all x, y ∈ V = h ⊕ h∗. − Idn 0

The group G(m, p, n) acts on V via its actions on h and h∗. The complex reflection group G(m, n, p) is a subgroup of Sp(V ) generated by symplectic reflections, rendering G(m, p, n) a symplectic reflection group.

0.2 Graded Hecke Algebras and Drinfeld algebras

k z }| { Given a C-vector space U, we deﬁne the kth tensor product T k(U) to be U ⊗ ... ⊗ U and the tensor algebra T (U) is

∞ M T (U) = T k(U), k=0

16 with multiplication deﬁned by the canonical isomorphism:

T l(U) ⊗ T k(U) → T k+l(U).

The symmetric algebra S(U) is the quotient of T (U) by the two sided ideal generated by

u1u2 − u2u1 for all u1, u2 ∈ U.

The alternating algebra Λ(U) is the quotient of T (U) by the ideal generated by the relations

u1u2 + u2u1 for all u1, u2 ∈ U.

Deﬁnition 0.2.1. Root datum is a tuple (X, Φ, X,ˇ Φˇ, ∆) such that X and Xˇ are free abelian groups with a perfect pairing

h, i : X × Xˇ → Z.

The roots Φ and coroots Φˇ are ﬁnite subsets of X and Xˇ which are in bijection

Φ ↔ Φˇ,

α ↔ α,ˇ

hα, αˇi = 2 for all α ∈ Φ.

The reﬂection sα must preserve Φ:

sα(Φ) = Φ for all α ∈ Φ,

+ The set ∆ ⊂ Φ is the set of simple roots such that if we deﬁn Φ = spanN ∆ and Φ− = {−α : α ∈ Φ+} then Φ = Φ+ ∪ Φ−.

∗ ˇ Define the C-vector spaces t = X ⊗Z C and t = X ⊗Z C and extend the pairing h, i to t × t∗. We may also denoted t by V . The Weyl group associated to (X, Φ, X,ˇ Φˇ, ∆) is the finite subgroup of GL(t) generated by the reflections sα for α ∈ Φ. Using the pairing h, i we can describe the reflection sα as sα(x) = x − hx, αˇiα.

Deﬁnition 0.2.2. Let W be a Weyl group associated to the root datum (X, Φ, X,ˇ Φˇ, ∆). A function c :Φ → C is a parameter function if it is W invariant, i.e.

c(α) = c(w(α)) for all w ∈ W, α ∈ Φ.

17 Definition 0.2.3. [38, Definition 0.1] Let (X, Φ, X,ˇ Φˇ, ∆) be root datum with associated Weyl group W and parameter function c. The graded Hecke algebra H associated to this data is the C-vector space S(t) ⊗ C[W ] with a structure of an associative C-algebra defined by the relations:

S(t) → H is an injective algebra homomorphism,

C[W ] → H is an injective algebra homomorphism, x − s (x) s · x − s (x)s = c(α) α . α α α α

Remark 0.2.4. One notes that x−sα(x) = x−x+hx, αˇiα = hx, αˇiα is a multiple of α.

x−sα(x) Hence α = hx, αˇi ∈ C and we can, and will, replace the relation sα ·x−sα(x)sα = x−sα(x) c(α) α by

sα · x − sα(x)sα = c(α)hx, αˇi.

Deﬁnition 0.2.5. For a parabolic subgroup WP of W one can deﬁne a parabolic subalgebra HP of the graded Hecke algebra H. This is the subalgebra generated by S(t) and the parabolic subgroup WP .

Proposition 0.2.6. [39, Proposition 4.5] The centre of H is the W invariant elements of S(V ), S(t)W .

For associative algebras A and B let A?B be the free product of A and B. The algebra A?B has a basis of words a1b1...anbn, where {aj} is a basis of A and {bk} is a basis of B. For a group algebra C[G] with representation (U, ρ) we deﬁne the semi direct product C[G] o T (U) by the quotient of the associative C-algebra C[G] ?T (U) by the relations: g · u · g−1 = ρ(g)(u) for all g ∈ G, u ∈ U.

Definition 0.2.7. Let G be a finite subgroup of GL(V ). Associate to every g ∈ G, a skew symmetric bilinear form bg : V × V → C. Define the algebra H(bg) associated to G and the set {bg} to be the quotient of the C-algebra C[G] o T (V ) by the relations X [u, v] = bg(u, v)g. g∈G

We define a filtration on h(bg), let v ∈ V have degree 1 and g ∈ G have degree 0, extend this to a filtration on H.

18 Definition 0.2.8. Let G be a finite subgroup of GL(V ) with skew symmetric forms bg for g ∈ G. The algebra H(bg) is a faithful Drinfeld algebra if it satisfies the PBW property, that is, the associated graded algebra of H is isomorphic to C[G] n S(V ).

g ⊥ Let G(b) = {g ∈ G : bg 6= 0} and let (V ) = {v − g(v): v ∈ V }.

Proposition 0.2.9. [21],[49, Theorem 1.9] The algebra H is a Drinfeld algebra if and only if the following properties hold:

(1) bh−1gh(u, v) = bg(h(u), h(v)), for all h ∈ G, u, v ∈ V, g (2) ker bg = V , for all g ∈ G(b) \ 1, (3) dim V g = dim V − 2, for all g ∈ G(b) \ 1,

(4) det(h|(V g)⊥ ) = 1, for all h ∈ ZG(g) .

Lemma 0.2.10. The graded Hecke algebra associated to (X, Φ, X,ˇ Φˇ, ∆) with parameter function c is isomorphic to the faithful Drinfeld algebra with W ∼= G ⊂ GL(X) with skew symmetric forms:

X ˇ ˇ bw(x1, x2) = c(α)c(β) hx1, αˇihx2, βi − hx2, αˇihx1, βi . + w=sαsβ :α,β∈Φ

Proof. Denote a degree one element in T (t) ⊂ H by x˜ and a degree one element in S(t) ⊂ H by x. We deﬁne an isomorphism from the Drinfeld algebra to the graded Hecke algebra φ : H → H by 1 X φ(˜x) = x − c(β)hx, βˇis for allx ˜ ∈ t ⊂ H, 2 β β∈R

φ(w) = w for all w ∈ W.

0.2.1 Rational Cherednik Algebras and the Dunkl-Opdam subalgebra

Let h be an n dimensional C-vector space, let h∗ be its dual. Define V = h ⊕ h∗ with symplectic form ω defined in Remark 0.1.32. The complex reflection group G(m, p, n) is a symplectic reflection group inside Sp(V ).

19 ∗ For a reﬂection s ∈ G(m, p, n) denote by αs and αˇs non-zero vectors in Im(Id −s)|h and Im(Id −s)|h. There exist λs 6= 1 a root of unity such that

−1 s(ˇαs) = λsαˇs, s(αs) = λs αs.

Let S ⊂ G be the set of symplectic reﬂections in G.

Deﬁnition 0.2.11. [22] Let t ∈ C, V = h ⊕ h∗, set G = G(m, p, n) ⊂ Sp(V ), set c : S → C be a G invariant parameter function. The rational Cherednik algebra Ht,c(G(m, p, n)) is the quotient of the algebra C[G(m, p, n)] o T (V ) by the relations

∗ [y1, y2] = 0 = [x1, x2] for all y1, y2 ∈ h, x1, x2 ∈ h ,

X ω(y, αz)ω(ˇαs, x) [y, x] = tω(y, x) − c(s) for all y ∈ h, x ∈ h∗. ω(ˇαs, αs) s∈S Remark 0.2.12. The rational Cherednik algebra is a particular type of symplectic reﬂection algebra deﬁned in [24]. It is also a Drinfeld algebra by considering G(m, p, n) ⊂ GL(h ⊗ h∗) and skew symmetric forms

tω(x, y) if g = 1,  b (x, y) = c(s) ω(x,αs)ω(ˇαs,y) if g = s ∈ S, g ω(ˇαs,αs) 0 otherwise, for x, y ∈ V = h ⊕ h∗.

For the rest of the section fix G = G(m, 1, n) and η to be an mth primitive root. We give a different definition for the rational Cherednik algebra of G(m, 1, n). Given

G(m, 1, n) acting on h let {xi} ⊂ h be a basis such that w(xi) = xw(i) for w ∈ Sn and

( −1 η xj if i = j, gi(xj) = . xj otherwise

∗ Let {y1, ..., yn} ∈ h be the dual basis to {x1, ..., xn}.

Remark 0.2.13. The rational Cherednik algebra for G(m, 1, n), Ht(G(m, 1, n)) can be formulated as the quotient of the C-algebra C[G(m, 1, n)] o T (V ⊕ V ∗) by the relations

[xi, xj] = [yi, yj] = 0, Pm−1 P −l l Pm−1 l −1 l [xi, yi] = t − l=1 i6=j c(αij)si,jgi gj − l=1 c(gi)(1 − η )gi, Pm−1 −l l [xi, yj] = c(αij) l=1 si,jgi gj.

20 The parameter function c is G invariant and hence is uniform on all αij. We will write c(α) for c(αij).

Definition 0.2.14. [22, Definition 3.7] For 0 < i ≤ n define Dunkl operators zi in

Ht(G(m, 1, n))

Pm−1 P −l l 1 Pm−1 l l 1 zi = yixi + c(α) l=1 i>j si,jgi gj + 2 l=1 c(gi)gi + 2 t, Pm−1 P −l l 1 Pm−1 l l 1 = xiyi − c(α) l=1 i

Deﬁnition 0.2.15. The Dunkl-Opdam subalgebra HDO(G(m, 1, n)) of the rational

Cherednik algebra is the subalgebra generated by G(m, 1, n) and zi for i = 1, ...n.

Remark 0.2.16. [40, Lemma 3.2] The following relations hold in HDO(G(m, 1, n))

[zi, zj] = 0 for i, j = 1, ..., n,

[zi, gk] = 0 for i, k = 1, ...n,

[zj, si,i+1] = 0 for j 6= i, i + 1,

zisi,i+1 = si,i+1zi+1 − c(α)ij for i = 1, ..., n − 1.

Pm−1 l −l Here ij = l=1 gigj .

In fact HDO(G(m, 1, n)) is isomorphic to the C associative algebra generated by zi and G(m, 1, n) subject to the relations stated in Remark 0.2.16.

0.3 Superalgebras

Deﬁnition 0.3.1. A vector superspace is a Z2-graded C-vector space V = V0 ⊕ V1. If dim V0 = n0 and dim V1 = n1 then sdim V = (n0, n1), dim V = n0 + n1.

Elements of V0 are called even and elements of V1 are called odd. Even and odd elements are called homogeneous and have degree 0 or 1 respectively. A subspace U of V is a super subspace of V if it is homogenous, that is:

U = (U ∩ V0) ⊕ (U ∩ V1).

Deﬁnition 0.3.2. A superalgebra is a vector superspace A which is an associative unital C-algebra such that AiAj ⊂ Ai+j for i, j ∈ Z2.

A superideal of A is an ungraded ideal which is also a super subspace.

Deﬁnition 0.3.3. Let A be a superalgebra. A (left) A supermodule is a superspace V which is a left A-module in the ungraded sense, such that AiVj ⊂ Vi+j for i, j ∈ Z2.

21 Deﬁnition 0.3.4. A homomorphism of A supermodules is a linear map f : V → W such that f(av) = (−1)deg(f) deg(a)af(v), for all a ∈ A, v ∈ V.

An isomorphism of supermodules is a bijective homomorphism of supermodules.

0.4 Dirac cohomology for Drinfeld algebras

Let V be a complex vector space and let G be a ﬁnite subgroup of GL(V ). We must assume there is a G invariant nondegenerate symmetric bilinear form on V , denoted (, ). Therefore G ⊂ O(V ) for the corresponding orthogonal group.

Deﬁnition 0.4.1. [41, 3.7.1] The Cliﬀord algebra C(V ) associated to V and (, ) is the quotient of the algebra T (V ) by the relations

v1v2 + v2v1 = ε2(v1, v2).

Where ε is either uniformly 1 or −1.

Note that since C(V ) is defined as a C-algebra then whether ε is 1 or −1 makes no difference on the structure, however it does alter a few equations up to sign. We will be clear in each chapter whether ε equals 1 or −1. Let v ∈ V have degree 1 then the Clifford algebra has a filtration Cn(V ) = {c ∈ C(V ): deg(c) ≤ n}. The associated graded algebra is isomorphic to Λ(V ). Set

C(V )0 = {c ∈ C(V ) : deg(c) = 0 mod 2} and

C(V )1 = {c ∈ C(V ) : deg(c) = 1 mod 2}.

The Cliﬀord algebra has a Z2 grading

C(V ) = C(V )0 ⊕ C(V )1.

The Clifford algebra can be considered as a super algebra with this Z2 grading. Definition 0.4.2. The transpose of C(V ) is the anti-involution defined by

t : C(V ) → C(V ):

vt = −v for all v ∈ V,

(ab)t = btat for all a, b ∈ C(V ).

22 Deﬁnition 0.4.3. Deﬁne an algebra automorphism : C(V ) → C(V ) such that = Id on C(V )0 and = − Id on C(V )1.

Deﬁnition 0.4.4. [41, Deﬁnition 3.8] The Pin group is a subgroup of the units of C(V ): P in(V ) = {a ∈ C(V )× : (a) · V · a−1 ⊂ V, at = a−1}.

The group P in(V ) is a double cover of O(V ) and the projection p : P in(V ) → O(V ) is given by p(a)(v) = (a)va−1, a ∈ P in(V ), v ∈ V.

Let W ⊂ O(V ) be a reﬂection group. That is it is generated by reﬂection sv for some (x,v) v ∈ V such that sv(x) = x − 2 (v,v) v. Then

−1 1 p (sv) = ± v ∈ P in(V ). p(v, v)

Deﬁnition 0.4.5. For a subgroup G ⊂ O(V ) deﬁne the double cover of G:

Ge = p−1(G) ⊂ P in(V ).

n Example 0.4.6. The symmetric group Sn acts on the vector space V = C , which −1 has basis ei. It is generated by reﬂections sαij for αij = ei − ej. The group p (Sn) is 1 generated by 2 αij ∈ C(V ). As an abstract group it is the group generated by {αi,i+1, z} subject to the relations :

2 2 z = e, αi,i+1 = z, αi,i+1αi+1,i+2αi,i+1 = αi+1,i+2αi,i+1αi+1,i+2, z is central.

There is a second non-trivial double cover of the symmetric group with presentation

0 2 02 0 0 0 0 0 0 {z, αi,i+1 : z = e, αi,i+1 = e, αi,i+1αi+1,i+2αi,i+1 = zαi+1,i+2αi,i+1αi+1,i+2z is central.}

These groups are not isomorphic but their group algebras over C are, one can construct √ 0 an isomorphism by sending αi,i+1 to −1αi,i+1.

Definition 0.4.7. [41, Definition 3.4] Let V be a complex finite-dimensional vector space with non-degenerate symmetric bilinear form (, ).A Z2-graded (resp. ungraded) spinor module S over C(V ) is a graded (resp. ungraded) module (S, ρ) such that

ρ : C(V ) → End(S) is an isomorphism of super (resp. ungraded) algebras.

23 Theorem 0.4.8. [41, Theorem 3.10] Let V be an n dimensional complex vector space with non-degenerate symmetric bilinear form. Suppose n is even then there is:

a unique isomorphism class of irreducible ungraded C(V ) spinor modules, a unique class of irreducible Z2-graded C(V ) spinor modules, two isomorphism classes of irreducible ungraded C(V )0-modules.

Suppose n is odd, then there is:

two isomorphism classes of ungraded C(V ) spinor modules, a unique isomorphism class of Z2-graded C(V ) spinor modules, two isomorphism classes of irreducible ungraded C(V )0 spinor modules.

0.4.1 The Dirac operator

Recall Deﬁnition 0.2.8 of a Drinfeld algebra. Let H denote a Drinfeld algebra associated to G ⊂ O(V ) with skew symmetric forms bg. Let us extend the automorphism : C(V ) → C(V ) to : H ⊗ C(V ) by setting to be the identity on H. The Drinfeld algebra is the quotient of the associate algebra T (V ) o C[G]. For v ∈ V we denote the corresponding element in T (V ) (and H) byv ˜.

i Deﬁnition 0.4.9. [13, 2.3] Let {vi} and {v } be dual bases of V , with respect to the form (, ). The Dirac element associated to H and C(V ) is

X i D = v˜i ⊗ v ∈ H ⊗ C(V ). i The deﬁnition of the Dirac element does not depend on the basis chosen.

Deﬁnition 0.4.10. Deﬁne a group homomorphism ∆ from the subgroup G˜ of P in(V ) to H ⊗ C(V ): ∆(˜g) = p(˜g) ⊗ g,˜ g˜ ∈ G.e

i Deﬁnition 0.4.11. Let {vi} and {v } be dual bases of V . Deﬁne

X i h = v˜iv˜ ∈ H. i

For every g ∈ G(b) = {g ∈ G : bg 6= 0}, deﬁne

X i i κg = bg(vi, v )v vj ∈ C(V ). i,j

24 Lemma 0.4.12. [13, Lemma 2.5] In H ⊗ C(V ) 1 X D2 = −h ⊗ 1 + g ⊗ κ . 2 g g∈G(b)

g Lemma 0.2.9 stipulated that dim V = dim V − 2 for all g ∈ G(b) = {g ∈ G : bg 6= 0}.

Lemma 0.4.13. [13, Lemma 2.6] Any element g ∈ G(b) can be written as a product g⊥ of two reﬂection g = sαsβ, where α, β ∈ V are linearly independent and (α, α) = (β, β) = 1.

Deﬁnition 0.4.14. Let g ∈ G(b), Lemma 0.4.13 states g = sαsβ, deﬁne

g˜ = αβ ∈ C(V ),

bg(α,β) cg˜ = 1−(α,β)2 , bg(α,β)(α,β) eg = 1−(α,β)2 .

Furthermore deﬁne,

P G ΩH = h − g∈G(b) egg ∈ H , Ω = P c g˜ ∈ [G]Ge ⊂ C(V ). Ge g∈G(b) g˜ C e Theorem 0.4.15. [13, Theorem 2.7] The square of the Dirac element equals, in H ⊗ C(V ): 1 D2 = −Ω + 1 ⊗ κ + ∆(Ω ). H 2 1 Ge 0.4.2 Vogan’s Dirac morphism

The Cliﬀord algebra C(V ) is Z2-graded, C(V ) = C(V )0 ⊕ C(V )1. The automorphism is Id on C(V )0 and − Id on C(V )1. We extend to be Id on H creating an automorphism of H ⊗ C(V ).

Deﬁnition 0.4.16. Deﬁne the linear map

d : H ⊗ C(V ) → H ⊗ C(V ) by d(a) = Da − (a)D, a ∈ H ⊗ C(V ).

Lemma 0.4.17. [13, Lemma 3.2] ∆(C[Ge]) ⊂ ker d.

25 P G Definition 0.4.18. Recall Definition 0.4.14 sets ΩH to be h − g∈G(b) egg ∈ H , then define 1 Ge Ωe = Ω ⊗ 1 − 1 ⊗ κ1 ∈ ( ⊗ C(V )) = ⊗ C(V ). H H 2 H H

Deﬁne the algebra A to be the centraliser of ΩeH,

A = ZH⊗C(V )(ΩeH).

Remark 0.4.19. If 1 ∈/ G(b), then

A = H ⊗ C(V ).

Lemma 0.4.20. [13, Lemma 2.4] For every g˜ ∈ Ge we have the invariance property:

∆(˜g)D∆(˜g−1) = det(p(˜g))D.

Remark 0.4.21. The operator D interchanges the G˜ det-isotypic space and the triv- isotypic space.

Proof. Lemma 0.4.20 proves that ∆(g˜)D∆(g˜−1) = det(p(g˜))D. Suppose D is acting triv ˜ on a H ⊗ C(V )-representation U and utriv ∈ U a G triv-isotypic space. Then

∆(˜g)utriv = utriv, by deﬁnition of an isotypic space. Let Dutriv be the image of D of u. We show det triv −1 Dutriv ∈ U . Since utriv ∈ U then ∆(˜g )utriv = utriv.

−1 ∆(˜g)Dutriv = ∆(˜g)D∆(˜g )utriv then Lemma 0.4.20 shows

∆(˜g)Dutriv = det(p(˜g))Dutriv

det det −1 −1 hence Dutriv ∈ U . Suppose udet ∈ U then ∆(˜g )udet = det(p(˜g ))udet.

−1 −1 ∆(˜g)Dudet = ∆(˜g)D∆(˜g ) det p(˜g ))udet then Lemma 0.4.20 shows

−1 ∆(˜g)Dudet = det(p(˜g)) det(p(˜g ))Dudet = Dudet.

triv hence Dudet ∈ U .

26 Remark 0.4.21 shows that D interchanges the det and triv-isotypic spaces of a H ⊗ C(V )-representation. Therefore d(a) = Da − (a)D interchanges the det and triv-isotypic spaces.

Deﬁnition 0.4.22. Set A = ZH⊗C(V )(ΩeH), then restrict the map d:

triv det det triv dtriv : A → A , ddet : A → A .

Theorem 0.4.23. [13, Theorem 3.5] The kernel of dtriv equals:

Ge ker dtriv = Im ddet ⊕ ∆(C[Ge] ).

This is proved for Drinfeld algebras in [13], for graded Hecke algebras in [4].

Theorem 0.4.24 (Vogan’s Dirac homomorphism). The projection ζ : ker dtriv → C[Ge]Ge deﬁned by Theorem 0.4.23 is an algebra homomorphism.

Remark 0.4.25. Suppose B is a ﬁnitely generated abelian subalgebra of A∩(H ⊗ C(V )0) such that ΩeH ∈ B ⊂ ker dtriv. The element D commutes with every element of B. The homomorphism ζ in Theorem 0.4.24 deﬁnes a morphism

∗ G ζ : Irr(Ge) = Spec C[Ge] e → Spec B.

If G(b) does not contain 1 then

∗ Ge ∼ ∗ W ζ : irr(Ge) = Spec C[Ge] → Spec S(V ) = S(V ) .

0.4.3 Dirac cohomology

Let (X, π) be an H-module and let (S, ρ) be an ungraded spinor module for C(V ) (Definition 0.4.7). The Dirac element D is an element in H ⊗ C(V ). Define the Dirac operator, denoted by D, to be the operator π ⊗ ρ(D) on X ⊗ S. Definition 0.4.26. [4, Definition 4.1] The Dirac cohomology of X (and S) is

ker(D) HD(X) = . ker(D) ∩ Im(D)

Remark 0.4.27. Lemma 0.4.20 states that ∆(g˜)D∆(g˜−1) = det(p(g˜))D. Therefore ker D is a Ge sgn-invariant subspace of X ⊗ S, similarly for Im D ∩ ker D. The Dirac cohomology HD(X) can be considered as a Ge-module.

27 0.5 Closed Linear Groups

Deﬁnition 0.5.1. We denote by GL(n, R) the group of nonsingular n by n real matrices with matrix multiplication. Similarly GL(n, C) denotes the group of nonsingular complex matrices.

The group GL(n, R) and GL(n, C) have topologies deﬁned by identifying them 2 2 with subsets of Rn and R2n respectively. The set of nonsingular matrices is the open subset deﬁned by the polynomial det being non-zero.

Deﬁnition 0.5.2. A closed linear group is a closed subgroup of GL(n, R). Example 0.5.3. The groups GL(n, C) and GL(n, R) are closed linear groups. For a matrix x write xt for its transpose and x∗ for its conjugate transpose. Some more examples of closed linear groups: SO(n) = {x ∈ GL(n, R): xxt = 1, det x = 1}, U(n) = {x ∈ GL(n, C): xx∗ = 1}, SU(n) = {x ∈ GL(n, C): xx∗ = 1, det x = 1}, SL(n, R) = {x ∈ GL(n, R) : det x = 1}, SL(n, C) = {x ∈ GL(n, C) : det x = 1}. Deﬁnition 0.5.4. A real (resp. complex) Lie algebra g is a R (resp. C) algebra with multiplication given by a Lie bracket [, ]: g × g → g such that:

[X,X] = 0 for all X ∈ g,

[[X,Y ],Z] + [[Y,Z],X] + [[Z,X],Y ] = 0 for all X,Y,Z ∈ g. Deﬁnition 0.5.5. Let G be a closed linear group. Deﬁne a smooth curve on G to be

c : R → G, such that c : R → Im c is continuously differentiable. Definition 0.5.6. Let G be a closed linear group. We define the linear Lie algebra g of G to be 0 g = {c (0)|c : R → G is a smooth curve with c(0) = 1} . Remark 0.5.7. The set g is closed under addition, scalar multiplication by R and becomes a real Lie algebra with Lie bracket [X,Y ] = XY − YX for all X,Y ∈ g. Definition 0.5.8. For G a closed Linear group and g its associated Lie algebra, the exponential function exp : g → G is: ∞ X 1 exp(X) = eX = Xn, for X ∈ g. n! n=0

28 0.6 Lie groups and Lie algebras

Assume a Lie algebra g is an algebra over k, where k = R or C unless otherwise stated. For any Lie algebra g there is a linear map: ad : g → Derk g by

(ad X)(Y ) = [X,Y ], for all X,Y ∈ g.

The subset of derivations of g given by ad X is the set of inner automorphisms Int g.

Deﬁnition 0.6.1. A Lie subalgebra h of g is a subspace such that [h, h] ⊂ h. An ideal j ⊂ g is a subspace such that [j, g] ⊂ j.

Example 0.6.2. Examples of Ideals:

Zg = centre of g = {X :[X,Y ] = 0 for all Y ∈ g}, [g, g] = commutator ideal, ker π where π : g → h is a Lie algebra homomorphism.

Deﬁnition 0.6.3. Let g be a Lie algebra, the commutator series of g is

g = g0 ⊇ g1 = [g, g] ⊇ g2 = [g1, g1] ⊇ ...

A Lie algebra g is solvable if gj = 0 for some j.

Proposition 0.6.4. [36, Proposition 1.12] If g is a ﬁnite dimensional Lie algebra then there exists a unique solvable ideal rad(g) containing all solvable ideals in g.

Definition 0.6.5. A finite dimensional Lie algebra g is simple if g is non-abelian and has no proper nonzero ideals. A finite dimensional Lie algebra g is semisimple if rad(g) = 0, i.e. it has no nonzero solvable ideals.

Deﬁnition 0.6.6. A Lie algebra g is reductive if every ideal a ⊂ g corresponds to an ideal b of g such that g = a ⊕ b.

Corollary 0.6.7. [36, Corollary 1.56] If g is reductive, then g = [g, g] ⊕ Zg with [g, g] semisimple and Zg abelian.

29 Example 0.6.8. We deﬁne some classical Lie algebras, which are all reductive.

so(n) = {X ∈ gl(n, R): X + Xt = 0, Tr X = 0}, u(n) = {X ∈ gl(n, C): X + X∗ = 0}, su(n) = {X ∈ gl(n, C): X + X∗ = 0, Tr X = 0},

sl(n) = {X ∈ gl(n, C) : Tr X = 0}, so(n, C) = {X ∈ gl(n, C): X + Xt = 0}, sp(n, C) = {X ∈ gl(2n, C): XtJ + JX = 0},

sl(n, R) = {X ∈ gl(n, R) : Tr X = 0}, t so(m, n) = {X ∈ gl(m + n, R): X Im,n + Im,nX = 0}. Here 0 Idn Idm 0 J = , and Im,n = . − Idn 0 0 Idn The ﬁrst three Lie algebras are semisimple when n > 3. The middle three are semisimple as complex Lie algebras, furthermore gR is semisimple as a real Lie algebra. The ﬁnal two are semisimple as real Lie algebras.

0.6.1 Field extensions and Lie algebras

In this section K will be a ﬁeld extension of k. In practice k = R ⊂ C = K.

Deﬁnition 0.6.9. Let V be a k-vector space, we denote the extension to K:

V K = V ⊗k K.

k If U is a K-vector space then the restriction to k, U is U regarded as a k-vector space.

Remark 0.6.10. There shouldn’t be confusion between V K and U k since in one case the ﬁeld is extended and in the other restricted. For R ⊂ C the operations are not inverses: √ (V C)R = V ⊕ −1V.

Deﬁnition 0.6.11. If U is a complex vector space and V is a real vector space such that √ U R = V ⊕ −1V, then we say V is a real form of U. A real form deﬁnes conjugation on U which is the √ identity on V and − Id on −1V .

30 Remark 0.6.12. Given a real Lie algebra g0, the complexiﬁcation of g0 is g = C g0 ⊗R C = g0 . Given a complex Lie algebra g a real form of g is a real Lie algebra g0 such that √ R g = g0 ⊕ −1g0.

Proposition 0.6.13. [36, Proposition 1.17] Let g be a Lie algebra over k, g is solvable if and only if gK is solvable.

Deﬁnition 0.6.14. Let g be a ﬁnite dimensional Lie algebra over k the Killing form is a bilinear form on g × g:

B(X,Y ) = trace(ad X ad Y ) ∈ k, for all X,Y ∈ g.

Theorem 0.6.15. Cartan’s criterion for semisimple Lie algebras [36, Theorem 1.45] The Lie algebra g over k ⊂ C is semisimple if and only if the Killing form of g is nondegenerate.

Theorem 0.6.16. The Lie algebra g is semisimple if and only if

g = g1 ⊕ ... ⊕ gm, with gj ideals that are each simple Lie algebras.

Deﬁnition 0.6.17. Let g be a complex Lie algebra then the universal enveloping algebra of g is U(g) = T (g)/J, where J is the two sided ideal generated by X ⊗ Y − Y ⊗ X − [X,Y ] for X,Y ∈ g.

Theorem 0.6.18. Poincar´e-Birkhoﬀ-Witt.Let {Xi}i∈A be a basis of g, with a total ordering on A, then the set of monomials

ji jn (Xi1 ) ...(Xin ) with i1 < ... < in and jk ≥ 0, is a basis of U(g).

31 0.6.2 Compact Lie groups

Deﬁnition 0.6.19. A compact group is a topological group whose underlying topology is compact Hausdorﬀ. A compact Lie algebra is such that it is a Lie algebra of a compact group.

Definition 0.6.20. A finite dimensional representation of a compact Lie group G is a group homomorphism ρ from G to GLC(V ), where V is a finite dimensional R-vector space.

A representation of a compact group G on a complex Hilbert space H is a continuous map ρ from G to GL(H).

Deﬁnition 0.6.21. If V has a positive deﬁnite Hermitian form h, i then V is a unitary representation if ρ(g) is unitary with respect to h, i, that is:

hρ(g)v1, ρ(g)v2i = hv1, v2i, for all v1, v2 ∈ V, g ∈ G.

Theorem 0.6.22. Peter-Weyl Theorem. [36, Theorem 1.12] Let ρ be a unitary representation of a compact group G on a Hilbert space H, then H splits into direct sums of orthogonal irreducible ﬁnite dimensional unitary representations of G.

0.6.3 Split, Compact forms and Cartan involutions

In this subsection, we uniformly denote real Lie algebras by g0 and drop the subscript for complex Lie algebras. Recall Remark 0.6.12, a real form of a complex Lie algebra g is a real Lie algebra g0 such that √ R g = g0 ⊕ −1g0.

Theorem 0.6.23. [36, Theorem 6.9] Let g be a complex semisimple Lie algebra, with h a Cartan subalgebra and Φ a set of roots. It is possible to choose root vectors

Xα ∈ gα, α ∈ Φ, such that

[Xα,X−α] = Hα, for all α ∈ Φ,

[Xα,Xβ] = Nα,βXα+β if α + β ∈ Φ,

[Xα,Xβ] = 0 if α + β 6= 0 and α + β∈ / Φ, with constants Nα,β such that:

Nα,β = −N−α,−β.

32 Deﬁnition 0.6.24. Let g be a complex Lie algebra, with root vectors as in Theorem

0.6.23, deﬁne h0 = {H ∈ h : α(H) ∈ R for all α ∈ Φ}. A real form of g that contains h0 is called a split real form. The real Lie algebra M g0 = h0 ⊕ RXα α∈Φ is a split real form.

Deﬁnition 0.6.25. A real form of a compact Lie algebra g is called a compact real form if g0 is compact.

Theorem 0.6.26. [36, Theorem 6.11] If g is a complex semisimple Lie algebra, then g has a compact real form u0

Deﬁne root vectors as in Theorem 0.6.23 then a real compact form for g is

X √ X X √ u0 = R −1(Hα) + R(Xα − X−α) + R −1(Xα − Xα). α∈Φ α∈Φ α∈Φ

Recall Deﬁnition 0.6.14 of the Killing form B(X,Y ).

Deﬁnition 0.6.27. Let g0 be a real Lie algebra, an involution θ on g0 is a Cartan involution if the bilinear symmetric form

Bθ(X,Y ) = −B(X, θY ) is positive deﬁnite.

Proposition 0.6.28. [36, Proposition 6.14] Let g be a complex Lie algebra with compact real form u0. Let θ be conjugation of g deﬁned by the real form u0, then θ is a Cartan involution.

Corollary 0.6.29. [36, Corollary 6.18] If g0 is real semisimple Lie algebra, then g0 has a Cartan involution.

Deﬁnition 0.6.30. The Cartan decomposition of a real Lie algebra g0 is a decomposition

g0 = k0 ⊕ p0 such that the Killing form of g0 is negative deﬁnite on k0 and positive deﬁnite on p0.

33 A Cartan decomposition determines a Cartan involution θ with +1-eigenspace k0 and −1-eigenspace p0. Similarly in the reverse, a Cartan involution determines a Cartan decomposition.

Deﬁnition 0.6.31. Let G be a semisimple Lie group with Lie algebra g0, let θ be a

ﬁxed Cartan involution with corresponding Cartan decomposition g0 = k0 ⊕ p0, deﬁne

K to be the analytic subgroup of G, K = exp(k0) , with Lie algebra k0.

