The Pennsylvania State UniversityThe Graduate School

Department of Mathematics

HECKE C*-ALGEBRAS

A Thesis inMathematics

byRachel W. Hall

c© 1999 Rachel W. Hall

Submitted in Partial Fulfillmentof the Requirements

for the Degree of

Doctor of Philosophy

December 1999

We approve the thesis of Rachel W. Hall.

Date of Signature

Nigel D. HigsonProfessor of MathematicsThesis AdviserChair of Committee

Svetlana KatokProfessor of MathematicsAssociate Chair for Graduate Studies, Mathematics

Edward W. FormanekProfessor of Mathematics

John A. YeazellAssistant Professor of Physics

Gary L. MullenProfessor of MathematicsHead of the Department of Mathematics

iii

Abstract

The purpose of this thesis is to introduce C∗-algebra techniques to the theory of Heckealgebras, especially to the representation theory of Hecke algebras. We will investigate when aHecke algebra may be completed to a Hecke C∗-algebra in a manner analogous to the completionof the group algebra to the group C∗-algebra, and we will explore the connections between theexistence of such a completion and the structure of the category of ∗-representations of the Heckealgebra.

The construction of the Hecke C∗-algebra first arose in Bost and Connes’ study of theHecke algebra H(P+

Z\P+Q/P+Z

) and the Riemann zeta function. We will focus our attentionon algebras quite closely related to their example. Our original motivation was to prove thatthe category of nondegenerate ∗-representations of the Hecke algebra H(S\G/S) is equivalentto the category of unitary representations of the group G generated by their S-fixed vectors,implying the existence of the Hecke C∗-algebra. However, the Hecke algebra formed from thegroup SL2(Qp) with the maximal compact subgroup SL2(Zp) shows that the Hecke C∗-algebradoes not exist in general. On the other hand, Borel proved an algebraic counterpart of thiscategory equivalence for the Hecke algebra H(B\G/B), where B is the Iwahori subgroup of asemi-simple group G over a non-Archimedean local field. In addition, Barbasch and Moy settledthe unitary case of the category equivalence by invoking the proof of the Deligne-Langlandsconjecture and examining all the representations of G. We will prove, in special cases, a similar,and in some ways stronger, result using C∗-algebra theory and the geometry of buildings. Wehave also considered cases relevant to Bost and Connes’ example.

iv

Table of Contents

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2. Hecke ∗-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Hecke Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.1.1 The Hecke pair (G,S), where S is normal in G . . . . . . . . 82.1.1.2 The Hecke pair (G,S), where S is finite . . . . . . . . . . . . 82.1.1.3 The Hecke pair (P+

Q, P+Z

) . . . . . . . . . . . . . . . . . . . . 92.1.2 Topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2.1 Example: The Hecke pair (SL2(Qp), SL2(Zp)) . . . . . . . . 102.2 The Algebra Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Functions on the double coset space . . . . . . . . . . . . . . . . . . . 13

2.2.2.1 The Convolution Product on C[S\G/S] . . . . . . . . . . . . 132.2.2.2 Involution on H(S\G/S) . . . . . . . . . . . . . . . . . . . . 15

2.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3.1 H(S\G/S) when S is normal . . . . . . . . . . . . . . . . . . 162.2.3.2 The Hecke algebra for the pair (SL2(Qp), SL2(Zp)) . . . . . 16

2.3 The Hecke C∗-algebra: a first look . . . . . . . . . . . . . . . . . . . . . . . . 182.3.1 The group C∗-algebra C∗(G) . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 The Hecke C∗-algebra C∗(S\G/S) . . . . . . . . . . . . . . . . . . . . 192.3.3 Example: (SL2(Qp), SL2(Zp)) . . . . . . . . . . . . . . . . . . . . . . 192.3.4 Hecke algebras with bounded representation . . . . . . . . . . . . . . . 192.3.5 Example: SL2(Qp) with an Iwahori subgroup . . . . . . . . . . . . . . 20

Chapter 3. Hecke Algebras and Representation Theory . . . . . . . . . . . . . . . . . . . 213.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Representations of groups and Hecke algebras . . . . . . . . . . . . . . . . . . 22

3.2.1 Representations of group algebras . . . . . . . . . . . . . . . . . . . . . 223.2.2 Representations of Hecke algebras . . . . . . . . . . . . . . . . . . . . 22

3.3 Representations of the Iwahori Hecke algebra . . . . . . . . . . . . . . . . . . 283.4 ∗-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4.1 A Hecke-algebra-valued bilinear form . . . . . . . . . . . . . . . . . . . 303.4.2 The product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.3 The Hecke pair (SL2(Qp), SL2(Zp)) . . . . . . . . . . . . . . . . . . . 33

3.5 The category equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Chapter 4. First Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 Lower directed Hecke pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Finite-dimensional Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . 40

v

Chapter 5. Trees, Buildings, and Hecke Algebras . . . . . . . . . . . . . . . . . . . . . . . 445.1 Coxeter groups and Coxeter complexes . . . . . . . . . . . . . . . . . . . . . . 44

5.1.1 Posets and simplicial complexes . . . . . . . . . . . . . . . . . . . . . . 445.1.2 Constructing Coxeter complexes . . . . . . . . . . . . . . . . . . . . . 44

5.2 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.1 Example: projective geometry . . . . . . . . . . . . . . . . . . . . . . . 485.2.2 Example: a tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.3 Consequences of the building axioms . . . . . . . . . . . . . . . . . . . 515.2.4 Buildings and Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 BN -pairs and buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.3.1 The Iwahori Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . 535.3.2 The link of a simplex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.4 Buildings associated to SLn(Qp) . . . . . . . . . . . . . . . . . . . . . . . . . 545.4.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.4.2 Affine buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Chapter 6. SL2(Qp), SL3(Qp), and Iwahori Hecke algebras . . . . . . . . . . . . . . . . 586.1 SL2(Qp) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.1.1 Properties of the product 〈〈, 〉〉 . . . . . . . . . . . . . . . . . . . . . . . 596.1.2 Positivity of 〈〈, 〉〉 for the Iwahori Hecke algebra . . . . . . . . . . . . . 60

6.2 SL3(Qp) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.3 The mysterious Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

vi

List of Tables

5.1 The association of simplicial complexes to posets of subspaces . . . . . . . . . . . 48

vii

List of Figures

5.1 A poset and its associated simplicial complex . . . . . . . . . . . . . . . . . . . . 455.2 Tessellation of the plane by equilateral triangles . . . . . . . . . . . . . . . . . . . 465.3 A folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4 The simplicial complex associated to the poset of subspaces of (Z/2Z)3 . . . . . 505.5 The fundamental apartment of ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.6 The 4-regular tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.7 The actions which form the generator X . . . . . . . . . . . . . . . . . . . . . . . 555.8 The result of applying X twice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.9 A plane tessellated by equilateral triangles . . . . . . . . . . . . . . . . . . . . . . 575.10 The link of a vertex in ∆3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.1 The fundamental apartment of ∆2 . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 The subtrees Γ and Γv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3 A tame extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646.4 The projection of a subcomplex formed by tame extensions . . . . . . . . . . . . 676.5 A nontame extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

viii

Acknowledgments

I wish to thank my adviser, Nigel Higson, for his guidance and support. In addition, Ithank the members of my committee—Ed Formanek, Svetlana Katok, and John Yeazell—fortheir helpful suggestions.

This thesis was formatted using a LATEX style file developed by Steve Simpson.

1

Chapter 1

Introduction

The purpose of this thesis is to introduce C∗-algebra techniques to the theory of Heckealgebras, especially to the representation theory of Hecke algebras. We will investigate when aHecke algebra may be completed to a Hecke C∗-algebra in a manner analogous to the completionof the group algebra to the group C∗-algebra, and we will explore the connections between theexistence of such a completion and the structure of the category of ∗-representations of the Heckealgebra.

The construction of the Hecke C∗-algebra first arose in Bost and Connes’ study [3] of theHecke algebra H(P+

Z\P+Q/P+Z

) and the Riemann zeta function. We will focus our attention onalgebras quite closely related to their example. Our original motivation was to prove that thecategory of nondegenerate ∗-representations of the Hecke algebra H(S\G/S) is equivalent to thecategory of unitary representations of the groupG generated by their S-fixed vectors, implying theexistence of the Hecke C∗-algebra. However, the Hecke algebra formed from the group SL2(Qp)with the maximal compact subgroup SL2(Zp) shows that the Hecke C∗-algebra does not exist ingeneral. On the other hand, Borel [2] proved an algebraic counterpart of this category equivalencefor the Hecke algebraH(B\G/B), where B is the Iwahori subgroup of a semi-simple groupG overa non-Archimedean local field. In addition, Barbasch and Moy [1] settled the unitary case of thecategory equivalence by invoking the proof of the Deligne-Langlands conjecture and examiningall the representations of G. We will prove, in special cases, a similar, and in some ways stronger,result using C∗-algebra theory and the geometry of buildings. We have also considered casesrelevant to Bost and Connes’ example.

A Hecke algebra is formed from a group with a subgroup which is called almost normal.Before we define the Hecke algebra, we will see how to construct a C∗-algebra from a group witha normal subgroup. Let G be a discrete group and S a subgroup of G. If S is normal in G, thenthe left coset space G/S is a group and we can form its group algebra C[G/S] with the product

f1 ∗ f2(g) =∑

γ∈G/Sf1(γ) f2(γ−1g)

where the notation “γ ∈ G/S” means “γ runs through a set of representatives for left cosets inG/S.” The group algebra has a natural involution

f∗(g) = f(g−1)

which makes it a ∗-algebra. Given a unitary representation ρ : G → U(V ), one forms a nonde-generate ∗-representation ρ : C[G/S]→ B(V S) by

ρ(f)v =∑

g∈G/Sf(g) ρ(g)v,

2

and in fact every nondegenerate ∗-representation of C[G/S] is obtained in this way. Therefore,the supremum norm

‖f‖ = sup{ ‖ρ(f)‖ : ρ is a nondegenerate ∗-representation of C[G/S]}

is bounded. The group C∗-algebra of G/S, C∗(G/S), is the completion of C[G/S] in the supre-mum norm.

Now suppose the subgroup S is not required to be normal, but almost normal, meaningthat each double coset SgS is the union of finitely many left cosets (instead of just one, in thecase of a normal subgroup). Some basic examples of almost normal subgroups include normalsubgroups and finite subgroups. Although the double coset space is not a group in the generalcase, the space of compactly-supported functions on the double coset space, C[S\G/S], with theproduct

f1 ∗ f2(g) =∑

γ∈G/Sf1(γ) f2(γ−1g)

and involutionf∗(g) = f(g−1)

is a ∗-algebra. Observe also that the convolution product extends to a product

C[G/S]×C[S\G/S] −→ C[G/S]

under which C[G/S] is a right C[S\G/S]-module.We will call C[S\G/S] with the product and involution above the Hecke algebra of the

pair (G,S), denoted H(S\G/S). We will also consider locally compact topological groups Gwith an open compact subgroup S. The left coset space G/S is discrete, and S is automaticallyalmost normal. The appropriate space to consider is Cc(S\G/S), the compactly supported S-biinvariant functions, with the product and involution defined in manner similar to the discretecase. This Hecke algebra will also be denoted H(S\G/S). We ask the natural question: Can weform C∗(S\G/S)? That is, define the supremum norm of an element f of the Hecke algebra by

‖f‖ = sup{ ‖ρ(f)‖ : ρ is a nondegenerate ∗-representation of H(S\G/S) }.

Is ‖f‖ <∞ for all elements of the Hecke algebra?The answer to the question is “no,” in general. The group SL2(Qp) with the almost nor-

mal subgroup SL2(Zp) forms a Hecke ∗-algebra which is ∗-isomorphic to the complex polynomialsin one variable C[y] via the Satake isomorphism. This allows us to designate a ∗-representationρa for every real number a. For every f ∈ H(S\G/S), let ρa(f) be the evaluation at a of thepolynomial in C[y] which corresponds to f . Therefore, if f is not a multiple of the identity,

sup{ ‖ρa(f)‖ : a ∈ R} =∞.

This example shows that not every Hecke ∗-algebra can be completed in the supremum norm.However, the question can be answered in the affirmative for some classes of Hecke algebras

for fairly elementary reasons. For instance, the Hecke algebra formed from the group SLn(Qp)with its so-called Iwahori subgroup has a finite set of generators. Each generator Xi of the Heckealgebra is self-adjoint and satisfies

X2i

= pI + (p− 1)Xi

where I is the identity of the algebra and i = 1, . . . , n. Therefore, the spectrum of each generatoris the set {−1, p} and this gives a bound on the supremum norm of every element in the algebra.

3

Let G be a locally compact topological group and S an open compact subgroup of G.Borel [2] constructed adjoint functors between the category of nondegenerate representations ofH(S\G/S) and the category of smooth representations of G which are generated by their S-fixedvectors (a representation ρ : G→ Aut(V ) is smooth if the stabilizer of each v ∈ V is open in G)1.Borel’s method was the following. The restriction V → V S of a smooth representation V of G toits S-fixed vectors takes smooth G-modules to representations of H(S\G/S). If ρ : G→ Aut(V )is a smooth representation of G, then

ρ(f)v =∫Gf(g) ρ(g)v dg

for f ∈ H(S\G/S) and v ∈ V S defines a representation of the Hecke algebra. If W is anondegenerate representation of H(S\G/S), then there are two ways to recover a G-module—the induced module I(W ), given by

I(W ) = C[G/S]⊗H(S\G/S) W,

and the produced module P (W ), given by

P (W ) = {f ∈ C(G/S,W ) : f ∗ h = ρ(h)f for h ∈ H(S\G/S)}.

In either case, the G-action arises from the action of G on C[G/S] given by g(f)(γ) = f(g−1γ)for f ∈ C[G/S] and g, γ ∈ G. Borel showed that if B is an Iwahori subgroup of a semi-simple group G over a non-Archimedean local field, then the induced and produced modulesare canonically isomorphic. Moreover, these constructions give a category equivalence betweensmooth G-modules which are generated by their B-fixed vectors and nondegenerate H(B\G/B)-modules.

If we wish to consider the Hecke algebra H(S\G/S) as a ∗-algebra, the appropriate ques-tion to ask is: Are the categories of unitary representations of G and nondegenerate ∗-represen-tations of H(S\G/S) equivalent? The pair (SL2(Qp), SL2(Zp)) shows that the answer to thisquestion must be “no.” Indeed, if ρ : G→ U(V ) is a unitary representation of G, then we obtaina nondegenerate ∗-representation ρ : H(S\G/S)→ B(V S) by

ρ(f)v =∑

g∈G/SΞ

12 (g) f(g) ρ(g)v,

for f ∈ H(S\G/S) and v ∈ V S , where Ξ(g) is the number of right cosets in SgS divided by thenumber of left cosets. Then

‖ρ(f)‖ ≤∑

g∈G/SΞ

12 (g) |f(g)| ≤ C‖f‖1,

where C is some constant depending on f only. If every nondegenerate representation ofH(S\G/S)arose in this way, then the supremum norm of every element in the Hecke algebra would be finite.As we have already seen, the pair (SL2(Qp), SL2(Zp)) provides the contradiction.

1Actually, Borel required representations of G to be admissible, meaning ρ : G→ Aut(V ) is smooth

and VU

is finite-dimensional for every open subgroup U ⊂ G. However, all of Borel’s results hold if weonly require smoothness [7, p. 583].

4

Barbasch and Moy [1] obtained the category equivalence in the unitary case when B is theIwahori subgroup of a semi-simple group G over a non-Archimedean local field. They invokedthe proof of the Deligne-Langlands conjecture and categorized all representations of the groupand the Hecke algebra to prove that the category of unitary representations of G is equivalent tothe category of nondegenerate ∗-representations of H(B\G/B).