Theorem 0.6.32. [36, Theorem 6.36] Let G be a semisimple Lie group with Lie algebra g0, let θ be a ﬁxed Cartan involution with corresponding Cartan decomposition g0 = k0 ⊕ p0, K is an analytic subgroup of G with Lie algebra k0. Then

(a) there exists a Lie group automorphism Θ of G with differential θ, (b) the subgroup of G fixed by Θ is K, (c) the mapping K × p0 → G given by (k, X) 7→ k exp X is a diffeomorphism onto G, (d) K is closed, (e) K contains the centre Z ⊂ G, (f) K is compact if and only if Z is finite, (g) when Z is finite, K is a maximal compact subgroup of G.

0.6.4 Iwasawa Decomposition

In this subsection, real Lie algebras will be denoted by g, if we mention a complex Lie algebra we will denote it by gC. Let G be a semisimple Lie group and let g be its Lie algebra. Then g has a Cartan decomposition g = k ⊕ p, with corresponding Cartan involution θ. Let B be the Killing form on g.

Deﬁnition 0.6.33. Let a be a maximal abelian subspace of p. For λ ∈ a∗ write

gλ = {X ∈ g : (ad H)X = λ(H)X for all H ∈ a}.

If gλ =6 0 and λ 6= 0, we call λ a restricted root of g or a root of (g, a). The set of restricted roots is denoted Σ. Any gλ 6= 0 is called a restricted root space.

Proposition 0.6.34. [36, Proposition 6.40] The restricted roots and restricted root spaces have the following properties: L (a) g is the orthogonal direct sum g = g0 ⊕ λ∈Σ gλ, (b)[gλ, gµ] ⊂ gλ+µ, (c) θgλ = g−λ, therefore − λ ∈ Σ for all λ ∈ Σ, (d) g0 = a ⊕ m orthogonally, where m = Zk(a).

34 Remark 0.6.35. In the restricted root decomposition, g0 denotes the 0-eigenspace of ad a. Not to be confused with previous sections where g0 was a real Lie algebra with complex Lie algebra g.

Fix an ordering of a∗ then we deﬁne positive restricted roots Σ+ = {λ ∈ Σ: λ > 0}.

Deﬁnition 0.6.36. For a real Lie algebra g deﬁne a subspace

M n = gλ. λ∈Σ+ The subspace n is a nilpotent Lie subalgebra of g.

Proposition 0.6.37. Iwasawa decomposition of a Lie algebra [36, Proposition 6.43]. With notation as in Proposition 0.6.34, g is a vector space direct sum

g = k ⊕ a ⊕ n.

Here a is abelian, n is nilpotent and a⊕n is a solvable Lie subalgebra of g, [a⊕n, a⊕n] = n.

Theorem 0.6.38. Iwasawa decomposition. [36, Theorem 6.46] Let G be a semisimple Lie group, let g = k ⊕ a ⊕ n be an Iwasawa decomposition of the Lie algebra g of G. Let K, A, N be the analytic subgroups of G generated by k,a,n respectively. Then the multiplication K × A × N → G

(k, a, n) 7→ kan, is a diﬀeomorphism onto G.

0.7 Real reductive linear groups

We will denote real Lie algebras by g0 and the corresponding complex Lie algebra C g0 = g0 ⊗R C by g.

Deﬁnition 0.7.1. By a real reductive group, we mean a real Lie group G, a maximal compact subgroup K of G, and a Cartan involution θ of g0 the Lie algebra of G,

35 satisfying the following conditions.

(a) The Lie algebra g0 is a real reductive Lie algebra. (b) For g ∈ G, Ad g is an inner automorphism of the complex Lie algebra g. (c) The fixed point set of θ is k0. (d) Write p0 for the − 1 eigenspace of θ; The map K × p0 → G, (k, X) 7→ k · exp(X) is a diffeomorphism. (e) G has a faithful finite dimensional representation. (f) The centralizer H of a Cartan subalgebra h0 is abelian.

Remark 0.7.2. A real reductive group need not be connected.

Deﬁnition 0.7.3. A representation of a real reductive group G is a pair (π, H), with H a complex Hilbert space and a continuous homomorphism π : G → GL(H). The representation (π, H) is irreducible if H 6= 0 and there is no proper closed invariant subspaces. It is called unitary if the operators π(g) are unitary for all g ∈ G.

Definition 0.7.4. Suppose K is a compact group, hence every irreducible representation is finite dimensional. Define the dual object Kˆ to be the set of equivalence classes of irreducible representations of H.

Deﬁnition 0.7.5. Let (π, H) be a representation of a real reductive group G. A vector v ∈ H is called K-ﬁnite if

dim span{π(k)v : k ∈ K} < ∞.

Set

HK = {v ∈ H : v is K-ﬁnite }.

Fix an irreducible K-representation (δ, Vδ) deﬁne the δ K-type of H: [ HK (δ) = L(Vδ).

L∈HomK (δ,π)

The G-representation (π, H) is admissible if

ˆ dim HK (δ) < ∞, for all δ ∈ K.

In this case the multiplicity of δ in π is m(δ, π) = dim HomK (Vδ,HK ).

36 Theorem 0.7.6. [28][57, Theorem 0.3.5] Let (π, H) be an admissible representation of G. If v ∈ HK and X ∈ g0, the limit 1 lim (π(exp(tX)))v − v) t→0 t exists and call it π(X)v. Furthermore

π(X)v ∈ HK and this deﬁnes HK as a representation of g0.

C The above g0-representation can be complexiﬁed to a representation of g = g0 .

Theorem 0.7.7. [28][57, Theorem 0.3.6] Any irreducible unitary representation of a real reductive group G is admissible.

Deﬁnition 0.7.8. [57, Deﬁnition 0.3.8] A (g,K)-module is a pair (π, X) with X a complex vector space and π a map

π : g ∪ K → End(X) satisfying:

(a) The map π|g is a Lie algebra homomorphism, and π|K is a group representation. (b) Every vector v ∈ HK is K ﬁnite. (c) The diﬀerential of π|K is the restriction of π to k0. (d) If X ∈ g, k ∈ K, then π(Ad(k)X) = π(k)π(X)π(k)−1.

Deﬁnition 0.7.9. The admissible (g,K)-module HK attached to an admissible group representation (π, H) by Theorem 0.7.6 is called the Harish-Chandra module of (π, H). We denote the category of Harish-Chandra modules by HC(G).

Deﬁnition 0.7.10. Let (π, X) and (π0,X0) be two (g,K)-modules. Deﬁne

0 0 0 Homg,K (π, π ) = {L : X → X : L is complex linear, and π L = Lπ}.

Deﬁnition 0.7.11. Two (g,K)-modules (π, X) and (π0,X0) are equivalent if there 0 is an invertible (g,K) map L ∈ Homg,K (π, π ) between them. We say two G- representations are inﬁnitesimally equivalent if their Harish-Chandra modules are equivalent.

37 Let G be a real reductive group with Lie algebra g0, K its maximal compact subgroup. Let a0 be a maximal abelian subalgebra of p0. We form restricted roots of ∗ + 1 P Φ = Φ(g0, a0), and ﬁx an ordering on a to deﬁne positive roots Φ , set ρ = 2 λ∈Φ+ λ. L Let n0 = λ∈Φ+ gλ. We have an Iwasawa decomposition of the Lie algebra

g0 = k0 ⊕ a0 ⊕ n0.

Let K, A, N be the corresponding Lie subgroups of G to k0,a0,n0. Then the Iwasawa decomposition of G is G = KAN.

Deﬁnition 0.7.12. Let G be a real reductive group, G = KAN an Iwasawa decomposition of G. Set

M = ZK (a0) = centraliser of a0 in K.

The minimal parabolic subgroup of G is

P = MAN.

The subgroup A is abelian with Lie algebra a0, we identify the irreducible representations of A with a∗ by

∗ ν(exp X) = exp(ν(X)) for X ∈ a0, ν ∈ a .

∗ Deﬁnition 0.7.13. Fix a representation (δ, Vδ) of M and a character ν ∈ a . Deﬁne a Hilbert space:

Hδ⊗ν = {f : G → Vδ : f is measurable, f|K is square integrable, f(gman) = a−(ν+ρ)δ(m−1)f(g) for all g ∈ G, m ∈ M, a ∈ A, n ∈ N} with norm deﬁned by Z ||f||2 = ||f(k)||2dk. K

The above space Hδ⊗ν is invariant by left translation in G. We deﬁne the principal series representation with parameters δ and ν by the representation πδ⊗ν of G on

Hδ⊗ν: −1 πδ⊗ν(g) · f(x) = f(g x), for all g, x ∈ G.

We write X(δ ⊗ ν) for the Harish-Chandra (g,K)-module of πδ⊗ν.

38 Lemma 0.7.14. [57, Lemma 4.1.5] Let G be a real reductive group with maximal ˆ compact subgroup K, let (δ, Vδ) ∈ M. Set

Hδ = {f : K → Vδ : f is measurable, f|K is square integrable, f(km) = δ(m−1)f(k) for all k ∈ k, m ∈ M},

K we denote this space by IndM (δ). Then

G ∼ ResK (Hδ⊗ν) = Hδ and ˆ HomK (µ, Hδ⊗ν) = HomM (µ|M , δ) for any µ ∈ K.

Theorem 0.7.15. Harish-Chandra’s subquotient theorem. [57, Theorem 4.1.9] Every irreducible admissible representation of a real reductive group G is inﬁnitesimally equivalent to a composition factor of a principal series representation of G.

0.8 Schur-Weyl Duality

0.8.1 Schur’s double centraliser result.

n ⊗k Let V = C . The symmetric group Sk naturally acts on its k fold tensor power V . This action commutes with the action of GLn(C) acting by matrix multiplication on ⊗k V . The vector space V is a GLn(C), C[Sk]-bimodule. Denote this structure by:

ρ ⊗k σ GLn(C) −→ EndC(V ) ←− C[Sk].

Theorem 0.8.1. [51] We have the following equalities:

⊗k ρ(GLn(C)) = EndSk (V ), ⊗k σ(C[Sk]) = EndGLn (V ). Furthermore if k ≤ n then the representation σ is faithful.

The above result holds for the enveloping algebra of the Lie algebra gln associated to GLn(C).

Theorem 0.8.2. Notation as in Theorem 0.8.1, replacing the representation ρ of GLn ⊗k with dρ : U(gln) → End(V ) leads to an alternative formulation of Schur’s result:

⊗k dρ(gln) = EndSk (V ), ⊗k σ(C[Sk]) = Endgln (V ).

39 0.8.2 Brauer Algebras

Deﬁnition 0.8.3. Deﬁne a set of diagrams such that each diagram has k top vertices and k bottom vertices with pairings represented by edges between the pair. Every vertex must be paired with exactly one other vertex.

Example 0.8.4. For k = 4, two examples of a diagram of pairings are: • • • •

• • • • , • • • •

• • • • .

Two diagrams d1 and d2 are concatenated by identifying the bottom vertices of d1 with the top vertices of d2. This gives rise to a new diagram which has the top vertices of d1 and the bottom vertices of d2 and pairings deﬁned by the connected pairings in the concatenated diagram of 3k vertices formed.

Example 0.8.5. Set • • • • d1 = • • • • , • • • • d2 = • • • • ,

then identifying the bottom vertices of d1 with the top vertices of d2: • • • •

• • • •

• • • • . Therefore the concatenated diagram is: • • • • d1 · d2 = • • • • .

Definition 0.8.6. [8, Section 5 (a)] Let k be a characteristic zero field, set k ∈ N and δ ∈ k. The Brauer algebra Brk[δ] is the associative unital k-algebra with basis consisting of the diagrams defined in Definition 0.8.3 of pairings of 2k vertices with multiplication defined to be

m(d1,d2) d1 ·Brk[δ] d2 = δ d1 · d2,

40 where m(d1, d2) counts the connected components of the concatenation of d1 and d2 which do not appear in d1 · d2.

Example 0.8.7. Set • • • • d1 = • • • • , • • • • d2 = • • • • , then the concatenation • • • •

• • • •

has one connected component not occurring in the product d1 · d2. Therefore the

Brauer product d1 ·Br4[δ] d2 would be

 • • • • 

d1 ·Br4[δ] d2 = δ   . • • • •

We give an alternative deﬁnition of the Brauer algebra Brk[δ]

Deﬁnition 0.8.8. [8] The rank k Brauer algebra Brk[δ], with parameter δ ∈ k, is the associative unital k-algebras generated by elements ti,i+1 and ei,i+1 for i = 1, ..., k − 1, subject to the conditions:

the subalgebra generated by ti,i+1 is isomorphic to C[Sk],

2 ei,i+1 = δei,i+1,

ti,i+1ei,i+1 = ei,i+1ti,i+1 = ei,i+1,

ti,i+1ti+1,i+2ei,i+1ti+1,i+2ti,i+1 = ei+1,i+2,

[ti+1, ej,j+1] = 0 for j 6= i, i + 1.

Proposition 0.8.9. Definition 0.8.6 and Definition 0.8.8 define isomorphic k-algebras.

41 Proof. One deﬁnes an isomorphism on the generators

i i + 1 • • ... • • ... • • ti,i+1 7→ , • • ... • • ... • • i i + 1 • • ... • • ... • • ei,i+1 7→ . • • ... • • ... • •

Lemma 0.8.10. If g = sp2n(C) or so2n+1(C), with matrix module V , then V ⊗ V decomposes as 2 2 Λ V ⊕ S V/1 ⊕ 1 for g = so2n+1,

2 2 Λ V/1 ⊕ S V ⊕ 1 for g = sp2n.

Let pr1 be the projection of V ⊗V onto the trivial submodule 1 in the decomposition above. Let pri,i+1 be the projection of Vi ⊗ Vi+1 onto the trivial submodule of Vi ⊗ Vi+1. Given the k fold tensor V ⊗k as mentioned in Theorem 0.8.1 the symmetric group

Sk naturally acts on this tensor. Denote by si,i+1 the transpositions (i, i + 1) in the symmetric group.

2n 2n+1 Lemma 0.8.11. [20] Let g be sp2n(C) or so2n+1(C). Let V = C or C be the deﬁning module of g. Then there exists δ ∈ Z such that there is a homomorphism ⊗k π : Brk[δ] → End(V ):

π(ti,i+1) = si,i+1,

π(ei,i+1) = id ⊗ ... ⊗ id ⊗ δ pri,i+1 ⊗id ⊗ .... ⊗ id.

For g = sp2n then is δ = −n and if g = son, δ = n.

Theorem 0.8.12. [8, Theorem 46] Let ρ denote the action of SO2n+1(C) or Sp2n(C) ⊗k ⊗rk on V and π denote the action of Brk[δ] on V . We have the following equalities:

⊗r ρ(G) = EndBrk (V ), ⊗r π(Brk[δ]) = EndG(V ). Furthermore if k ≤ n then the representation π is faithful.

42 ⊗k Corollary 0.8.13. Let dρ denote the action of U(so2n+1(C)) or U(sp2n(C)) on V ⊗k and π denote the action of Brk[δ] on V . We have the following equalities:

⊗k dρ(U(g)) = EndBrk (V ), ⊗k π(Brk[δ]) = Endg(V ).

43 Chapter 1

Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

44 Abstract

We derive an explicit description of the genuine projective representations of the symmetric group Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. In [15] Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of Sn are related to the characters of graded elliptic modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties Be of g. We use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan’s morphism for Hecke algebras in type A using spectrum data of the Jucys-Murphy elements. 1.1 Introduction

Schur (cf. [54]) initially described the characters of the projective representations of the symmetric group. Nazarov [43] produced an orthogonal form for the symmetric group [44]. Okounkov and Vershik [45] developed an approach using Jucys-Murphy elements. This approach was later applied to the projective representations of Sn by Vershik and Sergeev [56] although there appears to be a flaw with their calculation of the spectrum data. We describe the action of the Jucys-Murphy elements using Dirac cohomology, providing an alternative proof for the spectrum calculation. We then ˜ describe the genuine projective representations, giving the action of Sn by matrices. Barbasch, Ciubotaru, and Trapa [4] developed Dirac cohomology for graded Hecke algebras. Ciubotaru [13] defined Dirac cohomology for symplectic reflection algebras. Ciubotaru and He [15] introduced extended Dirac cohomology which relates the tempered modules of the graded Hecke algebra with the irreducible representations of W˜ . Notably in [15] the functor taking modules to their extended Dirac index is exact. Combining this with combinatorial results from Garcia and Procesi [26] on the restriction of the cohomology groups of the Lie algebra g associated to Sn we are able to deduce the branching graph for the irreducible genuine projective representation of

Sn.

˜ Theorem 1.4.9. The branching rules of the genuine projective irreducible Sn-supermodules are:

Here λ is always a strict partition, λ(i) depends on a reduction rule, deﬁned in [26], which gives partitions λ(i) from λ.

We prove that the Casimir element Ω introduced in [4] for S˜ acts by the second S˜n n power polynomial in the Jucys-Murphy elements.

Lemma 1.5.12. On genuine projective representations, Ω = 1 Pn M 2. Here M S˜n 2 i=1 k k ˜ denotes the Jucys-Murphy elements for Sn. In section 1.5, we combine this lemma, our branching graph result and a result from ˜ [4] to prove inductively that the spectrum data for Sn on projective representations is equivalent to content data on shifted Young diagrams. Therefore we give a diﬀerent proof of the result claimed in [56], utilising Dirac cohomology and combinatorics from Garcia and Procesi [26].

Sspec(n, λ) = Scont(n, λ).

In section 1.6 we describe how one gets from the spectrum data to an explicit ˜ ˜ description of the genuine projective supermodules of Sn and the action of Sn in matrix form. In Section 1.7, we use the results from Section 1.5 on the action of the Jucys-Murphy elements to give an explicit description of the Vogan morphism for graded Hecke algebras of type A introduced in [4].

1.2 Deﬁnitions

We fix a real root system {V, Φ,V ∨, Φ∨}, where Φ is the set of roots inside the real vector space V , we have a perfect bilinear pairing B( , ): V × V ∨ → R. The roots are in bijection with the coroots such that B(α, α∨) = 2. We define reflections

∨ sα : V → V, sα(v) = v − B(v, α )α.

Let W be the subgroup of GL(V ) generated by the reﬂections sα for all α ∈ Φ. Fix a W -invariant bilinear form ( , ) on V , a set of positive (resp. simple) roots Φ+ (resp.

Π). The reﬂections sα for α ∈ Π generate W .

Deﬁnition 1.2.1. We let δ denote the automorphism of W given by the conjugation by w0 on V , where w0 is the longest element in W . Then let W# be the extension of W by δ. Explicitly

2 W# = hW, δ : δ = 1, δwδ = δ(w) for all w ∈ W i.

47 Example 1.2.2. Let W be the symmetric group Sn. The roots live in the permutation representation V = spanC{ξ1, . . . , ξn} and we ﬁx the simple roots Π = {αi = ξi − ξi+1 : + i = 1, .., n − 1}, and positive roots Φ = {ξi − ξj : i < j}. The bilinear form is

(ξi, ξj) = δij. We have the presentation;

m(i,j) Sn = hs1, . . . , sn−1 :(sisj) = 1i.

Here m(i, i) = 1 and m(i, j) = (αi, αj)(αj, αi) + 2 if i =6 j [42]. The automorphism δ comes from the action of −w0 on Φ hence δ(αi) = αn−i, and δ(sα) = sδ(α).

Deﬁnition 1.2.3. The graded Hecke algebra H associated to a Weyl group W , V and ( , ) is the associative unital C-algebra generated by {ξ ∈ V } and {w ∈ W } such that there exist injections, induced by the inclusion of generators,

S(V ) ,→ H,

C[W ] ,→ H. Furthermore, for every simple root α ∈ Π, there exist cross-relations,

ξ · sα − sα · sα(ξ) = (α, ξ).

Note that since we are interested in Sn we do not introduce parameters for H. The automorphism δ ( Definition 1.2.1) is an involution of W defined by conjugation by w0. We extend δ to H(W ) as conjugation by w0 on all of H. We define H# to be an extension of the graded Hecke algebra by δ. Explicitly,

2 H# = hH, δi = hH, δ : δ = 1, δhδ = δ(h) for all h ∈ Hi.

As a vector space H# is isomorphic to H ⊕ H. Recall Deﬁnition 0.4.1 of the Cliﬀord algebra C(V ) with relations:

ξξ0 + ξ0ξ = 2(ξ0, ξ) for all ξ, ξ0 ∈ V.

Note that this differs from some sources, [4] in particular, where the Clifford algebra is defined with a negative form. The theory is identical over C as one can just √ multiply the generators by −1.

Example 1.2.4. For Sn, the vector space V has the Euclidean norm and is of dimension n. In this case we denote the Cliﬀord algebra for the rank n case by C(n).

48 The Cliﬀord algebra has a natural ﬁltration by Z. Each generator is given degree one and then C(V )n is the span of all elements of C(V ) with degree n or lower. C(V ) also has a Z/2Z grading,

C(V ) = C(V )even ⊕ C(V )odd where C(V )even is the span of all homogeneous elements of even degree and likewise for C(V )odd. Define the transpose of C(V ) to be the antiautomorphism defined by ξt = −ξ for all elements of V and let : C(V ) → C(V ) be the involution which is the identity on C(V )even and acts as −1 on C(V )odd. Recall Definition 0.4.4:

Pin(V ) = {g ∈ C(V )×|gt = g−1 and (g)ξg−1 ∈ V, for all ξ ∈ V }.

This group is a double cover of the orthogonal group on V with projection p : Pin(V ) → O(V ), p(g)(ξ) = (g)ξg−1.

Since the Weyl group is a subgroup of O(V ) we can ﬁnd a double cover of W in the Pin group. Let W˜ be the preimage of W via p,

W˜ = p−1(W ).

Theorem 1.2.5. [42, 3.2] The double cover W˜ of W admits a presentation akin to Coxeter presentations for a Weyl group,

˜ 2 m(α,β) m(α,β)−1 W = hz, s˜α for all α ∈ Π: z = 1, (˜sαs˜β) = z , z is central i where m(α, α) = 1 and m(α, β) = hα, βihβ, αi + 2.

Remark 1.2.6. Morris also showed that one can deﬁne a presentation of W˜ with generators for each positive root [42, 3.2],

˜ + 2 2 W = hz, s˜α for all α ∈ Φ :s ˜α = 1, s˜βs˜αs˜β = zs˜γ, γ = sα(β), z = 1, z is centrali.

As a subgroup of O(V ), we have W# = hW, δi is equal to hW, −1i. Hence we can ˜ −1 also deﬁne W# = p (W#). The non scalar element in the preimage of −1 is ω which can be formulated in terms of an orthonormal basis {ξ1, . . . , ξn} of V , as

ω = ξ1ξ2 . . . ξn ∈ P in(V ).

49 ˜ 0 ˜ 0 We deﬁne W and W# to be ( W˜ ∩ C(V ) , dim V even, W˜ 0 = even W,˜ dim V odd,

and ( hW˜ ∩ C(V ) , ωi, dim V even, W˜ 0 = even # ˜ W#, dim V odd. In Section 1.3 we utilise the extended Dirac index, introduced by Ciubotaru and He [15]. In [15] it was crucial that the spinor module considered had a positive and negative part. Since [15] considers ungraded modules, in the even dimensional case, the ungraded spinor module does not have a positive and negative part. Hence to manufacture a module with a positive and negative part, one must restrict the groups ˜ ˜ ˜ 0 ˜ 0 W and W#. This is why we have introduced W and W#. However, we show in Corollary 1.3.5 that when one considers supermodules, one no longer needs to restrict W˜ . In the supertheory case, the C(V )-module S always has a positive and negative ˜ part. We are interested in the representation theory of Sn.

Deﬁnition 1.2.7. [55, p. 303] Let Tn be the twisted group algebra of Sn over C. Explicitly Tn is the unital associative C-algebra generated by τi for i = 1 . . . n − 1 such that 2 m(i,j) m(i,j)−1 τi = 1, (τiτj) = (−1) .

Remark 1.2.8. Following the presentation from Theorem 1.2.5 we could label the generators with the simple roots, α ∈ Π.

Then Tn becomes the unital associative C-algebra with presentation:

m(α,β) m(α,β)−1 hτα, α ∈ Π:(τατβ) = (−1) i.

Finally we have

+ 2 Tn = hτα for all α ∈ Φ : τα = 1, τατβτα = −τγ, γ = sα(β)i.

By the above presentation, similarly to the symmetric group, one has for each positive + root, α ∈ Φ , a ‘pseudo-transposition’ τα. Note that this τα is equal to the the ‘transposition’ [ij] deﬁned in [9, 3.1]. We switch between labellings of the generators whenever it clariﬁes the explanation.

50 ˜ The genuine projective irreducible representations of Sn are the irreducible repre- ˜ sentations that do not descend to a representation of Sn. The group algebra for Sn decomposes as

˜ ∼ ˜ ˜ ∼ C[Sn] = C[Sn]/(z − 1) ⊕ C[Sn]/(z + 1) = C[Sn] ⊕ Tn.

˜ If we are interested in the projective representations of Sn, i.e. those that do not occur as representations of Sn, then we just need to study the representations of Tn. ˜ Since the supertheory for Sn is cleaner than that of the ungraded representation ˜ theory, we focus on the supermodules of Sn. The Cliﬀord algebra is a superalgebra, by considering the generators to have degree one. Similarly Tn is a superalgebra. Again, the generators all have degree one. Recall the deﬁnition 0.3.4 of supermodules and supermodule homomorphisms. When one forgets the grading, all irreducible supermodules are either still irreducible or decompose into two irreducible parts. It is possible to recover the original nongraded representation theory from the super representation theory if we keep track of whether each irreducible supermodule is reducible as an ungraded module.

Deﬁnition 1.2.9. Let U be an A-supermodule. Then U is of type M if, when one forgets the grading, U is an irreducible A ungraded module. The module U is of type Q if it is reducible as an A ungraded module.

As an ungraded algebra, the Cliﬀord algebra C(V ) has either one or two isomorphism classes of irreducible modules, depending on dim(V ). However when we consider supermodules, C(V ) is supersimple; it always has exactly one simple supermodule, S [35, 12.2.4]. When dim(V ) is odd, then S is of type Q and when dim(V ) is even the supermodule is of type M.

The supertrace on the representation U = U0 ⊕ U1 is deﬁned to be zero on odd elements and, on even elements,

strV (a) = tr0(a) − tr1(a) where tri(a) is the trace of the matrix corresponding to a restricted to Ui. r We use three different types of partitions throughout. Let λ = (λ1, .., λr) ∈ N . Pr We define λ to be a partition of n if i=1 λi = n and λ1 ≥ λ2 ≥ ... ≥ λr. We denote the set of partitions of n by P(n). We define the length of a partition, l(λ), to be

#{λi : λi > 0}. We say a partition λ ` n is strict if λ1 > λ2 > . . . > λr.

51 Deﬁnition 1.2.10. For a partition λ ` n we associate to it a set of boxes forming the Young diagram. For λ = (λ1, . . . , λr), we deﬁne the Young diagram to be

λ = {(i, j) : 1 ≤ i ≤ l(λ) and 1 ≤ j ≤ λi}.

Here (i, j) denotes a cell in the ith row and jth column.

Deﬁnition 1.2.11. The set of shifted partitions of n is the same as the set of strict partitions of n. However, we consider a diﬀerent Y oung diagram for a shifted partition than the associated Young diagram of a normal partition. We write λ n to denote a shifted partition of n and SP(n) is the set of shifted partitions of n.

Deﬁnition 1.2.12. Given a shifted partition λ = (λ1, . . . , λr) n we associate a shifted Young diagram,

λ = {(i, j) : 1 ≤ i ≤ l(λ) and i ≤ j ≤ i + λi − 1}.

A Young tableau of shape λ is a numbering of 1 to n of the Young diagram associated to λ. We write [λ] for a Young tableau of shape λ. Similarly [[λ]] denotes a numbering of the shifted Young diagram associated to λ n. A standard Young tableau (resp. shifted) is a Young tableau (resp. shifted) such that along each row and column the numbers increase.

Example 1.2.13. The partition (3, 1) ` 4 has three standard Young tableaux;

1 2 3 , 1 2 4 , 1 3 4 . 4 3 2

When (3, 1) is considered as a shifted partition of 4 it has two standard shifted Young tableaux; 1 2 3 , 1 2 4 . 4 3 Given a partition λ we write λ0 for its conjugate partition, which is the reﬂection of the Young diagram in its main diagonal.

Example 1.2.14. If λ = (3, 1) then λ0 = (2, 1, 1),

λ = , λ0 = .

52 1.3 Extended Dirac cohomology

In [15] Ciubotaru and He introduced a notion of extended Dirac cohomology as a variation of Dirac cohomology deﬁned in [4]. An important feature is that the functor from an H#-module to its extended Dirac cohomology is exact. Using supertheory, we show the extended Dirac cohomology could be considered as the supertheory analogue of Dirac cohomology. ˜ Let ρ be the diagonal embedding of W# into H# ⊗ C(V ); ˜ ρ(w) = p(w) ⊗ w, for all w ∈ W#.

Hence, for an H#-module X and a spinor module S, we can consider X ⊗ S as a ˜ W#-module via the map ρ.

Recall w0 is the longest element in W . In our case, with Sn and our choice of generators, w0 is the element which takes (1, 2 . . . , n) to (n, n − 1,..., 1).

Deﬁnition 1.3.1. Let ∗ be the linear anti-automorphism on H deﬁned by

∗ −1 ∗ w = w , ξ = w0δ(ξ)w0, for all w ∈ W and ξ ∈ V.

We can extend ∗ to H# by deﬁning it to ﬁx δ. Given ξ, let 1 ξ˜ = (ξ − ξ∗). 2

Deﬁnition 1.3.2. Let {ξ1, . . . , ξn} be an orthonormal basis of V . The Dirac element

D# in H# ⊗ C(V ) is deﬁned to be, n X ˜ D# = ξi ⊗ ξi. i=1

This is independent of the choice of orthonormal basis. For any H#-module X and

C(V )-module S the Dirac element D# deﬁnes an operator D# : X ⊗ S → X ⊗ S.

As a superalgebra we consider H# to be concentrated in even degree and C(V ) to have its usual grading. With this grading H# ⊗ C(V ) is a superalgebra. Furthermore, since D# is odd, the operator D# interchanges the even and odd spaces of X ⊗ S,

± ± ∓ D# : X ⊗ S → X ⊗ S .

As a supermodule we consider X to be concentrated in degree 0 and S splits into homogeneous parts S+,S−.Therefore X ⊗ S+ (resp. X ⊗ S−) is the even (resp. odd) space of X ⊗ S.

53 The extended Dirac cohomology of X (with respect to S) is

D H# (X) = ker(D#)/ ker(D#) ∩ Im(D#).

˜ ˜ This is a W#-representation since W# sign-commutes with D. We are also interested ± in the restriction of D# to the even and odd spaces. The subspace X ⊗ S is always ˜ 0 a W#-module. Let,

D± ± ± ∓ H# (X) = ker(D#)/ ker(D#) ∩ Im(D#).

± Note that when dim V is even X ⊗ S is not an H# ⊗ C(V )-module, just an H# ⊗ D± ˜ 0 C(V )even-module. The space H# (X) is a W#-module.

Deﬁnition 1.3.3. The extended Dirac index is

D+ D− I#(X) = H# (X) − H# (X)

˜ 0 deﬁned as a virtual W#-module.

Proposition 1.3.4. [15, 5.10] Let X be an H#-module. The Dirac index can be + − expressed as a tensor, I#(X) = X ⊗ (S − S ). Hence it is exact. Furthermore for w˜ ∈ W˜ 0, + − tr(w, ˜ I#(X)) = tr(p(w ˜),X) tr(w, ˜ S − S ),

tr(wω, ˜ I#(X)) = tr(p(w ˜)w0δ, X) tr(w, ˜ S).

Note in Proposition 1.3.4 the spinor modules S+ and S− are ungraded, in the following corollary S is the Z2 graded spinor module. ˜ We make the observation that one does not need to split W# when dim V is even if one considers the Grothendieck group of supermodules. This leads to a more elegant formulation of Proposition 1.3.4 using supertheory.

Corollary 1.3.5. Let X be an H#-supermodule, concentrated in degree zero, and

S be a C(V )-supermodule.We endow I#(X) with a superspace structure with even D+ D− ˜ part H# (X) and odd part H# (X). Then I#(X) is a W#-supermodule and in the Grothendieck group,

I#(X) = X ⊗ S.

In particular:

˜ 0 str(w, ˜ I#(X)) = str(p(w ˜),X) str(w, ˜ S) for all w˜ ∈ W#.

54 ˜ Proof. From Proposition 1.3.4 we can describe the trace of w ∈ W on I#(X) as a product of the trace on X and the diﬀerence of the trace on S+ and S−. However this is just the product of the supertrace of the modules X and S;

+ − tr(w, I#(X)) = tr(p(w),X) tr(w, S − S ) = str(p(w),X) str(w, S).

In light of this corollary, one could consider the extended Dirac index as the Dirac supercohomology. When one considers supermodules, as opposed to ungraded modules, the extended Dirac cohomology is the same as considering the supertheory analogue of the original Dirac cohomology deﬁned in [4]. Introducing a certain graded module will be useful. Let g be a complex semisimple Lie algebra with Weyl group W . Then, for an element e in the nilpotent cone N , we let Be denote the variety of Borel subalgebras of g containing e. Springer [53] deﬁned a j j W action on the cohomology groups H (Be). The cohomology H (Be) vanishes unless j is even.

Deﬁnition 1.3.6. Let e be an element in the nilpotent cone N ⊂ g, Be the variety of

Borel subalgebras containing e and de be the dimension of Be. The sign character of W is denoted by sgn. Let q be a formal symbol. Springer [53] constructed an action j of W on H (Be). Deﬁne the q-graded W -module.

X de−i 2i Xq(e) = q H (Be) ⊗ sgn . i≥0

From here on we ﬁx W to be Sn and g = sln. Because of this, we take results from

[15] and restrict them to Sn to avoid surplus deﬁnitions. Note that the correspondence from [15, Appendix A], applies to almost any Weyl group, but outside of type A there is an extra complication involving the component group of the centraliser of e. In the full generality of [15] the graded modules involved are labelled by a nilpotent element e and a representation of this component group. However, for Sn this component j j group is always trivial. The modules H (Be) and H (Be0 ) are isomorphic if and only if e and e0 are in the same conjugacy class.

0 Remark 1.3.7. Two q-graded Sn-modules Xq(e) and Xq(e ) are isomorphic if and 0 only if e is in the same conjugacy class as e . Hence the isomorphism classes of Xq(e) can be labelled by nilpotent orbits, these are equivalent to partitions of n. We write

55 the isomorphism class of Xq(e) as Xq(λ) where e has Jordan form corresponding to a partition λ.

We can specialize Xq(e) to a Z/2Z-graded module:

X de−i 2i X−1(e) = (−1) H (Be) ⊗ sgn . i≥0

The W -module X−1(e) can be extended to several different H#-modules. See [15, Section 5] for more details, the H#-modules are denoted Ee,h. All of these extensions are isomorphic to X−1(e) as a W -module. We fix an extension of X−1(e) to H# such ∼ that as ungraded modules X−1(e) = Xλ (Definition 1.3.15), the standard module parabolically induced from the Steinberg module. We will denote this graded H# module by X−1(λ).

Proposition 1.3.8. [15, A.3] For every strict partition λ of n, let S be an ungraded spinor module and X−1(λ) be the graded module of Sn corresponding to λ. Deﬁne the ˜ Sn-module, 1 τ˜λ = X−1(λ) ⊗ S, aλ where  l(λ) 2 2 n even and l(λ) even,  l(λ)−1 aλ = 2 2 l(λ) odd,  l(λ)−2 2 2 n odd and l(λ) even. ˜ If n − l(λ) is even then τ˜λ is an irreducible sgn self-dual Sn-module. If n − l(λ) is odd + − ± then τ˜λ =τ ˜λ +τ ˜λ , with τ˜λ irreducible modules which are sgn dual to each other.

Corollary 1.3.9. For a strict partition λ, when one lets S be a super simple spinor ˜ module, τ˜λ is always an irreducible Sn-supermodule. If n − l(λ) is even then τ˜λ is type

M and if n − l(λ) is odd then τ˜λ if of type Q.

Remark 1.3.10. The extended Dirac index of X−1(λ), I#(X−1(λ)), is non-zero if and only if λ is strict. This follows from [15, 6.1] and the fact that nilpotent elements that are δ quasi-distinguished have Jordan form associated to a strict partition.

For ease of explanation later on in this chapter, and motivated by Remark 1.3.10, we setτ ˜λ = 0 if λ is not a strict partition. Finally, when looking at the action of the Jucys-Murphy elements in Section 1.4 we will need a result about the action of the Casimir elements on the Dirac cohomology.

56 Deﬁnition 1.3.11. The Casimir element for H is

n X 2 ΩH = ξi , i=1 where {ξi} is an orthonormal basis for V .

The element ΩH is independent of the choice of orthonormal basis, it is central in H [4, 2.4]. Hence it acts by a scalar for any irreducible representation of X. Furthermore, let (π, X) be an irreducible H-module. It has an associated central character χν which can be deﬁned in terms of ν ∈ H. It is shown in [4, 2.5] that

π(ΩH) = (ν, ν).

˜ Deﬁnition 1.3.12. [4, 3.4]The Casimir element for Sn is 1 X Ω ˜ = |α||β|τατβ, Sn 4 α,β∈Φ+ sα(β)<0

˜ where τα is the generator of Sn corresponding to α. In [4] ΩS˜ is negative but we deﬁne it as positive since we have the positive form on C(V ).

The Casimir Ω is central in [S˜ ]. S˜n C n

Theorem 1.3.13. [4, 3.5] As elements of H ⊗ C(V ),

D2 = Ω ⊗ 1 − ρ(Ω ). H S˜n

Recall ρ is the diagonal embedding of W˜ into H ⊗ C(V ). Note that this statement has a parity difference from the one in [4], but this is due to the sign difference of the Clifford algebra in Definition 0.4.1.

Deﬁnition 1.3.14. [14] We introduce the one dimensional H-module, called the

Steinberg module St. Here Sn acts on St by the sgn character and ξi ∈ S(V ) act by −n+1+2i 2 .

Deﬁnition 1.3.15. Let Xλ be the parabolically induced H-module from the Steinberg module for Hλ1 × ... × Hλk ,

H Xλ = Ind ×...× (Stλ1 ⊗ ... ⊗ Stλk ). Hλ1 Hλk

57 In [4, 5.8] it is shown that, inside the Dirac cohomology of Xλ, one can ﬁnd the genuine projective irreducible representation Vλ of isomorphism class associated to 2 a shifted partition λ n. Since HD(X) is a quotient of ker(D), D acts by zero on

HD(X).

Corollary 1.3.16. On the Dirac cohomology (resp. extended Dirac index), or any submodule found inside it, namely Vλ (resp. τ˜λ),

ΩH ⊗ 1 = ρ(ΩW˜ ).

Proof. This follows from D2 acting by zero and the equation given in Theorem 1.3.13.

1.4 Branching graph for S˜n

˜ In section 1.5 we need the branching graph of Sn. Mainly, we need to understand the modules that occur in the restriction of the irreducible representations τ˜λ. In this section we provide arguments for the whole branching graph of the genuine projective ˜ representations of Sn, this is the branching graph of Tn. We derive this graph from Theorem [15, A.4] and a branching result from Garcia and Procesi [26] on certain graded Sn-modules.

We know we can ﬁnd τ˜λ in S ⊗ X−1(λ). Tensoring with spinor modules is exact, and since the restriction rules of spinor modules are straightforward, all that is left to understand is the restriction of X−1(λ). Garcia and Procesi [26] studied a very similar graded module. It is the module

Xq(λ) but with the reverse q grading.

λ Deﬁnition 1.4.1. [26, I.7] Let p (q) be the character of the graded Sn-module

X i 2i q H (Bλ). i≥0

In [26] a reduction rule for partitions is deﬁned. Given a partition λ ` n this rule outputs a set, potentially with multiples, of partitions λ(i) ` n − 1.

0 Deﬁnition 1.4.2. [26, 1.1] Let λ = (λ1, . . . , λr) be a ﬁxed partition of n and λ = 0 0 (λ1, . . . , λr) be its conjugate.

58 For 1 ≤ i ≤ l(λ) deﬁne the integer ai by the condition

λ0 ≥ i > λ0 . ai ai+1

Given this integer, let λ(i) be the partition created when one removes a block from the bottom of column ai of the Young diagram associated to λ.

Note that here the inequality diﬀers from [26] but this is due to the fact that Garcia and Procesi deﬁne their partitions to increase. n Given λ ` n, let Θn−1(λ) be the set of partitions of n − 1 which one can create from the Young diagram of λ by removing a single box.

Lemma 1.4.3. For a strict partition λ = (λ1, . . . , λr) ` n we have

(i) λ = (λ1, . . . , λi−1, λi − 1, λi+1, . . . , λr).

(i) n Hence {λ : i = 1, . . . , l(λ)} is equal to the set Θn−1(λ).

Proof. Because λ is strict, the column of λ which is ﬁrst larger than i, λ0 , will ai always be the column that has a block from λ1, . . . , λi and no others. So removing a block from row λ is equivalent to removing a block from column λ0 . Hence i ai (i) λ = (λ1, . . . , λi−1, λi − 1, λi+1, . . . , λr). The second statement follows since we have an explicit description of {λ(i) : i = 1, . . . , l(λ)}.

˜ n Given a module or character A of either Sn, Sn or C(n), let Resn−1(A) denote the ˜ restriction to the rank (n − 1) object, that is the restriction to Sn−1, Sn−1 or C(n − 1).

λ Theorem 1.4.4. [26, 3.3] The restriction of the Sn q-graded character p (q) to Sn−1 is a sum of qi−1pλ(i) (q);

l(λ) n λ X i−1 λ(i) Resn−1 p (q) = q p (q). i=1

Remark 1.4.5. If we let χq(λ) be the character of Xq(λ) then

λ 1 n(λ) χq(λ) = p ( )q q

Pl(λ) where n(λ) = i=1(i − 1)λi.

We can combine Remark 1.4.5 with Theorem 1.4.4 to understand the branching rules for Xq(λ).

59 Lemma 1.4.6. Let λ be a strict partition. The restriction of the Sn-module Xq(λ) to

Sn−1 is l(λ) n M 2i−2 (i) Resn−1Xq(λ) = q Xq(λ ). i=1

λ 1 n(λ) Proof. By Remark 1.4.5 one can write χq(λ) = p ( q )q . Then, we can use the restriction rules given by Theorem 1.4.4 to restrict χq(λ) in terms of the characters λ(i) (i) p (q). Remark 1.4.5 can be used again to rewrite this in terms of characters χq(λ ). The coeﬃcient one gets is qi−1+n(λ)−n(λ(i)). Writing out the deﬁnition of n(λ);

l l(λ(i)) (i) X X (i) n(λ) − n(λ ) = (λ)(j − 1)λj − (j − 1)λj . j=1 j=1

However, by Lemma 1.4.3, λ and λ(i) differ by only one entry for λ strict. This difference is in the ith entry. Therefore n(λ) and n(λ(i)) differ by i − 1. Hence i − 1 + n(λ) − n(λ(i)) = 2i − 2.

Lemma 1.4.7. Let λ be a strict partition. The restriction of the graded Sn-module

X−1(λ) is, l(λ) n M (i) Resn−1X−1(λ) = X−1(λ ). i=1 Proof. This is the specialisation of Lemma 1.4.6 to q = −1.

Recall the deﬁnition ofτ ˜λ [4, A.3]; 1 τ˜λ = X−1(λ) ⊗ S. aλ

The set {τ˜λ : λ is strict} is a transversal for the irreducible genuine projective supermodules of Sn.

Remark 1.3.10 states that tensoring with the spinor kills X−1(µ) if µ is not strict. So we can describe which modules occur in the restriction of τ˜λ. Since aλτ˜λ = S ⊗ X−1(λ)

(i) (i) Lemma 1.4.8. Let λ be a strict partition. For a ﬁxed i, if λ is strict then τ˜λ is a n summand of Resn−1(˜τλ).

0 Proof. We note that aλτ˜λ = X−1(λ) ⊗ S. Here, S (resp. S ) is the spinor supermodule ˜ for the Cliﬀord algebra C(n) (resp. C(n − 1)). Hence restricting aλτ˜λ to Sn−1 is equivalent to restricting X−1(λ) to Sn−1 and S to C(n − 1). Therefore

60 n n n aλResn−1τ˜λ = Resn−1S ⊗ Resn−1X−1(λ) l(λ) 0 M (i) = cS ⊗ X−1(λ ). i=1

M 0 (i) = cS ⊗ X−1(λ ). λ(i) which are strict Here, c ∈ Z \{0} and will be determined in Theorem 1.4.9. Hence, for every strict (i) ˜ λ the Sn supermoduleτ ˜λ(i) occurs as a summand.

With the information we have, it is possible to describe the explicit branching graph for the supermodules of Tn and hence the branching graph for genuine projective ˜ supermodules of Sn. This result could be deduced from the crystal graph theory in [35].

Recall that if λ is not a strict partition we setτ ˜λ = 0 .

˜ Theorem 1.4.9. The branching rules of the genuine projective irreducible Sn-supermodules are:

Here λ is always a strict partition, λ(i) depends on a reduction rule, deﬁned in [26], which gives partitions λ(i) from λ.

Proof. By Lemma 1.4.7 we already know which summands occur. Hence calculating the multiplicities is all that is required. We know the multiplicities of the restriction of the spinor S; 2 if n is even, 1 otherwise. Also, by Lemma 1.4.7 the multiplicities in the restriction of X−1(λ) are always 1. Hence, we can ﬁnd the multiplicities for τ˜λ 1 1 by comparing a and a from Proposition 1.3.8. This is a simple calculation on the λ λ(i)

61 eight cases

 1 1 n even, l(λ) even and l(λ(i)) = l(λ),  2 a (i)  λ  1 1 (i)  2 a n even, l(λ) even and l(λ ) = l(λ) − 1,  λ(i)  1 (i)  a n even, l(λ) odd and l(λ ) = l(λ),  λ(i)  1 1 (i) 1  2 a n even, l(λ) odd and l(λ ) = l(λ) − 1, = λ(i) a 1 (i) λ 2 a n odd, l(λ) even and l(λ ) = l(λ),  λ(i)   1 n odd, l(λ) even and l(λ(i)) = l(λ) − 1,  a (i)  λ  1 (i)  a n odd, l(λ) odd and l(λ ) = l(λ),  λ(i)  1 (i)  a n odd, l(λ) odd and l(λ ) = l(λ) − 1. λ(i)

(i) Note that l(λ ) = l(λ) − 1 occurs once if and only if λr = 1, in which case it is the partition λ(r) which has shorter length.

1.5 Spectrum data for S˜n

In Section 1.6 we construct the genuine projective representations. However, the raw information that we need to do this is the action of the Jucys-Murphy elements squared; this is what we call the spectrum data. We prove that this is equivalent to a function on the contents of the Young tableaux for λ. + As described before Tn is generated as an associative algebra by τα for α ∈ Φ .

Deﬁnition 1.5.1. [9, 3.1] The Jucys-Murphy elements in Tn for i = 1, . . . , n, are: X Mi = τα. + α∈Φ :(α,ξi)<0

This is the same construction as the Jucys-Murphy elements for Sn. We have just replaced transpositions with pseudo-transpositions.

Note that Brundan and Kleschev define the Jucys-Murphy elements with different terminology. They use the generators τi and replace τα with [ij] when α = ξi − ξj P and then define Mi = j

Remark 1.5.2. The Jucy-Murphy elements anti-commute [9, 3.1], that is

MkMl = −MlMk if k 6= l.

62 Lemma 1.5.3. [9, 3.2] The even centre Z(Tn)0 of Tn is spanned by the set of symmetric polynomials of the squares of the Jucys-Murphy elements.

˜ Schur (cf. [54]) deﬁned all of the genuine projective characters for Sn and showed that these correspond to the set of shifted partitions λ ∈ SP(n).

˜ Deﬁnition 1.5.4. Let Vλ denote any genuine projective Sn-module which has character corresponding to λ ∈ SP(n).

It is useful, for notation, to introduce a function q : Z → Z such that q(m) = m(m + 1).

Deﬁnition 1.5.5. [56] We say α = (α1, . . . , αn) is in Sspec(n) if there exists a vector ˜ v in an irreducible genuine projective representation V of Sn such that: 1 M 2v = q(α )v for all k = 1, . . . , n. k 2 k

Sspec(n, λ) is the restriction of Sspec(n) by considering vectors in any Vλ, for a ﬁxed λ ∈ SP(n).

Deﬁnition 1.5.6. Let λ ∈ SP(n) be a shifted partition.The content of a box (i, j) contained in a shifted Young diagram λ is its distance from the main diagonal.

cont(i, j) = j − i

Deﬁnition 1.5.7. We introduce the set of shifted content vectors Scont(n). A vector

β = (β1, . . . , βn) is associated to a standard shifted tableaux [[λ]] if, for all i = 1, . . . , n,

βi is equal to the content of the box labelled i in [[λ]]. Scont(n) is the set of vectors which are associated to a standard shifted tableau of size n. Similarly Scont(n, λ), is the set of vectors associated to standard shifted tableaux of shape λ.

In this section, our goal is to prove that

Sspec(n) = Scont(n) and Sspec(n, λ) = Scont(n, λ). Recall the Casimir elements for and S˜ ,Ω and Ω respectively. Our technique H n H Sñ for describing the set Sspec(n) is largely based on using two different descriptions of the action of the Casimir element Ω . The first one, descending from Dirac cohomology, Sñ

63 states that Ω = Ω on τ˜ . The second is linked to the Jucy-Murphy elements. Using S˜n H λ these descriptions we are able to show Sspec(n) = Scont(n) inductively.

Recall the deﬁnition of Xλ;

H Xλ = Ind ×..,× (Stλ1 ... Stλk ). Hλ1 Hλk

A central character is a map Z(H) = S(V )W → C. The standard representations W Xλ have central characters χλ. The vector space Hom(S(V ) , C) can be associated with V ∗/W via evaluating polynomials in S(V )/W at an element in V ∗/W . Hence a central character, χ, corresponds to an element ν ∈ V ∗/W .

Lemma 1.5.8. Let Stm be the Steinberg module for Hm. Let νStm be the element in ∗ ∗ S(V ) /W corresponding to χStm . Let e1, . . . , em ∈ V be a a dual basis of the basis x1, .., xm of V . The element νStm is m m − 1 m − 3 m − 3 m − 1 X 2i − m ν = − e + − e + ... + e + e = e . Stm 2 1 2 2 2 m−1 2 m 2 i i=1

Corollary 1.5.9. The element νλ deﬁning the central character of Xλ is

l(λ) X ν = ν . Stλi i=1

Lemma 1.5.10. [4, 2.5] On the representation Xλ of the graded Hecke algebra H,

ΩH = (ν, ν).

Lemma 1.5.11. On the representation τ˜λ corresponding to λ = (λ1, . . . , λr),

l(λ) λi X X 2j − λi Ω = ( )2. H 2 i=1 j=1 Furthermore, this can be reformulated in terms of the content of λ, 1 X Ω = q(cont(i, j)). H 4 (i,j)∈λ Proof. The ﬁrst part can be proved by noticing that, when one forgets the grading, the module X−1(λ) is just the parabolically induced module Xλ, see [7]. Hence, one can use Lemma 1.5.10 to show how ΩH acts. The constant (ν, ν) is exactly the sum we stated. The second part follows by a simple induction. One can show Pλi 2 Pλi Pλk j=1(2j − λi) = j=1 j(j + 1) = j=1 q(cont(k, j)), splitting the cases when λi is odd or even. This is covered in more detail in Section 1.7.

64 Lemma 1.5.12. On genuine projective representations, Ω = 1 Pn M 2. Here M S˜n 2 i=1 k k ˜ denotes the Jucys-Murphy elements for Sn.

Proof.

1 X Ω ˜ = |α||β|τατβ, Sn 4 α,β∈Φ+ sα(β)<0 1 X 1 X = |α||β|τ τ = ( |α|τ )2, 4 α β 4 α α,β∈Φ+ α∈Φ+ n n 1 X 1 X = ( |α|M )2 = M 2. 4 k 2 k k=1 i=1

In the second equality, we use τατβ = −τβτsα(β) for α and β such that sα(β) > 0. Pn P The third equality uses the fact that every i=1 Mk = α∈Φ+ τα and the last equality uses the fact that the Jucys-Murphy elements anti-commute in Tn [9, 3.1]. Hence the square of the sum is equal to the sum of the squares.

Sspec(n, λ) = Scont(n, λ).

Proof. We prove the theorem by induction, the base case being trivial. Suppose for every shifted partition µ ∈ SP(n − 1), Scont(n − 1, µ) = Sspec(n − 1, µ). Let us ﬁx a shifted partition λ of n, and consider τ˜λ the irreducible representation of Tn associated to λ.

Let α = (α1, . . . , αn) ∈ Sspec(n, λ). By deﬁnition there exists a vector vα ∈ τ˜λ 2 such that Mj (vα) = q(αj)vα for all j. Ll(λ) The restriction of τ˜λ to Tn−1 is the direct sum, i=1 τ˜λ(i) . Also vα is an eigenvector for the Jucy-Murphy elements corresponding to (α1, . . . , αn), when restricted to Tn−1 vα corresponds to (α1, . . . , αn−1). We know, by the inductive hypothesis, (α1, . . . , αn−1) ∈ Sspec(n − 1, λ(j)) = Scont(n − 1, λ(j)). Explicitly, this means there exists a standard (j) (j) numbering [[λ ]] of λ which corresponds to (α1, . . . , αn−1). Therefore (α1, . . . , αn−1) (j) is in one shape namely λ . Hence vα ∈ τ˜λ(j) ⊂ τ˜λ for some ﬁxed j.

65 Now using Lemma 1.5.11, Lemma 1.5.12 and Theorem 1.3.13, we can describe the ˜ ˜ action of the Casimir elements of Sn and Sn−1 on vα. Since vα is inτ ˜λ ,

n 1 X 2 1 X M vα = Ω ˜ vα = Ω vα = cont(i, j)(cont(i, j) + 1)vα. 2 k Sn Hn 4 i=1 (i,j)∈λ

However vα is also contained inτ ˜λ(j) ⊂ τ˜λ so,

n−1 1 X 2 1 X M vα = Ω ˜ vα = Ω vα = cont(i, j)(cont(i, j) + 1)vα. 2 k Sn−1 Hn−1 4 i=1 (i,j)∈λ(j)

1 2 The diﬀerence of these two Casimir elements is 2 Mn. Hence   1 X 1 X M 2v = cont(i, j)(cont(i, j) + 1) − cont(i, j)(cont(i, j) + 1) v n α 2 2  α (i,j)∈λ (i,j)∈λ(j)

1 = cont(k, l)(cont(k, l) + 1)v , 2 α where (k, l) is the block of λ not included in λ(i). Taking the standard shifted tableau (j) [[λ ]](α1,...,αn−1) associated to (α1, . . . , αn−1) and adding the block (k, l) labelled with the number n creates a standard shifted tableau [[λ]]α of shape λ. By the above argument this standard shifted tableau has content vector equal to α ∈ Sspec(n, λ). Hence α ∈ Scont(n, λ). For the reverse direction, we use an almost identical argument. Suppose β =

(β1, . . . , βn) ∈ Scont(n, λ) corresponds to a standard shifted tableau [[λ]]β. By restriction and inductive hypothesis, there exists a vector v(β1,...,βn−1) ∈ τ˜λ(j) ⊂ τ˜λ such that 2 2 Mk v(β1,...,βn−1) = q(βk)v(β1,...,βn−1). Again using the fact that Mn acts by the diﬀerence of the Casimir elements ΩSn and ΩSn−1 , we get,

2 1 M v(β ,...,β ) = 2(Ω ˜ − Ω ˜ ) = q(cont(n))v(β ,...,β ), n 1 n−1 Sn Sn−1 2 1 n−1 where cont(n) is the content of the box labelled by n in [[λ]]β. Hence β ∈ Spec(n, λ).

66 1.6 Explicit representation from spectrum data

This is, in essence, the same as the last part of [56]. We describe how to explicitly construct the genuine irreducible representations from the spectrum data we calculated in the previous section. ˜ Let hM1,...,Mni be the subalgebra of C[Sn] generated by the Jucys-Murphy elements Mi. ˜ Lemma 1.6.1. Let V be a genuine irreducible projective representation of Sn. Then the restriction of V to hM1,...,Mni is a direct sum of the common eigenspaces of 2 Mi .

2 Proof. The Mi all commute so we can separate V into its common eigenspaces. Now

Mi ﬁx these spaces so the common eigenspaces are hM1,...,Mni submodules of V .

Lemma 1.6.2. Let U be an hM1,...,Mni-supermodule which occurs as a common 2 eigenspace of Mi in some V . Then the the representation U of hM1,...,Mni factors 2 through the Cliﬀord algebra of rank r, where r = |{Mi with non zero eigenvalue}| = n − l(λ).

Proof. The set of Jucys-Murphy elements anti-commute and on U, by Deﬁnition 1.5.5, 2 p Mi = q(αi) for some α = (α1, . . . , αn) ∈ Sspec(n). By sending Mi to q(αi)ci for

αi =6 0, where ci are the Clifford generators for the Clifford algebra, the action of hM1,...,Mni on U factors through the Clifford algebra of rank #{αi 6= 0}. ˜ Corollary 1.6.3. Let τ˜λ be the genuine irreducible supermodule of Sn, associated to λ. 2 Then the common eigenspaces for Mi are labelled by the spectrum data α ∈ Sspec(n, λ).

Hence, when considered as a hM1,...,Mni-supermodule, M τ˜λ = Uα. α∈Sspec(n,λ) ˜ If we take any genuine projective irreducible representation V of Sn then when we restrict it to the subalgebra hM1,...,Mni, generated by the Jucys-Murphy elements, 2 V decomposes as the eigenspaces of Mi . Furthermore these eigenspaces are irreducible hM1,...,Mni-modules which can be considered as spinor modules for the Cliﬀord algebra of rank n − l(λ). ˜ We build a genuine irreducible representation Vλ of Sn which is isomorphic to τ˜λ. n Since the set Sspec(n) is a subset of N then Sn naturally acts on this set, further,

67 Sn preserves Sspec(n, λ). We understand how Mi act on an irreducible Tn-module corresponding to λ. Motivated by studying how τi acts on τ˜λ, we will add in the action of τi on Vλ.

Mi−Mi+1 Lemma 1.6.4. Using the decomposition in Corollary 1.6.3, the subspace (τi− )Uα q(ai)−q(ai+1) of τ˜λ is ﬁxed by hMji and Mi − Mi+1 τi − Uα = Usi(α). q(ai) − q(ai+1)

Proof. One can see that for every u ∈ Uα, Mi − Mi+1 Mi − Mi+1 Mj τi − u = − τi − Msi(j)u. q(ai) − q(ai+1) q(ai) − q(ai+1)

Mi−Mi+1 Therefore we can conclude that τi − Uα is the subspace we labelled q(ai)−q(ai+1)

Mi−Mi+1 Us (α). Note that if si(α) ∈/ Sspec(n, λ) then τi − Uα = 0 hence, in this i q(ai)−q(ai+1) Mi−Mi+1 case, τi = on Uα. q(ai)−q(ai+1)

Deﬁnition 1.6.5. If α and si(α) are in Sspec(n, λ) then deﬁne Psi : Vα → Vsi(α) to be the vector space isomorphism such that

M 2 (P (v)) = P M v for all v ∈ V . si(j) si si j α

Note that Psi is not an hM1, .., Mni isomorphism since it interchanges the action ˜ of diﬀerent Jucys-Murphy elements. Now we can describe Vλ as a Sn-module.

P ˜ Deﬁnition 1.6.6. Let Vλ = α∈Sspec(n,λ) Vα as supermodules of hM1,...,Mni ⊂ Sn.

Let α = (α1, . . . , αn) ∈ Sspec(n, λ). The action of τi on Vα is deﬁned in the following way. If si(α) ∈ Sspec(n) then,

1 − 2 Mi − Mi+1 q(ai) + q(ai+1) τi = − 1 − 2 Psi . q(ai) − q(ai+1) (q(ai) − q(ai+1))

q(ai)+q(ai+1) If si(α) ∈/ Sspec(n, λ) then (1 − 2 )) = 0 and on Vα, (q(ai)−q(ai+1)

Mi − Mi+1 τi = . q(ai) − q(ai+1)

Theorem 1.6.7. For the set of shifted tableaux SP(n), {Vλ : λ ∈ SP(n)} is a full set ˜ of representatives of the irreducible genuine projective supermodules of Sn.

68 Proof. All we need to show is that the supermodules Vλ are isomorphic to τ˜λ. Corollary

1.6.3 shows that they are isomorphic as hM1,...,Mni-supermodules. The following arguments show that the action of τi agree on both Vλ and τ˜λ. If si(α) ∈/ Sspec(n, λ) then Lemma 1.6.4 shows that τi acts on both supermodules identically. In the case

Mi−Mi+1 that (τi − ) interchanges Vα and Vs (α) one observes, that applying this q(ai)−q(ai+1) i operator twice gives 2 Mi − Mi+1 q(ai) + q(ai+1) τi − = 1 − 2 ) Id q(ai) − q(ai+1) (q(ai) − q(ai+1)

q(ai)+q(ai+1) on Vα. Let p = 1 − 2 . Furthermore, again by Lemma 1.6.4 (q(ai)−q(ai+1))

Mi − Mi+1 1 τi − = −√ Psi . q(ai) − q(ai+1) p Rearranging this equation gives the result,

1 − 2 Mi − Mi+1 q(ai) + q(ai+1) τi = − 1 − 2 Psi . q(ai) − q(ai+1) (q(ai) − q(ai+1))

So again the action of τi is identical on both modules.

Lemma 1.6.8. [50, 1.2] The hook length formula for shifted tableaux gives an explicit description for the number of standard shifted tableaux for a shifted partition λ =

(λ1, . . . , λr} n, r n! Y λi − λj #{ standard shifted tableaux of shape λ} = . λ ! . . . λ ! λ + λ 1 r i=1 i j

Corollary 1.6.9. The dimension of Vλ is

n−l(λ) n! λi − λj d 2 e r 2 Πi=1 . λ1! . . . λr! λi + λj

Proof. We built Vλ as a direct sum of simple supermodules Sn−l(λ), which have d n−l(λ) e dimension 2 2 . The number of Sn−l(λ) in the sum is equal to the number of standard shifted tableaux, which is n! Qr λi−λj . λ1!...λr! i=1 λi+λj

Corollary 1.6.9 shows that the ungraded representations σ˜λ in [15] have dimension r b n−l(λ) c n! Y λi − λj 2 2 . λ ! . . . λ ! λ + λ 1 r i=1 i j This is stated but not proved in [15, Example 6.9]. We illustrate, with an example, how one can create an explicit model of a representation corresponding to a shifted partition λ.

69 ˜ Example 1.6.10. Take S4 and the shifted tableau λ = (3, 1) = . There are two standard shifted tableaux, namely,

1 2 3 1 2 4 . 4 3 with content vectors

α1 = (0, 1, 2, 0), α2 = (0, 1, 0, 2).

In this case n − l(λ) = 2 so our building blocks are C(2)-supermodules. Let S be 2 the (1,1) dimensional supermodule of C(2) = hc1, c2 : c1c2 = −c2c1, ci = 1i where √ 0 1 0 −1 c = and c = √ . 1 1 0 2 − −1 0

As a superspace V = S1 ⊕ S2, where Sj ∼= S as a C(2)-module. For the action of (3,1) √ √ √ the Jucys-Murphy elements, M1 = 0, M2 = ( 2c1, 2c1), M3 = ( 6c2, 0) and M4 = √ (0, 6c2). Finally, to calculate the action of τi, when i = 1, 2, si(αj) ∈/ Sspec(4, (3, 1)) 1 2 therefore τ1 and τ2 ﬁx S and S . Explicitly, using Deﬁnition 1.6.6: √   √0 2 0 0  2 0 0 0  τ1 =  √  ,  0 0√ 0 2 0 0 2 0 √ √  0 1 ( −6 − 2) 0 0  √ 2 1 √ − 2 ( −6 + 2) 0 0 0  τ2 =  √  .  0 0√ 0 2 0 0 2 0

Now τ3 interchanges α1 and α2, hence again using Deﬁnition 1.6.6

√  −6 q 2  0 3 3 0  √ q  − −6 0 0 2   3 3  τ3 =  q √  .  2 −6   3 0 0 3   √  q 2 −6 0 3 − 3 0

This gives an explicit realisation of of the supermodule V(3,1) on the four dimensional space spanned by {v1, v2, v3, v4}, by the above matrices. Note that the even space is 0 1 V(3,1) = span{v1, v4} and the odd space is V(3,1) = span{v2, v3}.

70 1.7 Description of Vogan’s morphism

This section is dedicated to describing Vogan’s morphism in type A. We use the description of the action of Z(Tn)0 on Vλ from Section 1.5 and the central character

χλ of Xλ describing the actions of Z(H) on Xλ. Vogan’s morphism for graded Hecke algebras is the algebra homomorphism ζ : ˜ ˜ Z(H) → Z(C[Sn]) deﬁned such that ζ(z) for z ∈ Z(H) is the unique element in Z(Sn) such that z ⊗ 1 = ρ(ζ(z)) + Da + aD as elements of H ⊗ C(V ), for some a ∈ H ⊗ C(V ). This morphism was introduced in [4] and is inspired by Vogan’s morphism for real reductive groups. It occurs naturally in Dirac theory for graded Hecke algebras but as far as the author is aware it has not been described unlike in Dirac theory for real reductive groups. We will describe

ζ : Z(H) → Z(Tn).

˜ This is ζ composed with the projection from C[Sn] to Tn. The following lemma is well known.

Lemma 1.7.1. Let A be a semisimple ﬁnite dimensional C-algebra and let a and b ∈ Z(A). Suppose for all irreducible representations σ of A,

σ(a) = σ(b), then a = b.

Since Tn is a semisimple ﬁnite dimensional algebra over C, we use Lemma 1.7.1 and our description of how the centres act on Vλ to describe ζ.

Let λ = (λ1, . . . , λk) be a partition. Recall the central character χλ of the H-module Sn Xλ is a function χλ : Z(H) → C. Since Z(H) = S(V ) any central character can be ∗ deﬁned by a Sn orbit of V . Let {yi} be a dual basis of {xi} then the element deﬁning

χλ is k λ 1 X Xi ν = (−λ + 2j − 1)y . λ 2 i λi+1+j i=1 j=1

71 Example 1.7.2. Take λ = (5, 3, 2), then the coeﬃcients of ν can be encoded in the following labelled diagram similar to a Young diagram for (5, 3, 2) but translated to be vertically symmetrical. −2 −1 0 1 2 −1 0 1 1 1 − 2 2

∗ Deﬁnition 1.7.3. The dual map ζ takes the irreducible representations of Tn to central characters of H. Let σ ∈ irr(Tn) and let p ∈ Z(H). ∗ The central character ζ (σ): Z(H) → C is

ζ∗(σ)(p) = σ(ζ(p)).

We can describe ζ∗ explicitly.

Lemma 1.7.4. Let (Vλ, σλ) be the Tn-representation described in Section 1.6 corresponding to a strict partition λ, let χλ be the central character of Xλ. Then as described ∗ ζ (Vλ) = χλ.

This is a corollary of [4, 4.4].

Lemma 1.7.5. Let PH ∈ Z(H) and PTn ∈ Z(Tn). If, for all λ strict,

Vλ(PTn ) = χλ(PH)

then ζ(PH) = PTn .

Proof. The set of Vλ for shifted partitions is a full set of irreducible representations ∗ of Tn. Also ζ (Vλ) = χλ. So the assumption is equivalent to, for all σ ∈ irr(Tn),

σ(PTn ) = (σ)(ζ(PH)). Then applying Lemma 1.7.1, ζ(PH) = PTn .