Our approach is the following. If V is a unitary G-module, then ρ : H(S\G/S)→ B(V S) isa nondegenerate ∗-representation of the Hecke algebra. Given a nondegenerate ∗-representationof H(S\G/S), form the G-module

C[G/S]⊗H(S\G/S) W

with the bilinear form〈f1 ⊗ w1, f2 ⊗ w2〉 = 〈w1, 〈〈f1, f2〉〉w2〉,

where〈〈, 〉〉 : C[G/S]×C[G/S] −→ H(S\G/S)

is defined by

〈〈f1, f2〉〉(g) =1

L(g)

∑γ∈G/Sg

f1(γ) f2(γg) Ξ(γ),

This is a well-defined bilinear form and the action of G given by

g(f ⊗ w) = Ξ−12 (g) g(f)⊗ w

satisfies〈g(f1 ⊗ w1), f2 ⊗ w2〉 = 〈f1 ⊗ w1, g

−1(f2 ⊗ w2)〉.

An element a in a ∗-algebra A is positive if a can be expressed as a finite sum a =∑i a∗iai

where each ai ∈ A.

1.1. Theorem. Let G be a (discrete or locally compact topological) group and S an almostnormal subgroup of G (or open compact subgroup of a topological group). If the bilinear form〈〈, 〉〉 is positive, then the category of nondegenerate ∗-representations of H(S\G/S) is equivalentto the category of unitary representations of G which are generated by their S-fixed vectors.

1.2. Corollary. If 〈〈, 〉〉 is positive, then C∗(S\G/S) exists.

We note that if G is SL2(Qp) and S is SL2(Zp), then there exist f ∈ C[G/S] such that〈〈f, f〉〉 is negative.

We will prove in Chapter 4 that 〈〈, 〉〉 is positive if G is finite or if (G,S) is lower directed.The Hecke pair studied by Bost and Connes in [3] is an example of a lower directed pair.

Our main result in the Iwahori case is the following theorem.

1.3. Theorem. Let G be SL2(Qp) and B be its Iwahori subgroup. The bilinear form 〈〈, 〉〉 ispositive.

Sketch of Proof. We use the fact that SL2(Qp) acts as type-preserving automorphismsof the (p + 1)-regular tree, the edges of which correspond to the left coset space G/B. Theproof is by induction. If 〈〈f, f〉〉 is positive for all f ∈ C[G/B] supported on the left cosetscorresponding to the edges of some finite subtree, then we show that 〈〈h, h〉〉 is positive for allh ∈ C[G/B] supported on the cosets corresponding to the larger subtree formed by adding allthe edges adjacent to one vertex in the subtree. The proof depends on the geometry of thetree. The first step in the inductive argument follows from the fact that a finite Hecke algebra

5

may be associated to subtree formed from all the edges in the tree emanating from one vertex.2

Let G be SL3(Qp) and B be its Iwahori subgroup. Then G acts on a (p + 1)-regularaffine building. Using similar methods we have shown that 〈〈f, f〉〉 is positive for all f ∈ C[G/B]supported on cosets associated to certain subcomplexes of the (p+ 1)-regular building.

1.4. Conjecture. Let G be SLn(Qp) and B its Iwahori subgroup. Then 〈〈f, f〉〉 is positive forall f ∈ C[G/B].

Chapter 2 outlines the background necessary to understand Hecke algebras, and providesour first examples of Hecke C∗-algebras. In Chapter 3, we examine the functorial correspondencebetween unitary representations of G and nondegenerate ∗-representations of H(S\G/S). Chap-ter 4 contains results about finite Hecke algebras and lower directed Hecke algebras. Chapter 5is an exposition of the theory of trees and buildings, which we will use in the proof of the Iwa-hori case for SL2(Qp) in Chapter 6. We also present our results relating to SL3(Qp) and someassociated open problems in Chapter 6.

6

Chapter 2

Hecke ∗-Algebras

In this chapter, we will provide the background necessary to the study of Hecke C∗-algebras. Section 2.1 contains the definition of a Hecke pair and introduces some of the exampleswhich will be used throughout the thesis. In Section 2.2 we review some of the basic theoryof Hecke algebras. We define the Hecke C∗-algebra in Section 2.3 and show that the HeckeC∗-algebra does not exist for every Hecke pair.

2.1 Hecke Pairs

The purpose of this section is to develop some basic definitions, algebraic properties,and examples of Hecke pairs which will come in handy in our study of Hecke algebras. Thispresentation is based on Krieg [12, pp. 1–11].

2.1. Definition. Let G be a discrete group and S a subgroup of G. The pair (G,S) is called aHecke pair if each double coset SgS can be written as a finite union of left cosets.

We will call S an almost normal subgroup of G, meaning S is finitely close to being normalin G. Of course, a normal subgroup S is almost normal, since left cosets and double cosets of anormal subgroup coincide.

2.2. Definition. For g ∈ G, let L(g) denote the number of distinct left cosets γS in the doublecoset SgS and R(g) denote the number of right cosets in SgS.

The Hecke pair condition means that each double coset SgS can be written

SgS =L(g)⋃i=1

giS.

We will express the above equation as

SgS =⋃

γ∈SgS/SγS

where “γ ∈ SgS/S” means “γ runs through a set of representatives of the left cosets giS in SgS.”Likewise, “γ ∈ G/S” means “γ runs through a set of representatives of left cosets gS in G.”

If each double coset consists of finitely many left cosets, will it also contain only finitelymany right cosets? Lemma 2.4 implies that we can use either right or left cosets to define aHecke pair.

2.3. Definition. For g ∈ G, let the subgroup Sg ⊂ S be defined by

Sg = {s ∈ S : sgS = gS} = gSg−1 ∩ S.

7

2.4. Lemma. Let G be a group and S a subgroup of G. For g ∈ G,

SgS =⋃

s∈S/SgsgS, and (2.1)

SgS =⋃

s∈S/Sg−1

Sgs. (2.2)

where the unions are disjoint.

Proof. Consider Equation 2.1. The union on the right is well-defined, since for a differentchoice of representative for the coset sSg, say sgs′g−1, we have sgs′g−1gS = sgS. Moreover, if

sgS = s′gS, then s−1s′ ∈ Sg, so sSg = s′Sg , which shows that the cosets are disjoint. The righthand side is contained in SgS by definition. Since every s ∈ S lies in some left coset sSg, everyelement sgs′ ∈ SgS can be written sgs′ = ss′′gs′ ∈ sgS for some s′′ ∈ Sg . Thus the left handside of Equation 2.1 is contained in the right.

Note that Sg−1 = g−1Sg ∩ S = {s ∈ S : Sgs = Sg}. By similar calculations we have

Equation 2.2. 2

2.5. Corollary. Suppose S is a subgroup of a group G. For any g ∈ G,

1. L(g) = [S : Sg] and R(g) = [S : Sg−1]

2. R(g) = L(g−1) 2

2.6. Corollary. Every double coset of S in G contains finitely many left cosets if and only ifevery double coset contains finitely many right cosets. 2

Is there a relationship between the number of left and right cosets in a double coset? Ofcourse, if the double coset is a finite set, then both numbers are equal to |SgS||S| . In general, thefollowing is the best we can say.

2.7. Theorem. Let (G,S) be a Hecke pair. The mapping Ξ : G −→ Q+ defined by

Ξ(g) =R(g)L(g)

is a homomorphism of groups, with S contained in the kernel of Ξ.

Proof. First we will need the following:

1. Two subgroups S1 and S2 of a group G are commensurable if their intersection has finiteindex in both. Let S1 and S2 be commensurable subgroups of G, and let T be any subgroupof finite index in S1 ∩ S2. Then

[S1 : T ][S2 : T ]

=[S1 : S1 ∩ S2][S2 : S1 ∩ S2]

.

In other words, the quantity [S1:T ][S2:T ]

, which we will denote [S1][S2]

, is independent of the choice

of T .

8

2. Let S1, S2, and S3 be subgroups of G which are pairwise commensurable. Then

[S1][S2]

· [S2][S3]

=[S1][S3]

since the subgroup S1 ∩ S2 ∩ S3 has finite index in S1, S2, and S3 (note that

[S1 : S1 ∩ S2 ∩ S3] = [S1 : S1 ∩ S2][S1 ∩ S2 : S1 ∩ S2 ∩ S3]

≤ [S1 : S1 ∩ S2][S2 : S2 ∩ S3] <∞

and similarly for S2 and S3).

3. Let S1 and S2 be commensurable subgroups. Then

[S1][S2]

=[gS1g

−1][gS2g

−1]

By Lemma 2.4, gSg−1 and S are commensurable if (G,S) is a Hecke pair. Now

Ξ(g) =[S : S

g−1 ]

[S : Sg]=

[S : S ∩ g−1Sg][S : Sg]

=[gSg−1 : gSg−1 ∩ S]

[S : Sg ]

=[gSg−1 : Sg ]

[S : Sg]=

[gSg−1][S]

.

So

Ξ(gh) =[ghSh−1g−1]

[S]

=[ghSh−1g−1]

[gSg−1]· [gSg−1]

[S]

=[hSh−1]

[S]· [gSg−1]

[S]= Ξ(h) Ξ(g).

Since for all s ∈ S, L(s) = R(s) = 1, S is contained in the kernel of Ξ. 2

2.1.1 Examples

2.1.1.1 The Hecke pair (G,S), where S is normal in G

Since L(g) = R(g) = 1 for all g ∈ G, the pair (G,S) is a Hecke pair, and Ξ is the trivialhomomorphism.

2.1.1.2 The Hecke pair (G,S), where S is finite

In this case each double coset is finite, and

L(g) = R(g) =|SgS||gS| <∞.

Therefore, the map Ξ is trivial.

9

2.1.1.3 The Hecke pair (P+Q, P+Z

)

This Hecke pair, studied by Bost and Connes in [3], provides our first example where Ξ isnontrivial. Let

G = P+Q

={(

1 a0 r

): a, r ∈ Q, a > 0

}S = P+

Z={(

1 n0 1

): n ∈ Z

}.

Then for any arbitrary g =(

1 a0 r

)∈ G, the left coset associated to g is

gS ={(

1 n+ a0 r

): n ∈ Z

},

and the double coset associated to g is

SgS ={(

1 n+ a+ rm0 r

): n,m ∈ Z

}.

Let g =(

1 a0 r

)as above. If q is the denominator of r (in other words, if r = p

q where p and qhave no common factors, and q > 0), then each double coset SgS may be written as the disjointunion of q single cosets giS by

SgS =q⋃i=1

{(1 n+ a+ ri0 r

): n ∈ Z

}

=q⋃i=1

{(1 a+ ri0 r

)(1 n0 1

): n ∈ Z

}

=q⋃i=1

giS,

where gi =(

1 a+ ri0 r

). Hence (P+

Q, P+Z

) is a Hecke pair. In addition, L(g) is the denominatorof r. But since (

1 a0 r

)−1=(

1 −ar0 1

r

),

it follows that R(g) is the numerator of r. Therefore the homomorphism Ξ associates to g ∈ Gits lower right hand entry r. In this case Ξ is clearly a group homomorphism.

2.1.2 Topological groups

Our definition of a Hecke pair extends a topological group G with an open subgroup S.The quotient space of a topological group by an open subgroup is discrete, so we can count leftcosets as before.

2.8. Definition. Let G be a topological group with S an open subgroup. Then (G,S) is a Heckepair if every double coset contains finitely many left cosets.

Observe that if the subgroup S is compact and open, the pair (G,S) is a Hecke pair, foressentially the same reason that finite subgroups are almost normal in the discrete case.

10

2.1.2.1 Example: The Hecke pair (SL2(Qp), SL2(Zp))

A good introduction to p-adic numbers can be found in [9]. Let p be a prime. Any nonzerorational number r can be written uniquely in the form r = apk, where neither the numerator northe denominator of a is divisible by p. Let the absolute value of r be |r| = p−k if r = apk. Wedefine the distance between r1, r2 ∈ Q by

d(r1, r2) = |r1 − r2|.

The completion of the rational numbers obtained from this metric is a locally compact, totallydisconnected field called the p-adic numbers Qp. The function | | is a non-Archimedean absolute

value on Qp. Every element r ∈ Q×p

has a unique p-adic expansion

r = akpk + ak+1p

k+1 + ak+2pk+2 + · · ·

where ai ∈ Z, 0 ≤ ai < p− 1, and p−k = |r|. Observe that the above series converges in Qp.The ring of p-adic integers Zp is the collection of r ∈ Qp for which |r| < 1. In other words,

Zp is the open unit ball in Qp. One can show that the ring of p-adic integers Zp is contained inQp as a compact open set. See Gouvea [9, pp. 58–63] for more details.

The matrix group SL2(Qp) is a locally compact, totally disconnected topological groupcontaining SL2(Zp) as a compact open subgroup. Therefore, (SL2(Qp), SL2(Zp)) is a Heckepair.

2.9. Lemma. Let G = SL2(Qp) and P = SL2(Zp), for p a prime. For all g ∈ G, the doublecoset PgP can be written uniquely in the form

PgP = P

(p−k 00 pk

)P

for k ∈ {0, 1, 2, . . .}. Moreover for any g ∈ G,

g ∈ P(p−k 00 pk

)P

if and only if −k is the minimum valuation of the entries of g.

Proof. Let g =(a bc d

)∈ SL2(Qp). Since

(0 −11 0

)lies in SL2(Zp), the double

coset PgP has a representative g′ whose entries are a rearrangement of the entries of g so that

g′ =(a′ b′c′ d′

)∈ SL2(Qp) where v(a′) = min{v(a′), v(b′), v(c′), v(d′)} (note that v(a′) ≤

0). Multiplying on the left and right by matrices(

1 n0 1

)∈ SL2(Zp) and

(m 00 m−1

)∈

SL2(Zp) gives a representative of the desired form. Thus

P(a bc d

)P = P

(p−k 00 pk

)P

where k ≥ 0 and −k = min{v(a), v(b), v(c), v(d)}.

11

Now we will prove these representatives give unique double cosets. Observe that PgP =PhP if and only if g−1sh ∈ P for some s ∈ P . Suppose

P

(p−k 00 pk

)P = P

(p−l 00 pl

)P for k, l ∈ Z

Then for some s =(a bc d

)∈ P, suppose

g−1sh =(pk 00 p−k

)(a bc d

)(p−l 00 pl

)=(

pk−la pk+lb

p−k−lc pl−kd

)∈ P.

If both k− l and k+ l are nonzero, v(det(s)) > 0, which is impossible. So either k = l or k = −l.Choosing k and l to be nonnegative integers gives a unique set of representatives for the doublecosets. 2

In the course of proving Lemma 2.9 above, we proved the following important fact:

2.10. Proposition. For g ∈ G, PgP = Pg−1P . 2

The following lemma describes a set of representatives for the left cosets. First, supposer ∈ Qp has terminating p-adic expansion

r = akpk + ak+1p

k+1 + · · ·+ ampm

where ai ∈ Z, 0 ≤ ai ≤ p − 1, and p−k = |r|. Define t(r) for r ∈ Q×p

to be the coefficient ofthe highest power of p in the expansion of r if the expansion is terminating. In this example,t(r) = m. If the p-adic expansion of r does not terminate, t(r) = +∞. Define t(0) = −∞.

2.11. Lemma. Let G = SL2(Qp) and P = SL2(Zp). For all g ∈ G, the left coset gP can bewritten uniquely in the form

gP =(pn a0 p−n

)P

where n ∈ Z and t(a) < n. Moreover, the left coset(pn a0 p−n

)P

belongs to the double coset

P

(p−N 00 pN

)P

where −N = min{−|n|, v(a)}.

Proof. Using similar calculations to Lemma 2.9 (except in this case one can only multiplyon the left by elements of SL2(Zp)), one can show that every left coset gP can be written in theform

gP =(pn b0 p−n

)P,

where n ∈ Z and b ∈ Qp. This representative can be multiplied on the left by matrices(1 q0 1

)∈ SL2(Zp)

to subtract off powers pn and higher in the p-adic expansion of b.

12

To prove uniqueness, it suffices to show that if

g =(pn a0 p−n

)and g′ =

(pm b0 p−m

)are matrices in SL2(Qp) such that t(a) < n and t(b) < m, then g−1g′ ∈ P if and only if g = g′.Now suppose g−1g′ ∈ P , where

g−1g′ =(p−n −a0 pn

)(pm b0 p−m

)=(pm−n bp−n − ap−m0 pn−m

).

Then m = n and bp−n − ap−m = bp−m − ap−n ∈ Zp. Since t(bp−m) < 0 and t(ap−n) < 0,

bp−m − ap−n = 0. This shows that the chosen set of representatives gives disjoint left cosets.The inclusion (

pn a0 p−n

)P ⊂ P

(p−N 00 pN

)P

where −N = min{−|n|, v(a)} follows from Lemma 2.9. 2

2.12. Corollary. For g ∈ G,

L(g) = R(g) = p2N−1(p+ 1).

Proof. Lemma 2.11 allows us to explicitly calculate L(g). For any g ∈ G,

P

(p−N 00 pN

)P =

⋃−N=min{−|n|,v(a)}

t(a)<n

(pn a0 p−n

)P

where the union is disjoint. Therefore

L(g) = R(g) =∣∣∣∣{( pn a

0 p−n

): |n| = N, t(a) < n, v(a) ≥ −N

}∣∣∣∣+

N−1∑n=−N+1

∣∣∣∣{( pn a0 p−n

): t(a) < n, v(a) = −N

}∣∣∣∣= 1 + p2N +

N−1∑n=−N+1

|{a ∈ Qp : t(a) < n, v(a) = −N}|

= 1 + p2N +

N−1∑n=−N+1

(p− 1)pn+N−1

= p2N−1 + p2N = p2N−1(p+ 1) 2

2.2 The Algebra Structure

2.2.1 Group algebras

Let X be a discrete space. We will use the notation C[X ] to denote the space of complex,compactly supported functions on X . If Y ⊂ X then the characteristic function of Y is denoted[Y ].

13

2.13. Definition. Let G be a discrete group. The group algebra C[G] is the space of complex,compactly supported functions on G with pointwise addition of functions and the convolutionproduct

f1 ∗ f2(g) =∑γ∈G

f1(γ) f2(γ−1g)

for g ∈ G and f1, f2 ∈ C[G].

The convolution product is directly related to multiplication in G: if g1, g2 ∈ G, then

[g1] ∗ [g2] = [g1g2].

If G is a locally compact, totally disconnected group with Haar measure m, let

f1 ∗ f2(g) =∫Gf1(γ) f2(γ−1g) dm(γ),

for g ∈ G and f1, f2 ∈ C∞c

(G), the space of compactly supported, locally constant functions inG. If U and V are compact subsets of G, then for all g ∈ G,

[U ] ∗ [V ](g) = m({u ∈ U : u−1g ∈ V }) = m(V ∩ U−1g).

In particular, the support of [U ] ∗ [V ] is the set U · V .

2.2.2 Functions on the double coset space

Suppose G is a discrete group with almost normal subgroup S. Let C[S\G/S] be thespace of finitely-supported complex functions on the double coset space S\G/S. Each functionin C[S\G/S] can be expressed

f =∑

g∈S\G/Sf(g) [SgS]

where f(g) ∈ C. Equivalently, define a G-action on C[G/S], the compactly supported functionson the left coset space G/S, by

g(f)(γ) = f(g−1γ)

for g, γ ∈ G and f ∈ C[G/S]. Then h ∈ C[S\G/S] can be thought of as a left-S-invariantelement of C[G/S]. To emphasize this relationship we can write

f =∑

g∈G/Sf(g) [gS],

where f(g1) = f(g2) whenever Sg1S = Sg2S.

2.2.2.1 The Convolution Product on C[S\G/S]

Let the product in C[S\G/S] be defined as follows:

f1 ∗ f2(g) =∑

γ∈G/Sf1(γ) f2(γ−1

g), (2.3)

for g ∈ G and f1, f2 ∈ C[S\G/S]. Note that the sum is finite, since the both f1 and f2 aresupported on finitely many left cosets in G/S.

14

2.14. Lemma. Equation 2.3 gives an associative product

C[S\G/S]×C[S\G/S]→ C[S\G/S].

That is, for all f1, f2, f3 ∈ C[S\G/S]:

1. f1 ∗ f2 ∈ C[S\G/S]

2. f1 ∗ (f2 ∗ f3) = (f1 ∗ f2) ∗ f3.

Proof. Observe that the expression∑γ∈G/S f1(γ) f2(γ−1g) is invariant under a different

choice of representatives for the left cosets G/S. That is, if there exists s ∈ S so that γ′ = γs,then

f1(γ′) f2((γ′)−1g) = f1(γs) f2(s−1

γ−1g) = f1(γ) f2(γ−1

g)

due to the right S-invariance of f1 and bi-S-invariance of f2. Hence the product is defined onC[S\G/S]×C[S\G/S]—and, in fact, on C[G/S]×C[S\G/S].

Right invariance of f1 ∗ f2 follows from

f1 ∗ f2(gs) =∑

γ∈G/Sf1(γ) f2(γ−1

gs) =∑

γ∈G/Sf1(γ) f2(γ−1

g) = f1 ∗ f2(g),

since f2 is bi-S-invariant. Left invariance follows from

f1 ∗ f2(sg) =∑

γ∈G/Sf1(γ) f2(γ−1sg) (2.4)

=∑

γ∈G/Sf1(s−1γ) f2((s−1γ)−1g) (2.5)

=∑

τ∈G/Sf1(τ) f2(τ−1g) (2.6)

= f1 ∗ f2(g). (2.7)

The substitution τ = s−1γ in Equation 2.6 is valid because {s−1γ : γ ∈ G/S} is a complete setof representatives for left cosets in G/S.

Statement 2 (associativity) follows from

f1 ∗ (f2 ∗ f3)(g) =∑

η∈G/Sf1(η)

(f2 ∗ f3(η−1g)

)(2.8)

=∑

η∈G/Sf1(η)

∑γ∈G/S

f2(γ) f3(γ−1η−1g)

(2.9)

=∑

η∈G/S

∑γ∈G/S

f1(η) f2(γ) f3(γ−1η−1g) (2.10)

=∑

η∈G/S

∑τ∈G/S

f1(η) f2(η−1τ) f3(τ−1g) (2.11)

=∑

τ∈G/S

∑η∈G/S

f1(η) f2(η−1τ)

f3(τ−1g) (2.12)

= (f1 ∗ f2) ∗ f3(g). (2.13)

15

The substitution τ = ηγ in Equation 2.11 is valid because {ηγ : γ ∈ G/S} is a complete set ofrepresentatives for left cosets in G/S. 2

The proof of Statement 2 does not depend on the fact that f1 lies in C[S\G/S]. All weneed is right-S-invariance; that is, f1 ∈ C[G/S]. If f2 ∈ C[S\G/S], then the first part of theproof of Statement 1 shows that f1 ∗ f2 ∈ C[G/S].

2.15. Lemma. The convolution product gives a mapping

C[G/S]×C[S\G/S] −→ C[G/S]

under which C[G/S] is a right C[S\G/S]-module. 2

Example The product [SgS]∗ [ShS] produces a more complicated formula than we mightexpect. We begin by computing

([gS] ∗ [ShS])(jS) =∑

γ∈G/S[gS](γ)[ShS](γ−1jS)

={

1 if j = gshs′ for s, s′ ∈ S.0 otherwise.

=∑

η∈ShS/S[gηS].

Then[SgS] ∗ [ShS] =

∑γ∈SgS/S

[γS] ∗ [ShS] =∑

γ∈SgS/S

∑η∈ShS/S

[γηS].

The product defined above works perfectly well for G a locally compact, totally discon-nected topological group and S an open, compact subgroup, since in this case the left coset spaceG/S is also discrete.

2.16. Definition. The Hecke algebra associated to the Hecke pair (G,S), denotedH(S\G/S), is the space C[S\G/S] with the algebra structure given by pointwise addition offunctions and convolution multiplication.

2.2.2.2 Involution on H(S\G/S)

Since our ultimate goal is to complete H(S\G/S) to a C∗-algebra, we actually need a∗-algebra structure on H(S\G/S). In the group algebra case, given f ∈ C[G], the involutionf → f∗ is defined by

f∗(g) = f(g−1).

The same definition works perfectly well for Hecke algebras.

2.17. Lemma. The mapping f → f∗ given by

f∗(g) = f(g−1)

is an involution on H(S\G/S) which is compatible with the algebra structure. That is, for allf, f1, f2 ∈ H(S\G/S),

1. f∗ ∈ H(S\G/S).

2. (f∗)∗ = f .

3. (f1 ∗ f2)∗ = f∗2∗ f∗

1.

16

Proof. The first statement follows from f∗(sg) = f(g−1s−1) = f(g−1) = f∗(gs) for allg ∈ G and s ∈ S. The second statement (f∗)∗ = f is clear from the definition. To prove thisinvolution is compatible with the product, we calculate

(f1 ∗ f2)∗(g) = f1 ∗ f2(g−1)

=∑

γ∈G/Sf1(γ) f2(γ−1g−1)

=∑

κ∈G/Sf1(g−1κ) f2(κ−1)

=∑

κ∈G/Sf∗1

(κ−1g) f∗2

(κ)

= f∗2∗ f∗

1(g).

These observations prove H(S\G/S) is a ∗-algebra with the product and involution described.2

2.2.3 Examples

2.2.3.1 H(S\G/S) when S is normal

If S is normal in G, G/S is a group. The Hecke ∗-algebra H(S\G/S) is identical to thegroup ∗-algebra C[G/S], since the formula

f1 ∗ f2(g) =∑

γ∈G/Sf1(γ) f2(γ−1g)

for f1, f2 ∈ H(S\G/S) agrees with the group algebra product, and involution was defined in thesame way for the group algebra and Hecke algebra.

2.2.3.2 The Hecke algebra for the pair (SL2(Qp), SL2(Zp))

Let G = SL2(Qp) and P = SL2(Zp). As stated before, the Hecke pair (G,P ) has theproperty that

SgS = Sg−1S

for all g ∈ G. Therefore, its Hecke algebra has the property f = f∗ and therefore is abelian,since

f1 ∗ f2 = (f1 ∗ f2)∗ = f2∗ ∗ f1

∗ = f2 ∗ f1

for f1, f2 ∈ H(P\G/P ).A special case of the Satake isomorphism proves useful in studying this Hecke algebra.

Let C[x, x−1] denote the polynomials in x and x−1 (that is, the Laurent polynomial ring) andC[y] the polynomials in y.

2.18. Theorem. Let G be SL2(Qp) and P be SL2(Zp) for p a prime. Then H(P\G/P ) is∗-isomorphic to the polynomials in one variable C[y].

Proof. Let Γ ⊂ G be those elements of the form(pn a0 p−n

)

17

where n ∈ Z and t(a) < n (recall that t(a) is the exponent of the highest power of p in the p-adicexpansion of a). We showed in Lemma 2.11 that Γ forms a complete set of representatives forthe left cosets of P in G. Define S : Γ→ C[x, x−1] by

S((pn a0 p−n

)) = p−nxn.

The map S extends to a map S : Cc(G/P )→ C[x, x−1] by

S(f) =∑γ∈Γ

f(γ)S(γ).

This map is well-defined, since Lemma 2.11 shows that each element of γ ∈ Γ belongs to a different

left coset γP and every element of G belongs to some γP . Moreover, for γ1 =(pn a0 p−n

)and γ2 =

(pm b0 p−m

), we have S([γ1γ2P ]) = S([γ1P ])S([γ2P ]) since

S([γ1γ2P ]) = S([(pn+m bpn + ap−m0 p−n−m

)P ])

= S([(pn+m c0 p−n−m

)P ]) = p−n−mxn+m = S([γ1P ])S([γ2P ])

where c is such that t(c) < n+m.We claim that S is a ∗-isomorphism from H(P\G/P ) to the algebra of polynomials in

C[x, x−1] symmetric in x and x−1.That is, for f, f1, f2 ∈ H(P\G/P ),

1. S(f1)S(f2) = S(f1 ∗ f2).

2. S(f) is symmetric in x and x−1.

3. The set {S([PgP ]) : g ∈ P\G/P} is a basis for the symmetric polynomials in C[x, x−1].

4. S is injective.

(1) For f1, f2 ∈ H(P\G/P ),

S(f1)S(f2) =∑γ∈Γ

f1(γ)S(γ)∑η∈Γ

f2(η)S(η)

=∑γ∈Γ

∑η∈Γ

f1(γ) f2(η)S(γ)S(η)

=∑γ∈Γ

∑η∈Γ

f1(γ) f2(η)S(γη)

=∑τ∈Γ

∑γ∈Γ

f1(γ) f2(γ−1τ)S(τ)

=∑τ∈Γ

(f1 ∗ f2)(τ)S(τ) = S(f1 ∗ f2)

18

(2) It suffices to show that for all g ∈ G, S([PgP ]) is symmetric in x and x−1. Lemma 2.11tells us that if −N is the minimum valuation of the entries of g,

[PgP ] =⋃

−N=min{−|n|,v(a)}t(a)<n

[(pn a0 p−n

)P ]

=∑

|n|=N, t(a)<nv(a)≥−N

[(pn a0 p−n

)P ] +

∑−N+1≤n≤N−1t(a)<n, v(a)=−N

[(pn a0 p−n

)P ]

So for N > 0,

S([PgP ]) = pNx−N + p2N (p−NxN ) +N−1∑

n=−N+1

(p− 1)pn+N−1(p−nxn)

= pNx−N + pNxN + (p− 1)pN−1N−1∑

n=−N+1

xn

which is clearly symmetric. For N = 0, [PgP ] = [P ] and S([P ]) = 1.(3) Let PN be defined to be S([PgP ]), where −N is the minimum valuation of the entries

of g. Then P0 = 1 and

PN = pNx−N + p−NxN + (p− 1)pN−1N−1∑

n=−N+1

xn

for N > 0. It is clear that {PN : N = 0, 1, 2, . . .} forms a basis for the symmetric polynomials inC[x, x−1].

(4) Moreover, S is injective since it is injective on the characteristic functions of doublecosets

[P(p−N 00 pN

)P ]

which form a basis for the Hecke algebra H(P\G/P ).The symmetric Laurent polynomials are generated by (x+ x−1) and hence isomorphic to

C[y] by (x+ x−1)→ y. Therefore, H(P\G/P ) is ∗-isomorphic to C[y]. 2

2.3 The Hecke C∗-algebra: a first look

We are now in a position to define the Hecke C∗-algebra of the Hecke pair (G,S). As inthe case of the Hecke algebra, the motivating example comes from groups.

2.3.1 The group C∗-algebra C∗(G)

A ∗-representation ρ of a ∗-algebra A on a Hilbert space V is nondegenerate ifρ(A)V = V . For any f ∈ A, define the supremum norm ‖f‖ to be

‖f‖ = sup{ ‖ρ(f)‖ : ρ is a nondegenerate ∗-representation of A}.

19

If A is the group algebra C[G] for some discrete group G, then ‖f‖ is bounded by the `1 normof f , as the following argument shows. Suppose ρ is a nondegenerate ∗-representation of C[G]on V . Then

‖ρ(f)‖ = ‖∑g∈G

f(g) ρ([g]) ‖ ≤∑g∈G|f(g)| ‖ρ([g])‖

where the sum is a finite sum. We also have

‖ρ([g])‖ = ‖ρ∗([g])‖ = ‖ρ([g−1])‖,

so ρ takes G to U(V ), the group of unitary operators on V , and

‖ρ(f)‖ ≤∑g∈G|f(g)| = ‖f‖1.

We now have a bound on the supremum norm of each f ∈ C[G].

2.19. Definition. Let G be a discrete group. The group C∗-algebra of G, C∗(G), is the com-pletion of C[G] in the supremum norm.

2.3.2 The Hecke C∗-algebra C∗(S\G/S)

2.20. Definition. Let (G,S) be a Hecke pair, and let f ∈ H(S\G/S). We define

‖f‖ = sup{ ‖ρ(f)‖ : ρ is a nondegenerate ∗-representation of H(S\G/S)}.

If ‖f‖ <∞ for all f ∈ H(S\G/S), the Hecke C∗-algebra of (G,S), C∗(S\G/S), is the completionof H(S\G/S) in the norm ‖ ‖.