Sn We deﬁne polynomials in S(V ) = Z(H) and elements in Z(Tn) which have the same action. Then we use Lemma 1.7.5 to describe ζ. Recall that the even centre of

Tn is spanned by symmetric polynomials in the squares of the Jucy-Murphy elements

Mi.

72 Deﬁnition 1.7.6. Let Mi be the Jucys-Murphy elements for Tn. We deﬁne, n 2m X 2m PTn = (2Mi) ∈ Z(Tn)even. i=1

The set {x1, .., xn} is an orthogonal basis of V . Deﬁne

m b c n X2 m X P 2m = (2x )2m−2j ∈ Z( ) . H 2j i H even j=0 i=1 Theorem 1.7.7. For m, n ∈ N and for all shifted λ, σ (P 2m) = χ (P 2m). λ Tn λ H We delay the proof of this theorem and ﬁrst state the consequences.

Corollary 1.7.8. We have ζ(P 2m) = P 2m. H Tn Proof. This follows from Lemma 1.7.5 and Theorem 1.7.7.

Hence we can describe how ζ acts on P 2m. Note that this set of polynomials spans H the even centre of H so we have described half of ζ.

Lemma 1.7.9. For all λ strict and n, m ∈ N, n X 2m+1 χλ( xi ) = 0. i=1

Proof. The character χλ is defined by νλ which for every positive coefficient has a negative coefficient of the same magnitude. Hence any symmetric odd polynomial evaluated on νλ is zero.

This gives a description of ζ on the odd part of the centre, Z(H)odd. Hence we have described the action of ζ on Z(H).

Corollary 1.7.10. The map ζ surjects onto the even part of the centre of Tn. Also ker(ζ) ⊃ S(V )odd.

Proof. The ﬁrst statement is clear since Z(T )even is generated by the symmetric polynomials in the squares of the Jucys-Murphy elements. The second statement follows from the fact that every symmetric homogeneous polynomial of odd degree is generated by monomials of power polynomials where there is at least one odd power polynomial. Hence ζ, since it is a homomorphism, will kill any homogeneous symmetric polynomial of odd degree.

73 Note that Kleschev and Brundan [9, 3.2] provide a basis for Z(Tn)even in the form of products of power polynomials of the M 2. The set {Πk P 2mi : Pk 2m + 1 ≤ n} i i=1 Tn i=1 i is a basis for Z(Tn)even. Therefore k X {ζ(Πk P 2mi ): 2m + 1 ≤ n} i=1 H i i=1 is a basis of Z(Tn). For a particular n it is possible to ﬁnd the even elements in the kernel of ζ, but this must be done on a case-by-case basis comparing the action on each Vλ and showing it to be zero.

1.7.1 Proof of Theorem 1.7.7

Definition 1.7.11. Define P1 to be the set of partitions, of any number, of length 1. 1 1 Pn is the singleton subset of P , consisting of the partition λ = (n). Our first step is to show Theorem 1.7.7 can be proved just by considering partitions of length one.

Theorem 1.7.12. Let µ ∈ P1 be a partition of length one then, for all m ∈ N, σ (P 2m) = χ (P 2m). µ Tn µ H Lemma 1.7.13. Theorem 1.7.12 implies Theorem 1.7.7, that is, it is enough to show the result on modules corresponding to the partition of length one.

Proof. Let λ = (λ1, λ2, . . . , λk) be a shifted partition. We are studying the action of power polynomials. Let Qj be the jth power polynomial in the x ’s, Pn xj. Let H i i=1 i

χλ1×...×λk be the central character of the Hλ1 × ... × Hλk -module Stλ1 ⊗ ... ⊗ Stλk and

χλl be the central character of the Hλl -module Stλl . We consider Hλl to be embedded in such that x , . . . , x are in the image of . Since Qj ∈ × ... × , H λl−1+1 λl Hλl H Hλ1 Hλk then k j j X j χλ(Q ) = χλ1×...×λk (Q ) = χλl (Q ). H H Hλl l=1 Similarly, for Qj = Pn M j, Tn i=1 i k j X j σλ(QT ) = σλl (QT ). n λl l=1 1 Therefore if we can prove the result of the theorem for every µ = (λl) ∈ P , then using the above decomposition of both χ (Qj ) and σ (Qj ) we can extend this to σ λ H λ Tn λ and χλ for every strict λ.

74 Lemma 1.7.14. Let λ = (λ1, .., λn) n be a shifted partition, with associated shifted

Young diagram λ = {(i, j): i = 1, .., l(λ) and j = i − 1, . . . , i − 1 + λi.}. Then

n n 2m X 2m X m m σλ(PTn ) = σλ( (2Mi) ) = (cont(i)) (cont(i) + 1) . i=1 i=1 Proof. This is a restatement of Theorem 1.5.13 where we described the action of 2 2 Mi on Vλ. Recall that the result showed that on a certain subspace Mi acted by cont(i)(cont(i) + 1).

Corollary 1.7.15. Fix a t ∈ Z and let µ = (µi) = (t) t. Then

t t X 2m X m m σµ( (2Mi) ) = (i − 1) (i) . i=1 i=1 Proof. This follows from Lemma 1.7.14 since for µ, cont(i) = i − 1.

Lemma 1.7.16. Fix a t ∈ Z Let µ = (µ1) = (t) t. Then

t t X 2m X 2m χµ( (2xi) ) = (2(i − 1) − (t − 1)) . i=1 i=1

Pt Proof. The central character χµ is deﬁned by νµ = i=1(2(i − 1) − (t − 1)yi. The Pt 2m result follows by evaluating i=1(2xi) at νµ.

Proof of Theorem 1.7.12. Fix m ∈ N. We prove this statement by induction on n with steps of length two. Note that for n = 0 and n = 1, all operators act by zero. Hence the base cases are trivial. Suppose the result is true for n − 2, so:

σ (P 2m ) = χ (P 2m). (n−2) Tn−2 (n−2) H

Now, considering that σ(n−2) is the restriction of σ(n) to Tn−2,

2m 2m 2m 2m σ(n)(PTn ) − σ(n−2)(PTn−2 ) = σ(n)((2Mn−1) + (2Mn) ).

By Corollary 1.7.15 this is equal to

75 t t−2 X X (i − 1)m(i)m − (i − 1)m(i)m = (t − 2)m(t − 1)m + (t − 1)m(t)m i=1 i=1 = (t − 1)m((t − 1 + 1)m + (t − 1 − 1)m) m X m = (t − 1)m( (t − 1)m−l(1l + (−1)l)) l l=0 X m = (t − 1)m 2 (t − 1)m−l l l∈0,...,m and even b m c X2 m = 2 (t − 1)2m−2j. 2j j=0

We know that χ (P 2m) − χ P 2m , (t) Ht (t−2) Ht−2 is just the action of the x and x . Expanding out P 2m and explicitly writing t t−1 Ht χ (P 2m) using Lemma 1.7.16, we have (t) Ht

b m c X2 m χ (P 2m) − χ P 2m = 2 (t − 1)2m−2j. (t) Ht (t−2) Ht−2 2j j=0 Therefore we have shown that the inductive step holds. Hence

σ (P 2m ) = χ (P 2m) (n) Tn−2 (n) H by induction.

We have proved Theorem 1.7.12 and hence have proved Theorem 1.7.7.

76 Chapter 2

Dirac cohomology and simple modules of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

77 Abstract

We define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This presentation uncovers the Dunkl-Opdam subalgebra as a Drinfeld algebra. The Dunkl-Opdam subalgebra is the first natural occurrence of a non-faithful Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We formalise generalised graded Hecke algebras and extend a Langlands classification to generalised graded Hecke algebras. We then describe an equivalence between the irreducibles of the Dunkl-Opdam subalgebra and direct sums of graded Hecke algebras of type A. This equivalence commutes with taking Dirac cohomology. 2.1 Introduction

We study the Dunkl-Opdam subalgebra, HDO [22], of the rational Cherednik algebra n associated to G(m, 1, n) = Sn o (Zm) . This subalgebra of the rational Cherednik algebra Ht(G(m, 1, n)) (Deﬁnition 2.4.1) is independent of the parameter t. In this chapter, we take a closer look at HDO and notice that it is similar to both a graded Hecke algebra and a Drinfeld algebra. We extend several results for Hecke algebras and faithful Drinfeld algebras to include the Dunkl-Opdam subalgebra.

We construct a new presentation of HDO:

Theorem 2.4.13. There exists a presentation of HDO given by elements {z˜i : i =

1, ..., n} and generators {si, gj : i ≤ i ≤ n − 1, 1 ≤ j ≤ n} in G(m, 1, n) such that:

−1 siz˜jsi = si(˜zj),

giz˜j =z ˜jgi ∀i, j = 1, ..., n,

[˜zi, z˜j] ∈ CG(m, 1, n), ∼ n where G(m, 1, n) = Sn o (Zm) . The group Sn is generated by transpositions si, th i = 1, ..., n − 1 and gj generates the j copy of Zm.

This presentation exposes HDO as a Drinfeld algebra. Drinfeld [21] initially defined these algebras (Definition 2.2.1) with the potential to have non-faithful representations. In the literature this has been largely forgotten, perhaps because there appeared to be no natural examples of a non-faithful Drinfeld algebra. The Dunkl-Opdam subalgebra is a naturally occurring non-faithful Drinfeld algebra. Ciubotaru [13] defined Dirac cohomology for faithful Drinfeld algebras and we extend this to non-faithful Drinfeld algebras. Dezéléeintroduced the idea of generalised graded Hecke algebra to look at the Dunkl-Opdam subalgebra. We concretely define a class of algebras, which contains Dezélée’sexamples. We extend Evens’ [25] Langlands classification to generalised graded Hecke algebras.

∼ n The Dunkl-Opdam subalgebra has a commutative subalgebra CT = (Zm) . We decompose representations into weight spaces (Deﬁnition 2.3.4), which then deﬁnes weights of a representation. The weights of an HDO-representation come in orbits (Lemma 2.3.10). We use these weights to highlight that irreducible representations of

HDO are pullbacks of H(Sai )-representations via speciﬁc quotients (Lemma 2.5.5).

∼ G irr(H(G(m, 1, n))) = irr(H(Sa0 )) ⊗ ... ⊗ irr(H(Sam−1 )). a∈A

We use this equivalence to describe the Dirac cohomology of an HDO-module X −1 in terms of Dirac cohomology of the associated H(Sai )-modules. Let F and F be functors displaying this equivalence.

Theorem 2.6.7. Given an irreducible -representation V then F (V ) as an ⊗ HDO HSa0 ... ⊗ -module is isomorphic to F (V ) ∼ X ⊗ ... ⊗ X . The Dirac cohomology HSam−1 = a0 am−1 of V is M c HD(Xa0 ) ⊗ ... ⊗ HD(Xam−1 ) , c∈C where HD(X) is the type A Dirac cohomology of the HSk -module X, P is the parabolic subset associated to a and C is a set of coset representatives of Sn/SP . Let HD(•) denote the functor taking the relevant module to its Dirac cohomology. We have the following commutative diagram:

80 G ( (G(m, 1, n))) F G (L ⊗ ... ⊗ ) 0 H 0 a∈A HSa0 HSam−1

HD(•) HD(•) L CG(^m, 1, n)-mod a∈A CSfa0 ⊗ ... ⊗ CS^am−1 -mod. Fg−1

Here G0(H(G(m, 1, n)) is th Grothendieck group of the isomorphism classes of representations of H(G(m, 1, n)).

In Section 2.2 we study Drinfeld algebras, we focus on the fact that these algebras can be defined with a non-faithful representation. We extend Dirac cohomology defined in [13] to the non-faithful case. In Section 2.3, we introduce the class of generalised graded Hecke algebras, and we extend Evens’ [25] Langlands classification to this class. In Section 2.4 we introduce the Dunkl-Opdam subalgebra. We highlight that it is a generalised graded Hecke algebra and by introducing a new presentation show that it is also a non-faithful Drinfeld algebra. Section 2.5 defines an equivalence between the irreducible modules of the Dunkl-Opdam subalgebra and direct sums of graded Hecke algebras associated to parabolic symmetric groups. In Section 2.6 we combine results on Dirac cohomology (Section 2.2) and the equivalence (Section 2.5) to describe Dirac cohomology of an HDO-module with Dirac cohomology of its associated module under this equivalence equivalence. This highlights that this equivalence of irreducibles behaves well with respect to Dirac cohomology.

2.2 Drinfeld algebras

In this section, we will deﬁne Drinfeld algebras as introduced by Drinfeld, that is, with a potentially non-faithful representation. Ciubotaru [13] deﬁned Dirac cohomology for faithful Drinfeld algebras, and we will extend Dirac cohomology to non-faithful Drinfeld algebras. The results in this section follow almost verbatim from the proofs in [13], so we will not write them here.

Given a ﬁnite group G, antisymmetric bilinear forms bg for g ∈ G and a representation (ρ, V ) of G, we construct an algebra

H = C[G] o T (V )/R.

Here R is the two sided ideal of C[G] o T (V ) generated by the relations,

g−1vg = ρ(g)(v) for all g ∈ G and v ∈ V,

81 and X [u, v] = bg(u, v)g for all v, u ∈ V. g∈G We deﬁne a ﬁltration on the algebra C[G] o T (V )/R, a vector v has degree 1 and a group element g ∈ G has degree 0.

Deﬁnition 2.2.1. [21] An algebra of the form H = C[G] o T (V )/R is a Drinfeld algebra if it satisﬁes a PBW criterion. That is the associated graded algebra is naturally isomorphic to C[G] o S(V ).

Here o denotes the semi-direct product with the natural action of G on V .

We state the conditions on the bilinear forms bg such that H is a Drinfeld algebra. This was originally stated in [21], and explained for the faithful case in [49]. Deﬁne

G(b) = {g ∈ G : bg 6= 0}.

Theorem 2.2.2. [21][49, Theorem 1.9] The algebra H is a Drinfeld algebra if and only if:

(1) For every g ∈ G, bg−1hg(u, v) = bh(ρ(g)(u), ρ(g)(v)) for every u, v ∈ V, ρ(g) ρ(g) (2) For every g ∈ G(b) \ Ker ρ, Ker bg = V and dim(V ) = dim V − 2, ρ(g)⊥ (3) For every g ∈ G(b) \ Ker ρ and h ∈ ZG(g), det(h|V ρ(g)⊥ ) = 1, where V = {v − ρ(g)(v): v ∈ V }.

The above statements follow immediately from the proofs given in [49] for the faithful case. The only variation is that the set Ker ρ replaces 1 in [49].

2.2.1 Non-faithful Drinfeld algebras

In the recent literature, Drinfeld algebras have predominately been considered with G a subgroup of GL(V ). However, Drinfeld originally expressed them with a potentially non-faithful representation. To address this disparity and to avoid confusion we will say that a Drinfeld algebra is a faithful Drinfeld algebra if the representation involved is faithful and we will state that a Drinfeld algebra is non-faithful if the representation is non-faithful. The class of Drinfeld algebras includes both faithful and non-faithful Drinfeld algebras.

82 2.2.2 The Dirac operator for (non-faithful) Drinfeld algebras

If V has a G-invariant symmetric bilinear form, then one can deﬁne a Dirac operator D. In [13] Dirac cohomology is deﬁned for any faithful Drinfeld algebra. Furthermore, an equation involving the square of the Dirac operator is proved [16, Theorem 2.7]. The extension of these theorems to the case of non-faithful representations is clear from the proofs. We will, however, give the equivalent formulation of the theorems in the non-faithful case. In this section, we will denote a Drinfeld algebra by H.

2.2.2.1 The Cliﬀord algebra

Let h, i be a G-invariant non-degenerate symmetric bilinear form on V . The Cliﬀord algebra C(V ) associated to V and h, i is the quotient of the tensor algebra T (V ) by the relations v · v0 + v0 · v = −2hv, v0i.

The Cliﬀord algebra has a ﬁltration by degrees and a Z/2Z-grading by parity of degrees.

In this grading C(V ) = C(V )0 ⊕C(V )1. We define an automorphism : C(V ) → C(V ) which is the identity on C(V )0 and minus the identity on C(V )1. Let us extend to be an automorphism of H ⊗ C(V ) by defining be the identity on H. We define an anti-automorphism, the transpose of C(V ): vt = −v for all v ∈ V . The Pin group is:

Pin(V ) = {a ∈ C(V )× : (a)V a−1 ⊂ V, at = a−1}.

Let (ρ, V ) be a complex representation of G with G-invariant form. This establishes ρ(G) as a subgroup of O(V ). The Pin group is a double cover of O(V ) with surjection p : Pin(V ) → O(V ). We deﬁne the pin double cover of ρ(G) ⊂ O(V ) as

ρ](G) := p−1(ρG) ⊂ Pin(V ).

Note that ρ](G) is not a double cover of G but it is a double cover of ρ(G). We construct a cover of G.

Deﬁnition 2.2.3. Consider the algebra C[G] ⊗ C(V ), we deﬁne G˜ to be the subgroup generated by elements g ⊗ p−1(ρ(g)) for all g ∈ G. That is

G˜ = {g ⊗ h : g ∈ G, h = p−1(ρ(g))∀g ∈ G}.

83 If Ker ρ is abelian then G˜ is the semi-direct product Ker ρ o ρ](G) with cross multiplication:

(h, g˜) · (h0, g˜0) = (hp(˜g−1)h0p(˜g), g˜g˜0), for allg, ˜ g˜0 ∈ ρ](G) and h, h0 ∈ Ker ρ.

By construction G˜ embeds in H ⊗ C(V ) via

∆ : G˜ → H ⊗ C(V ).

For more information on the Cliﬀord algebra see [29] and [41].

2.2.2.2 The Dirac element

i Given any basis {vi} of V and dual basis {v } with respect to h, i we deﬁne the Dirac element X i D = vi ⊗ v ∈ H ⊗ C(V ). i We give a formula for D2. This is equivalent to [13, Theorem 2.7]. The only variation being that Ker ρ replaces 1. For every g ∈ G(b) set,

X j i kg = bg(vi, v )v vj ∈ C(V ), i,j and X i h = viv ∈ H. i The commutation relation deﬁned for a Drinfeld algebra shows: 1 X D2 = −h ⊗ 1 + g ⊗ k . 2 g g∈G(b)

This result is [13, Lemma 2.5]. Recall G(b) = {g ∈ G : bg 6= 0}, we write Ge(b) for the cover of this subset.

Lemma 2.2.4. Similarly to [13, Lemma 2.6] every element g in G(b) \ Ker ρ can be expressed a a product of two reﬂections. Every element in G(b) \ Ker ρ can be written as a coset representative of G(b)/ Ker ρ conjugated by an element in Ker ρ. Therefore given g ∈ G(b) \ Ker ρ, there exists an h ∈ Ker ρ and α, β ∈ V such that −1 ρ(g) ⊥ g = h sαsβh and the roots α, β span the space (V ) . We scale α and β such that hα, αi = hβ, βi = 1. This gives α and β up to sign, we choose all α and β for every g to be such that g˜1g˜2 = g1˜g2 for all g1 and g2 ∈ G/ Ker ρ. There are two ways to do this due to the (G/ Ker˜ ρ) being a double cover, we ﬁx a particular choice.

84 Proof. See proof of [13, Lemma 2.6] with G replaced by G/ Ker(ρ).

For every coset representative g ∈ G(b)/ Ker(ρ) deﬁne

b (α, β) b (α, β)hαβi g˜ = αβ ∈ C(V ), c = g ∈ , e = g ∈ . g˜ 1 − hα, βi2 C g 1 − hα, βi2 C

Every x ∈ Ge(b) can be written as h−1gh where g is a coset representative of G˜(b)/ Ker ρ and h ∈ Ker ρ. Lemma 2.2.4 gives g = sαsβ and g˜ = αβ ∈ C(V ). We deﬁne, for x = h−1gh ∈ G˜,

−1 x˜ =g ˜ = αβ ∈ C(V ), cx˜ = cg˜C , ex = hegC h .

˜ Let us deﬁne the Casimir elements, ΩH in H and ΩG˜ in G.

X G ΩH = h − egg ∈ H , g∈G(b)/ Ker ρ

X −1 ˜ G˜ ΩG˜ = h ghc˜ g˜ ∈ C[G] . h∈Ker ρ g∈G(b)/ Ker ρ Theorem 2.2.5. [13, c.f. Theorem 2.7] The square of the Dirac element can be expressed as a sum of the two Casimir elements plus terms from the kernel;

1 X D2 = −Ω ⊗ 1 + ∆(Ω ) + ⊗ k . H G˜ 2 g g∈ker ρ

2.2.2.3 Vogan’s Morphism

˜ 1 P Let ΩH = ΩH − 2 ⊗ h∈Ker ρ kh, deﬁne

˜ G˜ A = ZH⊗C(V )(ΩH) ⊂ (H ⊗ C(V )) .

If Ker ρ ∩ G(b) = ∅ then A = H ⊗ C(V ). Deﬁne a derivation

d : H ⊗ C(V ) → H ⊗ C(V ),

d(a) = Da − (a)D.

The Dirac operator D interchanges the trivial and det G˜ - isotypic spaces of A. We ˜ deﬁne dtriv and ddet to be the restriction of d to the trivial and det G - isotypic spaces. We state the theorems in [13] but note that the proofs apply verbatim to this case.

85 Theorem 2.2.6. [13, c.f. Theorem 3.5] The kernel of dtriv equals:

˜ G˜ Ker dtriv = Im ddet ⊕ ∆(C[G] ).

Since d is a derivation then Ker dtriv is an algebra. The following theorem is the statement about Vogan’s Dirac homomorphism in the non-faithful Drinfeld case

˜ G˜ Theorem 2.2.7. [13, c.f. Theorem 3.8] The projection ζ : Ker dtriv → C[G] deﬁned in Theorem 2.2.6 is an algebra homomorphism.

Note that since the image of ζ is an abelian algebra the morphism must factor through the abelianisation of Ker dtriv. Recall that when bg = 0 for all g ∈ Ker ρ then

Z(H) ⊗ 1 is contained in Ker dtriv. With this extra condition we can consider the dual of ζ which relates the representations of G˜ with characters of Z(H).

∗ G˜ ζ : irr(G˜) = Spec C[G˜] → Spec Z(H).

Here Spec denotes the algebra of characters on a given algebra.

2.3 Generalised graded Hecke algebras

Ram and Shepler [49] show that there does not exist a faithful Drinfeld algebra n associated to the complex reflection group G(m, 1, n) = Sn o (Zm) . However, they define a candidate for an algebra that is similar to a graded Hecke algebra. Dezélée [19] introduced the term generalized graded Hecke algebra. In Section 2.4 we show that these algebras are non-faithful Drinfeld algebras. We define a larger group of algebras denoted generalised graded Hecke algebras or GGH for short. Set A?B to be the free product of unital associative complex algebras.

Definition 2.3.1. Let W be a Weyl group generated by simple reflections sα, α ∈ Π. The group W acts on a finite commutative group T , t is a faithful complex W - representation. The form h, i is a W -invariant pairing between the vector spaces t∗ and t. We define a parameter function

c˜ :Π → C[T ] which is invariant under the action of W . The generalised graded Hecke algebra GH(W o T ) is the quotient of the algebra

C[W o T ] ?S(t)

86 by the relations

sαt = sα(t)sα + hα, tic˜(α), ∀t ∈ t, α ∈ Π

[h, t] = 0 ∀t ∈ t, h ∈ T.

When T is the trivial group then GH(W o 1) is a graded Hecke algebra. In this form, the relations look very similar to the graded Hecke algebras except that the parameter function takes values in C[T ] instead of C. In the following section, we will prove a Langlands classiﬁcation for generalised graded Hecke algebras. This proof follows Evens’ [25] proof of the Langlands classiﬁcation for graded Hecke algebras.

2.3.1 Preliminaries for the Langlands classiﬁcation

Let {X,R,Y, R,ˇ Π} be root datum, where X and Y are free finitely generated abelian groups and there exists a perfect bilinear pairing between them. The roots R ⊂ X and coroots Rˇ ⊂ Y are finite subsets with a bijection between them. Let Π denote ˇ+ the simple roots {α1, ..., αl}. Positive roots R+ (resp. R ) are the N span of Π (respectively αˇ for α ∈ Π). Let t = X ⊗ C and t∗ = Y ⊗ C be dual vector spaces, similarly let t , t∗ be the real spans of X and Y . Let T be a finite abelian group such R R that W acts on T . Let c˜ be a function from Π to CT which is invariant on conjugacy classes,c ˜(g−1hg) = g−1c˜(h)g = g(˜c(h)). We will denote a generalised graded Hecke algebra by GH. In the case of the n generalised graded Hecke algebra associated to G(m, 1, n), we set W = Sn, T = (Zm) . Pm−1 l −l The function c˜ from Π to CT is defined by c˜(i+1 − i) = l=0 gigi+1. The element gi th n is the i generator of (Zm) .

Deﬁnition 2.3.2. Given a subset ΠP of Π we can deﬁne a parabolic subgroup WP of W generated by sα for α ∈ ΠP . The corresponding parabolic subalgebra of the generalised graded Hecke algebra GHP is generated by sα for α ∈ Π and C[T ] ⊗ S(t). This is the generalised graded Hecke algebra associated to Wp o T .

Deﬁnition 2.3.3. Deﬁne a to be the vector space {x ∈ t : αˇ(x) = 0, α ∈ Π}. Given any parabolic subalgebra GHP , set aP = {x ∈ t : αˇ(x) = 0, α ∈ ΠP }. Let as be the ∗ ∗ perpendicular subspace to aP under the pairing of t and t .

87 ∼ Then GHP = GHPs (WP o T ) ⊗ S(aP ), where GHPs (WP o T ) is constructed (Deﬁnition 2.3.1) as the quotient of the following algebra:

∼ GHPs (WP o T ) = C[WP o T ] ?S(as).

The commutative subalgebra A = CT ⊗ S(t) is a subalgebra of all parabolic subalgebras. For every A -module V we can consider a weight space decomposition.

Definition 2.3.4. Let A = CT ⊗ S(t) and A ∗ denote characters on this algebra. Given an A -module V , on which S(t) acts locally finitely, and a character µ ⊗ λ ∈ A ∗ = CT ∗ ⊗ S(t)∗ define the subspace:

Vµ⊗λ = {v ∈ V : y ⊗ x(v) = µ(y) ⊗ λ(x)(v) for all y ⊗ x ∈ A }.

We can decompose V into weight spaces: M V = Vµ⊗λ. λ⊗µ∈A ∗

∗ The weights of V are the µ ⊗ λ ∈ A such that Vµ⊗λ is non-zero.

Deﬁnition 2.3.5. Given simple roots α1, ..., αn, the fundamental coweights xi ∈ t are such that ∗ αˇj(xi) = δij and ν(xi) = 0 for all ν ∈ a .

∗ ∗ Example 2.3.6. Let W = Sn, t = span{1, ..., n}, t = span{e1, ..., en}, a = span{e1 + e2 + ... + en}. Let the simple roots be αi = i − i+1 for i = 1, ..., n − 1. Then the fundamental coweights are X i x = − ( + + ... + ) . i j n 1 2 n j≤i

∗ The modiﬁcation by 1 + ... + n is required so that a is perpendicular to xi. If we had ∗ deﬁned t to be span{1, ..., n}/ span{1 + ... + n} this would not be required as a = 0.

Deﬁnition 2.3.7. An irreducible GH-module V is essentially tempered if for all weights µ ⊗ λ : CT ⊗ S(t) → C of V , Re(λ(xi)) ≤ 0, for all fundamental coweights xi. The module V is tempered if V is essentially tempered and Re(λ| ) = 0. Here a is aR R the real span of x ∈ X perpendicular to the coroots.

Let ∗+ ∗ aP = {ν ∈ aP : Re(ν(α)) > 0, α ∈ Π − ΠP }.

88 2.3.2 The Langlands classiﬁcation for generalized graded Hecke algebras

Firstly, we state a couple of lemmas of Langlands and a technical lemma about orbits of weights. Let Z be a real inner product space of dimension n. Let {αˇ1, ..., αˇn} be a basis such that (αˇi, αˇj) ≤ 0 whenever i =6 j. Let {β1, .., βn} be a dual basis. For a subset F of Π, let

X X SF = { cjβj − diαˇi : cj > 0, di ≤ 0}. j∈ /F i∈F

Lemma 2.3.9. [6, IV, 6.11] Let x ∈ Z. Then x ∈ SF for a unique subset F = F (x). P P P If x ∈ Z then let x0 = j∈ /F cjβj, where x ∈ SF and x = j∈ /F cjβj − i∈F diαˇi.

It is clear that if x0 = y0 then F (x) = F (y). Deﬁne a partial order on Z by setting x ≥ y if x − y = P t αˇ . ti≥0 i i

Lemma 2.3.10. [6, IV, 6.13] If x, y ∈ Z and x ≥ y then x0 ≥ y0.

Lemma 2.3.11. Given an irreducible GHP module V , the set of weights {λ ⊗ µ} are all in the same WP orbit.

Proof. The group WP is the only part of GHP which does not act by eigenvalues on Vµ⊗λ. For any GHP module U and a weight µ ⊗ λ the subspace M Uw(λ)⊗w(µ)

w∈WP is a GHP submodule of U.

89 Proof of (i). The simple coroots αˇ1, ..., αˇn will have a dual basis β1, ..., βn in tR, relative to the Killing form. Let V be an irreducible GH-representation. Let µ ⊗ λ be an

A weight of V which is maximal among Re(λ). Let ΠP = F = F (Re(λ)). Let as ∗ ∗ ∗ (respectively as) be the elements of t (respectively t ) perpendicular to aP (respectively ∗ aP ). The space t splits ∗ ∗ ∗ t = aP ⊕ as.

We can restrict characters of t to aP (respectively as) by considering the projection of ∗ ∗ ∗ the character in t to aP (respectively as). ∗+ Let ν = λ|aP . Since λ was considered maximal then by construction ν ∈ aP . Let

U be an irreducible representation of GHP appearing in V such that S(aP ) acts by ν. ∼ Let µ ⊗ φ be a CT ⊗ S(as) weight of U. Since CT ⊗ S(t) = CT ⊗ S(as) ⊗ S(aP ) then µ ⊗ φ ⊗ ν = µ ⊗ (φ + ν) is a CT ⊗ S(a) weight of V . X X Re(φ + ν) = cjβj − ziαˇi, cj > 0, j∈ /F j∈F while X X Re(λ) = cjβj − diαˇi, cj > 0, di ≥ 0. j∈ /F j∈F

To prove U is a tempered representation of GHPs it is suﬃcient to prove that zi ≥ 0. P Let F2 = {i ∈ F : zi < 0} and F1 = F − F2. Then Re(φ + ν) ≥ j∈ /F cjβj − P z αˇ . Thus by Lemma 2.3.10 Re(φ + ν) ≥ P c β = Re(λ) . But Re(λ) ≥ i∈F1 i i 0 j∈ /F j j 0

Re(φ + ν), hence Re(λ)0 = Re(φ + ν)0. Therefore F (Re(λ)0) = F (Re(φ + λ)0). Thus

φ + ν is in SF and zi ≥ 0 for all i. The inclusion of GHP -modules U ⊂ V induces a nonzero map π : GH ⊗GHP U → V given by π(h ⊗ w) = h.w. Since V is irreducible,

V is a quotient of GH ⊗GHP U.

This argument is very similar to the argument given by Evens for the case of graded Hecke algebras. Note that this argument implies that every weight µ0 ⊗ λ0 of U has F (Re(λ0)) = F .

Proof of (ii). The space U is naturally embedded in GH ⊗GHP U. U is a GHP -module therefore it is invariant under WP . Lemma 2.3.11 implies that the weights of GH⊗GHP U are w(µ) ⊗ w(λ) where w ∈ W and µ ⊗ λ is a weight of U. Considering the weights of

(GH ⊗GHP U)/U, these are w(µ) ⊗ w(λ) where w =6 1 and is a coset representative of P + + W/WP , alternatively w ∈ W = {w ∈ W : w(RP ) ⊂ R }. Note that for all w ∈ W P P one can write w as a product of w ∈ W and wP ∈ WP .

90 Let µ ⊗ λ be a weight of U, and write

X X Re(λ) = cjβj − diαˇi, cj > 0, di ≥ 0. j∈ /F j∈F

P P P Then if w ∈ W , Re(wλ) = j∈ /F cjwβj − j∈F diwαˇi. Define ρ : t → C by ρ(αˇ) = ˇ + 1, αˇ ∈ Π. Since w :ΠP → R then ρ(w(αˇi)) ≥ ρ(αˇi), for i ∈ F . Since βj is a fundamental weight, w(βj) ≤ βj, with equality if and only if each expression of w as a product of simple reflections is such that each simple reflection fixes βj. If we make P this requirement for all j∈ / F then this implies w ∈ WP . Hence w ∈ WP ∩ W = {1}, therefore w = 1. Thus we can assume that if w ∈ W P \1 then ρ(Re(w(λ))) < ρ(Re(λ)). Fix a weight µ ⊗ λ such that ρ(Re(λ)) is maximal, then µ ⊗ λ can not occur as a weight of (GH ⊗GHP U)/U. This implies that if a submodule Z of GH ⊗GHP U contains µ ⊗ λ then Z contains U and hence is GH ⊗GHP U. Define Imax to be the sum of all submodules of GH ⊗GHP U which do not contain µ ⊗ λ then Imax is maximal and (GH ⊗GHP U)/Imax is the unique irreducible quotient.

Proof of (iii). Suppose π : J(P,U) ∼= J(P 0,U 0). Let µ ⊗ λ (respectively µ0 ⊗ λ0) be a weight of U (respectively U 0) which is maximal with respect to ρ. Suppose F (Re(λ)) 6= F (Re(λ0)). Then it follows that µ ⊗ λ is not a weight of U 0 and µ0 ⊗ λ0 is 0 0 not a weight of U. Therefore µ⊗λ is a weight of (GH⊗GHP U )/U which suggests that ρ(Re(λ)) < ρ(Re(λ0)). However exchanging λ with λ0 suggests ρ(Re(λ0)) < ρ(Re(λ)), which can not be the case. Hence F (Re(λ)) = F (Re(λ0)) and P = P 0. Since J(P,U) ∼= J(P 0,U 0) is irreducible Lemma 2.3.11 implies there exists a w ∈ W 0 0 such that w(µ) ⊗ w(λ) = µ ⊗ λ . If we suppose that w∈ / WP then w has part of its decomposition in W P , by the proof of (ii) this suggests that

ρ(Re(w(λ))) = ρ(Re(λ0)) < ρ(Re(λ)).