The Hecke C∗-algebra will not exist for every Hecke pair, as the following example shows.

2.3.3 Example: (SL2(Qp), SL2(Zp))

Let G be SL2(Qp) and P be SL2(Zp). Theorem 2.18 shows that the Hecke algebraH(P\G/P ) is isomorphic to C[y]. This allows us to designate a representation ρa for every realnumber a. For every f ∈ H(S\G/S), let ρa(f) be the evaluation at a of the polynomial in C[y]which corresponds to f . Then for any f ∈ H(P\G/P ) which is not a multiple of the identity [P ],

‖f‖ ≥ sup{ ‖ρa(f)‖ : a ∈ R} =∞.

So we now know

2.21. Proposition. The Hecke C∗-algebra C∗(S\G/S) does not exist for every Hecke pair(G,S). In particular, it does not exist for the Hecke pair (SL2(Qp), SL2(Zp)). 2

The full strength of the Satake isomorphism is needed to tackle the Hecke pair (SLn(Qp), SLn(Zp)).In this case, the Hecke algebra will be isomorphic to an algebra of polynomials in several vari-ables. We expect that an analog of Proposition 2.21 will be true for (SLn(Qp), SLn(Zp)) forthe reasons outlined above.

2.3.4 Hecke algebras with bounded representation

In the group algebra construction, what was needed in order for completion to C∗(G) tobe possible was a bound on the representations of each [g]. The same is true for Hecke algebras.

20

2.22. Lemma. If the Hecke pair (G,S) has the property that the nondegenerate ∗-representa-tions of each double coset are bounded, then C∗(S\G/S) exists.

Proof. For f ∈ H(S\G/S),

sup{ ‖ρ(f)‖ : ρ is a nondegenerate ∗-representation of H(S\G/S)}

≤∑

g∈S\G/S|f(g)| ‖ [SgS] ‖ <∞

since ‖ [SgS] ‖ is finite for all g ∈ G. 2

2.3.5 Example: SL2(Qp) with an Iwahori subgroup

Let G be SL2(Qp) and B the subgroup of SL2(Zp) given by

B ={(

a bc d

)∈ SL2(Zp) : c = 0 (mod p)

}.

We will discuss this example in much more detail in Section 6.1. For the moment, it sufficesto state that (G,B) is a Hecke pair, and B is called an Iwahori subgroup of G. Moreover, it isknown that H(B\G/B) is has two generators, X and Y , which satisfy

1. X∗ = X and Y ∗ = Y

2. X2 = p[B] + (p− 1)X and Y 2 = p[B] + (p− 1)Y

Since we can rewrite the second relation as (X − p[B]) ∗ (X + [B]) = 0, the spectral radius of X(and Y ) is p, so the norms of ∗-representations of X and Y are bounded by p.

2.23. Proposition. The Hecke algebra of SL2(Qp) with an Iwahori subgroup can be completedto the Hecke C∗-algebra. 2

In general, let G be SLn(Qp) and B its Iwahori subgroup. Then H(B\G/B) has nself-adjoint generators Xi which satisfy

X2i

= p[B] + (p− 1)Xi.

By the same reasoning, we have

2.24. Proposition. The Hecke algebra of SLn(Qp) with an Iwahori subgroup can be completedto the Hecke C∗-algebra. 2

21

Chapter 3

Hecke Algebras and Representation Theory

3.1 Preliminaries

Suppose G is a discrete or locally compact, totally disconnected group and S is an almostnormal subgroup (or open compact subgroup of a topological group). For all f in the Heckealgebra H(S\G/S), the supremum norm ‖f‖ is defined

‖f‖ = sup{‖ρ(f)‖ : ρ is a nondegenerate ∗-representation of H(S\G/S)}.

If ‖f‖ is finite for all f ∈ H(S\G/S), the Hecke C∗-algebra C∗(S\G/S) is the completion ofH(S\G/S) in the supremum norm (Definition 2.20). The Hecke pair(SL2(Qp), SL2(Zp)) is an example for which the Hecke C∗-algebra does not exist (see Sec-tion 2.3). However, we have also presented examples for which the Hecke C∗-algebra does exist(for instance, Hecke algebras with bounded representations). We will now study further thequestion of the existence of C∗(S\G/S).

The Hecke algebra arises naturally as the commutant of the left regular representation ofG on C[G/S] (that is, the representation

g(f)(γ) = f(g−1γ)

for f ∈ C[G/S] and g, γ ∈ G). For h ∈ H(S\G/S), let Rh : C[G/S]→ C[G/S] be defined by

Rh(f) = f ∗ h

for all f ∈ C[G/S]. Then Rh commutes with the action of G on C[G/S]. Let

H(S\G/S)op ∼= H(S\G/S)

be the algebra of elements in H(S\G/S) with the “opposite” product: f1 ◦ f2 = f2 ∗ f1. In fact,H(S\G/S)op is the full intertwining algebra for the left regular representation of G on C[G/S].

The form of the supremum norm in the definition of C∗(S\G/S) suggests that we shouldinvestigate the representation theory of H(S\G/S). However, for this purpose it is not appro-priate to view H(S\G/S) as an intertwining algebra. Rather, we will make use of a constructionfrom the world of Hilbert modules to form a correspondence between certain unitary represen-tations of G and the nondegenerate ∗-representations of H(S\G/S).

Representations of a group algebra are in one-to-one correspondence with representationsof the group itself, as explained in Section 3.2. The Hecke algebra situation is more subtle. Sup-pose V is a left G-module and V S is the subspace of vectors fixed by a subgroup S ⊂ G. We saythat a representation ρ : G→ Aut(V ) is generated by its S-fixed vectors if ρ(G)V S spans a densesubspace of V . Our original motivation was to investigate the following question: Let (G,S) bea Hecke pair. Is the category of nondegenerate ∗-representations ofH(S\G/S) equivalent to the category of unitary representations of G which are generated by theirS-fixed vectors?If the answer to this question were “yes”, then ‖f‖ would be finite for all f ∈ H(S\G/S).

22

Therefore, the Hecke pair (SL2(Qp), SL2(Zp)) provides an example where the answer is nega-tive. However, if we make some additional assumptions about the group and its almost normalsubgroup, we can prove a limited version of this category equivalence.

In Section 3.2 we consider the relationship between (not necessarily unitary) representa-tions of a groupG which are generated by their S-fixed vectors and nondegenerate representationsof the associated Hecke algebra H(S\G/S). Section 3.3 is an exploration of the history of theabove question as approached through representation theory. Finally, in Section 3.4 we considerH(S\G/S) as a ∗-algebra. In this case, we show that under certain circumstances the categoryof unitary representations of G which are generated by their S-fixed vectors is equivalent to thecategory of nondegenerate ∗-representations of H(S\G/S).

3.2 Representations of groups and Hecke algebras

Suppose A is an algebra and V a vector space. We will denote a representation ρ : A →End(V ) by the pair (V, ρ). If G is a group, a representation ρ : G→ Aut(V ) will also be denotedby the pair (V, ρ). For the time being we will consider algebras without involution.

3.2.1 Representations of group algebras

Representations of a discrete group G are in one-to-one correspondence with nondegener-ate representations of its group algebra C[G]. Indeed, suppose (V, ρ) is a representation of G ona vector space V . Then for f ∈ C[G], the formula

ρ(f) =∑g∈G

f(g) ρ(g)

defines a nondegenerate representation of C[G]. On the other hand, suppose (W,π) is a nonde-generate representation of C[G]. Then the formula

π(g) = π([g])

defines a representation of G on W . This correspondence extends to locally compact non-discretegroups, though in this case some extra hypotheses are needed. For instance, if G is totallydisconnected, C[G] should be replaced by C∞

c(G) (the locally constant, compactly supported

functions on G), and the representations under consideration should be “smooth” [8, p. 176].

3.2.2 Representations of Hecke algebras

Let G be a discrete or locally compact, totally disconnected group with an open almostnormal subgroup S. In general, representations of G do not correspond bijectively with represen-tations of H(S\G/S). The situation is more promising when we focus on the subspace of S-fixedvectors.

Given a representation (V, ρ) of G, we can define a representation (V S , ρ) of H(S\G/S)as follows.

3.1. Definition. Let (V, ρ) be a representation of a group G with almost normal subgroup S.Let ρ : H(S\G/S)→ End(V ) be determined by

ρ(f)v =∑

γ∈G/Sf(γ) ρ(γ)v

for f ∈ H(S\G/S) and v ∈ V S .

23

Note that the expression f(γ) ρ(γ)v depends only on γS, since f(γs) = f(γ) and ρ(γs)v =ρ(γ)v for v ∈ V S . The pair (V S , ρ) is a nondegenerate representation of H(S\G/S). Indeed, fors ∈ S, f ∈ H(S\G/S), and v ∈ V S ,

ρ(s)(ρ(f)v) =∑

γ∈G/Sf(γ) ρ(s)ρ(γ)v = ρ(s(f))v = ρ(f)v,

and hence ρ(f)v ∈ V S . One checks that ρ(f1 ∗ f2) = ρ(f1) ◦ ρ(f2). Moreover, ρ([S]) is theidentity on V S , so the representation is nondegenerate.

Suppose that (V, ρ) and (W,π) are representations of G. A linear transformation Φ : V →W is a G-intertwiner if π(g)Φ(v) = Φ ρ(g)(v) for all g ∈ G and v ∈ V .

3.2. Lemma. Let (G,S) be a Hecke pair and (V, ρ) and (W,π) be representations of G. IfΦ : V →W is a G-intertwiner, then Φ maps V S to WS and this restricted map intertwines theactions of H(S\G/S) on V S and WS .

Proof. Let ΦS be the restriction of Φ to V S . The image of V S under ΦS lies in WS ,since for all s ∈ S and v ∈ V S , we have

π(s)ΦS(v) = π(s)Φ(v) = Φ ρ(s)(v) = ΦS(v).

In addition, for f ∈ H(S\G/S), π(f) ΦS = ΦS ρ(f) follows from π(g)Φ = Φρ(g) for all g ∈ G.Therefore, ΦS intertwines the actions of H(S\G/S) on V S and WS . 2

We have seen that to a representation of a group G there corresponds naturally a nonde-generate representation of the Hecke algebra H(S\G/S) by the assignment V → V S . Given anondegenerate representation of the Hecke algebra, we would like to recover a representation ofthe group. Recall thatH(S\G/S)op is the intertwining algebra of the leftG-action on C[G/S]. Inparticular, C[G/S] has the structure of a right H(S\G/S)-module. If (W,π) is a nondegeneraterepresentation of H(S\G/S), we can form the tensor product

C[G/S]⊗H(S\G/S) W,

meaning the quotient of C[G/S]⊗W by the vector space span of the set

{(f ∗ h⊗ w)− (f ⊗ π(h)w) : f ∈ C[G/S], h ∈ H(S\G/S), and w ∈W}.

The left regular representation of G on C[G/S] gives a G-action π onC[G/S]⊗H(S\G/S) W by

π(g)∑i

(fi ⊗ wi) =∑i

g(fi)⊗ wi

for g ∈ G, fi ∈ C[G/S], and wi ∈ W .Since

π(G)(C[G/S]⊗H(S\G/S) W ) = span(π(G)([S]⊗H(S\G/S) W ))

and[S]⊗H(S\G/S) W ⊂ (C[G/S]⊗H(S\G/S) W )S ,

the vector space C[G/S]⊗H(S\G/S) W is generated by its S-fixed vectors.

24

If Ψ : V →W is an intertwiner for the actions of H(S\G/S) on V and W , then let

Ψ : C[G/S]⊗H(S\G/S) V −→ C[G/S]⊗H(S\G/S) W

be defined by

Ψ

(∑i

fi ⊗ vi

)=∑i

fi ⊗Ψ(vi)

for fi ∈ C[G/S] and vi ∈ V . Since

Ψ(f ∗ h⊗ v) = (f ⊗ π(h) Ψ(v)) = (f ⊗Ψ(ρ(h)v)) = Ψ(f ⊗ ρ(h)v)

for f ∈ C[G/S], h ∈ H(S\G/S), and v ∈ V , Ψ is a well-defined map fromC[G/S]⊗H(S\G/S) V to C[G/S]⊗H(S\G/S) W .

3.3. Lemma. Let (G,S) be a Hecke pair, and let (V, ρ) and (W,π) be nondegenerate represen-tations of H(S\G/S). If Ψ : V → W is an intertwiner for the actions of H(S\G/S) on V andW , then Ψ intertwines the G-actions on C[G/S]⊗H(S\G/S) V and C[G/S]⊗H(S\G/S) W .

Proof. For all∑i fi ⊗ vi ∈ C[G/S]⊗H(S\G/S) V and g ∈ G,

π(g)Ψ(∑i

fi ⊗ vi) =∑i

g(fi)⊗Ψ(vi) = Ψ ρ(g)(∑i

fi ⊗ vi).

2

The following function will be useful in the proof that

W ∼= (C[G/S]⊗H(S\G/S) W )S .

3.4. Definition. Define A : C[G/S]→ H(S\G/S) by

Af(g) =1

L(g)

∑γ∈SgS/S

f(γ)

for f ∈ C[G/S] and g ∈ G.

Think of A as the “averaging function” from C[G/S] to H(S\G/S). Note that Af = f ifand only if f ∈ H(S\G/S).

Let Γ ⊂ G be a set of representatives for the disjoint left cosets of S in G. For f ∈ C[G/S],define a set Γf ⊂ Γ to be

Γf = {γ ∈ Γ : f(γ) 6= 0}.

In other words, Γf is a (finite) set of coset representatives for the left cosets on which f issupported. Let the subgroup SΓf

⊂ S be given by

SΓf= S ∩

⋂γ∈Γf

γSγ−1 =⋂γ∈Γf

Sγ .

Then [S : SΓf] is finite, since [S : Sγ ] is finite for each γ and Γf is a finite set.

25

3.5. Lemma. For all f ∈ C[G/S],

Af =1

[S : SΓf]

∑s∈S/SΓf

s−1(f). (3.1)

Proof. The proof of Equation 3.1 amounts to showing that if f ∈ C[G/S], then

1L(g)

∑γ∈SgS/S

f(γ) =1

[S : SΓf]

∑s∈S/SΓf

f(sg) (3.2)

for all g ∈ G. Fix g ∈ G. If f(sg) = 0 for all s ∈ S, then both sides of Equation 3.2 equal zero.Otherwise, the right hand side of Equation 3.2 can be written

1[S : SΓf

]

∑s∈S/SΓf

f(sg) =1

[S : SΓf]

∑s∈S/SΓf

∑γ∈SgS/S

f(γ) [γS](sg)

=∑

γ∈SgS/S

1[S : SΓf

]f(γ)

∑s∈S/SΓf

[γS](sg)

=∑

γ∈SgS/S

1[S : SΓf

]f(γ) · [Sγ : SΓf

] =1

L(g)

∑γ∈SgS/S

f(γ).

2

3.6. Lemma. Let (G,S) be a Hecke pair and let (W,π) be a nondegenerate representation ofH(S\G/S). Then

W ∼= (C[G/S]⊗H(S\G/S) W )S

as vector spaces.

Proof. We claim that the map from W to (C[G/S]⊗H(S\G/S) W )S given by

w → [S]⊗ w

and the map from (C[G/S]⊗H(S\G/S) W )S to W given by∑i

fi ⊗ wi →∑i

π(Afi)wi

for∑i fi ⊗ wi ∈ (C[G/S] ⊗H(S\G/S) W )S are inverse to one another. Applying the first map

and then the second to w ∈W , we have

w 7−→ [S]⊗ w 7−→ A[S]w = w.

Now suppose∑i fi ⊗ wi ∈ C[G/S]⊗H(S\G/S) W , consider the mapping∑

i

fi ⊗ wi 7−→∑i

π(Afi)wi 7−→ [S]⊗∑i

π(Afi)wi.

26

Let the subset SΓ⊂ S be given by

SΓ

=⋂i

SΓfi.

Since {i ∈ I} is a finite set, [S : SΓ

] is finite. Then

1[S : S

Γ]

∑s∈S/S

Γ

∑i

s−1(fi)⊗ wi =∑i

Afi ⊗ wi.

On the other hand,∑i fi ⊗ wi is an S-fixed vector, so∑i

s−1(fi)⊗ wi = s−1∑i

fi ⊗ wi =∑i

fi ⊗ wi

and hence∑i fi ⊗ wi =

∑iAfi ⊗ wi. Therefore, the mapping∑

i

fi ⊗ wi 7−→ [S]⊗∑i

π(Afi)wi =∑i

Afi ⊗ wi

is the identity on (C[G/S]⊗H(S\G/S) W )S . This proves the lemma. 2

Lemma 3.6 also proves the following:

3.7. Corollary. Let G be a group with almost normal subgroup S. Then every nondegeneraterepresentation of the Hecke algebra H(S\G/S) is obtained from a representation (V, ρ) of thegroup G by the assignment (V, ρ)→ (V S , ρ). 2

3.8. Definition. If (V, ρ) and (W,π) are representations of G, let HomG(V,W ) be the set ofG-equivariant homomorphisms from V to W—that is

HomG(V,W ) = {Φ ∈ Hom(V,W ) : π(g)Φ(v) = Φ(ρ(g)v)}

for all g ∈ G and v ∈ V . Likewise HomH(S\G/S)(V,W ) denotes the elements of Hom(V,W )which are intertwiners for the action of H(S\G/S).

The following theorem appears as Proposition I.2.5 in Borel [2], though his notation isslightly different.

3.9. Theorem (Borel, 1976). Let (V, ρ) be a representation of G and (W,π) a nondegeneraterepresentation of H(S\G/S). Then there is an isomorphism

HomG(C[G/S]⊗H(S\G/S) W, V ) ∼= HomH(S\G/S)(W,V S)

for which a G-equivariant homomorphism Φ : C[G/S] ⊗H(S\G/S) W → V is mapped to the

homomorphism which sends w ∈ W to Φ([S]⊗ w) ∈ V S .

Proof. We will construct maps

HomG(C[G/S]⊗H(S\G/S) W, V )σ−→←−τ

HomH(S\G/S)(W,V S)

and show that σ and τ are inverses.

27

If Φ ∈ HomG(C[G/S]⊗H(S\G/S) W, V ), then

ΦS ∈ HomH(S\G/S)((C[G/S]⊗H(S\G/S) W )S , V S), by Lemma 3.2. Since

W ∼= (C[G/S]⊗H(S\G/S) W )S ,

by ι : w 7→ ([S]⊗ w) (Lemma 3.6), we can form

σ(Φ) = ΦS ◦ ι ∈ HomH(S\G/S)(W, V S),

whereσ(Φ) : w 7−→ ΦS([S]⊗ w) = Φ([S]⊗ w)

for w ∈W .Now suppose Ψ ∈ HomH(S\G/S)(W,V S). Then

Ψ ∈ HomG(C[G/S]⊗H(S\G/S) W, C[G/S]⊗H(S\G/S) VS),

by Lemma 3.3. Define η : C[G/S]⊗H(S\G/S) VS → V by

η(f ⊗ v) = ρ(f)v,

for f ∈ C[G/S] and v ∈ V S . Then we set

τ(Ψ) = η ◦Ψ ∈ HomG(C[G/S]⊗H(S\G/S) W, V ),

whereτ(Ψ) : (f ⊗ w) 7−→ ρ(f)Ψ(w),

for f ∈ C[G/S] and w ∈W .Now for f ⊗ w ∈ C[G/S]⊗H(S\G/S) W ,

τ ◦ σ(Φ) : f ⊗ w 7−→ ρ(f)σ(Φ)w = ρ(f)(Φ([S]⊗ w)).

Since Φ is G-equivariant, ρ(f)(Φ([S]⊗ w)) = Φ(f ⊗w), and τ ◦ σ is the identity. Given w ∈W ,

σ ◦ τ(Ψ) : w 7−→ τ(Ψ)([S] ⊗ w) = ρ([S])Ψ(w) = Ψ(w),

which completes the proof. 2

In the language of category theory we have proved the following:

3.10. Corollary. The functor from nondegenerate representations of H(S\G/S) to representa-tions of G given by

(W,π) 7→ (C[G/S]⊗H(S\G/S) W,π) and Ψ 7→ Ψ

is the left adjoint of the functor from representations of G to nondegenerate representations ofH(S\G/S) given by

(V, ρ) 7→ (V S , ρ) and Φ 7→ ΦS . 2

28

3.3 Representations of the Iwahori Hecke algebra

The following is based on work of Borel [2]. Let G be a semi-simple group over a non-Archimedean local field, with S a compact open subgroup of G. Let ρ : G → Aut(V ) be arepresentation of G. We say that the representation ρ is smooth if Gv = {g ∈ G : gv = v} isopen in G for all v ∈ V . Borel proved that every representation of H(S\G/S) occurs as therepresentation (V S , ρ) obtained from a smooth representation (V, ρ) of G, where ρ is defined asin Section 3.2.1

Let S be a compact open subgroup of G. Given a representation (W,π) of H(S\G/S),there are two smooth G-modules which arise from W :

• the induced module I(W ) = Cc(G/S)⊗H(S\G/S) W

• the produced module P (W ) defined byP (W ) = {f ∈ C(G/S,W ) : f ∗ h = π(h)f for all h ∈ H(S\G/S) }.

where C(G/S,W ) is the set of functions on G/S valued in W . Borel proved that both I(W )S

and P (W )S are isomorphic to W . If B is an Iwahori subgroup of G, and (W,π) is a H(B\G/B)-module, he proved that I(W ) ∼= P (W ) and I(W ) and P (W ) are irreducible if and only if W is(see [2, p. 249]), which implies the following theorem:

3.11. Theorem. Let B be an Iwahori subgroup of a semi-simple group G over a non-Archimedeanlocal field. The category of representations of H(B\G/B) is equivalent to the category of smoothrepresentations of G which are generated by their B-fixed vectors.

3.4 ∗-Representations

Suppose A is a ∗-algebra and V a Hilbert space. We will denote a representation ρ : A →B(V ) by the pair (V, ρ). The representation ρ is a ∗-representation if ρ(a∗) = (ρ(a))∗ for a ∈ A.If G is a group, a unitary representation ρ : G→ U(V ) will also be denoted by the pair (V, ρ).

If V is a Hilbert space, then V S is a Hilbert space, since it inherits an inner product fromV and the limit of S-invariant vectors is S-invariant. Given a unitary representation (V, ρ) of Gwe wish to define a nondegenerate ∗-representation (V S , ρ) of H(S\G/S). The formula we usedin Section 3.2,

ρ(f)v =∑

γ∈G/Sf(γ) ρ(γ)v

for f ∈ H(S\G/S) and v ∈ V S , defines a homomorphism from H(S\G/S) to B(V S). But thisis not in general a ∗-representation of H(S\G/S), as the following example illustrates. For allg ∈ G, v, w ∈ V S ,

〈ρ([SgS])v, w〉 =∑

s∈S/Sg〈ρ(sg)v, w〉 =

∑s∈S/Sg

〈ρ(g)v, ρ(s−1)w〉 = L(g)〈ρ(g)v, w〉

(recall that Sg = S ∩ gSg−1 and [S : Sg] = L(g)), while

〈v, ρ∗([SgS])w〉 = 〈v, ρ([Sg−1S])w〉 = L(g−1)〈v, ρ(g−1)w〉 = R(g)〈ρ(g)v, w〉.

1Actually, Borel required representations of G to be admissible, meaning ρ : G→ Aut(V ) is smooth

and VU

is finite-dimensional for every open subgroup U ⊂ G. However, all of Borel’s results hold if weonly require smoothness [7, p. 583].

29

The problem lies with the fact that the number of left cosets in a double coset doesnot equal the number of right cosets, in general. However, we can make use of the functionΞ : G→ Q+ defined by

Ξ(g) =R(g)L(g)

,

which is a homomorphism of groups (Theorem 2.7).

3.12. Definition. Let (V, ρ) be a unitary representation of G. For f ∈ H(S\G/S), let ρ(f) begiven by

ρ(f)v =∑

γ∈G/Sf(γ) Ξ

12 (γ) ρ(γ)v (3.3)

for all v ∈ V S .

Observe that the sum is finite and ρ(f) does not depend on the choice of left coset repre-sentatives γ. Indeed, if γ′ ∈ γS, then γ′ = γs for some s ∈ S, and

f(γ′) Ξ12 (γ′) ρ(γ′)v = f(γs) Ξ

12 (γs) ρ(γs)v = f(γ) Ξ

12 (γ) ρ(γ)v,

since S is contained in the kernel of Ξ and v is fixed by S. Therefore, Equation 3.3 determinesa well-defined linear map from H(S\G/S) to B(V S).

3.13. Lemma. If (V, ρ) is a unitary representation of a group G, then Equation 3.3 defines anondegenerate ∗-representation (V S , ρ) of H(S\G/S).

Proof. First we will show that if f1, f2 ∈ H(S\G/S) , then ρ(f1 ∗ f2) = ρ(f1) ρ(f2). Forall v ∈ V S ,

ρ(f1 ∗ f2)v =∑

γ∈G/S(f1 ∗ f2)(γ) Ξ

12 (γ) ρ(γ)v

=∑

γ∈G/S

∑η∈G/S

f1(η) f2(η−1γ) Ξ12 (γ) ρ(γ)v

=∑

η∈G/S

∑κ∈G/S

f1(η) f2(κ) Ξ12 (ηκ) ρ(ηκ)v = ρ(f1) ρ(f2)v,

so ρ is a representation of H(S\G/S). In addition, ρ is nondegenerate becauseρ([S])V S = V S .

Moreover, ρ is a ∗-representation. For g ∈ G, we defined the subgroup Sg to be Sg =

gSg−1 ∩ S (Definition 2.3), and recall that [S : Sg] = L(g). If v, w ∈ V S and if f ∈ H(S\G/S),then

〈ρ(f)v, w〉 =∑

γ∈G/S〈f(γ) Ξ

12 (γ) ρ(γ)v, w〉

=∑

g∈S\G/S

∑s∈S/Sg

〈f(sg) Ξ12 (sg) ρ(sg)v, w〉

=∑

g∈S\G/SL(g)〈v, f(g) Ξ

12 (g) ρ(g−1)w〉

30

=∑

γ∈S\G

L(γ)R(γ)

〈v, f(γ) Ξ12 (γ) ρ(γ−1)w〉.

If {γ ∈ G/S} is a set of representatives for the left cosets, then {γ−1 : γ ∈ G/S} is a completeset of representatives for the right cosets. Hence

〈ρ(f)v, w〉 =∑

η∈G/SΞ−1(η−1)〈v, f(η−1) Ξ

12 (η−1) ρ(η)w〉

=∑

η∈G/S〈v, f∗(η) Ξ

12 (η) ρ(η)w〉 = 〈v, ρ(f∗)w〉.

This proves the lemma. 2

In fact, Definition 3.12 and the following lemma allow us to construct a functor fromunitary representations of G to nondegenerate ∗-representations of H(S\G/S).

3.14. Lemma. Let (G,S) be a Hecke pair and (V, ρ) and (W,π) be unitary representations ofG. If Φ : V →W is a G-intertwiner then Φ maps V S to WS and this restricted map intertwinesthe actions of H(S\G/S) on V S and WS .

Proof. Lemma 3.13 shows that (V S , ρ) is a nondegenerate ∗-representation of H(S\G/S).The rest of the proof is essentially the same as Lemma 3.2. 2

3.4.1 A Hecke-algebra-valued bilinear form

We have seen that a unitary representation of a group G corresponds to a nondegen-erate ∗-representation of the Hecke algebra H(S\G/S). Given a nondegenerate ∗-representa-tion of the Hecke algebra, we would like to recover a unitary representation of the group. If(W,ρ) is a nondegenerate ∗-representation of H(S\G/S), we can again form the tensor productC[G/S]⊗H(S\G/S)W . In order to recover a unitary representation ofG on C[G/S]⊗H(S\G/S)W

we need two things:

1. An inner product on the vector space C[G/S]⊗H(S\G/S) W which makes it a pre-Hilbertspace.

2. A representation of G on C[G/S] ⊗H(S\G/S) W so that 〈g(h1), g(h2)〉 = 〈h1, h2〉 for allg ∈ G and all h1, h2 ∈ C[G/S]⊗H(S\G/S) W .

Let f1, f2 ∈ C[G/S] and v, w ∈W . We shall define an “inner product” on C[G/S]⊗H(S\G/S)W by

〈f1 ⊗ v, f2 ⊗ w〉 = 〈v, ρ(〈〈f1, f2〉〉)w〉 (3.4)

where 〈〈, 〉〉 is a bilinear form on C[G/S] valued in H(S\G/S). We shall want 〈〈, 〉〉 to satisfy fourrequirements:

G1 If f1, f2 ∈ C[G/S] and h ∈ H(S\G/S), then

〈〈f1, f2 ∗ h〉〉 = 〈〈f1, f2〉〉 ∗ h.

If this holds then 〈〈, 〉〉 is compatible with the relation f ∗h⊗ v = f ⊗ρ(h)v for f ∈ C[G/S],h ∈ H(S\G/S), and v ∈W , in the sense that

〈f1 ∗ h⊗ v, f2 ⊗ w〉 = 〈f1 ⊗ ρ(h)v, f2 ⊗ w〉.

31

G2 If f1, f2 ∈ C[G/S], then〈〈f1, f2〉〉 = 〈〈f2, f1〉〉

∗.

This implies that the form (3.4) satisfies 〈v, w〉 = 〈w, v〉 for v, w ∈ W .

G3 For all g ∈ G and f1, f2 ∈ C[G/S],

〈〈Ξ−12 (g) g(f1), Ξ−

12 (g) g(f2)〉〉 = 〈〈f1, f2〉〉,

and therefore g(f ⊗ w) = Ξ−12 (g) g(f)⊗ w is a unitary G-action on

C[G/S]⊗H(S\G/S) W .

G4 For all f ∈ C[G/S], the product 〈〈f, f〉〉 is a positive element of H(S\G/S), in the sense of∗-algebras. An element a of a ∗-algebra A is positive if it can be expressed as a finite sumof the form a =

∑i a∗iai for ai ∈ A. For emphasis, we will say that a is positive, in the

sense of ∗-algebras or write a ≥ 0. Positivity of 〈〈, 〉〉 implies that

〈f ⊗ v, f ⊗ v〉 = 〈v, ρ(〈〈f, f〉〉)v〉 ≥ 0

for all f ∈ C[G/S] and for all v ∈ W .

3.15. Definition. Let V be a vector space with an inner product, and let V 0 be the subspace ofvectors v ∈ V such that ‖v‖ = 0. Then we define the Hilbert space completion of V to be thecompletion of V/V 0 in the norm ‖v‖ = 〈v, v〉. This completion is a Hilbert space which we willdenote V . We will call V a pre-Hilbert space.

3.16. Proposition. If the bilinear form 〈〈, 〉〉 satisfies properties G1–G4, then for any nondegen-erate ∗-representation (W,π) of H(S\G/S), the form (3.4) on the vector space C[G/S]⊗H(S\G/S)W makes it a pre-Hilbert space, and C[G/S]⊗H(S\G/S) W is a unitary G-module. 2

If 〈〈, 〉〉 satisfies G1–G4, we will denote the Hilbert space C[G/S]⊗H(S\G/S) W by HW .

3.4.2 The product formula

Let 〈〈, 〉〉 be given by

〈〈f1, f2〉〉(g) =1

L(g)

∑γ∈G/Sg

f1(γ) f2(γg) Ξ(γ), (3.5)

for f1, f2 ∈ C[G/S]. Observe that for all σ ∈ Sg = S ∩ gSg−1,

f1(γσ) f2(γσg) Ξ(γσ) = f1(γ) f2(γg) Ξ(γ),

so the expression f1(γ) f2(γg) Ξ(γ) is unchanged by a different choice of coset representative.The following fact will prove useful in the proof of Lemma 3.17. Suppose G is a group and

T is a subgroup of G. Then for any g ∈ G, the set {γg−1 : γ ∈ G/T } is a set of representativesof left cosets of the subgroup gTg−1 in G.