However if this is the case then λ is not maximal with respect to ρ. Therefore w(µ) = µ0 0 0 where w ∈ WP . π(U) = U since U (respectively U ) is the unique GHP submodule which has a weight µ1 ⊗ λ1 such that ρ(Re(λ)) = ρ(Re(λ1)) and µ is in the same WP 0 0 orbit as µ1. Similarly ρ(Re(λ )) = ρ(Re(λ1)) and µ is in the same WP orbit as µ1. ∼ 0 Hence U = U as GHP submodules.

91 2.4 Dunkl-Opdam subalgebra

In this section, we study the Dunkl-Opdam subalgebra HDO, defined in [22]. The algebra HDO is a subalgebra of the rational Cherednik algebra associated to the complex reflection group G(m, 1, n). This subalgebra exists for any parameter t and its existence is independent of the parameters c1, ..., cm−1. We show that HDO is a naturally occurring example of a non-faithful Drinfeld algebra. Section 2.2.2 endows HDO with a Dirac operator. From the defining presentation given by [22], this subalgebra is a generalized Hecke algebra. Therefore we have a Langlands classification for HDO. This then sets up Section 2.5, in which we describe the representation theory of HDO as blocks corresponding to multi-partitions and representations of the graded Hecke algebra of type A.

2.4.1 The rational Cherednik algebra

Dunkl and Opdam [22] introduced the rational Cherednik algebra. Let G ⊂ GL(V ) be a complex reflection group with reflections S. Let h, i be the natural pairing of V and ∗ ∗ −1 V . Let αs ∈ V be a λ eigenvector for s ∈ S and let vs ∈ V be a λ eigenvectors for s ∈ S such that λ 6= 1, and hαs, vsi = 1. For every reflection s ∈ S introduce the 0 parameters t, cs ∈ C such that cs = cs0 if s and s are in the same conjugacy class. The rational Cherednik algebra is defined as the quotient of the associative C-algebra

∗ T (V ⊕ V ) n C[G] by the relations

[x, x0] = [y, y0] = 0, for all x, x0 ∈ V, y, y0 ∈ V ∗,

X hαs, yihx, vsi ∗ [x, y] = thx, yi − cs s ∀x ∈ V, y ∈ V , hαs, vsi s∈S g−1vg = g(v), ∀v ∈ V or V ∗.

If one restricts to rational Cherednik algebras associated to classical complex groups G(m, p, n) then [22] shows there is a set of commuting operators inside the rational Cherednik algebra. The main part of these operators is quadratic on a special basis of V and V ∗. We give a particular presentation of the rational Cherednik algebra ∼ n associated to G(m, 1, n) = Sn o (ZM ) .

92 Deﬁne a generating set for G(m, 1, n) consisting of the reﬂections {si,i+1 : i =

1, ..., n − 1} in Sn and the reﬂections {gi : i = 1, ..., n} which have order m, we may th write si for si,i+1. Let η be a primitive m root of unity. Given G(m, 1, n) acting on

V let xi ∈ V be the vectors such that w(xi) = xw(i) for w ∈ Sn and

( −1 η xj if i = j, gi(xj) = xj otherwise.

∗ Let {y1, ..., yn} ∈ V be the dual basis to {x1, ..., xn}. For G = G(m, 1, n) there are m + 1 conjugacy classes of reflections, for reflections in the conjugacy class of s1,2 l then let k ∈ C denote their parameter. For reflections conjugate to g1 denote the parameter by cl ∈ C.

Deﬁnition 2.4.1. The rational Cherednik algebra for G(m, 1, n) and parameters ∗ k, cl, t, Ht(G(m, 1, n)) is the quotient of the C-algebra T (V ⊕ V ) n C[G(m, 1, n)] by the relations

[xi, xj] = [yi, yj] = 0, m−1 m−1 X X −l l X l [xi, yi] = t − k si,jgi gj − clgi, l=1 i6=j l=1 m−1 X −l l [xi, yj] = k si,jgi gj, l=1 g−1vg = g(v), for all v ∈ V or V ∗.

2.4.2 Dunkl-Opdam quadratic operators

Deﬁnition 2.4.2. For i ≤ n deﬁne elements in Ht(G(m, 1, n))

m−1 m−1 X X 1 X 1 z = y x + k s g−lgl + c gl + t, i i i i,j i j 2 l i 2 l=1 i>j l=1

m−1 m−1 X X 1 X 1 = x y − k s g−lgl − c gl − t. i i i,j i j 2 l i 2 l=1 i

Deﬁnition 2.4.3. The Dunkl-Opdam subalgebra HDO(G(m, 1, n)) of the rational

Cherednik algebra is the subalgebra generated by G(m, 1, n) and zi for i = 1, ...n.

93 Remark 2.4.4. The following relations hold in HDO(G(m, 1, n))

[zi, zj] = 0 for i, j = 1, ..., n,

[zi, gk] = 0, ∀i, k = 1, ...n,

[zj, si,i+1] = 0 for j 6= i, i + 1,

zisi,i+1 = si,i+1zi+1 − kij.

Pm−1 l −l Here ij = l=0 gigj .

In fact HDO(G(m, 1, n)) is isomorphic to the associative C[k]- algebra generated by zi and G(m, 1, n) subject to the relations stated in Remark 0.2.16.

2.4.3 Dunkl-Opdam subalgebra admits a non-faithful Drin- feld presentation

In this section, we derive a new presentation of HDO which demonstrates that HDO is a non-faithful Drinfeld algebra. Thus, simultaneously showing that one can associate a Drinfeld algebra to G(m, 1, n), and also uncovering a natural example of a non-faithful Drinfeld algebra. We introduce Jucys-Murphy elements for G(m, 1, n), these are well known. How- ever, the tool that we use here is that we consider two diﬀerent sets of Jucys-Murphy elements.

Deﬁnition 2.4.5. We deﬁne Jucys-Murphy elements for G(m, 1, n).

m−1 X X −s s Mi = sk,igk gi , k

m−1 X X −s s Mi = si,kgi gk. k>i s=0

The commutator [Mi,Mj] = 0 = [Mi,M j] by a standard argument using the fact P that j≤i Mi is in the centralizer of the subgroup generated by {sk−1,k, gk : k ≤ i}. It should be noted that [Mi,Mj] 6= 0 for i > j. Furthermore

m−1 X −s s siMi = Mi+1si − gi gi+1 s=0

94 and m−1 X −s s siM i = M i+1si + gi gi+1. s=0

If we adjust zi by −kMi or kM i, zî = zi − kMi (zî = zi + kMi respectively) the −1 elements zî satisfy the relations sizîsi = zî+1 and gjzîgj = zî. Hence we obtain an action of G(m, 1, n) on the set zî. However the set {zî} no longer commutes. This presentation was given in [22, Corollary 3.6] where the symbol Tixi denotes zî. We provide an exposition of the presentation with {zî} using a simple automorphism of

HDO.

Corollary 2.4.6. [22, Corollary 3.6] Let zî = zi − kMi then zî and G generate HDO and the following relations define HDO(G(m, 1, n)):

sjzˆi =z ˆsj (i)sj,

[gi, zˆj] = 0, m−1 X −s s [ˆzi, zˆj] = k(ˆzi − zˆj) ssi,j gi gj . s=0

Lemma 2.4.7. Let Φ: HDO → HDO such that

Φ(zi) = −zn+1−i,

Φ(si) = sn−i,

Φ(gi) = gn+1−i.

The map Φ is an automorphism of HDO. Furthermore Φ(Mi) = M n+1−i.

Proof.

Φ(sizi − zi+1si − ki,i+1) = −sn−izn+1−i + zn−isn−i − kn−i,n+1−i.

We formally define Φ as a map from CG o S(V ) → HDO then Φ takes the set of defining relations in HDO to itself. Φ is surjective since it takes generators to generators. Hence we can define Φ as an automorphism of HDO.

Using Φ we deﬁne the presentation of HDO with generators {zˆi}.

95 Lemma 2.4.8. Let zˆi = zi + kMi, then the set zˆi and G generate HDO. Furthermore, the following relations hold:

m−1 X −s s [ˆzi, zˆj] = −k(ˆzi − zˆj) ssi,j gi gj , s=0

sizˆi =z ˆi+1si,

[si, zˆj] = 0, ∀j 6= i, i + 1,

[gi, zˆj] = 0.

Proof. Φ(zî) = Φ(zi − kMi) = −zn+1−i − kM n+1−i = −zˆn+1−i. Hence zî and G generate HDO since they are images of a generating set under the automorphism Φ. From the definition of Φ,

[ˆzi, zˆj] = Φ([−zˆn+1−i, −zˆn+1−j]) = Φ([ˆzn+1−i, zˆn+1−j])

Expanding [ˆzn+1−i, zˆn+1−j] with Corollary 2.4.6 one obtains:

m−1 !! X −s s [ˆzi, zˆj] = Φ k(ˆzn+1−i − zˆn+1−j) s(n+1−i,n+1−j)gn+1−ign+1−j s=0

m−1 ! X −s s = −k(ˆzi − zˆj) ssi,j gi gj . s=0 Similarly

sizˆi = −Φ(s(n+1−i,n−i)zˆn+1−i) = −Φ(ˆzn−is(n+1−i,n−i))

=z ˆi+1si.

The relations we gave are images of relations in the second presentation under Φ. Furthermore, the generators and relations are exactly the images of the generators and relations of a presentation. Hence 2.4.8 gives another presentation of HDO.

We will now work towards a fourth presentation of HDO. We are aiming for a Drinfeld presentation of HDO. We require the commutator [zˆi, zˆj] to be an element of the group algebra. We now work through several Lemmas to prove that we can alter k zi by 2 (Mi − M i) to give the Drinfeld presentation we expect.

We observe that the commutators of the set {zˆi}, and similarly {zˆj} can be expressed as commutators of the Jucys-Murphy elements and zi.

96 Lemma 2.4.9. The commutator of the operators zˆi and zˆi are such that:

[ˆzi, zˆj] = k([zj,Mi] − [zi,Mj]).

Similarly for zˆi and Mi.

[ˆzi, zˆj] = k([zi,M j] − [zj,M i]).

Proof.

2 [ˆzi, zˆj] = [zi − kMi, zi − kMj] = [zi, zj] − k[zi,Mj] + k[zj,Mi] + k [Mi,Mj],

= k([zj,Mi] − [zi,Mj]), since [Mi,Mj] = [M i.M j] = [zi, zj] = 0.

Lemma 2.4.10. For operators zˆi and zˆj,

[ˆzi, zˆj] + [ˆzi, zˆj] ∈ CG.

Proof. Using Corollary 2.4.6 and Lemma 2.4.8 we can expand the commutators:

[ˆzi, zˆj] + [ˆzi, zˆj]

m−1 m−1 X −s s X −s s = (ˆzi − zˆj) si,jgi gj − (ˆzi − zˆj) ssi,j gi gj . s=0 s=0

Writing outz ˆi andz ˆi in terms of the commuting operators zi one obtains

m−1 m−1 X −s s X −s s = (zi − kMi − zj + kMj) si,jgi gj − (zi + kMi − zj − kM j) ssi,j gi gj . s=0 s=0

Cancelling out the operators zi and zj we arrive at the element of the group algebra

m−1 X −s s [ˆzi, zˆj] + [ˆzi, zˆj] = k(Mj + M j − Mi − M i) si,jgi gj ∈ CG. s=0

k k Lemma 2.4.11. The commutators of zi − 2 Mi + 2 M i are in the group algebra, explicitly: k k k k [z − M + M , z − M + M ] ∈ G. i 2 i 2 i j 2 j 2 i C 97 Proof. Expanding out the commutator linearly: k k k k [z − M + M , z − M + M ] i 2 i 2 i j 2 j 2 j k = [z , z ] + [z ,M ] − [z ,M ] + [z ,M ] − [z ,M ] i j 2 j i i j i j j i k2 + [M ,M ] − [M ,M ] − [M ,M ] + [M ,M ] 2 i j i j i j i j Using Lemma 2.4.9:

1 k2 = [ˆz , zˆ ] + [ˆz , zˆ ] − [M ,M ] + [M ,M ] ∈ G, 2 i j i j 2 i j i j C

m−1 2 k X k = M + M − M − M s g−sgs − [M ,M ] + [M ,M ] ∈ G. 2 j j i i si,j i j 2 i j i j C s=0

Lemma 2.4.12. s (z − k M + k M )s−1 = (z − k M + k M ). j i 2 i 2 i j sj (i) 2 sj (i) 2 sj (i)

Proof. The result follows from compiling these three relations:

sizi = zi+1si + ki,i+1,

siMi = Mi+1si + i,i+1,

siM i = M i+1si − i,i+1.

Theorem 2.4.13. There exists a presentation of HDO given by elements {z˜i : i =

1, ..., n} and generators {si, gj : i ≤ i ≤ n − 1, 1 ≤ j ≤ n} in G(m, 1, n) such that:

−1 siz˜jsi = si(˜zj),

giz˜j =z ˜jgi ∀i, j = 1, ..., n,

[˜zi, z˜j] ∈ CG(m, 1, n), ∼ n where G(m, 1, n) = Sn o (Zm) . The group Sn is generated by transpositions si, th i = 1, ..., n − 1 and gj generates the j copy of Zm.

98 1 k Proof. Let z˜i = 2 (zî + zˆj) = zi − 2 (Mi − M i) then the first two relations follows from Lemma 2.4.12 and by Lemma 2.4.11 their commutant is in CG. One may be worried that we have defined an algebra that surjects onto HDO but does not procure an injection. However performing the above arguments in reverse setting k zi = z˜i + 2 (Mi − M i) shows that the original relations follow from these relations.

Definition 2.4.14. Give V a basis {vi} and recall that Sn act on this basis by permutations. Let π be the homomorphism of G(m, 1, n) onto Sn. (V, φ) is the standard representation of Sn, now define (V, ρ) to be the representation of G via the projection onto Sn, that is, ρ : G → GL(V ) via ρ(g) = φ(π(g)). We define skew-symmetric forms on V for elements in G(m, 1, n):

2 bsij sjk (vp, vq) = k (hi − j, vpihj − k, vqi − hi − j, vqihj − k, vpi) ,

for 0 < i < j < k ≤ n,

0 0 0 b l −l l −l = bsij sjk for all l, l = 0, ..., m − 1 sij gigj sjkgj lgk

bg = 0 otherwise.

Theorem 2.4.15. The algebra HDO is a Drinfeld algebra. More concretely HDO is isomorphic to CG o T (V ) with the relations: X [u, v] = bg(u, v)g ∀u, v ∈ V, g∈G where bg are skew-symmetric forms on V deﬁned in Deﬁnition 2.4.14.

Proof. Conjugating bg by gi must ﬁx bg since gi acts trivially on V . Taking the quotient of HDO(G(m, 1, n) by the two sided ideal generated by the elements {gi −1 : 1 ≤ i ≤ n} produces an algebra isomorphic to the graded Hecke algebra of type A, H(Sn). Hence the forms bg must agree with, under the quotient, the forms that construct the Drinfeld presentation of the graded Hecke algebra for Sn. The forms bsij sjk descend to the forms, labelled by the same element, deﬁning the graded Hecke algebra as a Drinfeld l algebra. Conjugating by various gi gives the forms b l −l l0 −l0 above. Since b1 = 0 sij gigj sjkgj lgk in H(Sn) then bk = 0 for all k ∈ ker ρ. There are no other elements of G(m, 1, n) such ρ that dim V (g) = dim V − 2, therefore the rest of the bg = 0 for all g not mentioned above.

99 2.4.4 Dunkl-Opdam subalgebra is a generalised graded Hecke algebra

Recall Deﬁnition 2.3.1 of the generalised graded Hecke algebra associated to the root Pm−1 l −l system of W , T and parameter function c˜. Let ij = l=1 gigj ∈ CG(m, 1, n). The algebra HDO is isomorphic to, as a vector space, C[G(m, 1, n)] ⊗ S(V ), with multiplication such that C[G(m, 1, n)] and S(V ) are subalgebras and the following cross relations hold;

[zi, gk] = 0, ∀i, k = 1, ...n,

[zj, si,i+1] = 0 for j 6= i, i + 1,

zisi,i+1 = (i, i + 1)zi+1 − kij.

∼ n If we substitute G(m, 1, n) = Sn o (Z/mZ) for W o T then HDO is a generalised Hecke algebra with parameter function

m−1 X l −l n c˜(sij) = kij = gjgj ∈ C[(Zm) ]. l=0

Since HDO is a generalised graded Hecke algebra we can apply the Langlands classiﬁcation from Section 2.3.2. Therefore we can construct every representation of

HDO as a quotient of the module induced from a tempered module of a parabolic subalgebra.

ˇ ˇ Corollary 2.4.16. Let U = U ⊗ Cν be such that U is an irreducible tempered HDO- ∗+ module and ν is a character of a . Every irreducible representation of HDO can be constructed as a quotient of a tempered module of an irreducible parabolic subalgebra

HP . That is it is a quotient of

HDO ⊗HP U.

2.5 Constructing the representations of H(G(m, 1, n)) from H(Sn)

In this section we prove the irreducible representations of H(G(m, 1, n)) can be built up from blocks of irreducible representations of the graded Hecke algebras associated to the symmetric group. We then get a Functor between the semisimpliﬁcation of the category of H(G(m, 1, n)) modules and the semisimpliﬁcation of the category of L N . Let K ( (G(m, 1, n))) denote he Grothendieck group of representation a∈A Hsai 0 H

100 of H(G(m, 1, n)), this is the smisimpliﬁcation of th category of H(G(m, 1, n)) modules.

This is very similar to how one can build the representations of W (Bn) from the pullback of two representations of symmetric groups, Sa and Sb, where a + b = n

We denote the usual graded Hecke algebra of type Ak−1 by H(Sk), η denotes a fixed primitive mth root of unity. We define N to include zero. Let A ⊂ Nm be the set of vectors such that the coordinates sum to n. Then let a = (a0, ..., am−1) be a vector in A, explicitly Pm−1 ∗ i=0 ai = n. We define the character µa ∈ C[T ] by

i µa(gj) = η

Pi−1 Pi where k=0 ak < j ≤ k=0 ak.

This character takes the first a0 reflections to 1 it then takes the following a1 2 reflections to η then the following a2 to η and continues in this way. Finally it takes m−1 the last am−1 reflections to η . The set A will become a parametrising set.

Example 2.5.1. Let n = 5 and m = 3, deﬁne ω to be a primitive 3rd root of unity. The character of C[(Z/3)5] associated to the vector (1, 1, 3) is such that;

µ(1,1,3)(g1) = 1, µ(1,1,3)(g2) = ω,

2 µ(1,1,3)(g3) = µ(1,1,2)(g4) = µ(1,1,3)(g5) = ω .

∗ If we take the Sn orbits of C[T ] then a representative set of these orbits is

{µa|a ∈ A}.

Let irr(H(G(m, 1, n)) be the set of isomorphism classes of irreducible modules for H(G(m, 1, n)). We deﬁne irr(H(G(m, 1, n))|µa) to be the subset of irr(H(G(m, 1, n)) ∗ consisting of representations that have a weight µa ⊗ λ for any λ ∈ S(V ) . Similarly, we will denote the set of irreducible representations of a complex algebra B by irr(B).

Lemma 2.5.2. The irreducible representations of H(G(m, 1, n)) split into disjoint sets labelled by A,

G irr(H(G(m, 1, n))) = irr(H(G(m, 1, n))|µa). a∈A

101 Proof. Since every irreducible representation of H(G(m, 1, n)) has at least one C[T ] weight then by Lemma 2.3.11 it must contain one and only one Sn orbit, hence it must contain exactly one µa. Therefore every irreducible representation V is in exactly one of the sets irr(H(G(m, 1, n))|µa).

Let Sa0 × Sa1 × ...Sam−1 be the parabolic subgroup of Sn generated by sα for

j−1 j X X Πa = {i − i+1| ak ≤ i < ak for some j} ⊂ Π. k=0 k=0

Fix a ∈ A. The stabiliser, stab(µa) ⊂ H(G(m, 1, n)), of the character µa is generated by C[T ], S(V ) and si ∈ Sa0 × Sa1 × ...Sam−1 ⊂ Sn. The set Πa is equivalent to the set of simple roots i − i+1 such that µa(gi) = µa(gi+1). This is the parabolic subalgebra associated to the subset Πa ⊂ Π deﬁned in Deﬁnition 2.3.2.

Lemma 2.5.3. The subalgebra stab(µa) which stabilises the character µa is isomorphic to H(G(a0, 1, m)) ⊗ H(G(a1, 1, m)) ⊗ ... ⊗ H(G(am−1, 1, m)).

Proof. The subalgebra generated by Sa0 × Sa1 × ...Sam−1 , C[T ] and S(V ) certainly contains H(G(ai, 1, m)) for every i = 0, ..., m − 1. The algebra H(G(ai, 1, m)) consists, as a vector space of S(Vi) ⊗ C[Ti] ⊗ Sai where Vi is the span of j, and C[Ti] is Pi−1 Pi generated by gj such that k=0 ak < j ≤ k=0 ak. We have V0 ⊕...⊕Vm−1 = V hence S(V0)⊗...⊗S(Vm−1) = S(V0⊕...⊕Vm−1) = S(V ). Similarly C[T0]⊗...⊗C[Tm−1] = C[T ], and C[Sa0 ] ⊗ ... ⊗ C[Sam−1 ] = C[Sa0 × ... × Sam−1 ]. Hence as a vector space:

m−1 O stab(µa) = S(Vi) ⊗ C[Ti] ⊗ C[Sai ]. i=0

Each H(G(ai, 1, m)) is a subalgebra and as vector spaces we have equality, one just needs to check that each subalgebra commutes with the other subalgebras. We already know Sai and Saj commute, for i =6 j, and Sai commutes with C[Tj] because Sai ﬁxes

Tj. Similarly Sai ﬁxes Vj so sαi ∈ Sai commutes with j ∈ S(Vj).

Lemma 2.5.4. The set of irreducible representations irr(H(G(m, 1, n))|µa) is in natural one-to-one correspondence with irr(stab(a))|µa). The bijection F is deﬁned by

−1 F : irr(stab(a)|µa) → irr(H(G(m, 1, n))|µa),

−1 H(G(m,1,n)) F (W ) = Indstab(a) W,

102 and

F : irr(H(G(m, 1, n))|µa) → irr(stab(a)|µa),

H(G(m,1,n)) F (U) = Unique irreducible submodule of Resstab(a) U with weight µa.

−1 For an irreducible module U in irr(H(G(m, 1, n))|µa), F (U) is the µa-weight space of U.

Proof. Let P be the corresponding partition deﬁned by the subset Πa ⊂ Π and set C to be the set of shortest length coset representatives of the parabolic group SP in Sn. For a stab(a)-module W the module W c, for c ∈ C, is isomorphic to W as a vector space with the action

b · W c = c−1bcW for all b ∈ stab(a).

−1 We must check that F (W ) for W ∈ irr(stab(µa)) is an irreducible H(G(m, 1, n))- module, the rest follows. Here C is a set of shortest length coset representative of the parabolic group SP in Sn. Since C becomes a basis for the quotient vector space H L H(G(m, 1, n))/ stab(a) then as a vector space Indstab(a) is c∈C cW . The space W is a stab(a)-submodule which can be taken as the first composition factor. Then cW with c of shortest length can be taken as the next submodule in the quotient Ind W/W . Continuing in this manner each composition factor is isomorphic to W c for the labeling set C. This statement about composition factors is also proved in [19, Proposition 3.5]. H(G(m,1,n)) c The stab(a)-composition series of Indstab(a) W consists of the stab(a)-modules W H(G(m,1,n)) where c ∈ C. We must show that Indstab(a) W is irreducible. The module W is an irreducible stab(a)-module. If W is an irreducible G(m, 1, a0) × .. × G(m, 1, am−1)- module then utilising Mackey’s criterion for finite groups we need to show that W c are c not isomorphic. However by construction W has only weights containing µa, and W will only have weights containing c(µa), and since SP is the stabiliser of µa in Sn then c for all c =6 1 we have c(µa) 6= µa. Therefore each W has a different set of weights and hence are not isomorphic. Hence if W is an irreducible G(m, 1, n)-module then using H(G(m,1,n)) Mackey’s irreducibility criterion Indstab(a) W is an irreducible G(m.1, n)-module and hence F (W ) is irreducible as an H(G(m, 1, n))-module. L The module W decomposes as a direct sum of irreducibles W = Vi as irreducible modules of G(m, 1, a0) × .. × G(m, 1, am−1). By the same argument as above the induction of each of these is an irreducible G(m, 1, n)-module. We have

(G(m,1,n)) M G(m,1,n) IndH W = Ind V stab(a) G(m1,1,n)×...×G(mm−1,1,n) i

103 H(G(m,1,n)) as a G(m, 1, n)-module. Suppose that Indstab(a) W is not irreducible as an 0 H(G(m, 1, n))-module then some direct sum of Ind Vi s is a submodule. Suppose L L i∈I IndVi is a H(G(m, 1, n)-submodule, then F ( i∈I Ind Vi) is a stab(a)-submodule L of W . Therefore i∈I Vi is a stab(a)-submodule of W . Since W is irreducible as a L stab(a)-module then I is the whole indexing set, i.e. W = i∈I Vi. Therefore the H(G(m,1,n)) only irreducible non-trivial submodule is the whole module. Hence Indstab(a) W is irreducible. It is easy to verify that F −1 · F (V ) = V using the universal property of induced modules and similarly F · F −1(W ) = W .

Given a representation of (V, π) ∈ irr(stab(a)|µa) we can explicitly describe how gi ∈ G(m, 1, n) acts. Since this algebra stabilises µa this is the only C[T ] weight occurring in V . Therefore

gi = µa(gi) Id .

Pm−1 l −l Let αi = i − i+1, if we study the relation sαi αi = sαi (αi)sαi + l=0 gigi+1 Pm−1 l −l Pm−1 l −l in H(G(m, 1, n), on (V, π), the element l=0 gigi+1 is equal to π( l=0 gigi+1) = Pm−1 l −l l=0 µa(gi) µa(gi+1) Id. This then equals m if µa is constant on gi and gi+1 and Pm−1 l −l l=0 gigi+1 is zero if µa(gi) 6= µa(gi+1). One can summarise, on any representation in irr(stab(a)|µa) m−1 ( X m if s − ∈ stab(µa), glg−l = i i−1 i i+1 0 otherwise. l=0

Recall from Lemma 2.5.3 that stab(a) is isomorphic to H(G(a0, 1, m))⊗...⊗H(G(am−1, 1, m)).

We have shown that if (V, π) ∈ irr(stab(a|µa) then the relations sαi αi = sαi (αi)sαi + Pm−1 l −l k l=0 gigi+1 become sαi αi = sαi (αi)sαi + km on V for all αi ∈ Πa and the relations Pm−1 l −l sαi αi = sαi (αi)sαi +k l=0 gigi+1 become sαi αi = sαi (αi)sαi +0 on V for all αi ∈/ Πa .

Hence we can conclude that if (V, π) ∈ irr(stab(a|µa)) then this representation factors through the algebra

H(Sa0 ) ⊗ ... ⊗ H(Sam−1 ), via the quotient by the ideal Ia =< gi − µa(gi) Id >. Explicitly

π : H(G(a0, 1, m)) ⊗ ... ⊗ H(G(am−1, 1, m)) H(Sa0 ) ⊗ ... ⊗ H(Sam−1 ) → GL(V ).

Where H(Sn) is the usual graded Hecke algebra associated to Sn, with parameter c(α) = mk.

104 Lemma 2.5.5. The set irr(H(G(a0, 1, m))⊗...⊗H(G(am−1, 1, m))|µa) is in one-to-one correspondence with irr(H(Sa0 )) ⊗ ... ⊗ irr(H(Sam−1 )).

Proof. The irreducible representations in irr(H(G(a0, 1, m))⊗...⊗H(G(am−1, 1, m))|µa) all occur as pullbacks of the irreducible representations of H(Sa0 ) ⊗ ... ⊗ H(Sam−1 ) via the speciﬁc quotient of H(G(a0, 1, m)) ⊗ ... ⊗ H(G(am−1, 1, m))|µa onto H(Sa0 ) ⊗ ... ⊗

H(Sam−1 ), with the ideal

Ia =< gi − µa(gi) Id |i = 1, ..., n > .

Furthermore given a representation U of H(Sa0 ) ⊗ ... ⊗ H(Sam−1 ) one can create a representation in irr(H(G(a0, 1, m)) ⊗ ... ⊗ H(G(am−1, 1, m))|µa) by pulling back the representation U from the quotient of Ia.

∼ G irr(H(G(m, 1, n))) = irr(H(Sa0 )) ⊗ ... ⊗ irr(H(Sam−1 )). a∈A

Since we have a bijection between irreducible module of H(G(m, 1, n)) and irreducible modules of L N then we have an equivalence of the Grothendieck a∈A Hsai groups of each algbra. When we consider tempered H(G(m, 1, n)) modules, that is V such that the C[T ] ⊗ S(t) weights µ ⊗ λ are such that Re(λ(xi)) ≤ 0 for all fundamental coweights and that Re(λ| ) = 0, we note this condition is only dependent on the S(t) weight λ. aR Therefore a tempered H(G(m, 1, n))-module corresponds to a tempered H(Sn)-module with weight λ. Every tempered H(Sn)-module V will correspond to m different tempered H(G(m, 1, n)-modules. Each tempered H(G(m, 1, n))-module will be a pullback of the module V . However the difference between the modules is that the j short reflections gi will act by η for fixed j = 1, ..., m.

105 This gives a method to parametrise the Langlands data for an irreducible H(G(m, 1, n))- module via tempered modules of H(Sa). Recall that every irreducible H(G(m, 1, n))- module can be realized as a quotient of

ˇ H(G(m, 1, n)) ⊗H(G(m,1,n))P U ⊗ Cν.

If we ﬁx an irreducible module X then using the above realization, we associate to it Langlands data (P,U).

Fix P = (p0, ..., pm−1) a partition of n with at most m parts. The tempered

H(G(m, 1, n))-modules are the pullbacks of tempered H(Sa0 ) ⊗ ... ⊗ H(Sam−1 )-modules. Pj Recall that the size of a partition λ = {x1, ..., xj} is i=1 xj. The tempered modules of graded Hecke algebra with real central character correspond to partitions

(e, ψ) where ψ is nilpotent [17, 3.3],[33]. In the case of W = Sn, e is always 1 and ψ is characterized by it’s Jordan form and hence corresponds to a partition. Hence the tempered modules of H(Sp0 ) ⊗ ... ⊗ H(Spm−1 ) with real central character will correspond to a set of m partitions {λ0, ..., λm−1} such that the sum of the sizes of the partitions λi is ai.

Theorem 2.5.7. Let P be a parabolic set associated to a = (a0, ..., am). We associate to P , a parabolic subalgebra HP ⊂ H(G(m, 1, n)). The tempered modules of the parabolic algebra HP are built up from tempered modules of each parabolic part H(G(m, 1, ai)). By above a tempered module of H(G(m, 1, ai)) corresponds to a tempered module of H(Sai ). The tempered modules of H(Sai ) with real central character are labelled by partitions of ai. Hence tempered modules of HP (G(m, 1, n)) with real central character are labelled by multipartitions {λ0, ..., λm−1} with m partitions such that the size of λi equals ai. Furthermore one can construct these tempered modules via the pullback of

φ H(G(m, 1, n)) → H(Sa0 )) ⊗ ... ⊗ (H(Sam−1 ) → GL(Vλ0 ⊗ ... ⊗ Vλm−1 ).

Where Vλi is the tempered module of H(Sai ) corresponding to the partition λi and φ is the quotient by the ideal Ia =< gi − µa(gi) Id |i = 1, ..., n >.

106 2.6 Dirac cohomology of the Dunkl-Opdam subalgebra

In this section, we will use the description of irreducible representations from Section 2.5 to describe how the Dirac operator for the Dunkl-Opdam subalgebra acts on irreducible modules. We show that the Dirac operator DDO for HDO descends to a relevant Dirac operator for a tensor of type A graded Hecke algebras. Then, we describe the Dirac operator in terms of Dirac operators for type A parabolic algebras. Let A be a Drinfeld algebra T (V ) o C[G]/R. We have an associated Clifford algebra C(V ), with respect to the G-invariant symmetric product <, > . Given a basis B of V with dual basis B∗ with respect to h, i, we defined the Dirac operator to be X b ⊗ b∗ ∈ A ⊗ C(V ). b∈B We have two presentations of the Dunkl-Opdam subalgebra, one producing the Lusztig presentation Definition 0.2.16 with commuting basis elements and the Drinfeld presentation used in Theorem 2.4.13 which shows that H(G(m, 1, n)) is a Drinfeld algebra. We used the Lusztig presentation to show the equivalence of the irreducibles of the Dunkl-Opdam subalgebra to irreducibles of a sum of tensors of type A graded Hecke algebras. This utilised parabolic subalgebras. However, the Dirac theory developed for the Dunkl-Opdam subalgebra uses the Drinfeld presentation. This Drinfeld presentation does not admit parabolic subalgebras. Let us recall that to transform from the Lusztig presentation to the Drinfeld presentation one takes the standard basis {z1, ..., zn} of the reflection representation of Sn which along with G(m, 1, n) gives the Lusztig presentation. Then to obtain the Drinfeld presentation we use the generators: m−1 m−1 k X X k X X k z˜ = z + s g−lgl − s g−lgl = z + (M − M ). i i 2 i,j i j 2 i,j i j i 2 i i i

Recall Mi and M i are Jucys-Murphy elements of G(m, 1, n) with reverse orderings. The Dirac element in terms of the Drinfeld presentation is: n X ∗ DDO = z˜i ⊗ zi . i=1

In terms of the Lusztig presentation {zi} the Dirac element is n m−1 m−1 ! X k X X k X X D = z + s g−lgl − s g−lgl ⊗ z∗. DO i 2 i,j i j 2 i,j i j i i=1 i

107 Deﬁnition 2.6.1. Given an H-module X and a spinor S of C(V ), then DDO : X ⊗ S → X ⊗ S. The Dirac cohomology of X with respect to S is deﬁned to be

Ker(DDO)/ Im(DDO) ∩ Ker(DDO).