3.17. Lemma. Equation 3.5 defines a product

〈〈, 〉〉 : C[G/S]×C[G/S]→ H(S\G/S)

which satisfies properties G1–G3 above.

32

Proof. First we will show that 〈〈f1, f2〉〉 is in H(S\G/S). For s ∈ S, 〈〈f1, f2〉〉(gs) =〈〈f1, f2〉〉(g) is clear. Note that Ssg = sgSg−1s−1 ∩ S = sSgs

−1 and hence

{γs−1 : γ ∈ S/Sg} forms a set of representatives for left cosets of Ssg in S. Now compute

〈〈f1, f2〉〉(sg) =1

L(sg)

∑γ∈G/Ssg

f1(γ) f2(γsg) Ξ(γ)

=1

L(g)

∑γ∈G/Sg

f1(γs−1) f2(γs−1sg) Ξ(γs−1)

=1

L(g)

∑γ∈G/Sg

f1(γ) f2(γg) Ξ(γ) = 〈〈f1, f2〉〉(g).

Therefore, 〈〈f1, f2〉〉 ∈ H(S\G/S).G1 For any g ∈ G and h ∈ H(S\G/S),

〈〈f1, f2 ∗ h〉〉(g) =1

L(g)

∑γ∈G/Sg

f1(γ) (f2 ∗ h)(γg) Ξ(γ)

=1

L(g)

∑γ∈G/Sg

f1(γ)∑

η∈G/Sf2(η)h(η−1γg) Ξ(γ)

=1

L(g)

∑κ∈G/S

∑γ∈G/Sg

f1(γ) f2(γκ) Ξ(γ)h(κ−1g).

Now we wish to compute the sum over G/(Sg ∩ Sκ). Note that [Sg : Sg ∩ Sκ] is finite and

equals [S ∩ gSg−1 : S ∩ gSg−1 ∩ κSκ−1] = [S : S ∩ κSκ−1] = L(κ). By similar reasoning,[Sκ : Sκ ∩ Sg ] = L(g). So

〈〈f1, f2 ∗ h〉〉(g) =1

L(g)

∑κ∈G/S

∑γ∈G/(Sg∩Sκ)

1L(κ)

f1(γ) f2(γκ) Ξ(γ)h(κ−1g)

=∑

κ∈G/S

1L(κ)

∑γ∈G/Sκ

f1(γ) f2(γκ) Ξ(γ)h(κ−1g)

=∑

κ∈G/S〈〈f1, f2〉〉(κ)h(κ−1g) = 〈〈f1, f2〉〉 ∗ h(g)

G2 For any g ∈ G,

〈〈f1, f2〉〉∗(g) = 〈〈f1, f2〉〉(g

−1)

=1

L(g−1)

∑γ∈G/S

g−1

f1(γ) f2(γg−1) Ξ(γ).

33

Now Sg−1 = S ∩ g−1Sg = gSgg

−1, so

〈〈f1, f2〉〉∗(g) =

1R(g)

∑κ∈G/Sg

f1(κg) f2(κgg−1) Ξ(κg)

=1

R(g)Ξ(g)

∑κ∈G/Sg

f2(κ) f1(κg) Ξ(κ) = 〈〈f2, f1〉〉(g).

G3 For g, γ ∈ G and f1, f2 ∈ C[G/S],

〈〈Ξ−12 (g)g(f1),Ξ−

12 (g)g(f2)〉〉(γ)

=1

L(γ)

∑η∈G/Sγ

f1(g−1η) f2(g−1ηγ) Ξ(g−1η) = 〈〈f1, f2〉〉(γ).

We see that the product we have defined satisfies all our requirements except G4, positivity.2

3.18. Corollary. If (G,S) is a Hecke pair such that G4 holds, and (W,ρ) is a nondegenerate∗-representation of H(S\G/S), then C[G/S]⊗H(S\G/S) W is a pre-Hilbert space and

ρ : G −→ U(HW )

given byρ(g)

∑i

(fi ⊗ wi) =∑i

g(fi ⊗ wi)

for g ∈ G, fi ∈ C[G/S], and wi ∈W is a unitary representation of G. 2

The following example shows that G4 does not hold for all Hecke pairs.

3.4.3 The Hecke pair (SL2(Qp), SL2(Zp))

Fix a prime p. Let G = SL2(Qp) and P = SL2(Zp), and let d =(p 00 1

p

)and

d′ =(p 10 1

p

). Suppose f ∈ C[G/P ] is given by

f = [dP ] + [d′P ].

Recall that H(P\G/P ) is abelian. Then

〈〈f, f〉〉 = 2[P ] + 2 〈〈[dP ], [d′P ]〉〉

= 2[P ] +2

L(d−1d′)[Pd−1d′P ] = 2[P ] +

2L(d)

[PdP ],

since d−1d′ =(

1 1p

0 1

). Now we apply the Satake isomorphism S : H(P\G/P )→ C[x, x−1]:

S([PdP ]) = px+ px−1 + 1.

34

Therefore, 〈〈f, f〉〉 corresponds to 2 + 2p(p+1)

(px + px−1 + 1) under the Satake isomorphism

(see Theorem 2.18). Evaluation of this polynomial at any real number is a representation ofH(P\G/P ); in particular, since we can evaluate at a negative number, 〈〈f, f〉〉 may be negative.

3.19. Proposition. The product 〈〈, 〉〉 has negative values for the Hecke pair(SL2(Qp), SL2(Zp)). 2

If we start with a representation of Hecke algebra which is actually the restriction V S ofa unitary representation V of the group, then G4 holds. The proof is contained in the proof ofthe following lemma.

3.20. Lemma. Let (G,S) be a Hecke pair and (V, ρ) a unitary representation of G. ThenC[G/S]⊗H(S\G/S) V

S is a pre-Hilbert space and

HV S∼= span(ρ(G)V S).

Proof. The form (3.5) satisfies all requirements for an inner product except positivity. Forf1, f2 ∈ C[G/S] and v, w ∈ V S ,

〈f1 ⊗ v, f2 ⊗ w〉 = 〈v, ρ(〈〈f1, f2〉〉)w〉

= 〈v,∑

g∈G/SΞ

12 (g) 〈〈f1, f2〉〉(g) ρ(g)w〉

= 〈v,∑

g∈G/S

∑γ∈G/Sg

1L(g)

Ξ(γ) Ξ12 (g) f1(γ) f2(γg) ρ(g)w〉

=∑

g∈G/S

∑γ∈G/Sg

〈Ξ12 (γ) f1(γ) ρ(γ)v,

1L(g)

Ξ12 (γg) f2(γg) ρ(γg)w〉

=∑

g∈G/S

∑η∈G/S

∑s∈S/Sg

〈Ξ12 (ηs) f1(ηs) ρ(ηs)v,

1L(g)

Ξ12 (ηsg) f2(ηsg) ρ(ηsg)w〉

=∑

η∈G/S

∑g∈G/S

∑κ∈SgS/S

Ξ12 (η) 〈f1(η) ρ(η)v,

1L(g)

Ξ12 (ηκ) f2(ηκ) ρ(ηκ)w〉

= 〈∑

η∈G/SΞ

12 (η) f1(η) ρ(η)v,

∑γ∈G/S

Ξ12 (γ) f2(γ) ρ(γ)w〉.

Extend ρ : G→ U(V ) to a mapping ρ : C[G/S]→ B(V S) by

ρ(f)v =∑

g∈G/SΞ

12 (g) f(g) ρ(g)v

35

where f ∈ C[G/S] and v ∈ V S . This mapping agrees with the representation (V S , ρ) ofH(S\G/S) ⊂ C[G/S], and, by the above calculation,

〈f1 ⊗ v, f2 ⊗ w〉 = 〈ρ(f1)v, ρ(f2)w〉.

In particular, this means that ‖f ⊗ v‖ = ‖ρ(f)v‖ and hence

〈∑i

fi ⊗ vi,∑j

fj ⊗ vj〉 = ‖∑i

ρ(fi)vi‖ ≥ 0,

so C[G/S]⊗H(S\G/S) VS is a pre-Hilbert space.

Moreover, we have shown that the mapping f⊗v 7→ ρ(f)v is an isometry from C[G/S]⊗H(S\G/S)

V S to the subspace ρ(C[G/S])V S = span(ρ(G)V S) ⊂ V and hence extends to an isomorphismof Hilbert spaces H

V S∼= span(ρ(G)V S). In other words, H

V Sis isomorphic to the unitary

G-module which is generated by the action of G on V S . 2

3.21. Corollary. If the representation (V, ρ) is generated by its S-fixed vectors,HV S

is isomorphic to V . 2

3.22. Corollary. Every unitary representation of G which is generated by its S-fixed vectors canbe obtained from a nondegenerate ∗-representation π : H(S\G/S) → B(W ) by the assignment(W,π) −→ (HW , π). 2

If Ψ : V → W is an intertwiner for nondegenerate ∗-representations of H(S\G/S) on Vand W , then let

Ψ : C[G/S]⊗H(S\G/S) V → C[G/S]⊗H(S\G/S) W

be defined, as in Section 3.2, by

Ψ

(∑i

fi ⊗ vi

)=∑i

(fi ⊗Ψ(vi))

for fi ∈ C[G/S] and vi ∈ V .

3.23. Lemma. Let (G,S) be a Hecke pair, and let (V, ρ) and (W,π) be nondegenerate ∗-repre-sentations of H(S\G/S). If Ψ : V → W is an intertwiner for the actions of H(S\G/S) on Vand W , then Ψ intertwines the unitary G-action onC[G/S]⊗H(S\G/S) V and C[G/S]⊗H(S\G/S) W given by

g(∑i

fi ⊗ vi) =∑i

Ξ−12 (g) g(fi)⊗ vi

for g ∈ G and∑i fi ⊗ vi in either C[G/S]⊗H(S\G/S) V or C[G/S]⊗H(S\G/S) W .

Proof. For all∑i fi ⊗ vi ∈ C[G/S]⊗H(S\G/S) V and g ∈ G,

π(g)Ψ(∑i

fi ⊗ vi) =∑i

Ξ−12 (g) g(fi)⊗Ψ(vi) = Ψ ρ(g)(

∑i

fi ⊗ vi).

2

Note that Ψ can be extended to a G-intertwiner HV → HW .

36

3.24. Lemma. Let (G,S) be a Hecke pair such that 〈〈, 〉〉 is positive and let (W,π) be a nonde-generate representation of H(S\G/S). Then

W ∼= (HW )S

as Hilbert spaces.

Proof. We showed in Lemma 3.6 that W ∼= (C[G/S]⊗H(S\G/S) W )S as vector spaces.In addition, the mapping

w 7−→ ([S]⊗ w)

for w ∈W preserves the inner product on the (pre-)Hilbert spaces, since

〈[S]⊗ v, [S]⊗ w〉 = 〈v, w〉

for all v, w ∈ W . Hence, W ∼= (HW )S as Hilbert spaces. 2

3.5 The category equivalence

3.25. Theorem. Let (G,S) be a Hecke pair such that the product 〈〈, 〉〉 is positive. The categoryof unitary representations of G generated by their S-fixed vectors is equivalent to the category ofnondegenerate ∗-representations of H(S\G/S).

Proof. We have shown

• (Lemma 3.20) If (V, ρ) is a unitary representation of G generated by its S-fixed vectors,then

V ∼= HV S

as Hilbert spaces, and

• (Lemma 3.24) If (W,π) is a nondegenerate ∗-representation of H(S\G/S), then

W ∼= (HW )S

as Hilbert spaces.

Let Φ : (V, ρ)→ (W,π) and Ψ : (W,π) → (Z, η) be morphisms of unitary representationsof G. Since ΦS(V S) ⊂ (Φ(V ))S , for all v ∈ V S ,

ΨS ◦ ΦS(v) = Ψ ◦ Φ(v) = (Ψ ◦ Φ)S(v).

Lemma 3.14 shows that ΦS intertwines the actions of H(S\G/S) on V S and WS . This allowsus to define the functor E from unitary representations of G generated by their S-fixed vectorsto nondegenerate ∗-representations of H(S\G/S) by

E(V, ρ) = (V S , ρ) and E(Φ) = ΦS

for (V, ρ) a unitary representation of G and Φ a morphism of unitary G-representations.Moreover, if Φ and Ψ are morphisms of nondegenerate ∗-representations of

H(S\G/S), we have Φ ◦Ψ = Φ ◦Ψ. Let F be the functor from nondegenerate ∗-representationsof the Hecke algebra to unitary representations of the group given by

F (W,π) = (HW , π) and F (Ψ) = Ψ

37

for (W,π) a representation of H(S\G/S) and Ψ a morphism of ∗-representations of H(S\G/S).Lemmas 3.20 and 3.24 show that E ◦ F and F ◦E are equivalent to the identity functors

on objects in their respective categories.

For each unitary G-module (V, ρ), let αV : F ◦ E(V )∼=→ V be the isomorphism described

in Lemma 3.20; that is, αV (∑i fi ⊗ vi) =

∑i ρ(fi)vi. Let Φ : (V, ρ) → (W,π) be a morphism

of unitary representations of G which are generated by their S-fixed vectors. We claim that thefollowing diagram commutes:

F ◦ E(V )αV−−−−→ V

F◦E(Φ)

y yΦ

F ◦ E(W )αW−−−−→ W

(3.6)

For all∑i fi ⊗ vi ∈ F ◦ E(V ) = H

V S, we have

Φ(αV (∑i

fi ⊗ vi)) = Φ(∑i

ρ(fi)vi).

On the other hand,

F ◦ E(Φ)(∑i

fi ⊗ vi) = ΦS(∑i

fi ⊗ vi) =∑i

fi ⊗ Φ(vi).

SinceαW (

∑i

fi ⊗ Φ(vi)) =∑i

π(fi) Φ(vi) = Φ(∑i

ρ(fi)vi),

the diagram (3.6) commutes.For each nondegenerate ∗-representation of the Hecke algebra (W,π), let βW : E◦F (W ) =

(HW )S∼=→ W be the isomorphism described in Lemma 3.24; that is, βW (

∑i fi ⊗ wi) =∑

i π(Afi)wi. Let Ψ : (W,π) → (Z, η) be a morphism of nondegenerate ∗-representations ofH(S\G/S). We claim that the following diagram commutes:

E ◦ F (W )βW−−−−→ W

E◦F (Ψ)

y yΨ

E ◦ F (Z)βZ−−−−→ Z

(3.7)

For all∑i fi ⊗ wi ∈ E ◦ F (W ), we have

Ψ(βW (∑i

fi ⊗ wi)) = Φ(∑i

π(Afi)wi).

On the other hand,

E ◦ F (Ψ)(∑i

fi ⊗ wi) = (Ψ)S(∑i

fi ⊗ wi) =∑i

Afi ⊗Ψ(wi).

SinceβZ (

∑i

Afi ⊗Ψ(wi)) =∑i

η(Afi) Ψ(wi) = Φ(∑i

π(Afi)wi),

38

the diagram (3.7) commutes. Therefore, the categories are equivalent. 2

39

Chapter 4

First Results

In this chapter we will investigate how our results in Chapter 3 apply to certain examplesof Hecke algebras. In particular, we will show that the form 〈〈, 〉〉 defined in Section 3.4 is positivefor lower directed Hecke algebras and finite Hecke algebras.

4.1 Lower directed Hecke pairs

This class of Hecke pairs is discussed by Brenken [4]. Let (G,S) be a Hecke pair. Firstwe will define the subset T ⊂ G of elements where left cosets and double cosets coincide.

4.1. Definition. If (G,S) is a Hecke pair let

T = {g ∈ G : L(g) = 1} = {g ∈ G : gS = SgS}.

We claim that T is a submoniod of G. Indeed, 1 ∈ T and for any α, β ∈ T , the product[SαS] ∗ [SβS] gives

[SαS] ∗ [SβS] = [αS] ∗ [SβS] =∑

b∈SβS/S[αbS] = [αβS].

But [SαS] ∗ [SβS] ∈ H(S\G/S), so [αβS] = [SαβS]. Hence αβ ∈ T .We will use T to define a pre-order on G. For any a, b ∈ G, define the relation a ≺∼ b by

a≺∼ b if and only if a−1b ∈ T .

This relation is

• reflexive, since 1 ∈ T .

• transitive, since a ≺∼ b and b ≺∼ c means that a−1b ∈ T and b−1c ∈ T . Therefore, a−1c ∈ T ,since T is closed under multiplication.

• invariant under left multiplication, since a ≺∼ b implies ga ≺∼ gb for all g ∈ G.

4.2. Definition. Let (G,S) be a Hecke pair. We say that (G,S) is lower directed if any twoelements of G have a common lower bound with respect to the relation ≺∼ (that is, for a, b ∈ G,there exists a c ∈ G so that c ≺∼ a and c ≺∼ b).

4.3. Lemma. The Hecke pair (G,S) is lower directed if and only if G = T −1T .

Proof. Suppose G is lower directed. Let l be a common lower bound for the pair a, b ∈ G.Then a−1b = a−1ll−1b ∈ T −1T . Setting a = 1 we get b ∈ T −1T for all b. If G = T −1T , thenfor a, b ∈ G, a−1b = s−1t for some s, t ∈ T . Then s−1 is a common lower bound for s−1t = a−1b

and 1. By left invariance, as−1 is a common lower bound for a and b. 2

40

Example The Hecke pair (P+Q, P+Z

) studied by Bost and Connes provides an example of alower directed pair. Let

G = P+Q

={(

1 a0 r

): a, r ∈ Q, a > 0

}S = P

+Z

={(

1 n0 1

): n ∈ Z

}.

Now we see why G is lower directed. If g1 =( 1 a1

0 r1

)and g2 =

( 1 a20 r2

), then

g−11g2 =

(1 a2 −

a1r2r1

0 r2r1

),

and g−11g2 ∈ T if and only if r2r1

∈ N. If g and h are two elements of G, then there exists anelement of G which clears the denominators in the lower left entries of g and h, and hence is acommon lower bound for g and h.

4.4. Lemma. Let (G,S) be a Hecke pair such that G = T −1T . Then the product 〈〈, 〉〉 is positivedefinite, in the sense of ∗-algebras.

Proof. Suppose f ∈ C[G/S]. We will use the unitary action of G on 〈〈, 〉〉 to show〈〈f, f〉〉 is positive. Let ` be a common lower bound for the elements of a set consisting of onerepresentative from each left coset in support(f). Then ` exists (since support(f) is finite) and`−1(f) ∈ H(S\G/S)—that is, left multiplication by `−1 shifts f to a function defined only onthose cosets gS such that gS = SgS.

Now we calculate 〈〈f, f〉〉.

〈〈f, f〉〉 = 〈〈Ξ12 (`) `−1(f),Ξ

12 (`) `−1(f)〉〉

= Ξ(`) (`−1(f))∗ ∗ `−1(f)

since `−1(f) ∈ H(S\G/S). Therefore, 〈〈f, f〉〉 is a positive element of H(S\G/S). Furthermore,this inner product is positive definite since `−1(f) = 0 if and only if f = 0. 2

4.5. Corollary. Let (G,S) be a lower directed Hecke pair. The category of nondegenerate ∗-representations of H(S\G/S) is equivalent to the category of unitary representations of G whichare generated by their S-fixed vectors. 2

4.6. Corollary. Let (G,S) be a lower directed Hecke pair. Then C∗(S\G/S) exists. 2

In particular, these results apply to the Hecke algebra discussed by Bost and Connes.

4.2 Finite-dimensional Hecke algebras

A Hecke algebra is finite-dimensional if H(S\G/S) is a finite-dimensional vector space.First, we will show that H(S\G/S) is isomorphic to a Hecke algebra associated to a finite group.

4.7. Lemma. Let (G,S) be a Hecke pair whose Hecke algebra is finite-dimensional. Then thereexists a finite group G and subgroup S such that

H(S\G/S) ∼= H(S\G/S).

41

Proof. Define the subgroup N by

N =⋂

g∈G/SgSg−1.

Since there are only finitely many left cosets of S in G, N is a subgroup of S of finite index inG. Moreover, N is normal in G, since for all h ∈ G we have

hNh−1 =⋂

g∈G/ShgSg−1h−1 =

⋂k∈G/S

kSk−1 = N.

Now set G = G/N and S = S/N , and let φ : G → G/N be the quotient homomorphism. Thenfor g ∈ G,

φ(S)φ(g)φ(S) = φ(SgS) = φ(⋃

g∈SgS/SgS) =

⋃g∈SgS/S

φ(g)φ(S)

which is a disjoint union, since N ⊂ S. Hence (G,S) is a Hecke pair, with L(φ(g)) = L(g).The group homomorphism φ extends to an isomorphism of the respective Hecke algebras byφ([SgS]) = [φ(SgS)]. One checks that φ(f1 ∗ f2) = φ(f1) ∗ φ(f2) for all f1, f2 ∈ H(S\G/S).2

4.8. Definition. Let Λ ∈ H(S\G/S) be the element given by

Λ =∑

g∈S\G/S

1L(g)

[SgS] ∗ [Sg−1S].

4.9. Lemma. Let (G,S) be a Hecke pair whose Hecke algebra is finite-dimensional. Then forall f ∈ C[G/S],

Λ ∗ 〈〈f, f〉〉 =∑

g∈G/S〈〈f, [gS]〉〉 ∗ 〈〈[gS], f〉〉.

Proof. For f ∈ C[G/S],∑g∈G/S

〈〈f, [gS]〉〉 ∗ 〈〈[gS], f〉〉

=∑

g∈G/S

∑γ∈G/S

∑η∈G/S

f(γ) f(η) 〈〈[γS], [gS]〉〉 ∗ 〈〈[gS], [ηS]〉〉

=∑

γ∈G/S

∑η∈G/S

f(γ) f(η)∑

g∈G/S〈〈[γS], [gS]〉〉 ∗ 〈〈[gS], [ηS]〉〉

Fixing γ, η ∈ G and using the unitary action of G (recall that Ξ is the trivial homomorphism forfinite-dimensional Hecke algebras), we have∑

g∈G/S〈〈[γS], [gS]〉〉 ∗ 〈〈[gS], [ηS]〉〉 = Ξ(γ)

∑g∈G/S

〈〈1, [γ−1gS]〉〉 ∗ 〈〈[γ−1gS], [γ−1ηS]〉〉

=∑

g∈G/S〈〈1, [gS]〉〉 ∗ 〈〈[gS], [γ−1ηS]〉〉

42

Since for γ ∈ G,

〈〈1, [gS]〉〉(γ) =1

L(γ)

∑h∈G/Sγ

[S](h) [gS](hγ) Ξ(γ)

which equals 1L(γ)

if γ ∈ SgS and 0 otherwise, we have

〈〈1, [gS]〉〉 =1

L(g)[SgS].

Then ∑g∈G/S

〈〈1, [gS]〉〉 ∗ 〈〈[gS], [γ−1ηS]〉〉

=∑

h∈S\G/S

1L(h)

[ShS] ∗∑

k∈ShS/S〈〈[kS], [γ−1ηS]〉〉

=∑

h∈S\G/S

1L(h)

[ShS] ∗ 〈〈[ShS], [γ−1ηS]〉〉

=∑

h∈S\G/S

1L(h)

[ShS] ∗ [Sh−1S] ∗ 〈〈[γS], [ηS]〉〉 = Λ ∗ 〈〈[γS], [ηS]〉〉.

Putting it all together,∑g∈G/S

〈〈f, [gS]〉〉 ∗ 〈〈[gS], f〉〉 =∑

γ∈G/S

∑η∈G/S

Λ ∗ (f(γ) f(η) 〈〈[γS], [ηS]〉〉)

= Λ ∗ 〈〈f, f〉〉,

which is what we intended to prove. 2

To prove that 〈〈, 〉〉 is positive, we will examine the element Λ. First we will show thatH(S\G/S) is a C∗-algebra, and then apply C∗-algebra techniques to Λ.

4.10. Lemma. If H(S\G/S) is a finite Hecke algebra, then H(S\G/S) ∼= C∗(S\G/S).

Proof. The space `2(S\G) is a Hilbert space, since the set of right cosets S\G is finite.Then π : H(S\G/S)→ B(`2(S\G)) given by π(h)f = h ∗ f for h ∈ H(S\G/S) and f ∈ `2(S\B)is a nondegenerate ∗-representation of H(S\G/S) (in this case, the product is computed usingright cosets). The norm

‖f‖ = ‖π(f)‖

for f ∈ H(S\G/S) is the C∗-algebra norm. If ρ : H(S\G/S)→ B(W ) is any other representationof the Hecke algebra, then ‖ρ(f)‖ is bounded by the C∗-norm of f , so

‖ρ(f)‖ ≤ ‖f‖ = ‖π(f)‖.

Since the Hecke algebra is finite, H(S\G/S) ∼= C∗(S\G/S). 2

More details of the following assertions about positive and invertible elements in a C∗-algebra may be found in Kadison-Ringrose [11, pp. 244–250].

The element Λ ∈ H(S\G/S) is positive—in fact, Λ− 1 is positive, since

Λ = 1 +∑

g∈S\G/S, g 6∈S

1L(g)

[SgS] ∗ [SgS]∗,

43

where 1 = [S] is the identity in H(S\G/S). Therefore, Λ is invertible in C∗(S\G/S) and hencein H(S\G/S), since its spectrum is contained in [1,∞), and Λ−1 is also positive. Moreover, wehave the following.

4.11. Lemma. Let (G,S) be a Hecke pair whose Hecke algebra is finite-dimensional. Then Λis in the center of H(S\G/S).

Proof. It suffices to prove that Λ commutes with [ShS] for all h ∈ G. We compute

Λ ∗ [ShS] =∑

g∈G/SL(g)〈〈1, [gS]〉〉 ∗ 〈〈[gS], 1〉〉 ∗ [ShS]

=∑

g∈G/SL(g) 〈〈1, [gS]〉〉 ∗ 〈〈[gS],

∑η∈ShS/S

[ηS]〉〉.

Since for all s ∈ S∑g∈G/S

〈〈1, [gS]〉〉 ∗ 〈〈[gS], [sηS]〉〉 =∑

g∈G/S〈〈1, [gS]〉〉 ∗ 〈〈[gS], [ηS]〉〉,

we have

Λ ∗ [ShS] =∑

g∈G/SL(g)L(h) 〈〈1, [gS]〉〉 ∗ 〈〈[gS], [hS]〉〉

=∑

g∈G/SL(g)L(h) 〈〈[h−1

S], [h−1gS]〉〉 ∗ 〈〈[h−1

gS], 1〉〉

=∑

g∈G/SL(g)

L(h)R(h)

〈〈[Sh−1S], [gS]〉〉 ∗ 〈〈[gS], 1〉〉 = [ShS] ∗ Λ.

2

4.12. Theorem. Let (G,S) be a Hecke pair whose Hecke algebra is finite-dimensional. Then〈〈, 〉〉 is positive, in the sense of ∗-algebras.

Proof. Lemma 4.9 shows that for f ∈ C[G/S],

Λ ∗ 〈〈f, f〉〉 =∑

g∈G/S〈〈f, [gS]〉〉 ∗ (〈〈f, [gS]〉〉)∗.

In other words, Λ ∗ 〈〈f, f〉〉 is positive for all f ∈ C[G/S]. Then Λ−1 is positive, and

〈〈f, f〉〉 = Λ−1 ∗∑

g∈G/S〈〈f, [gS]〉〉 ∗ (〈〈f, [gS]〉〉)∗.

But Λ−1 is in the center of H(S\G/S), since Λ is, and the spectrum of the product of twocommuting elements of a C∗-algebra is contained in the product of their spectra. 2

44

Chapter 5

Trees, Buildings, and Hecke Algebras

Several important examples of Hecke algebras are associated to automorphism groupsof simplicial complexes called buildings. The theory of Coxeter groups and Coxeter complexessummarized in Section 5.1 is a prerequisite to the study of buildings. Section 5.2 is an introductionto the theory of buildings. In Section 5.3, we will explore automorphism groups of buildings andtheir connection to Hecke algebras.

5.1 Coxeter groups and Coxeter complexes

Coxeter groups can be thought of as “generalized reflection groups.” For example, thegroup of symmetries of a regular polygon is a Coxeter group, as is the symmetry group of atiling of the plane. One way to study a Coxeter group is to construct a simplicial complex,called a Coxeter complex, on which the group acts. Before defining Coxeter groups and Coxetercomplexes, we will establish a connection between simplicial complexes and partially orderedsets.

5.1.1 Posets and simplicial complexes

There is a canonical association between partially ordered sets (posets) which satisfy prop-erties S1 and S2 below and simplicial complexes. Since we will construct both Coxeter complexesand buildings by associating a simplicial complex to a poset, it is worthwhile to examine the gen-eral construction.

Let P be a poset with partial ordering ≤, and suppose that

S1 Any two elements of P have a greatest lower bound.

S2 For any element A of P , the poset of elements P≤A = {B ∈ P : B ≤ A} is isomorphic tothe poset of subsets of {1, . . . , r} for some r ≥ 0, ordered by the inclusion relation.

The integer r in S2 is called the rank of A. We construct the simplicial complex associated to Pby associating a vertex to each rank 1 element of P , an edge to each rank 2 element of P , andso on. The empty simplex is associated to a rank 0 element. The inclusion of one simplex inanother is determined by the ordering on P . For instance, for A ∈ P , the poset P≤A correspondsto the poset of faces of the simplex associated to A. In particular, the vertices of this simplexcorrespond to rank 1 elements of P≤A. On the left of Figure 5.1 is a poset P , where for A,B ∈ P ,we have A ≤ B if A is connected to B by an ascending path. On the right of Figure 5.1 is thesimplicial complex associated to P . See also [5, pp. 27–28] for more details.

5.1.2 Constructing Coxeter complexes

5.1. Definition. A pair (W,S) is a Coxeter system if W is a group, S is a set of generatorsfor W of order two, and W may be presented

W = 〈S; (sisj)mij = 1〉,

where mij is the order of sisj and there is one relation for each pair si, sj with mij finite.

45

c

Σ

a b c

hgf

d

e

iP

i

h

dfb

ge

a

o

Fig. 5.1. A poset and its associated simplicial complex

Given a Coxeter system (W,S) we can construct the Coxeter complex Σ associated to(W,S) by the following procedure. Every subset S′ ⊆ S generates a so-called special subgroupof W , denoted by 〈S′〉. A special coset will be any left coset w〈S′〉 of a special subgroup. LetΣ(W,S) be the poset of special cosets in W , ordered by the opposite of the inclusion relation.That is, if A and B are special cosets, then A ≤ B if and only if B ⊆ A. Then Σ(W,S) satisfiesS1 and S2. Let Σ be the simplicial complex associated to Σ(W,S).

Suppose n is the cardinality of S. Each singleton set {w} ⊂W is associated to a (n− 1)-dimensional simplex. These will be the maximal simplices in Σ, called chambers. The faces ofthe chamber associated to {w} correspond to the special cosets which contain the element w. Inparticular, its vertices are associated to cosets of the form w〈S\{s}〉 for some s ∈ S.

Galleries and distance In any simplicial complex, two distinct maximal simplices are adjacentif they share a codimension-one face. A gallery is a sequence of maximal simplices

C0, C1, C2, . . . , Cn

such that Ci is adjacent to Ci−1 for i = 1, 2, . . . n. We say that the gallery above is a galleryfrom C0 to Cn which has length l. The distance between C0 and Cn, d(C0, Cn), is defined tobe the minimum of the lengths of all galleries from C0 to Cn. A simplicial complex is called achamber complex if all of its maximal simplices are of the same dimension, and any two can beconnected by a gallery. Every Coxeter complex is a chamber complex [5, p. 34].