Since DDO sgn-commutes with the group Ge then the Dirac cohomology is naturally a Ge-module.

From Section 2.5, if an irreducible V contains a weight µa corresponding to a = {a0, ..., am−1} then we will write Vµa for the C[T ]-weight space corresponding to the weight a. We can decompose V into C[T ] weight spaces,

M M c V = Vµa = Vµa . a∈A c∈C

Lemma 2.5.4 shows that Vµa is in the image of the functor F . It is a stab(µa)-module and is the pullback of a tensor of -modules. A problem that occurs is that the Hsai ∼ Nm−1 Dirac operator DDO does not sit in the subalgebra stab(µa) = i=1 HDO(G(m, 1, ai)). We will look at the Dirac operators already given for the standard type A graded Hecke algebra.

Deﬁnition 2.6.2. [4] For the graded Hecke algebra H(Sk) the Dirac operator is ! X mk X mk X D = z + s − s ⊗ z∗. Sk i 2 i,j 2 i,j i i=1,...,k i

Remark 2.6.3. We abuse notation here as zi in this context denotes the same basis as we have used in the deﬁnition of HDO but of course it is not in the same algebra.

We justify this since all surjections of HDO onto HSn preserve this notation. We have used the parameter mk as opposed to k for HSn because naturally our map sends HDO to HSn with parameter mk.

Recall that the weight space V is naturally a Nm−1 -module, via the functor µa i=1 HSai F defined in the Lemma 2.5.4. We extend Definition 2.6.2 to define a Dirac operator for Nm−1 : i=1 HSai D = D ⊗ ... ⊗ D . Sa0 ×...×Sam−1 Sa0 Sam−1 Written out explicitly this is   m−1 j=ai X X mk X mk X ∗ DSa ×...×Sa = zj + sj,k − sj,k ⊗ zj . i m−1 2 2 i=0 j=ai−1 j

108 Nm−1 Here we have associated i=1 S(Vi) with S(⊕Vi). Similarly we have substituted N C(⊕Vi) = C(Vi), using the graded tensor product. Initially this looks like the Dirac operator for HSn , however one should notice that not all of the reﬂections are involved in this Dirac operator. We highlight this with an example.

Example 2.6.4. Let n=3. The Dirac operator for HS3 is mk mk mk mk z − (1, 2) − (1, 3) ⊗ z∗ + z + (1, 2) − (2, 3) ⊗ z∗ 1 2 2 1 2 2 2 2

 mk mk + z + (1, 3) + (2, 3) ⊗ z∗. 3 2 2 3

However the Dirac operator for Hs1 ⊗ HS2 ⊂ HS3 is mk mk z ⊗ z∗ + z − (2, 3) ⊗ z∗ + z + (2, 3) ⊗ z∗. 1 1 2 2 2 3 2 3

One can see that there are four reﬂections in the DH(S3) not involved in DH(S1) ×DH(S2).

Viewing Vµa as a stab(µa)-module, V is a sum of twists of F (V ). Let us look at the [T ]-invariant element of stab(µ ) which maps to D , this is: C a Sa0 ×...×Sam−1

m−1 j=a  m−1 m−1  X Xi k X X k X X z + g−lgl s − g−lgl s ⊗ z∗.  j 2 j k j,k 2 k j j,k j i=0 j=ai−1 l=1 j

Written this way one notices that this looks very similar to the HDO Dirac operator however it excludes the reﬂections that are not in the parabolic subgroup that stabilises

µa. The following lemma shows that the diﬀerence vanishes on Vµa .

Lemma 2.6.5. Given an irreducible module Va with C[T ] weight a then on the subspace F (V ) the Dirac operator for acts by the Dirac operator D . a HDO Sa0 ×...×Sam−1

Proof. Recall that since Vµa only has one C[T ] weight, namely a we can explicitly Pm−1 −l l describe how l=0 gi gj acts on this subspace.

m−1 ( X m Id if µa(gi) = µa(gi), g−lgl = i j 0 if µ (g ) 6= µ (g ). l=0 a i a i This parametrisation of pairs {i, j} can be described in another way. If the transposition Pm−1 −l l si,j stabilises the character µa then l=0 gi gj = m. However if si,j is not in Pm−1 −l l stab(µa) then l=0 gi gj = 0 on the µa-weight space. Ultimately this means that −1 the Dirac operator DDO ∈ HDO ⊗ CL(V ) preserves the subspace F (V ) ⊗ S since

109 the transpositions included in DDO which do not preserve Vµa are preceded by the Pm−1 −l l element l=0 gi gj which acts by zero in this case. Finally since DDO preserves Vµa it equals an element inside stab(a) ⊗ S. This is the pull back of D and Sa0 ×...×Sam−1 hence D agrees with D on the µ -weight space of V . DO Sa0 ×...×Sam−1 a

We have described how the Dirac operator acts on the a weight space of V . Since

DDO is G(m, 1, n) invariant we can describe how it acts on the rest of the weight spaces. As discussed in Lemma 2.5.4, the other weight spaces are twists of this space by the coset representatives, c ∈ C of the parabolic subgroup SP in Sn. The group S ﬁxes the a weight space. Therefore if D acts by D on V then D P DO Sa0 ×...×Sam−1 µa acts by cD c−1 on cV . Hence Ker(D ) ⊂ V ⊗ S is Sa0 ×...×Sam−1 µa DO M c Ker D . Sa0 ×...×Sam−1 c∈C

Similarly since D acts by D on stab(a) ⊗ S then DO Sa0 ×...×Sam−1

Im D = ⊕ c Im D . DO c∈C Sa0 ×...×Sam−1

We can describe the Dirac cohomology of an irreducible module X in terms of the Dirac cohomology of its corresponding ⊗ ... ⊗ -module. HSa0 HSam−1

˜ ˜ m Deﬁnition 2.6.6. The group G for H(G(m, 1, n)) is isomorphic to Sn n (Zn) . Since ˜ m this is a semidirect product irreducible representations of Sn n (Zn) decompose into m ∗ blocks corresponding to characters µa ∈ (C(Zn) ) and under this correspondence

˜ m ∼ ˜ ˜ irr(Sn n (Zn) |µa} = irr(Sa0 × ... × Sam−1 ).

There is a corresponding equivalence between the semisimplification of the module ˜ L N ˜ ˜ ˜−1 category of G and a∈A CSa0 ⊗ ... ⊗ CSam−1 . Define Fe and F to be the functors L L exhibiting this equivalence between G(^m, 1, n) and a∈A a∈A CSfa0 ⊗ ... ⊗ CSâm−1 , similarly to Lemma 2.5.4.

110 where HD(X) is the type A Dirac cohomology of the HSk -module X, P is the parabolic subset associated to a and C is a set of coset representatives of Sn/SP . Let HD(•) denote the functor taking the relevant module to its Dirac cohomology. We have the following commutative diagram:

G ( (G(m, 1, n))) F G (L ⊗ ... ⊗ ) 0 H 0 a∈A HSa0 HSam−1

HD(•) HD(•) L CG(^m, 1, n)-mod a∈A CSfa0 ⊗ ... ⊗ CS^am−1 -mod. Fg−1

Here G0(H(G(m, 1, n)) is th Grothendieck group of the isomorphism classes of representations of H(G(m, 1, n)).

111 Chapter 3

Duality between non-spherical principal series representations

112 Abstract

We define an extension of the affine Brauer algebra called the type B/C affine Brauer algebra. This new algebra contains the hyperoctahedral group ⊗k and it naturally acts on EndK (X ⊗ V ). We study functors Fµ,k from the category of admissible O(p, q) or Sp2n(R)-modules to representations of the type B/C affine Brauer algebra. Furthermore these functors take non-spherical principal series modules to principal series modules for the graded Hecke algebra of type Dk, Cn−k or Bn−k. 3.1 Introduction

Let G be an odd real orthogonal group or symplectic group whose complexiﬁcation is

GC, G is O(p, q) for p + q = 2n + 1 or Sp2n(R). Let K denote a maximal compact subgroup of G. Let g0 be the real Lie algebra of G. Define its complexification g = g0 ⊗R C. Let X be an admissible G-module and let V be the defining matrix module of the linear group G. The papers [1, 59, 18, 10, 23] study the C-algebra ⊗k ⊗k EndG(X ⊗V ) of operators on X ⊗V commuting with G. For G = Sp2n(R),O(p, q) ⊗k there is a homomorphism [20] of the affine Brauer algebra to EndG(X ⊗ V ). In this chapter we focus on the larger algebra

⊗k EndK (X ⊗ V ).

θ We define an extension of the affine Brauer algebra, Bk, by operators related to the θ ⊗k Cartan involution θ of G. This new algebra Bk acts on X ⊗ V and commutes with the action of K. It is the analogue of the affine Brauer algebra for operators commuting with K. The extension contains the Weyl group of type B/C, the hyperoctahedral group. This new algebra’s module category is a natural image for the functors defined by Ciubotaru and Trapa [16]:

⊗k Fµ,k(X) = HomK (µ, X ⊗ V ).

We show that the functors Fµ,k take the category of admissible O(p, q) or Sp2n(R)- θ modules to Bk-modules. Unlike previous functors, for G = O(p, q) or Sp2n(R), both categories are related to the hyperoctahedral group. Let G = KAN be the Iwasawa decomposition of G, and P = MAN be the minimal parabolic subgroup. ν ν For characters δ of M and e of A, the principal series representation Xδ (Definition 3.6.1) is: ν G ν Xδ = IndMAN (δ ⊗ e ⊗ 1). For split real orthogonal or symplectic groups this covers all of the principal series n q modules. When G = O(p, q) or Sp2n(R) then M = (Z2) or O(p − q) × (Z2) . Denote the character of M which is triv (or det) on O(p − q), −1 on the first k generators k k and 1 on the remaining n − k or q − k by δtriv (resp. δdet). For Sp2n(R) we drop the subscript det and triv. The graded Hecke algebra Hk(c) (Definition 3.4.1) is the graded Hecke algebra associated to the hyperoctahedral group W (Bk) with a certain parameter function related to c ∈ R. For G = Sp2n(R), the functors Ftriv,k and ν Fdet,n−k take principal series modules Xδk to principal series modules for the graded Hecke algebra Hk(0) and Hn−k(1) respectively. For G = O(p, q) the functors Ftriv ⊗ det,k ν and Fdet ⊗ triv,q−k take principal series modules X k to principal series modules for δtriv the graded Hecke algebra Hk(0) and Hq−k(1) respectively. A similar result holds for ν X k and functors Ftriv ⊗ triv,k and Fdet ⊗ det,q−k. Given a particular character δ of M δdet we associate to it K-characters µ, and µ (Table 3.6.1) with scalars cµ and cµ (Table 3.7.1). We prove that under these conditions principal series representations map to principal series representations.

ν Theorem 3.8.14. For G = Sp2n(R) or O(p, q) p + q = 2n + 1, the module Fµ,k(Xδ ) is isomorphic to the Hk(cµ) principal series module

Hk(cµ) X(νk) = Ind νk. S(ak)

ν The module Fµ,n−k(Xδ ) is isomorphic to the Hn−k(cµ) principal series module

Hk(cµ) X(¯νn−k) = Ind ν¯n−k. S(¯an−k)

This extends the results of Ciubotaru and Trapa [16] to non-spherical principal series modules. Importantly, if G is a split real orthogonal or symplectic group, we can describe the Hecke algebra module of the image of every principal series modules resulting from functors Fµ,k and Fµ,n−k. Furthermore using Casselman’s subrespresention theorem, for these split groups we have a correspondence of irreducible Harish-Chandra modules of G and graded Hecke algebra modules.

Fµ,k(X) and Fµ,n−k(X) are submodules of the Hk(cµ) and Hn−k(cµ)-modules

X(νk) and X(¯νn−k).

θ We deﬁne two anti-involutions on Bk which descend to the usual anti-involutions on the graded Hecke algebra [5]. Furthermore we show that if X is a Hermitian

(resp. unitary) module of G = Sp2n(R) then the image of X under the functor Fµ,k θ is a Hermitian (resp. unitary) module for Bk[m]. We also show that the Langlands quotient is preserved.

115 ν Theorem 3.9.26. Let Xδ be a principal series module for G = O(p, q) or Sp2n(R). ν ν The Langlands quotient X = X / radh, i ν is mapped by F , to the Langlands δ δ Xδ µ,k quotient of the (c )-module, X(ν ) = X(ν )/ radh, i . Similarly, Xν is mapped Hk µ k k Xνk δ by Fµ,n−k, to the Hn−k(cµ)-module X(¯νn−k).

We then give a non-unitary test for principal series modules.

This result gives a functorial proof of the nonunitarity criterion proved by Barbasch, Pantano, Paul and Salamanca-Riba [3, 46]. We also obtain a non-unitary test for any Harish Chandra module; in the split case one could check unitarity of Hecke algebra modules however in the non-split case one would have to work with aﬃne type B/C Brauer algebra modules.

Theorem 3.9.30. [Non-unitary test for Harish-Chandra modules] Let X be a Harish

Fµ,k(X), Fµ,n−k(X) is not unitary as a Hecke algebra module then X is not unitary as a G-module.

θ In Section 3.3, we define the type B/C affine Brauer algebra Bk and show that it acts on X ⊗ V ⊗k and commutes with the action of K. Section 3.4 defines particular θ quotients of Bk isomorphic to the graded Hecke algebras Hk(c). In Section 3.5, we θ introduce the functors, defined in [16], Fµ,k : HC(G) → Bn-mod. These functors θ naturally create Bk-modules. In Section 3.7, we show that the functors restricted to principal series modules define Hecke algebra modules. Section 3.8 describes ν ν the isomorphism classes of Fµ,k(Xδk ) and Fµ,n−k(Xδk ) as principal series modules of graded Hecke algebras Hk(cµ) and Hn−k(cµ). In Section 3.9, we prove that functors

Fµ,k preserve unitarity and invariant Hermitian forms.

116 3.2 Preliminaries

Let G be O(p, q), p + q = 2n + 1 or Sp2n(R). Let g0 be its Lie algebra, with complexification g = g0 ⊗R C. We uniformly denote a real Lie algebra by g0, for a complex Lie algebra we drop the subscript. We fix a Cartan involution θ of g0 and extend to g, let Θ be the corresponding involution of G. A maximal compact subgroup of G is K, the fixed space of Θ. The Lie algebra g0 decomposes as k0 ⊕ p0. The subspace p0 is the −1 eigenspace of θ, the subalgebra k0 is the +1 eigenspace of θ and the Lie algebra of K. Similarly, g = k ⊕ p. Let a0 be a maximal commutative Lie subalgebra of p0. Let M be the centralizer of a0 in K under the adjoint action. Define the Lie algebra of M to be m0.

t Example 3.2.1. For A ∈ GLn(R) or GLn(C) let A denote its transpose. Let 0 Idn Jn = and set − Idn 0

t Sp2n(R) = {A ∈ GL2n(R): A JnA = Jn}.

The algebra t sp2n(R) = {g ∈ M2n(R): g Jn + Jng = 0}, is the Lie algebra of Sp2n(R). It is a split real form of its complexiﬁcation

t sp2n(C) = {g ∈ M2n(C): g Jn + Jng = 0}.

The maximal compact Lie group K of Sp2n(R) is U(n); the real Lie algebra k0 of K is un embedded into sp2n(R) as AB : A + iB ∈ u . −BA n

The complexiﬁcation of sp2n(R) is sp2n(C), the corresponding complex Lie algebra of un is gln(C). For the complex Lie algebra sp2n(C) the complex Lie subalgebra k is gln(C) embedded into sp2n(C) as AB : A, B ∈ gl ( ),At = −A, Bt = B . −BA n C

The subspace p for sp2n(C) is the set of matrices 0 B such that B ∈ gl ( ) and Bt = B . B 0 n C

117 In this form, the Cartan involution of sp2n(C) is equivalent to conjugation by the involutory matrix 0 i Id ξ = n . −i Idn 0

When a speciﬁc basis and formulation of sp2n(C) is needed, we will use the above example. We will also denote sp2n(C) by sp2n.

0 0 Example 3.2.2. Let Ip,q be a diagonal matrix with p 1 s followed by q −1 s, we ﬁx p ≥ q and p + q = 2n + 1. We deﬁne the real orthogonal group O(p, q) to be

t O(p, q) = {A ∈ GLn(R): A Ip,qA = Ip,q}.

Definition 3.2.3. For G equal to O(p, q) or Sp2n(R) write V0 for the defining matrix module. That is ρ : G → GL(V0) is the injection defining G as a linear group. Write

V = V0 ⊗R C for the complexiﬁcation of V0.

2n 2n+1 If G = Sp2n(R) then V = C and if G = O(p, q) then V = C . When G = Sp2n(R), let e1, ..., e2n be the standard matrix basis of V , then define a new basis 0 1 fi = ei +en+i for i = 1, .., n and fi = ei −en+i for i = 1, .., n. We also label fi by fi and 0 −1 fi by fi . When G = O(p, q) then V has basis e1, ..., e2n+1, we let fi = ep−i+1 + ep+i 0 and fi = ep−i+1 − ep+i. Following [39, Section 1.1], let {R,X, R,ˆ X,ˆ ∆} be root datum where R is the set of roots, Rˆ is the set of coroots and X and Xˆ are free groups that contain R and Rˆ respectively. There is a perfect pairing h, i between X and Xˆ which defines a pairing between R and Rˆ. The simple roots ∆ are a subset of R. Let t equal the complexification of X, and similarly ˆt is the complexification of Xˆ. The Lie algebra g decomposes as k ⊕ p. Let a be a maximal abelian Lie subalgebra of p. The restricted roots Σ of g are given by the eigenvalues of a acting on g. The nilpotent Lie subalgebra n is the sum of positive root spaces of the restricted roots of a.

Deﬁnition 3.2.4. [36, Proposition 6.46], [32] The Iwasawa decomposition of the complex vector space g is g = k ⊕ a ⊕ n.

The Iwasawa decomposition of G [36, Theorem 6.46] is

G = KAN.

118 Let M be the centralizer of a in K and denote by NK (a) the normalizer of a in K.

Let m0 be the Lie algebra of M with complexiﬁcation m. The Weyl group associated to G is the group

WG = NK (a)/M.

Example 3.2.5. For G = Sp2n(R), a maximal abelian subalgebra a of p is 0 B a = : B is diagonal,B ∈ gl ( ) . B 0 n C

The algebra a has dimension n, this is the real rank of Sp2n(R). Let Ei,j be the matrix with 1 in the (i, j) position and zero elsewhere. Let k ∈ {0, ..., n} The subspace ak is the span of Ei,n+i + En+i,i for i = 1, ..., k. The subspace a¯n−k ⊂ a is the span of the vectors Ek+i,n+k+i + En+k+i,k+i for i = 1, ..., n − k. Note that

a = ak ⊕ a¯n−k.

∗ Deﬁne a basis of a as εi = Ei,n+i + En+i,i and let i be the dual basis in a [31]. The roots of Sp2n(R) are

∗ {±2i, ±j ∓ k, ±j ± k : for i, j, k = 1, ..., n} ⊂ a .

Fix the simple roots

∆ = {2n, i − i+1 : for i = 1, ..., n − 1}.

In this example

R = {±2i, ±j ∓ k, ±j ± k : for i, j, k = 1, ..., n}, P ∗ X = λii ∈ a , λi∈Z ∗ t = X ⊗Z C = a , ˆ R = {±εi, ±εj ∓ εk, ±εj ± εk : for i, j, k = 1, ..., n}, ˆ P X = λiεi ∈ a , λi∈Z ˆ ˆ t = X ⊗Z C = a,

The Dynkin classiﬁcation [30] labels this the root system of type Cn. For G = Sp2n(R) the subgroup M of K is the diagonal matrices with 1 or −1 on the diagonal. This group is generated by n 2-cycles,

∼ n M = (Z2) .

119 By the ith generator of M we mean the diagonal matrix with −1 in the ith position and 1 on all the others:

M = hM1, ..., Mni, i z}|{ Mi = Diag(1, 1, ..., 1 −1 , 1, ..., 1).

n A character for M can be represented by an ordered set in (Z2) . The trivial character would be denoted {1, 1, ..., 1} and the determinant character restricted to M would correspond to the sgn character {−1, −1, ..., −1}. Let δk denote the character that takes the ﬁrst k generators of M to −1 and the last n − k generators to 1. The character δk corresponds to the ordered set

k times n−k times {−z 1, −}|1..., −1{, 1z, 1}|, ..., 1{}

For G = O(p, q) or Sp2n(R), g is so2n+1 or sp2n. It is well known that the associated Weyl group is isomorphic to the hyperoctahedral group,

∼ ∼ n WO(p,q) = WSp2n = Sn n (Z2) .

Let us denote the hyperoctahedral group of rank n ∈ N by W (Bn), i.e.

n W (Bn) = Sn n (Z2) .

Definition 3.2.6. Given a finite dimensional complex Lie algebra g with basis B and dual basis B∗ with respect to the Killing form, we define the Casimir element in the enveloping algebra U(g) to be

X Cg = bb∗ ∈ U(g). b∈B

h P ∗ For a subalgebra h ⊂ g we denote the Casimir element of h in g by C = b∈B∩h bb where the basis B is taken such that B ∩ h is a basis of h.

Deﬁnition 3.2.7. Let A be a group or a C-algebra with a complex representation U. We deﬁne the commutator of A in End(U) as

EndA(U) = {f ∈ EndC(U):[f, a] = 0 for all a ∈ A}.

For a complex vector space Y we denote by T (Y ) and S(Y ) the tensor algebra and symmetric algebra of Y .

120 3.3 Brauer Algebras

For a given g-module X and the matrix module V , the endomorphism ring EndC(X ⊗ V ⊗k) has been thoroughly studied. Most attention ([59, 18, 20, 10, 23]) has been on ⊗k understanding the subalgebra of EndC(X ⊗ V ) commuting with G:

⊗k EndG(X ⊗ V ).

⊗k In the case of g = gln, Endgln (X ⊗ V ) has a map from the graded Hecke algebra associated to the symmetric group [1]. However in Type B [20], with g = so2n+1, the relevant algebra is the affine Brauer algebra with parameter n. In Type C, with g = sp2n, the corresponding algebra is the affine Brauer algebra with parameter −n. In this section, we define the affine type B/C Brauer algebra as an extension of the affine Brauer algebra. We endow it with a natural action on X ⊗ V ⊗k and prove that it commutes with the action of K.

Deﬁnition 3.3.1. [8] The rank k Brauer algebra Bk[m], with parameter m ∈ C, is the associative C-algebra generated by elements ti,i+1 and ei,i+1 for i = 1, ..., k − 1, subject to the conditions:

the subalgebra generated by ti,i+1 is isomorphic to C[Sn],

2 ei,i+1 = mei,i+1,

ti,i+1ei,i+1 = ei,i+1ti,i+1 = ei,i+1,

ti,i+1ti+1,i+2ei,i+1ti+1,i+2ti,i+1 = ei+1,i+2,

[ti,i+1, ej,j+1] = 0 for j 6= i, i + 1.

Deﬁnition 3.3.2. Let U be a vector space with basis x1, ..., xk. The rank k aﬃne

Brauer algebra Bk[m], with parameter m ∈ C is as a vector space equal to

∼ Bk[m] = Bk[m] ⊗ S(U).

The multiplication satisﬁes the following conditions:

ti,i+1xi − xi+1ti,i+1 = 1 + ei,i+1,

[ti,i+1, xj] = 0, j 6= i, i + 1,

ei,i+1(xi + xi+1) = 0 = (xi + xi+1)ei,i+1,

121 [ei,i+1, xj] = 0, j 6= i, i + 1,

[xi, xj] = 0, the subalgebra generated by ti,i+1, ei,i+1 is isomorphic to Bk[m]. Let us consider X and V as U(g)-modules then X ⊗ V ⊗k has a U(g)⊗k+1-module structure. We deﬁne operators that form the action of the Brauer algebra.

⊗k+1 ⊗k Deﬁnition 3.3.3. Given the action of U(g) on X ⊗ V we write (g)i for the action of g on the i + 1st tensor in U(g)⊗k+1,

i times k−i times z }| { z }| { (g)i = id ⊗ ... ⊗ id ⊗g ⊗ id ⊗ ... ⊗ id .

By construction we start counting from zero. Hence (g)0 = g ⊗ id ⊗ ... ⊗ id ∈ U(g)⊗k+1.

Definition 3.3.4. Fix a basis B such that B = (B ∩ k) S(B ∩ p). Let B∗ be the dual basis with respect to the Killing form of g. For 0 ≤ i < j ≤ k, define Ωij to be the operator X ∗ ⊗k+1 Ωij = (b)i ⊗ (b )j ∈ U(g) . b∈B k p Similarly we define Ωij and Ωij as

k X ∗ ⊗k+1 Ωij = (b)i ⊗ (b )j ∈ U(g) , b∈B∩k

p X ∗ ⊗k+1 Ωij = (b)i ⊗ (b )j ∈ U(g) . b∈B∩p k p Lemma 3.3.5. The operators Ωij,Ωij and Ωij are independent of the choice of basis of g, k and p respectively.

2 g P ∗ Proof. It is suﬃcient to prove the statement for Ω12 ∈ U(g) . Let C = b∈B bb ∈ U(g) be the Casimir element and ∆ : U(g) → U(g) × U(g) denote comultiplication.

The operator Ω12 can be written as:

g g g Ω12 = ∆(C ) − C ⊗ 1 − 1 ⊗ C .

g The Casimir element C is independent of the choice of basis therefore Ω12 is also k independent. Similarly replacing g with the Lie subalgebra k,Ω12 is independent of p choice of basis. Finally Ω12 is independent as it is the diﬀerence of the other two,

k p Ωij − Ωij = Ωij.

122 ⊗k Let the symmetric group on k elements Sk act on X ⊗ V by permuting the factors of V . Explicitly the simple reﬂection si,i+1 acts by

si,i+1(x0 ⊗ v1 ⊗ .. ⊗ vi ⊗ vi+1 ⊗ .... ⊗ vk) = x0 ⊗ v1 ⊗ .. ⊗ vi+1 ⊗ vi ⊗ .... ⊗ vk.

Lemma 3.3.6. If g = sp2n or so2n+1 then V ⊗ V decomposes as

2 2 Λ V ⊕ S V/1 ⊕ 1 for so2n+1,

2 2 Λ V/1 ⊕ S V ⊕ 1 for sp2n. Here 1 denotes the trivial module of g.

Let pr1 be the projection of V ⊗V onto the trivial submodule 1 in the decomposition above. Let pri,i+1 be the projection onto the trivial submodule of Vi ⊗ Vi+1.

Lemma 3.3.7. [23, Theorem A] Let G be O(p, q) or Sp2n(R). Let X be a complex G-representation and V the deﬁning matrix module of G. Then there exists m ∈ N ⊗k such that there is a homomorphism π : Bk[m] → End(X ⊗ V ): X π(xi) = Ωji, j

π(ti,i+1) = si,i+1,

π(ei,i+1) = id ⊗ ... ⊗ id ⊗ m pri,i+1 ⊗id ⊗ .... ⊗ id.

p+q For G = Sp2n(R) the parameter is m = −n and if G = O(p, q) then m = b 2 c.

Theorem 3.3.8. For G = O(p, q) or Sp2n(R), the aﬃne Brauer algebra with the action on X ⊗ V ⊗k deﬁned in Lemma 3.3.7 commutes with the action of U(g) on X ⊗ V ⊗k.

Lemma 3.3.9. [16, Lemma 2.3.1] Let 0 < i < j ≤ k and G = O(p, q) or Sp2n(R) . As operators on X ⊗ V ⊗k

Ωij = sij + m pri,i+1,

p+q where m = b 2 c or −n respectively.

123 Proof. One only needs to consider the operator Ω12 on V ⊗ V . By Lemma 0.8.10 V ⊗ V decomposes as

2 2 Λ V ⊕ S V/1 ⊕ 1 for gC = so2n+1(R),

2 2 Λ V/1 ⊕ S V ⊕ 1 for gC = sp2n(C).

On V ⊗ V s12 = prS2V − prΛ2V . Then using the fact that Ω12 = ∆(C) − C ⊗ 1 − 1 ⊗ C we ﬁnd the operators

Ω12 and s12 + me12, act by the same scalars on the irreducible decomposition of V ⊗ V .

⊗k For G = GLn the commutator EndGLn (X ⊗ V ) contains the same type Weyl group, the symmetric group ([1]). One might expect that in type B and C this may ⊗k ⊗k be the case too. However EndSp2n(R)(X ⊗ V ), EndO(p,q)(X ⊗ V ) and the affine Brauer algebra, do not contain a copy of the hyperoctahedral group. We look to establish a theory that has this type symmetry reflected in the commutator. We introduce the type B/C affine Brauer algebra which acts on X ⊗ V ⊗k and commutes with the action of K for G = Sp2n(R) or O(p, q). Crucially the type B/C affine Brauer algebra contains the Weyl group of Type B/C, W (Bk). Recall the hyperoctahedral group is generated by simple reflections si−i+1 and sk .

θ Definition 3.3.10. The type B/C affine Brauer algebra Bk[m] is generated by the affine Brauer algebra Bk[m] and reflections θj, for j = 1, ..., k, such that the subalgebra generated by ti,i+1, for i = 1, .., k − 1 and θj is isomorphic to the group algebra of the th k hyperoctahedral group C[W (Bk)] and the following relations hold;

[ei,i+1, θj] = 0 for all j,

ei,i+1θiθi+1 = ei,i+1 = θiθi+1ei,i+1 for i = 1, ..., k − 1,

[θn, xj] = 0 for j 6= k.

The Lie algebra g decomposes as eigenspaces of a Cartan involution θ that is g = k ⊕ p. For G = O(p, q) or Sp2n(R) there is a semisimple involutive ξ ∈ g such that θ is equal to conjugation by ξ. For G = Sp2n(R), ξ is given in Example 3.2.1.

For O(p, q) ξ is given by the matrix Ip,q.

124 θ ⊗k Lemma 3.3.11. The type B/C affine Brauer algebra Bk[m] acts on X ⊗ V . This action is defined by extending the action π of the affine Brauer algebra to the extra ⊗k+1 θ generators θi. The generators θi act by (ξ)i ∈ U(g) . Extend π to Bk[m] by ⊗k+1 ⊗k π(θi) = (ξ)i ∈ U(g) ⊂ End(X ⊗ V ). That is

θ ⊗k π : Bk[m] −→ EndK (X ⊗ V ),

π(θi) = (ξ)i.

i k−i z }| { z }| { ⊗k+1 Explicitly, (ξ)i = id ⊗ ... ⊗ id ⊗ξ ⊗id ⊗ ... ⊗ id ∈ U(g) . The constant m equals p+q b 2 c when G = O(p, q) and m = −n if G = Sp2n(R).

⊗k Proof. Since we know that the aﬃne Brauer algebra Bk[m] acts on X ⊗ V we only need to check the action of θj and the relations involving it in Deﬁnition 3.3.10. First, we show that (ξ)j and sij generate the hyperoctahedral group. Since ξ is an involution 2 then (ξ)j = id. For any g ∈ g we have that

⊗k+1 sij(g)i = (g)jsij for sij ∈ Sk, (g)i ∈ U(g) and i, j 6= 0.

Hence sij(ξ)ksij = (ξ)sij (k). Therefore (ξ)j and sij generate the hyperoctahedral k+1 group. If i 6= j, then (g)i and (h)j commute in U(g) . Deﬁnition 3.3.7 states P π(xk) = i

[θj, ei,i+1] = 0 for all j 6= i, i + 1.

We show the relations involving ei,i+1 and θi, θi+1 on the kernel and image of ei,i+1 separately. Recall that ei,i+1 acts by m pri,i+1; the projection onto the trivial submodule k+1 of Vi ⊗ Vi+1. The image of pri,i+1 is a U(g) trivial submodule, (ξ)i(ξ)i+1 ∈ U(g) act by the identity on this module. The operator pri,i+1 is an idempotent, hence on the trivial module pri,i+1 = id. On this submodule the relations

ei,i+1θi = ei,i+1θi+1 = ei,i+1 = θiei,i+1 = θi+1ei,i+1 become id = id. On the kernel of ei,i+1, by deﬁnition ei,i+1 is the zero operator and the relations are immediately true.

125 θ This verifies the relations in Definition 3.3.10 of Bk[m]. Hence π is a well defined action: θ ⊗k π : Bk[m] → End(X ⊗ V ).

Theorem 3.3.12. Let G = O(p, q) or Sp2n(R) and X a Harish-Chandra module. The θ ⊗k type B/C Brauer algebra Bk[m] acts on X ⊗ V and commutes with the action of K on X ⊗ V ⊗k.

Proof. The action of Bk[m] commutes with g and by restriction with K. The algebra θ θ Bk[m] = hBk[m], θj : for j = 1, ..., ki. Therefore, to verify that Bk[m] commutes with the action of K, one only needs to check that π(θj) = (ξ)j commutes with the action of K. Conjugation by ξ is the Cartan involution: ξ−1Kξ = Θ(K). By deﬁnition, Θ is the identity on K. Hence ξk − kξ = 0 for k ∈ k. Therefore:

k+1 X [(ξ)i, k] = (ξ)i − (k)j(ξ)i(k)j = 0. j=0

Hence the action of (ξ)i and K commute.

3.4 Quotients of the aﬃne type B/C Brauer alge- θ bra Bk[m]

In Section 3.5 we introduce functors, defined in [16], from the category HC(G)-mod θ to the category of Bk[m] modules. However, we are aiming at graded Hecke algebra θ modules. In this section, we look at particular ideals and quotients of Bk[m] which are isomorphic to graded Hecke algebras. This will set up Section 3.6 in which we focus on principal series modules and show that via the quotients defined in this section, the functors defined in Section 3.5 descend to take principal series modules to graded Hecke algebra modules. Recall that W (R) denotes the Weyl group associated to a root datum (R,X, R,ˆ X,ˆ ∆) and h, i : X × Xˆ → C is the pairing between dual spaces. Define the C-spaces ∗ ˆ t = X ⊗Z C, t = X ⊗Z C.

Deﬁnition 3.4.1. [39] The graded Hecke algebra HR(c) associated to the root system (R,X, R,ˆ X,ˆ ∆) and parameter function c from ∆ to C, is as a vector space

R ∼ H (c) = S(t) ⊗ C[W (R)],

126 such that as an algebra S(t) and C[W (R)] are subalgebras and the following cross relations hold,

sα − sα()sα = c(α)hα, ˆi, for ∈ t and α ∈ ∆.

If the parameter function c : ∆ → C is taken to uniformly be 1, then in this case the graded Hecke algebra is entirely deﬁned by the root system (W, R, ∆). For a Hecke

Rk algebra determined by the uniform parameter we denote it by H where Rk is the root system. For example HDk denotes the graded Hecke algebra associated to the root system Dk with the parameter function c : δ → C such that c(α) ≡ 1.

We ﬁx the set of simple reﬂections of the hyperoctahedral group W (Bk) to be

{si,i+1, θk : i = 1, ..., k − 1}. We also associate to the hyperoctahedral group a k dimensional vector space t with basis 1, ..., k and subset ∆ = {i − i+1 and k : i = 1, ..., k − 1}. Then for c ∈ C we deﬁne the parameter c : ∆ → C as ( 1 if α = − , c(α) = i i+1 c if α = k.

We denote the graded Hecke algebra associated to the Weyl group W (Bk) with the parameter c as Hk(c).

Lemma 3.4.2. The graded Hecke algebra of type Bk (resp. type Ck) is isomorphic 1 to Hk(1) (resp. Hk( 2 ))) and the algebra Hk(0) is isomorphic to an extension of the

Hecke algebra of type Dk, ∼ Dk Hk(0) = H o Z2.

Bk Proof. The isomorphism of Hk(1) and the graded Hecke algebra H is apparent from the deﬁnitions. The space tDk is equal to the space t in Hk(0). The Weyl group

Dk W (Dk) is naturally a subgroup of W (Bk). The generator t ∈ Z2 acts on H by interchanging roots k−1 − k and k−1 + k and acts by conjugation by sk ∈ W (Bk) on W (Dk) ⊂ W (Bk).

θ We deﬁne two ideals in the type B/C aﬃne Brauer algebra Bk[m]. We then show θ that the quotient of Bk[m] by these ideals is isomorphic to a graded Hecke algebra.

127 θ Deﬁnition 3.4.3. Let Ie be the two sided ideal in Bk[m] generated by the idempotents,

{ei,i+1 : for i = 1, ..., k − 1}.

r Let c ∈ C and r ∈ Z, deﬁne Ic to be the two sided ideal,

r Ic = hθkxk + xkθk − 2c + 2rθki.

The ideal Ie can be generated by any idempotent since they are all in the same

Sk conjugation orbit. By using c ∈ C we have abused notation; however the two occurrences of c will correspond to the same constant.

θ r Lemma 3.4.4. The quotient of the algebra Bk[m] by the ideal generated by Ie and Ic is isomorphic to the graded Hecke algebra

θ r ∼ Bk[m]/hIe,Ic i = Hk(c).

Proof. Consider the presentation in Deﬁnition 3.3.10 with generators

xi, θj, ti,i+1, ei,i+1 and relations

2 2 3 4 θj = 1, si,i+1 = 1, (si,i+1si+1.i+2) = 1, (sk−1,kθk) = 1,

ti,i+1xi − xi+1ti,i+1 = 1 + ei,i+1,

[ti,i+1, xj] = 0, j 6= i, i + 1,

ei,i+1(xi + xi+1) = 0 = (xi + xi+1)ei,i+1,

[ei,i+1, xj] = 0, j 6= i, i + 1,

[xi, xj] = 0,

[ei,i+1, θj] = 0 for all j,

ei,i+1θiθi+1 = ei,i+1 = θiθi+1ei,i+1 for i = 1, ..., k − 1,

[θn, xj] = 0 for j 6= k.

r Under the quotient by Ie and Ic the generators ei,i+1 and the relations ei,i+1 = 0 cancel out. Furthermore we add another relation: xkθk + θkxk − 2c + 2rθk. Hence the presentation has generators

xi, θj, ti,i+1

128 with relations

2 2 3 4 θj = 1, si,i+1 = 1, (si,i+1si+1.i+2) = 1, (sk−1,kθk) = 1,

ti,i+1xi − xi+1ti,i+1 = 1,

[ti,i+1, xj] = 0, j 6= i, i + 1,

[xi, xj] = 0,

[θn, xj] = 0 for j 6= k,

xkθk + θkxk − 2c + 2rθk.

This is a presentation of the Hecke algebra Hk(c); it is the modiﬁcation of the presentation in Deﬁnition 3.4.1 by i 7→ xi + r. Since we have shown that the θ r presentation of Bk[m]/hIeIc i is identical to the presentation of Hk(c) then these algebras are isomorphic.

Remark 3.4.5. We could have chosen to quotient by the ideal generated by θkxk + xkθk − c without the 2rθk part. This quotient would also be isomorphic to Hk(c) with

i mapping to xi. However, we need the modification of the affine parts by the scalar r to enable our results regarding images of principal series modules descending to Hecke algebra modules. One can think of this modification by r as an analogue of the ρ shift.

θ 3.5 Functors from HC(G)-mod to Bk-mod.

In this section, we introduce functors, deﬁned in [16]. We show these functors take θ Harish-Chandra modules to modules of the Bk algebra.

Definition 3.5.1. [16, (2.8)] Let n be the real rank of G. If G = Sp2n(R) the real rank is n. Uf G = O(p, q) then n = q = min(p, q). Let µ be an irreducible K-module, fix an integer k ≤ n. The space V is the matrix module of G. We define the functor

Fµ,k to be: θ Fµ,k : HC(G)-mod −→ Bk-mod

⊗k X 7→ HomK (µ, X ⊗ V ),

129 ⊗k and on morphisms f : X → Y and g ∈ Homk(µ, X ⊗ V ),

⊗k Fµ,kf(g): µ → Y ⊗ V ,

⊗k Fµ,kf(g)(µ) = f ⊗ id g(µ).

θ ⊗k Remark 3.5.2. Lemma 3.3.11 gives an action of Bk on X ⊗ V . Since this action θ ⊗k commutes with the action of K then Bk naturally acts on HomK (µ, X ⊗ V ) from the inherited action on X ⊗ V ⊗k.

Lemma 3.5.3. For any irreducible K-module µ and k ≤ n, the functor Fµ,k deﬁned in Deﬁnition 3.5.1 is exact.

Proof. Tensoring with a ﬁnite dimensional module is exact. The module V ⊗k is ﬁnite dimensional hence the functor taking X to X ⊗ V ⊗k is exact. Furthermore, µ is an irreducible K-module. Therefore the functor which takes Y to HomK (µ, Y ) is exact.

The functor Fµ,k is the composition of these two exact functors, hence the result follows.

3.6 Restricting functors to principal series modules

θ The functors (Deﬁnition 3.5.1) take any Harish-Chandra module to a Bk-module. In this section, given a principal series module we give a basis for the image of the functors Fµ,k and Fµ,n−k for particular characters µ, µ depending on the principal series modules. ∼ ∼ n Let G = Sp2n(R). As in Example 3.2.1 K = U(n), M = (Z2) . The Cartan involution θ is equal to conjugation by the matrix

 0 i Id ξ = n . −i Idn 0

The subspace a has dimension n with basis εi and corresponds to the subgroup A under the exponential map. We label a character of a by ν ∈ a∗ and characters of A by eν. ∼ 2n 1 1 −1 −1 The matrix module V = C has two bases: {e1, ..., e2n} and {f1 , .., fn, f1 , ..., fn }, η where fi = ei + ηen+i. Recall that the Iwasawa decomposition of G is

G = KAN,

130 n also, that M is the centraliser of a0 in K, which is isomorphic to Z2 . The character k n δ is deﬁned to be the character of M which takes the ﬁrst k generators of Z2 to −1 and the last n − k to 1. We write 1 for the trivial character of N. If G = O(p, q) then K ∼= O(p) × O(q), M = O(p − q) × O(1)q embedded into O(p, q) as the block matrix

(O(p − q), x1, x2, ..., xq, xq, .., x1)

k k where xi ∈ O(1). We denote characters of M, δtriv and δdet to be

k k q−k q δtriv = triv ⊗(sgn ) ⊗ triv on O(p − q) ⊗ O(1) ,

k k q−k q δdet = det ⊗(sgn ) ⊗ triv on O(p − q) ⊗ O(1) . The Cartan involution θ is equal to conjugation by the matrix Id 0 ξ = p . 0 − Idq

Deﬁnition 3.6.1. [58] Let G = KAN (resp. g0 = k0 ⊕ a0 ⊕ n0) be the Iwasawa decomposition of G (resp. g0) and let M be the centraliser of a0 in K. Given a character eν of A and the character δ of M we deﬁne the minimal principal series representation; ν G ν Xδ = IndMAN (δ ⊗ e ⊗ 1).

In the non-split case principal series representations may be induced from irreducible representations of M which are not one dimensional. In this chapter we will only 1ν study principal series modules that are induced from a character of M. We write δ for the vector spanning the representation space of the character δ ⊗ eν ⊗ 1. Hence

ν G 1ν Xδ = IndMAN δ .

ν ν For G = Sp2n(R), we can calculate the dimension of Ftriv,k(Xδk ) and Fdet,n−k(Xδk ).

Note that if we want to describe the trivial isotypic component we must take Ftriv,k and if we wish to look at the det isotypic component then we must take the functor

Fdet,n−k.

For G = O(p, q), we can calculate the dimension of Ftriv ⊗ sgn,k andFtriv ⊗ triv,q−k. ν Similarly for X k , we take the functors Fsgn ⊗ triv,k and Fsgn ⊗ sgn,q−k. δdet To enable us to succinctly discuss all of the above cases we will associate a character µ and µ to each principal series modules. Note δ is a K-character and µ, µ are characters of M.

131 ν k n−k G = Sp2n(R) Xδ , δ = (triv) ⊗ (sgn) µ = triv µ = det ν k q−k G = O(p, q) Xδ , δ = trivp−q ⊗(triv) ⊗ (sgn) µ = triv ⊗ det µ = triv ⊗ triv ν k q−k G = O(p, q) Xδ , δ = detp−q ⊗(triv) ⊗ (sgn) µ = det ⊗ triv µ = sgn ⊗ sgn

Table 3.6.1: Characters µ, µ associated to particular principal series module.

ν Lemma 3.6.2. Let G = Sp2n(R) or G = O(p, q). If Xδ is a minimal principal series ν ν module, then Fµ,k(Xδ ) and Fµ,n−k(Xδ ) are ﬁnite dimensional. with dimensions:

ν k dim(Fµ,k(Xδk )) = k!2 = |W (Bk)|,

Similarly, ν n−k dim(Fµ,n−k(Xδk )) = (n − k)!2 = |W (Bn−k)|.

This is an extension of [16, Lemma 2.5.1] to non-spherical principal series modules and we use the same arguments.

Proof. We explicitly calculate a basis for

ν ν ⊗k Fµ,k(Xδ ) = HomK (µ, Xδ ⊗ V ).

ν 1ν Since Xδ is an induced module from δ and K is a compact group, by Frobenius reciprocity this is equal to,

ν 1ν ⊗k Fµ,k(Xδ ) = HomM (µ|M , δ ⊗ V |M ).

One can tensor by µ∗ to get a space ﬁxed by M, hence

ν ∗ 1ν ⊗k M Fµ,k(Xδ ) = (µ ⊗ δ ⊗ V ) .

ni We first prove the result for G = Sp2n(R). The module V has basis {fi : i = th δij ni 1, ..., n and ni = ±1} and the j generator of M acts by −1 on fi . Therefore if we require M to act trivially on u ∈ Xν ⊗ V ⊗k the generators M , ..., M must act by 1. δk 1 n ν ⊗k Let us first calculate all of the elementary tensors in Xδk ⊗ V which are fixed by M. 1ν The generators M1, ..., Mk act by −1 on δk , hence must act by −1 on the tensor part ⊗k 1 −1 contributed by V . To satisfy this we need to have fi or fi feature in the tensor of u, for every i = 1, ..., k,. Since there can only be k elements tensored together in

⊗k ⊗k n1 nk V then the contribution of u from V must be f1 , ..., fk in some order. The

132 ⊗k set of elementary tensors in V which feature all the required fi is the Sk orbit of ν ⊗k f1 ⊗ ... ⊗ fk. Considering not necessarily elementary tensors in v ∈ Xδk ⊗ V , X v = x0 ⊗ v1 ⊗ ... ⊗ vk,

lj nl th δ where vi ∈ {fl : l = 1, ..., n and nl = ±1}. The j generator of M, Mj, acts by −1 on fl. Since every elementary tensor in this basis is an eigenvector of the action of P M then if M fixes v = x0 ⊗ v1 ⊗ ... ⊗ vk then M fixes each elementary tensor in v. ν ⊗k Hence every M fixed vector in Xδk ⊗ V is in the subspace ( ) X 1ν n1 nk span δk ⊗ fw(1) ⊗ ... ⊗ fw(k) : ni = ±1 . w∈Sk

k k The size of the basis is |Sk| × 2 = k!2 = |W (Bk)|. The proof is almost identical for ν dim(Fdet,n−k(Xδk )). One needs to note that all of the generators of M must act by −1 on the det isotypic space, since det |M = sgn. Using Frobenius reciprocity one can show, ν ⊗n−k Fdet,n−k(Xδk ) = HomM (sgn, δk ⊗ V ), which has a basis:

    ν X 1ν nk+1 nn Fdet,n−k(Xδk ) = span δk ⊗ fw(k+1) ⊗ ...fw(n) : ni = ±1 . w∈Sn−k 

ith Lq z}|{ For G = O(p, q) note that V |M = Vp−q i=1 triv ⊗... ⊗ sgn ⊗... ⊗ triv and µ|M = q ni 1 −1 trivp−q ⊗ sgn . Recall the notation fi = ep−i+1 + niep+i, the vectors fi and fi are ith the two eigenvectors of M with character triv ⊗ triv ... ⊗ z}|{sgn ⊗... ⊗ triv. I.e. the ith generator of O(1)q in M acts by −1. ν We will prove that Ftriv ⊗ sgn,k(X k ) has basis δtriv ( ) X ν n1 nk 1 k ⊗ f ⊗ ...f : ni = ±1 . δtriv w(k) w(k) w∈Sk The other four calculations are almost identical. Note that this is equivalent to giving a basis for ν k M ((triv ⊗ sgn)|M ⊗ 1 k ⊗ V ) δtriv

133 which is equal to, as a vector space,

ν M k M 1 q 1 ( trivp−q ⊗ sgn ⊗ k ⊗ (Vp−q triv ... ⊗ sgn ⊗... sgn) ) . δtriv

ν 1 q 1 The vector trivp−q ⊗ sgn ⊗ k ⊗ f1 ⊗ ... ⊗ fq is fixed by M since O(p − q) acts trivially δtriv on each tensor. Furthermore for i = 1, ..., k the ith generator of O(1)q in M acts by ν th 1 q 1 −1 on trivp−q ⊗ sgn , 1 on k , and −1 on f1 ⊗ ... ⊗ fq. For i = k + 1, ...q the i δtriv q ν 1 q 1 generator of O(1) in M acts by −1 on trivp−q ⊗ sgn −1 on k and 1 on f1 ⊗ ... ⊗ fq. δtriv Hence every generator of M acts by 1. An identical argument shows that the orbit ν 1 q 1 of trivp−q ⊗ sgn ⊗ k ⊗ f1 ⊗ ... ⊗ fq by W (Bq) is also fixed. Any elementary tensor δtriv fixed by M must be of this form; if it is not, one of the generators will act by −1. Finally suppose that another vector v is fixed by M, then v is a sum of elementary tensors which are all eigenvalues for O(1)q, hence every elementary tensor involved must be fixed. This concludes that v is in the span of the vectors ( ) X ν n1 nk 1 k ⊗ f ⊗ ...f : ni = ±1 . δtriv w(1) w(k) w∈Sk

We state the basis for Fµ,k and Fµ,n−k Let G = O(p, q) n o ν P ν n1 nk F (X k ) = span 1 k ⊗ f ⊗ ...f : n = ±1 , triv ⊗ det,k δ w∈Sk δ w(1) w(k) i triv n triv o ν P ν nk+1 nq F (X k ) = span 1 k ⊗ f ⊗ ...f : n = ±1 , triv ⊗ triv,q−k δ w∈Sq−k δ w(k+1) w(q) i triv n triv o ν P ν n1 nk F (X k ) = span 1 k ⊗ f ⊗ ...f : n = ±1 , det ⊗ triv,k δ w∈Sk δ w(1) w(k) i det n det o ν P 1ν nk+1 nq Fdet ⊗ det,q−k(X k ) = span w∈S k ⊗ fw(k+1) ⊗ ...fw(q) : ni = ±1 . δdet q−k δdet

3.7 Images of principal series modules

θ We write the Type B/C affine Brauer algebra as Bk and omit m. We show that on minimal principal series representations the functors (Definition θ 3.5.1) which take admissible O(p, q) or Sp2n-modules to Bk-modules naturally descend p−q to graded Hecke algebra Hk(c)-modules, for c equal to 0, 1 or 2 . In Section 3.4 Lemma 3.4.4, we proved that the type B/C affine Brauer algebra has quotients isomorphic to the Hecke algebra Hk(c) with parameter c ∈ R. This quotient was defined by the relations ei,i+1 = 0 and θkxk + xkθk = 2c − 2rθk. Hence ν to show that Fµ,k(Xδ ) descends to an Hk(cµ)-module we must prove ei,i+1 = 0 and

134 ν ν θkxk + xkθk = 2cµ − 2rµθk as operators on Fµ,k(Xδ ). Similarly to show Fµ,n−k(Xδ ) is an Hn−k(rµ)-module then we must show ei,i+1 = 0 and θn−kxn−k + xn−kθn−k = ν 2cµ − 2rµθn−k on Fµ,n−k(Xδ ). The scalars rµ and cµ will be deﬁned in Table 3.7.1. The arguments of this section are inspired and very similar to [16, Proposition 2.4.5, Lemma 2.7.2]. We extend these results to non-spherical principal series modules. We also utilise an approach from the Brauer algebra perspective not used in [16].

θ θ ν Lemma 3.7.1. c.f. [16, 2.4.5] On the Bk (resp. Bn−k) module Fµ,k(Xδ ) (resp. ν Fµ,n−k(Xδ )) the idempotents ei,i+1 uniformly act by zero.

ν 1ν n1 nk Proof. Lemma 3.6.2 states that the basis of Fµ,k(Xδ ) is given by δ ⊗ fw(1) ⊗ ... ⊗ fw(k) for w ∈ Sk. The idempotents ei,i+1 act by the projection onto the trivial component of Vi ⊗ Vi+1. The trivial component of V ⊗ V is one dimensional with spanning vector Pn 0 1ν n1 nk i=1 fi ∧ fi . The vector δ ⊗ fw(1) ⊗ ... ⊗ fw(k) is in the subspace perpendicular to Pn 0 i=1 fi ∧ fi given in Lemma 0.8.10. Therefore it is in the kernel of the projection pri,i+1.

P ∗ k+1 k P Recall Deﬁnition 3.3.4, Ωi,j = b∈B(b)i ⊗(b )j ∈ U(g) , and Ωi,j = b∈B∩k(b)i ⊗ ∗ (b )j. Lemma 3.3.7 gives xk = Ω0,k + Ω1,k + ... + Ωk−1,k. ν As operators on Fµ,k(Xδ ): P P θkxk + xkθk = θk i

Hence, P P θkxk + xkθk = 2 i

X k θkxk + xkθk = 2θk Ωi,k. i

ν Similarly on Fµ,n−k(Xδ )

X k θn−kxn−k + xn−kθn−k = 2θn−k Ωi,n−k. i

135 θ ν Lemma 3.7.2. c.f. [16, 2.7.2] On the Bk-module Fµ,k(Xδ ), ! X ∗ k θkxk + xkθk = 2ξ µ(b)b − C , b∈B∩z k where z is the centre of g. P Proof. Recall Deﬁnition 3.3.4, Ωij = b∈B∩k(b)i ⊗ (ξb)k. Writing θkxk + xkθk as ν operators on Fµ,k(Xδ ),

P k θkxk + xkθk = 2θk i

X ∗ X ∗ θkxk + xkθk = 2θk (b )kb − (bb )k. b∈B∩k b∈B∩k

ν ν By deﬁnition Fµ,k(Xδ ) is the µ isotypic component of Xδ , hence

X ∗ X ∗ θkxk + xkθk = 2θk (b )kµ(b) − (bb )k. b∈B∩k b∈B∩k

P ∗ k th The operator b∈k(bb )k is the Casimir operator C on the k tensor V . We have µ(b) = 0 unless b is in the centre of U(k) for any character µ. Let z denote the centre of g. Therefore,

P ∗ k θkxk + xkθk = 2θk b∈B∩z µ(b)(b )k − (C )k , P ∗ k = 2 ξ( b∈B∩z µ(b)b − C ) . k

In order to calculate the action of θkxk + xkθk we must understand the operator ! X ∗ k Qµ = 2ξ µ(b)b − C b∈B∩z acting on the kth tensor of V .

θ ν Lemma 3.7.3. On the Bn−k-module Fµ,n−k(Xδ ); ! X ∗ k θn−kxn−k + xn−kθn−k = 2 ξ( µ(b)b − C ) . b∈B∩z n−k

136 G = Sp2n(R) µ = triv rtriv = 0 ctriv = −n G = Sp2n(R) µ = det rdet = 1 cdet = −n p+q p−q G = O(p, q) µ = triv ⊗ det rµ = 2 cµ = 2 p+q p−q G = O(p, q) µ = triv ⊗ triv rµ = 2 cµ = 2 p+q p−q G = O(p, q) µ = det ⊗ triv rµ = 2 cµ = 2 p+q p−q G = O(p, q) µ = det ⊗ det rµ = 2 cµ = 2

Table 3.7.1: Values of cµ and rµ for particular K-characters µ.

Replacing µ with µ, this follows the same way as Lemma 3.7.2.

P ∗ k Lemma 3.7.4. On the module V the operator Qµ = 2ξ b∈B∩z µ(b)b − C (resp. P ∗ k Qµ = 2ξ b∈B∩z µ(b)b − C ) is equal to 2rµ + 2cµξ (resp. 2rµ + 2cµξ), where rµ and cµ are scalars given below. In fact for G = O(p, q), rµ and cµ are independent of µ.

∼ Bk 1 ∼ Ck Recall Lemma 3.4.2, we have isomorphisms: Hk1) = H , Hk( 2 ) = H and Hk(0) is congruent to an extension of the type D graded Hecke algebra HDk . Hence when 1 G is split, that is G = O(n + 1, n) or Sp2n(R) then cµ = 1, 2 or 0 and we obtain correspondences between principal series modules of split real orthogonal Lie groups with graded Hecke algebras of type C and split real symplectic groups with graded Hecke algebras of type B and D.

Proof. We prove the result ﬁrst for G = Sp2n(R), in this case g = sp2n and k = gln. The Casimir Ck acts by the scalar n on V . The character triv is zero uniformly on P ∗ k hence triv(b) = 0 for all b and there is no contribution from b∈B∩z triv(b)b . For P ∗ the operator b∈B∩z det(b)b , we note that the centre of k = gln(C) is the span of the identity matrix, also the character det of U(n) diﬀerentiated to k is the trace character of gln. Taking the spanning vector Idn of the centre z of gln then on V , P ∗ b∈B∩z det(b)b is equal to

P ∗ ∗ b∈B∩z det(b)b = trace(Idn) Idn 1 = n n Idn,

= Idn .

Referring to Example 3.2.1, since Idn is symmetric, the identity matrix in U(k) embedded into g is 0 i Id n . −i Idn 0

137 The matrix ξ, deﬁned by the Cartan involution of Sp2n(R) is equal to 0 i Id ξ = n . −i Idn 0

Hence X det(b)b∗ = ξ, b∈B∩z as operators on V .

Now let G = O(p, q) p + q = 2n + 1, then g = so2n+1 and k = sop ⊕ soq. Any character µ of K diﬀerentiated and then restricted to z is zero. Hence for any µ, X µ(b)b∗ = 0. b∈z We are left to calculate Ck on V . Ck acts by

p Id 0 p . 0 q Idq

For G = O(p, q) the semisimple element deﬁning θ is

Id 0 ξ = p . 0 − Idq

Hence for G = O(p, q)

X p + q p − q Q = 2ξ( µ(b)b∗ − Ck) = 2ξ(− Id − ξ) = (q − p) Id −(p + q)ξ. µ 2 2 n b∈z

ν Corollary 3.7.5. For G = O(p, q) or Sp2n, consider the principal series module Xδ θ ν for particular µ and µ given in Table 3.6.1. On the Bk-module Fµ,k(Xδ ), the following equality holds:

θn−kxn−k + xn−kθn−k = 2rµ − 2cµθn−k.

ν Hence by Lemma 3.4.4, Fµ,k(Xδ ) is an Hk(cµ)-module via the quotient deﬁned by the ν relations ei,i+1 = 0 and θn−kxn−k + xn−kθn−k = 2rµ + 2cµθn−k. Similarly Fµ,n−k(Xδ ) is an Hn−k(cµ)-module.

ν We have shown that the image of Xδ under the functor Fµ,k naturally descends to a module of the graded Hecke algebra Hk(cµ).

138 ν Theorem 3.7.6. Let Xδ be a minimal principal series module of G = Sp2n(R) or

O(p, q). Let µ and µ be the particular characters in Table 3.6.1 and rµ, cµ be particular θ ν scalars in Table 3.7.1. Let π denote the homomorphism from Bk[m] to End(Fµ,k(Xδ )) ν in Lemmas 3.3.7 and 3.3.11. The graded Hecke algebra Hk(cµ) acts on Fµ,k(Xδ ), by the homomorphism, ν ψ : Hk(cµ) → End(Fµ,k(Xδ )),

i 7→ π(xi − rµ),

si,i+1 7→ π(si,i+1),

si 7→ π(θi). ν Hence Fµ,k(Xδ ), can be considered as an Hk(cµ)-module. θ ν Let π denote the homomorphism from Bn−k[m] to End(Fµ,n−k(Xδ )). The graded ν Hecke algebra Hn−k(cµ) acts on Fµ,n−k(Xδ ), by the homomorphism,

ν ψ : Hn−k(cµ) → End(Fµ,n−k(Xδ )),

i 7→ π(xi − rµ),

si,i+1 7→ π(si,i+1),

si 7→ π(θi).

ν Hence Fµ,n−k(Xδ ), can be considered an Hn−k(cµ)-module.

θ ν It should also be noted that as a Bk-module Fµ,k(Xδ ) is essentially an Hk(cµ)- θ ν module. That is, there is no element in Bk that has a non-trivial action on Ftriv,k(Xδ ) that does not correspond to an element in the Hecke algebra.

For G = O(n + 1, n) or Sp2n(R), every principal series module is induced from a character on M. Therefore for split real orthogonal or symplectic groups we can entirely describe the Hecke algebra modules resulting from functors Fµ,k and Fµ,n−k on principal series modules. Casselman [12] states that every irreducible representation in HC(G) is a subrepresentation of a principal series module. Therefore if X is a ν subrepresentation of Xδ then Fµ,k(X) also descends to a Hecke algebra module.

Theorem 3.7.7. Let G be a split real Lie group of type B or C. Let X be an irreducible Harish-Chandra G-module. Hence X is a subrepresentation of a principal ν θ θ series module Xδ , then the Bk and Bn−k-modules

Fµ,k(X) and Fµ,n−k(X) naturally descend to Hk and Hn−k-modules.

139 Proof. Let X be an irreducible Harish-Chandra module. Casselman’s theorem shows ν that X is a submodule of some principal series module, let Xδ be such a principal series modules containing X as a submodule. Note that this principal series module ν may not be unique. Then since Fµ,k(X) is exact and X is a submodule of Xδ then ν Fµ,k(X) is a submodule of Fµ,k(Xδ ) which is a Hk module. Therefore Fmu,k(X) is a Hk module. Similarly for µ and n − k.

Therefore for every Harish Chandra module of O(n + 1, n) and Sp2n(R) we can deﬁne two corresponding Hecke algebra modules.

3.8 Principal series modules map to principal series modules

ν In this section we take a closer look at the H(cµ)-modules obtained from Xδ under the functors Fµ,k and Fµ,n−k. We fully classify these as graded Hecke algebra principal series representations related to ν.

Recall that Hk(c), deﬁned in 3.4.1 is the graded Hecke algebra associated to W (Bk) with parameter function c : ∆ → C such that

ci−i+1 = 1 and c2i = 2c.

The algebra Hk(c) contains the group algebra, C[W (Bk)], of the hyperoctahedral group. Recall the labeling of vectors in X ⊗V ⊗k; we label the tensor product starting at ⊗k zero. A general elementary tensor in X ⊗ V would be written x0 ⊗ v1 ⊗ v2 ⊗ ... ⊗ vk.

We begin by restricting to the action of the Weyl group W (Bk) inside H(c) and computing the resulting C[W (Bk)]-modules isomorphism class. Fix a M-character δ and recall the K-characters µ and µ depending on δ from Table 3.6.1.

Lemma 3.8.1. As a C[W (Bk)]-module

ν ∼ Fµ,k(Xδ ) = C[W (Bk)], and as a C[W (Bn−k)]-module

ν ∼ Fµ,n−k(Xδ ) = C[W (Bn−k)].

140 ν Proof. From Lemma 3.6.2 we have an explicit basis of Fµ,k(Xδ );

⊗k X 1ν n1 nk HomK (µ, Xδk∗ ⊗ V ) = span{ δ ⊗ fw(1) ⊗ ...fw(k)}. w∈Sk

The symmetric group C[Sk] ⊂ C[W (Bk)] acts by permuting the tensor product. The 0 reﬂections in C[W (Bk)] related to 2i act by id ⊗ ... ⊗ θi ⊗ .. ⊗ id. They take fi to fi on the ith factor of the tensor product. 1ν ν Take the vector δ ⊗f1 ⊗...⊗fk, the C[W (Bk)] submodule of Ftriv,k(Xδk ) generated by 1δ ⊗ f1 ⊗ ... ⊗ fk is the subspace spanned by

1ν n1 nk { δ ⊗ fw(1) ⊗ ...fw(k) : w ∈ C[Sk]},

1ν The only group element of C[W (Bk)] that ﬁxes δ ⊗ f1 ⊗ ... ⊗ fk is the identity, k hence this module has dimension equal to k!2 , the dimension of C[W (Bk)]. The ν dimension is equal to the dimension of Ftriv,k(Xδ ), therefore we have equality. An ν isomorphism between the C[W (Bk)]-module C[W (Bk)] and Ftriv,k(Xδk ) can be deﬁned 1ν by sending the identity element e ∈ C[W (Bk)] to δ ⊗ f1 ⊗ .. ⊗ fk. ν The decomposition of Fµ,n−k(Xδ ) follows in exactly the same way, sending e ∈ 1ν C[W (Bn−k)] to δ ⊗ fk+1 ⊗ ... ⊗ fn.

ν We have a description of Fµ,k(Xδk ) as a C[W (Bk)]-module. We would like to describe it as an H(cµ)-module. The algebra H(cµ) is generated by C[W (Bk)] and the aﬃne operators 1, ..., k. Our calculation reduces to calculating the action of the ν ⊗k aﬃne operators i. The operators i ∈ S(ak) act on Xδ ⊗ V by X Ωji + rµ. 0

We deﬁne principal series representations for Hk(c). Then we show that the ν image of Xδ is isomorphic to a principal series representation deﬁned by a particular character.

The subspace ak ⊂ a deﬁned in Example 3.2.5 is a dimension k subspace of a.

Deﬁnition 3.8.2. [37] Let λ be a character for S(ak) ⊂ Hk(cµ), we deﬁne a principal series representation X(λ) for Hk(cµ):

X(λ) = IndHk(c) λ. S(ak)

141 1 We write λ for a fixed vector in the image of the character λ : S(a) → C. The symmetric algebra S(ak) is generated by the affine operators 1, ..., k. The principal 1 series representation can be described as a representation generated by, λ, a C[W (Bk)]- cyclic vector on which i acts by the scalar λ(i). We prove that the C[W (Bk)]-module, ν Fµ,k(Xδk ) is as a Hk(cµ)-module isomorphic to a principal series module for the correct character λ. k+1 We fix a specific basis for sp2n and so(p, q). Since the operators Ωij ∈ U(g) are defined in terms of, although independent of, a basis for g. This basis allows us to explicitly calculate Ω0j. It should be emphasized that the following basis is a decomposition of g into reduced root spaces under the adjoint action of a. Recall that a ⊂ sp2n(R) is 0 B : B is diagonal . B 0

Deﬁnition 3.8.3. Recall the decomposition of the Lie algebra g0 as

+ − g0 = n0 ⊕ a0 ⊕ n0 ,

+ where a0 is the maximal abelian subalgebra of p0 and n0 is the span of the positive root spaces with respect to the restricted root decomposition. Let Bn+ ,Bn− ,Ba be ﬁxed + − bases for n0 ,n0 and a0. The restricted roots Σ are ±i ± j,±j. We will denote a + vector in the positive root space λ ∈ Σ by nλ and the negative root space will be nˆλ. + − For example ni−j for i < j is in n . And nˆi−j ∈ n0 . We will scale nˆλ such that

nˆλ = n−λ = θ(nλ).

Hence nλ +n ˆλ is θ-invariant and hence in k.

Deﬁnition 3.8.4. For 1 ≤ s, t ≤ n, the matrix Es,t is the matrix with a 1 in the s, t position and zero elsewhere. Let i < j. Set

ni−j = Ei,j + Ei,n+j − Ej,i + Ej,n+i + En+i,j + En+i,n+j + En+j,i − En+j,n+i,

nˆi−j = −Ei,j + Ei,n+j + Ej,i + Ej,n+i + En+i,j − En+i,n+j + En+j,i + En+j,n+i,

ni+j = −Ei,j + Ei,n+j − Ej,i + Ej,n+i − En+i,j + En+i,n+j − En+j,i + En+j,n+i,

nˆi+j = Ei,j + Ei,n+j + Ej,i + Ej,n+i − En+i,j − En+i,n+j − En+j,i − En+j,n+i,

ni = Ei,i − Ei,n+i + En+i,i − En+i,n+i,

nˆi = −Ei,i − Ei,n+i + En+i,i + En+i,n+i,

ai = Ei,n+1 + En+i,i. + − These vectors give a reduced root space decomposition for sp2n(R) = n0 ⊕ a0 ⊕ n0 + − where ai ∈ a0, n ∈ n0 and nˆ ∈ n0 .

142 + Example 3.8.5. Let g = sp4. We give the basis given in Deﬁnition 3.8.4 for n ,

 0 1 0 1 −1 0 1 0 n1−2 =   ,  0 1 0 1 1 0 −1 0

 0 −1 0 1 −1 0 1 0 n1+2 =   ,  0 −1 0 1 −1 0 1 0

1 0 −1 0 0 0 0 0 n1 =   , 1 0 −1 0 0 0 0 0

0 0 0 0  0 1 0 1  n2 =   . 0 0 0 0  0 −1 0 −1

Deﬁnition 3.8.6. Let g0 = so(p, q) we follow [36, VI, pg. 371 Example so(p, q)].

ni−j = Ep−j+1,p−i+1 + Ep−j+1,p+i − Ep−i+1,p−j+1 + Ep−i+1,p+j −Ep+i,p−j+1 − Ep+i,p+j − Ep+j,p−i+1 + Ep+j,p+i,

ni+j = Ep−j+1,p−i+1 − Ep−j+1,p+i − Ep−i+1,p−j+1 + Ep−i+1,p+j −Ep+i,p−j+1 + Ep+i,p+j + Ep+j,p−i+1 − Ep+j,p+i,

nˆi−j = Ep−j+1,p−i+1 − Ep−j+1,p+i − Ep−i+1,p−j+1 − Ep−i+1,p+j +Ep+i,p−j+1 − Ep+i,p+j + Ep+j,p−i+1 + Ep+j,p+i,

ni+j = Ep−j+1,p−i+1 + Ep−j+1,p+i − Ep−i+1,p−j+1 − Ep−i+1,p+j +Ep+i,p−j+1 + Ep+i,p+j − Ep+j,p−i+1 − Ep+j,p+i.

The root space for i is p − q dimensional. Let l = 1, ..., p − q then

nl = E − E − E − E . i l,p−i+1 l,p+i p−i+1,l p+i,l

Finally

ai = Ep−i+1,p+i + Ep+i,p−i+1.

+ Example 3.8.7. Let g0 = so(3, 2). We give the basis given in Deﬁnition 3.8.6 for n0 ,

143 0 0 0 0 0  0 0 1 1 0    n1−2 = 0 −1 0 0 1  ,   0 −1 0 0 −1 0 0 −1 1 0

0 0 0 0 0 0 0 1 −1 0   n1+2 = 0 −1 0 0 1 ,   0 −1 0 0 1 0 0 1 −1 0

 0 0 1 −1 0  0 0 0 0 0   n1 = −1 0 0 0 0 ,   −1 0 0 0 0 0 0 0 0 0

 0 1 0 0 −1 −1 0 0 0 0    n2 =  0 0 0 0 0  .    0 0 0 0 0  −1 0 0 0 0

0 2n Lemma 3.8.8. For G = Sp2n, recall the basis fi = ei +en+i, fi = ei −en+i of V = C . 0 For G = O(p, q) we recall that fi = ep−i+1 + ep+i, fi = ep−i+1 − ep+i. Then by left multiplication of the given matrix in Deﬁnitions 3.8.4 and 3.8.6 we can calculate the following actions on fi:

ni+j (fk) = 0 for all k,

( 0 0 0 0 2fj if fk = fj, n + (f ) = i j k 0 otherwise.

ni (fk) = 0 for all k, ( 0 0 0 2fk if fk = fi , n (f ) = i k 0 otherwise, ( 2fi if fk = fj, n − (fk) = i j 0 otherwise,

( 0 0 0 2fj if fk = fi, n − (f ) = i j k 0 otherwise,

144  f if f = f ,  i k j (ni−j +n ˆi−j )(fk) = −fj if fk = fi, 0 otherwise.

Proof. This follows from left multiplication of the elements of sp2n and so(p, q) on the 0 deﬁning module V with elements fi and fi in the basis of V .

To prove that the C[W (Bk)]-module is in fact isomorphic to a principal series Hk(cµ)-module we need to ﬁnd a C[W (Bk)] cyclic vector such that the i act by scalars 1ν on this cyclic vector. The cyclic vector is δ ⊗ f1 ⊗ ... ⊗ fk.

1ν Lemma 3.8.9. On the vector δ ⊗ f1 ⊗ ... ⊗ fk the operator Ω0l acts by X X ν(l) − (stl + id) − id. tl

P ∗ Proof. Recall that Ω0l is defined to be b∈B(b)0 ⊗ (b )l for a given basis B of g0. We choose to use the fixed basis defined in Definition 3.8.3. The subspace a is the Lie algebra of the subgroup A ⊂ G. The basis of a defined in 3.8.4 and 3.8.6 is such that ai (fj) = δijfj. Furthermore ai acts on the cyclic vector 1ν ν δ of Xδ by ν(xi). Therefore the contribution from a ⊂ g is

(ai )0 ⊗ (ai )l = δilν(xil ).

ν ν The module Xδ is induced from the character δ ⊗ e ⊗ 1 of MAN, which is the + trivial character on N. The space n0 is the Lie algebra of N. The diﬀerential of the + 1ν + trivial character to n0 is zero. Therefore (n)0 acts by zero on δ for all n ∈ n0 . Hence the contribution from n+ is:

∗ + (n)0 ⊗ (n)l = 0, for n ∈ n .

+ 1ν + Since n ∈ n0 annihilates δ then (n)0 ⊗ (b)l = 0 for any b ∈ g2n, n ∈ n , a fact we ∗ 1 will use later in this proof. The operator (nˆi+j )l is equal to 2 (ni+j )l which is zero on any fk hence; ∗ (ˆni+l )0 ⊗ (ˆni+j )l = 0.

Similarly ni is zero on any fk therefore;

(ˆn ) ⊗ (ˆn∗ ) = 0. i 0 i l

145 The only remaining basis elements to consider are those of the form ni−j from − ν n0 ⊂ g0. We utilise the trick that as a K-module Fµ,k(Xδk ) is just the µ isotypic ν ⊗k component of Xδk ⊗ V . The contribution fromn ˆi−j is:

(ˆn ) ⊗ (ˆn∗ ) . i−j 0 i−j l We can add the operator (n ) ⊗ (nˆ∗ ) which since n ∈ n+, by above, acts i−j 0 i−j l i−j by zero. Therefore we are not modifying the original operator,

1 (ˆn ) ⊗ (ˆn∗ ) = (ˆn + n ) ⊗ (n ) . i−j 0 i−j l 2 i−j i−j 0 i−j l

The vector nî−j + ni−j is θ-invariant, hence is in k. Recall that for k ∈ k acting ⊗k Pk on the tensor X ⊗ V that k = i=0(k)i. Since we are working with the µ-isotypic space, we replace nî−j + ni−j ∈ k by µ(nî−j + ni−j ) and subtract the difference to find,

(ˆn ) ⊗ (ˆn∗ ) i−j 0 i−j l 1 1 X = µ(ˆn − n ) ⊗ (n ) − (ˆn − n ) ⊗ (n ) . 2 i−j i−j i−j l 2 i−j i−j m i−j l m>0 The character µ (or µ) diﬀerentiated to a is zero (or the trace character) hence

µ(nˆi−j + ni−j ) = 0. Lemma 3.8.8 gives the explicit action of ni−j on fk, using this one can determine the action;  −s − id if f = f and f = f ,  tl it i il j (ˆn ) ⊗ (ˆn∗ ) = i−j 0 i−j l −id if fil = fi, 0 otherwise.

The only non-zero terms are contributed by al , andn ˆi−l andn ˆl−i . Which act, on the cyclic vector, by ν(l), −stl − id and −id respectively. Summing these up gives,

X X Ω0l = ν(l) − (stl + id) − id, tl 1ν on the C[W (Bk)] cyclic vector δ ⊗ f1 ⊗ ... ⊗ fk.

ν The equivalent statement for Fµ,n−k(Xδ ) is below. It follows from the proof of Lemma 3.8.9.

146 1ν Lemma 3.8.10. On the vector δ ⊗ fk+1 ⊗ ... ⊗ fn the operator Ω0l acts by X X ν(k+l) − (sk+t,k+l + id) − id, tl for l = 1, ..., n − k.

P Corollary 3.8.11. The operator l = i

Deﬁnition 3.8.13. Example 3.2.5 deﬁnes subspaces ak and a¯n−k of a such that

a = ak ⊕ a¯n−k.

Let ν be a character of a. Deﬁne νk to be the restricted character

ν|ak

and ν¯n−k to be ν|¯an−k .

ν ν For a principal series module Xδ we have shown that as a W (Bk)-module Fµ,k(Xδ ) is isomorphic to C[W (Bk)] and as a Hecke algebra module it is a principal series module induced from a character of S(V ) ⊂ Hk(cµ).

ν Theorem 3.8.14. For G = Sp2n(R) or O(p, q) p + q = 2n + 1, the module Fµ,k(Xδ ) is isomorphic to the Hk(cµ) principal series module

Hk(cµ) X(νk) = Ind νk. S(ak)

ν The module Fµ,n−k(Xδ ) is isomorphic to the Hn−k(cµ) principal series module

Hk(cµ) X(¯νn−k) = Ind ν¯n−k. S(¯an−k)

For spherical principal series representations, this recovers the results of [16, Theorem 3.0.4].

147 1ν Proof. One deﬁnes an isomorphism by taking the given cyclic vector δ ⊗f1 ⊗...⊗fk ∈ ν 1 Ftriv,k(Xδk ) to the cyclic vector νk of X(νk). Both vectors are C[W (Bk)] cyclic. By

Corollary 3.8.11 the aﬃne operators i act on both vectors by νk(i), therefore this is a well-deﬁned isomorphism. ν Lemma 3.6.2 gives a basis of Fdet,n−k(Xδk ):

1ν n1 nn−k { δ ⊗ fw(1)+k ⊗ ... ⊗ fw(n−k)+k : w ∈ Sn−k}.

ν For Fµ,k(Xδ ) and X(ν¯n−k), both modules are C[W (Bn−k)] cyclic and Corollary 3.8.12 shows that the aﬃne operators i for i = 1, ..., n − k, act by the same scalar on on the 1ν 1 cyclic vector δ ⊗ fk+1 ⊗ ... ⊗ fn and ν¯n−k , respectively.

Casselman’s theorem [12] states that every irreducible representation in HC(G) is a subrepresentation of a principal series module. If G is a split real orthogonal or symplectic group then M is abelian and every principal series module is induced from a character.

Fµ,k(X) and Fµ,n−k(X) are submodules of the Hk(cµ) and Hn−k(cµ)-modules

X(νk) and X(¯νn−k).

3.9 Hermitian forms

θ In this section we deﬁne two star operations on Bk. Through the quotients deﬁned in Lemma 3.4.4 these star operations descend to the usual star operations on the graded

Hecke algebras Hk(c)[5]. We then show that a Harish-Chandra module X ∈ HC(G) θ with invariant Hermitian form gets mapped, by Fµ,k, to a Bk-module with invariant Hermitian form. This extends the results in [16] to any Harish-Chandra module. θ Furthermore, if X is a unitary module, then it maps to a unitary module for Bk. In this section we assume that µ is a character of K.

148 Definition 3.9.1. Let G be O(p, q) p+q = 2n+1 or Sp2n(R), let g0 be its Lie algebra, with complexification g = k ⊕ p. Conjugation¯: g → g is defined by the real form g0. Define the star operation as the conjugate linear map ∗ : g → g such that:

g∗ = −g¯ for all g ∈ g.

Deﬁne the operation • : g → g by:

p• =p ¯ for all p ∈ p.

k• = −k¯ for all k ∈ k.

Recall Deﬁnition 3.4.1 of the Hecke algebra Hk(c). We deﬁne the Drinfeld presentation of Hk(c).

Deﬁnition 3.9.2. Let R be a root system with pairing h, i : V × V → C, simple roots δ, and a parameter function c : ∆ → C. Denote the Weyl group of R by W (R). The Drinfeld Hecke algebra HR(c) is a quotient of the algebra

C[W (R)] o T (V ), by the relations wαw˜ −1 = w](α) for all w ∈ W (R), α ∈ V, ˜ X ˜ ˜ [˜α, β] = c(γ)c(δ)(hα,˜ γihβ, δi − hβ, γihα,˜ δi)sγsδ. γ,δ∈∆ Lemma 3.9.3. The Drinfeld Hecke algebra and the graded Hecke algebra are deﬁned by a root system and a parameter on simple roots. If the deﬁning root systems and parameters are equal then these algebras are isomorphic.

Proof. One deﬁnes an isomorphism φ :∈ HR(c) to HR(c) by 1 X φ(α − c(γ)hγ, αis ) =α, ˜ 2 γ γ∈∆ φ(w) = w, ∀w ∈ W (R).

Given that the graded Hecke algebra and the Drinfeld Hecke algebra are isomorphic we omit the diﬀerent notation and denote the graded Hecke algebra by HR(c). We uniformly denote a generator in the Drinfeld presentation by α˜ and α denotes a generator in the Lusztig presentation (Deﬁnition 3.4.1).

149 Deﬁnition 3.9.4. Let ∗ : Hk(c) → Hk(c) be the antihomomorphism such that:

α˜∗ = −α˜ for all α ∈ ˜t,

∗ −1 g = g for all g ∈ W (Bk).

• Let : Hk(c) → Hk(c) be the antihomomorphism such that:

α• = α for all α ∈ t (equivalently α˜• = α˜),

• −1 g = g for all g ∈ W (Bk). Here v is the complex conjugate of v.

2 Let w0 be the longest element in W (Bk), it is an involution and is generated by k simple reﬂections. It is in the centre of W (Bk). On the space of roots w0 acts by −1.

Lemma 3.9.5. The longest element w0 can be written as

w0 = θ1θ2...θk.

∗ It is well known that the longest element w0 relates the two star operations and • in Hk(c).

Lemma 3.9.6. ∗ • h = w0h w0 for all h ∈ Hk(c).

Lemma 3.9.7. The longest element w0 is central in the ﬁnite Brauer algebra Brk[m].

Proof. The element w0 is central in W (Bk), therefore it is suﬃcient to prove that w0 commutes with the idempotents ei,i+1. The reﬂections θl commute with ei,j

[ei,j, θl] = 0 for all i, j, l.

We have,

w0ei,jw0 = θ1...θkei,jθk..θ1 = ei,j.

Hence w0 is central in the ﬁnite Brauer algebra.

⊗k Since w0 = θ1θ2...θk then as an operator on X ⊗ V

π(w0) = (ξ)1(ξ)2...(ξ)k = id ⊗ ξ ⊗ ξ... ⊗ ξ.

k p We calculate how w0 and Ωij, Ωij, Ωij interact.

150 Lemma 3.9.8. As operators on X ⊗ V ⊗k,

k k w0(Ωij)w0 = Ωij for all 0 ≤ i < j ≤ n,

( p p Ωij for all 0 < i < j ≤ n, w0(Ωij)w0 = p −Ω0j when i = 0. ( g if g ∈ k, Proof. Recall ξgξ = Therefore one ﬁnds that π(w0) = id ⊗ ξ ⊗ ...ξ −g if g ∈ p. k P ∗ p commutes with Ωij = b∈B∩k(b)i ⊗ (b )j. For Ωij we have:

p p w0(Ωij)w0 = (id ⊗ ξ ⊗ ξ ⊗ ... ⊗ ξ)Ωij (id ⊗ ξ ⊗ ... ⊗ ξ) , P ∗ = (id ⊗ ξ ⊗ ξ ⊗ ... ⊗ ξ) b∈B∩p(b)i ⊗ (b )j (id ⊗ ξ ⊗ ... ⊗ ξ) , (P b∈B∩p(b)i ⊗ (ξbξ)j if i = 0, = P b∈B∩p(ξbξ)i ⊗ (ξbξ)j if i 6= 0, (P b∈B∩p(b)i ⊗ (−b)j if i = 0, = P b∈B∩p(−b)i ⊗ (−b)j if i 6= 0, ( P − b∈B∩p(b)i ⊗ (b)j if i = 0, = P b∈B∩p(b)i ⊗ (b)j if i 6= 0.

• θ θ Deﬁnition 3.9.9. Let : Bk → Bk be the conjugate linear antihomomorphism deﬁned on the generators as follows: • zi = zi

• −1 g = g for g ∈ W (Bk)

• ei,i+1 = ei,i+1.

Remark 3.9.10. To check this antihomomorphism is well defined one must just check that the relations in Definition 3.3.10 are fixed.

∗ θ θ Deﬁnition 3.9.11. Let : Bk → Bk by the antihomomorphism such that,

∗ • b = w0b w0.

∗ −1 Remark 3.9.12. Since w0 is central in the ﬁnite Brauer algebra then g = g for ∗ g ∈ W (Bk) and ei,j = ei,j.

151 Lemma 3.9.13. Under the quotients in Lemma 3.4.4 the antihomomorphisms ∗ : θ θ • θ θ ∗ Bk → Bk and : Bk → Bk descend to the antihomomorphisms : Hk(c) → Hk(c) • and : Hk(c) → Hk(c) respectively.

• Proof. The operation ﬁxes ei,i+1 and

θkzk + zkθk = 2c − 2rθk.

• θ Therefore on Bk descends to Hk(c). On the generators of Hk(c) it ﬁxes the aﬃne generators and is the inverse antihomomorphism on the group W (Bk). Hence the • θ • operation on Bk descends to the antihomomorphism on Hk(c). Since

∗ • h = w0h w0,

θ ∗ θ ∗ in both Bk and Hk(c) then the star operation on Bk descends to on Hk(c).

θ We give a new set of generators for Bk.

Deﬁnition 3.9.14. Deﬁne z − w z w z˜ = i 0 i 0 , for i = 1, ..., k, i 2 then θ ∼ Bk = hz˜i, sj,j+1, ej,j+1, θii.

θ The operators z˜i form a Drinfeld type presentation for Bk, they descend to the Drinfeld presentation of Hk(c) under the quotients deﬁned in 3.4.4. As operators on X ⊗ V ⊗k:

1 π(˜zi) = 2 π(zi − w0ziw0) 1 P P = 2 j

θ ∗ Remark 3.9.15. With this presentation of Bk the operation is deﬁned as

∗ z˜i = −z˜i, ∗ −1 g = g for all g ∈ w(Bk), ∗ ei,i+1 = ei,i+1.

152 Deﬁnition 3.9.16. Let X be a complex vector space, a Hermitian form h, iX on X is a map h, iX : X × X → C such that

0 0 0 0 0 hλ1x1 + λ2x2, x iX = λ1hx1, x iX + λ2hx2, xiX for all x1, x2, x ∈ X, λ1, λ2 ∈ C,

0 0 ¯ 0 ¯ 0 0 0 hx, λ1x1 + λ2x2iX = λ1hx, x1iX + λ2hx, x2iX for all x1, x2, x ∈ X, λ1, λ2 ∈ C.

Deﬁnition 3.9.17. Let X be a HC(G)-module. A Hermitian form h, iX is ∗-invariant if:

∗ hg(x1), x2iX = hx1, g (x2)i, for all x1, x2 ∈ X and g ∈ g.

Deﬁnition 3.9.18. Let U be an Hk(c)-module. A Hermitian form h, iU on U is invariant with respect to ∗ if:

∗ hh(x1), x2iX = hx1, h (x2)i, for all x1, x2 ∈ U and h ∈ Hk(c).

θ ∗ Similarly for U a Bk-module, a Hermitian form h, iU on U is -invariant if

∗ θ hb(x1), x2iX = hx1, b (x2)i, for all x1, x2 ∈ U and b ∈ Bk.

Deﬁnition 3.9.19. A HC(G)-module X is unitary if there exists a positive deﬁnite invariant Hermitian form on X.

Similarly, an Hk(c)-module U is unitary if U has an invariant positive deﬁnite θ Hermitian form and a Bk-module is unitary if it has a positive deﬁnite invariant Hermitian form.

Recall V is the deﬁning matrix module of G. Let h, iV be a non-degenerate Hermitian form on V such that

−1 hkv1, v2i = hv1, k v2i for all v1, v2 ∈ V, k ∈ K,

hpv1, v2i = hv1, pv2i for all v1, v2 ∈ V, p ∈ p.

This makes V unitary with respect to •.

Deﬁnition 3.9.20. c.f. [16, (4,.4)] Let X be in HC(G) with a ∗-invariant Hermitian ⊗k form h, iX then we endow X ⊗ V with a Hermitian form deﬁned on elementary tensors by

0 0 0 0 0 0 hx0 ⊗ v1 ⊗ ... ⊗ vk, x0 ⊗ v1 ⊗ ... ⊗ vkiX⊗V ⊗k = hx0, x0iX hv1, v1iV ...hvk, vkiV ,

153 then extended to a Hermitian form. For µ a character of K, deﬁne a Hermitian form ⊗k on Fµ,k(X) = HomK (µ, X ⊗ V ) by:

⊗k ⊗k hφ, ψiFµ,k = hφ(1), ψ(1)iX⊗V , for all φ, ψ ∈ HomK (µ, X ⊗ V ).

k Remark 3.9.21. If X is a unitary space then h, iX⊗V ⊗k endows X ⊗ V as a unitary space.

Lemma 3.9.22. Let V be the complex matrix module of G = O(p, q) or Sp2n(R) and pr12 be the projection of V ⊗ V onto its trivial G submodule. Deﬁne h, iV ⊗V on V ⊗ V by 0 0 0 0 hv1 ⊗ v2, v1 ⊗ v2iV ⊗V = hv1, v1iV hv2, v2iV , and extend to a Hermitian form. Then

0 0 0 0 hpr12(v1 ⊗ v2), v1 ⊗ v2iV ⊗V = hv1 ⊗ v2, pr12(v1 ⊗ v2)iV ⊗V .

Proof. It is suﬃcient to prove that the trivial submodule in V ⊗ V and its complement are orthogonal with the form h, iV ⊗V . The Peter-Weyl Theorem [48, Theorem 1.12] states that a unitary module of a compact group decomposes as an orthogonal direct sum of irreducibles. Considering V ⊗ V as a • unitary K-module , we have that the trivial submodule of V ⊗ V is orthogonal to its complement with respect to h, iV .

∗ Lemma 3.9.23. Suppose X ∈ HC(G) with a -invariant Hermitian form h, iX then θ ∗ Fµ,k(X) ∈ Bk-mod has a -invariant Hermitian form h, iFµ,k . Furthermore if X is unitary then Fµ,k(X) is unitary.

Ciubotaru and Trapa [16] prove this result for spherical principal series modules mapping to graded Hecke algebras. We extend this to any Harish-Chandra module which requires considering the image as a Type B/C aﬃne Brauer algebra module.

Proof. We show that the Hermitian form is invariant under the generators z˜i, si,i+1, ∗ p ∗ θj and ei,i+1. For z˜i, z˜i = −z˜i and π(z˜i) = Ω0i. The form h, iX is a -invariant form • ⊗k on X and h, iV is a -invariant form on V . Let φ, ψ ∈ Fµ,k(X) = HomK (µ, X ⊗ V ), then

hπz˜i(φ), ψiF µ,k(X) = hπ(˜zi)(φ(1)), ψ(1)iX⊗V ⊗k .

154 p P ∗ Since π(˜zi) = Ω0i = b∈B∩p(b)0 ⊗ (b )i,

∗ p ∗ hπz˜i (φ), ψiF µ,k(X) = h(Ω0i) φ(1), ψ(1)iX⊗V ⊗k ,

p = −hΩ0iφ(1), ψ(1)iX⊗V ⊗k , * ! + X = − (b)0 ⊗ (b)i φ(1), ψ(1) . b∈B∩p X⊗V ⊗k P P 0 0 0 Denote φ(1) by x0 ⊗ v1 ⊗ ... ⊗ vk and ψ(1) by x0 ⊗ v1 ⊗ ... ⊗ vk substituting in the deﬁnition of h, iX⊗V ⊗k then

X X 0 0 0 0 hπz˜i(φ), ψiF µ,k(X) = h−(b)0(x0), x0iX hv1, v1iV ...h(b)ivi, viiV ...hvk, vkiV . b∈B∩p

∗ • 0 The form h, iX is -invariant for g and h, iV is -invariant for g, hence h−bx0, x0iX = 0 0 0 hx0, bx0iX and hbvi, viiV = hvi, bviiX for all b ∈ p:

X X 0 0 0 0 hπz˜i(φ), ψiF µ,k(X) = h(x0), (b)0x0iX hv1, v1iV ...hvi, (b)iviiV ...hvk, vkiV . b∈B∩p Reversing through the deﬁnitions, we show

∗ hπ(˜zi) φ, ψiFµ,k = hφ, π(˜zi)ψiFµ,k .

0 0 The element θj acts by (ξ)j where ξ ∈ k, hence hξv, v iV = hv, ξv iV . Therefore

0 0 0 0 0 0 0 0 hx0, x0iX hv1, v1iV ...h(ξ)vj, vjiV ...hvk, vkiV = hx0, x0iX hv1, v1iV ...hvj, (ξ)vjiV ...hvk, vkiV .

Similarly for si,i+1

0 0 0 hsi,i+1(x0 ⊗ v1 ⊗ ...vk), x0 ⊗ v1 ⊗ ... ⊗ vkiX⊗V ⊗k 0 0 0 = hx0 ⊗ v1 ⊗ ... ⊗ vi+1 ⊗ vi ⊗ ... ⊗ vk, x0 ⊗ v1 ⊗ ... ⊗ vkiX⊗V ⊗k 0 0 0 0 = hx0 ⊗ v1 ⊗ ... ⊗ vk, x0 ⊗ v1 ⊗ ... ⊗ vi+1 ⊗ vi ⊗ ... ⊗ vkiX⊗V ⊗k 0 0 0 = hx0 ⊗ v1 ⊗ ...vk, si,i+1(x0 ⊗ v1 ⊗ ... ⊗ vk)iX⊗V ⊗k .

The projection ei,i+1 acts on elementary tensors x0 ⊗ v1 ⊗ ... ⊗ vk by

ei,i+1(x0 ⊗ v1 ⊗ ... ⊗ vk) = mx0 ⊗ v1 ⊗ ... ⊗ vi−1 ⊗ pr12(vi ⊗ vi+1) ⊗ .., ⊗vk.

Then

0 0 0 hei,i+1(x0 ⊗ v1 ⊗ ... ⊗ vk), x0 ⊗ v1 ⊗ ... ⊗ vkiX⊗V ⊗k , 0 0 0 = mhx0 ⊗ v1 ⊗ ... ⊗ vi−1 ⊗ pr12(vi ⊗ vi+1) ⊗ ... ⊗ vk, x0 ⊗ v1 ⊗ ... ⊗ vkiX⊗V ⊗k , 0 0 0 0 0 = hx0, x0iX hv1, v1iV ...hpr12(vi ⊗ vi+1), vi ⊗ vi+1iV ⊗V ...hvk, vkiV .

155 Using Lemma 3.9.22 ,

0 0 0 0 0 0 = hx0 ⊗ v1 ⊗ ... ⊗ vk, x0 ⊗ v1 ⊗ ... ⊗ vi−1 ⊗ pr12(vi ⊗ vi+1) ⊗ .., ⊗vk, iX⊗V ⊗k ,

Therefore

hei,i+1(φ), ψiFµ,k = hφ, ei,i+1(ψ)iFµ,k .

∗ The module Fµ,k(X) has induced Hermitian form h, iFµ,k which is -invariant on the θ ∗ ⊗k generators of Bk. Hence h, iFµ,k is a -invariant form. If X is unitary then h, iX⊗V is positive deﬁnite. Hence h, iFµ,k is a positive deﬁnite invariant Hermitian form and

Fµ,k(X) is unitary.

θ Deﬁnition 3.9.24. Let X ∈ HC(G), Bk-mod or Hk(c)-mod module with invariant

Hermitian form h, iX . We deﬁne the Langlands quotient X to be:

X = X/ radh, iX , where radh, i is the radical of the form h, i.

Lemma 3.9.25. Let X be in HC(G)-mod with Hermitian invariant form h, iX and

Langlands quotient X. The form h, iFµ,k is the endowed hermitian form of Fµ,k(X) from Deﬁnition 3.9.20 then

Fµ,k(X) = Fµ,k(X) = Fµ,k(X)/ radh, iFµ,k .

Proof. One can construct an exact sequence:

0 −→ radh, iX −→ X −→ X −→ 0.

Exactness of the functors Fµ,k is given by Lemma 3.5.3. Hence there is an exact sequence:

0 −→ Fµ,k(radh, iX ) −→ Fµ,k(X) −→ Fµ,k(X) −→ 0.

For the result it is suﬃcient to prove that Fµ,k(radh, iX ) = radh, iFµ,k . Since h, iFµ,k is an invariant form on Fµ,k(X) and a non-degenerate form on Fµ,k(X) then Fµ,k radh, iX = radh, iFµ,k .

156 ν Theorem 3.9.26. Let Xδ be a principal series module for G = O(p, q) or Sp2n(R). ν ν The Langlands quotient X = X / radh, i ν is mapped by F , to the Langlands δ δ Xδ µ,k quotient of the (c )-module, X(ν ) = X(ν )/ radh, i . Similarly, Xν is mapped Hk µ k k Xνk δ by Fµ,n−k, to the Hn−k(cµ)-module X(¯νn−k).

Deﬁnition 3.9.27. We deﬁne subsets of a∗:

∗ ν Uδ = {ν ∈ a : Xδ is a unitary Harish-Chandra module}.

Similarly deﬁne

1 ∗ Uk( ) = {λ ∈ ak : Xλ is a unitary Hk(cµ) module} and 1 ¯ ∗ ¯ Un−k( ) = {λ ∈ a¯n−k : X(λ) is a unitary Hn−k(cµ) module}.

¯ ∗ ∗ Since a = ak ⊕ a¯n−k we can associate a pair (λk, λn−k) ∈ a × a¯n−k to a character of a via: ¯ (λk, λn−k): a → C ¯ (λk, λn−k)(ak) = λk(ak), ¯ ¯ (λk, λn−k)(a¯n−k) = λn−k(a¯n−k).

ν Theorem 3.9.26 shows that the Langlands quotients of Xδ map under Fµ,k and Fµ,n−k to Langlands quotients of principal series modules for Hk(cµ) and Hn−k(cµ) hence we can formulate the following non-unitary test.

Lemma 3.9.28. We have an inclusion of sets:

Uδ ⊆ Uk(1) × Un−k(1).

This inclusion of sets states that if we take a minimal principal series module X and ﬁnd that, under the functor Fµ,k, the Langlands quotient of the image is not unitary then the Langlands quotient of X is not unitary.

157 This provides a functorial proof of the results of Barbasch, Pantano, Paul and Salamanca-Riba [3, 46] on unitarity of principal series module using petite K-types. The above working does not require the image to be a Hecke algebra module. Therefore, we have also proved the following theorem.

Theorem 3.9.30. [Non-unitary test for Harish-Chandra modules] Let X be a Harish

Fµ,k(X), Fµ,n−k(X) is not unitary as a Hecke algebra module then X is not unitary as a G-module.

We ﬁnish with a toy example.

∼ Example 3.9.31. Let G = Sp2(R) = SL2(R). Then principal series modules are ν labelled by δ = ±1 and ν ∈ C. The principal series modules Xδ are unitary exactly when ν = ib for b ∈ R, that is ν is entirely imaginary. In this case all principal series modules are spherical principal series modules. The root system associated to Sp2 has one root and the Weyl group is Z2 which acts by −1 on . Here H(c) will be the graded Hecke algebra associated to type B1 with parameter c. The algebra H(c) is generated by and s ∈ Z2 such that s = −s + c. We note that s∗ = s and ∗ = − + cs. Our theorem gives that

F (Xν) ∼ IndH(c) 1 . triv,1 1 = C ν

Note that IndH(c) 1 is two dimensional with basis {1 , s1 } we will denote the C ν ν ν module IndH(c) 1 by Y . Let h, i be a hermitian form on Y , for Y to be unitary we C ν ν ν ν require hs(u), vi = hu, s∗(v)i = hu, s(v)i and h(u), vi = hu, ∗(v)i = hu, [− + cs](v)i.

Letting u = 1ν and v = 1ν, then the above requirement implies

νh1ν, 1νi = h(1ν), 1νi = h1ν, [− + cs]1νi = −ν¯h1ν, 1νi + h1ν, s1νi.

158 For the above equation to hold ν = −ν¯ and h1ν, s1νi = 0. Hence for Yν to be unitary ν must be purely imaginary. Furthermore if ν is purely imaginary then we can construct a Hermitian non-degenerate form on Yν such that it is a unitary form. Therefore in the case of Sp2(R) our non-unitary test becomes an equivalence:

Xν is unitary if and only if F (Xν) ∼ IndH(c) 1 is unitary. δ triv,1 δ = C ν

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