Labelling A labelling of a chamber complex is a map from the set of vertices of the complexto some index set I which is a bijection on the vertices of each chamber. We can define a map λon cosets of Σ(W,S) of the form w〈S\{s}〉 by

λ(w〈S\{s}〉) 7−→ s,

which gives a labelling of Σ (in this case S becomes the index set).

The W -action The action of W on Σ(W,S) by left multiplication defines a natural action ofW on the Coxeter complex associated to Σ(W,S). This action is transitive on the chambers of W(meaning that for any two chambers there exists an element of W which maps one to the other).Moreover, the action of W preserves the labelling λ defined above. We say that such an actionis type-preserving. If we choose the fundamental chamber to be the chamber corresponding to

46

{1}, it is a fundamental domain for the action of W . Moreover, the W -action on Σ preserves theadjacency relation and hence preserves distances.

Examples If S = {s1, s2} consists of elements of order two and W can be presented

W = 〈S; (s1s2)m = 1〉

then the Coxeter complex associated to (W,S) is a 2m-gon. If s1s2 has infinite order, the Coxetercomplex is a line.

If S = {s1, s2, s3} consists of elements of order two and W can be presented

W = 〈S; (sisj)2 = 1 for all si 6= sj〉

then Coxeter complex associated to (W,S) is the tessellation of the Euclidean plane by equilateraltriangles (Figure 5.2). Each element of S acts on the fundamental chamber corresponding to {1}

Fig. 5.2. Tessellation of the plane by equilateral triangles

by reflecting it in one of its sides. Note also that the labelling of the vertices indicates their orbitsunder the action of W .

For an idea of the other possible types of Coxeter complexes, consult [6, p. 35].

Foldings The existence of endomorphisms called foldings on a Coxeter complex Σ will allowus to establish some properties which will be useful later. An idempotent endomorphism φ on Σis called a folding if for every chamber C ∈ φ(Σ), there is exactly one chamber C′ in Σ such thatφ(C′) = C but C′ /∈ φ(Σ). Figure 5.3 suggests why φ is called a “folding.” Let Φ = φ(Σ) andΦ′ be the “completion” of the complement of Φ (meaning that Φ′ includes the faces of all of itschambers). Then the following are true (see [5, pp. 67–68]).

• φ preserves adjacency.

• Φ and Φ′ are convex chamber complexes (a subcomplex is convex if it contains every minimalgallery between any two of its chambers).

• If the chambers C ∈ Φ and C′ ∈ Φ′ are adjacent, then φ(C′) = C.

47

φ

CC′

Φ′ Φ

Fig. 5.3. A folding

• The set of chambers of Φ is equal to the set of chambers in Σ which are strictly closer to Cthan C′. Likewise, the set of chambers of Φ′ is exactly those chambers which are strictlycloser to C′ than C.

Note that the last item implies that no chamber is equidistant from C and C′.The following lemma gives a useful way to construct foldings.

5.2. Lemma. Given any pair C,C′ of adjacent chambers in a Coxeter complex, there exists aunique folding φ such that φ(C′) = C.

Sketch of Proof. There exists some s ∈ S such that C′ = sC. Let φ be the map

φ(wC) ={

wC if `(sw) > `(w)swC if `(sw) < `(w).

See [5, p. 48] for the details of this construction and [5, p. 69] for the proof of uniqueness.2

Walls Let φ be a folding and define Φ and Φ′ as above. The intersection H = Φ ∩Φ′ is calleda wall. If C ∈ Φ and C′ ∈ Φ′ are adjacent chambers, we say that H is the wall separating C andC′. It can be shown that the minimal galleries between chambers C and D are precisely thosewhich cross each wall between C and D once [5, p. 73].

5.3. Definition. For any vertex v in a Coxeter complex, define the neighborhood of v, nbhd(v),to be the set of chambers sharing the vertex v.

5.4. Lemma. Let v be a vertex in Σ and let {C1, . . . , Cr} be the chambers in nbhd(v). Thenthe set of chambers of Σ is the disjoint union of sets C1, . . . ,Cr, where

Ci = {D ∈ chambers(Σ) : d(D,Ci) < d(D,Cj) for i 6= j}.

Sketch of Proof. Let H be the set of walls separating adjacent elements of nbhd(v), andlet Ci be the set of chambers in Σ which may be connected to Ci by a gallery which does notcross any wall in H. Then every element in ∆ is contained in some Ci. Moreover, if D ∈ Ci and

48

Cj ∈ nbhd(v)\{Ci}, then there exists a set of foldings φ1, . . . , φn determined by a subset of thewalls in H so that

φ1 ◦ · · · ◦ φn(Σ) = Ci

and φ1 ◦ · · · ◦φn(Cj) = Ci. If chambers C and D lie on opposite sides of a wall, then the foldingφ determined by the wall has the property that d(C,D) > d(φ(C), φ(D)). Since we can fold Cjonto Ci, D is closer to Ci than Cj . Set Ci = Ci. 2

5.5. Definition. A subcomplex of Σ will be called a cone of the vertex v if it is of the form Cias defined in Lemma 5.4.

So we have shown that every Coxeter complex is the disjoint union of the cones of any ofits vertices.

5.2 Buildings

The following exposition is based on Brown [5]. A building ∆ is a chamber complex whichsatisfies the building axioms below. Jacques Tits [14] was the first to define buildings.

5.6. Definition. A building is a finite-dimensional simplicial complex ∆ for which there existsa system of subcomplexes Σ, called apartments, such that

1. Each apartment is a Coxeter complex of the same dimension as ∆.

2. For any two chambers in the building, there is an apartment containing both of them.

3. If two apartments Σ and Σ′ have two chambers in common, there is an isomorphism Σ→ Σ′

which fixes Σ ∩ Σ′ pointwise.

Note that a building is a chamber complex (since every two simplices lie in an apartmentand hence can be connected by a gallery). A chamber complex is called thin if every codimension-one simplex is a face of exactly two chambers. It is thick if every codimension-one simplex isa face of three or more chambers. A Coxeter complex is a thin building. It contains only oneapartment.

5.2.1 Example: projective geometry

Let k be a field and let V = kn be the n-dimensional vector space over k. To the poset ofproper non-zero subspaces of V , ordered by inclusion, we associate a simplicial complex ∆ as inTable 5.1.

vertices ←→ proper non-zero subspaces of V

higher simplices ←→ chains of subspaces V1 < V2 < · · · < Vl

Table 5.1. The association of simplicial complexes to posets of subspaces

49

Observe that each chamber of ∆ has n − 1 vertices and hence is a (n − 2)-dimensionalsimplex.

The projective geometry associated to V corresponds to the set of proper non-zero sub-spaces of V , equipped with the so-called incidence relation: two subspaces are incident if oneof them is contained in the other. The one-skeleton of the building ∆ is the incidence graphwhose vertices are the subspaces of V . In the example V = k3, we call the one-dimensionalsubspaces points and the two-dimensional subspaces lines. If a point is contained in a line, thentheir corresponding vertices are connected.

We also see that there is a natural map from the vertices of each chamber of ∆ onto theintegers 1, . . . , n−1—namely, a vertex is mapped to the dimension of its corresponding subspace.This map is a labelling of the vertices of ∆. Note also that matrix multiplication of a subspace ofV by an element of GLn(k) gives us a natural type-preserving (that is, label-preserving) actionof GLn(k) on the vertices of the building which partitions the vertices into n− 1 orbits.

Given a basis B = {e1, . . . , en} of V , one can form the subcomplex Σ ⊂ ∆ associatedto the poset of subspaces of V spanned by proper subsets of B. Then Σ is isomorphic to thebarycentric subdivision of the boundary of an (n − 1)-simplex [6, p. 3]. If we define the set ofall such Σ to be a system of apartments for ∆, it is possible to verify that ∆ is a building [5,pp. 84–85]. The following example is one of the most basic non-trivial examples of this type ofbuilding.

Example Let V = (Z/2Z)3. The vector space (Z/2Z)3 contains seven one-dimensional sub-spaces, each generated by one of the following vectors:

(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1).

In addition, V contains seven two-dimensional subspaces, represented by

{(a, b, 0)}, {(a, 0, b)}, {(0, a, b)}, {(a, b, b)}, {(b, a, b)}, {(b, b, a)}, {(a, b, a+ b)},

where a and b are elements of Z/2Z. Figure 5.4 shows the simplicial complex ∆ associated to theposet of proper non-zero subspaces of (Z/2Z)3. Black vertices correspond to one-dimensionalsubspaces and white vertices to two-dimensional subspaces. A black vertex is connected to awhite one if there is an inclusion of the corresponding subspaces.

If we think of ∆ as the incidence graph of the projective geometry associated to (Z/2Z)3,we see that

• Any two points are contained in exactly one line, and any two lines intersect in exactly onepoint.

• Each point is contained in exactly three lines, and each line contains exactly three points.

The apartments of the building ∆ are hexagons representing the inclusion of three linesin three planes. Pictured in Figure 5.5 is the apartment generated by the standard basis vectorsof V , which we will call the fundamental apartment.

The general linear group GL3(Z/2Z) acts transitively on subspaces of V of the samedimension. If we choose the fundamental chamber to be the edge corresponding to the inclusion〈(1, 0, 0)〉 ⊂ 〈(1, 0, 0), (0, 1, 0)〉, the stabilizer of the fundamental chamber is the group of uppertriangular matrices in GL3(Z/2Z).

5.2.2 Example: a tree

A tree is a graph T with no loops. If every vertex of T is contained in at least two edges,then T is a building. The apartments are lines. Indeed it is readily verified that for any two

50

Fig. 5.4. The simplicial complex associated to the poset of subspaces of (Z/2Z)3

〈(0, 0, 1)〉

〈(0, 1, 0)〉

〈(0, 1, 0), (0, 0, 1)〉〈(1, 0, 0), (0, 0, 1)〉

〈(1, 0, 0)〉

〈(1, 0, 0), (0, 1, 0)〉

Fig. 5.5. The fundamental apartment of ∆

51

edges in the tree, there is a line containing both of them, and if two lines L and L′ have two edgesin common, there is an isomorphism from one line to the other which fixes L ∩ L′ pointwise, asrequired by the axioms for a building. The 4-regular tree is pictured in Figure 5.6.

Fig. 5.6. The 4-regular tree

5.2.3 Consequences of the building axioms

Here are several consequences of the building axioms which we will need. One may consultChapter IV of Brown [5] for the proofs of these statements.

• All apartments of ∆ are isomorphic, and which type of Coxeter complex they are is deter-mined by ∆.

• There is a unique maximal system of apartments.

• Every apartment Σ is a retract of ∆. Briefly, choose a chamber C in Σ. For any D inthe building, there exists an apartment Σ′ containing both C and D and an isomorphismρ

Σ′,D : Σ′ → Σ which fixes Σ′∩Σ. Proceeding in this manner, we can construct a retraction

ρ of ∆ onto Σ, where the restriction of ρ to any apartment Σ containing a chamber D isρ

Σ,D.

• Every apartment is convex in ∆.

• ∆ is labellable. Choose an apartment Σ. It is a Coxeter complex, and therefore labellable.The retraction ∆→ Σ determines a labelling for ∆.

5.2.4 Buildings and Hecke algebras

Several Hecke algebras may be constructed from a building ∆. We will require that theaction of a group G of type-preserving automorphisms be strongly transitive, meaning that Gacts transitively on the set of pairs (Σ, C), where Σ lies in some system of apartments A and Cis a chamber in Σ, and ∆ satisfies the condition that each chamber is adjacent to only finitely

52

many chambers. Choose an apartment which will be called the fundamental apartment, and afundamental chamber contained in the fundamental apartment. Let W ⊂ G be the group of type-preserving automorphisms of Σ, and let S ⊂ W be the set of reflections in the codimension-onefaces of C. Then (W,S) is a Coxeter system and Σ is its associated Coxeter complex [5, p. 99].

Let B ⊂ G be the stabilizer of the fundamental chamber C. Since G acts transitively, thechambers in ∆ are in one-to-one correspondence with the elements of the left coset space G/Bby the assignment gC ↔ gB. Moreover, one can show that there is a notion of adjacency in G/Bunder which left cosets gB and g′B are adjacent in G/B if and only if gC and g′C are adjacentin ∆. Adjacency is preserved by the left action of G on G/B, and hence there is a G-invariantdistance on G/B [5, p. 102].

The left cosets gB contained in the double coset BgB correspond to all translates of thechamber gC by elements of B. Since for all b ∈ B we have

d(C, gC) = d(C, bgC),

the chambers {bgC : b ∈ B} all lie within distance d(C, gC) of the fundamental chamber. Theset of chambers within a fixed distance of the fundamental chamber is finite. But the chambersbgC are in one-to-one correspondence with the left cosets bgB, so BgB contains finitely manyleft cosets for each g ∈ G and (G,B) is a Hecke pair. By a similar construction, one can showthat if P is the stabilizer of any simplex in ∆, then (G,P ) is a Hecke pair.

5.7. Definition. The link of a simplex v in a building is the set of all opposite faces (that is,faces which are disjoint from v) of simplices which contain v. We will denote the link of v bylink(v).

5.8. Lemma. The link of any simplex in a building is itself a building.

Sketch of Proof. First consider the link of a simplex v in the Coxeter complex associatedto Σ(W,S). Let w〈S′〉 be the special coset corresponding to v, where S′ ⊂ S. If W ′ = 〈S′〉 then(W ′, S′) is a Coxeter system whose Coxeter complex is isomorphic to the link of v [5, p. 60].Now let v be a simplex in the building ∆. Then linkΣ(v), the restriction of link(v) to Σ ∈ A,is a Coxeter complex. One can show that {linkΣ(v) : Σ ∈ A} forms a system of apartmentsfor link(v) and the restriction of the action of G to link(v) provides the isomorphisms needed tosatisfy the building axioms [5, p. 79]. 2

5.3 BN-pairs and buildings

The original approach to buildings was group theoretic. We have seen that, given aCoxeter system (W,S), there is a canonical way to construct a Coxeter complex on which W actstransitively as type-preserving automorphisms. We would like to know if a similar constructioncan be extended to other groups. The answer to this question is provided by the theory ofBN -pairs.

5.9. Definition. A BN -pair consists of a group G, together with subgroups B and N and asubset S such that the following axioms hold

1. The subgroups B and N generate G, and their intersection B ∩N is normal in N .

2. The set S generates W = N/T .

3. BsB · BwB ⊆ BwB ∪BsB for all s ∈ S and w ∈W .

4. For all s ∈ S, sBs−1 6⊆ B.

53

The quadruple (G,B,N, S) is also called a Tits system.

The group W = N/(B∩N) is called the Weyl group of the BN-pair. It can be shown that(W,S) is a Coxeter system (see Brown [5, pp. 107–109]). If G admits a BN -pair, then G can bewritten

G =∐w∈W

BwB

which is known as the Bruhat decomposition [5, p. 104].

Construction of the building As the construction of the Coxeter complex suggests, thebuilding on which G (the group of a BN -pair) acts will be a chamber complex associated to aposet of some system of subgroups of G. Let S′ be a proper subset of S and W ′ = 〈S′〉. Thenaxiom (3) above implies that BW ′B is a subgroup. We will call all subgroups of the form BW ′Bspecial subgroups. A left coset of a special subgroup is called a special coset.

5.10. Definition. Let ∆(G,B) be the simplicial complex associated to the poset of special cosetsof G, ordered by the opposite of the inclusion relation.

Given a BN -pair, we say that a subgroup of G is parabolic if it is a conjugate of a specialsubgroup. Equivalently, we define ∆(G,B) as the simplicial complex associated to the poset ofparabolic subgroups, ordered by the opposite of the inclusion relation.

5.11. Theorem. Let G be a group with a BN -pair. Then ∆(G,B) is a thick chamber complex.Let Σ be the subcomplex of ∆(G,B) corresponding to special cosets of the form wP , where w ∈Wand P is a special subgroup. Then Σ is isomorphic to the Coxeter complex associated to (W,S)and the set {gΣ : g ∈ G} is a system of apartments for ∆(G,B). With this system of apartments,∆(G,B) is a building. The action of G on ∆(G,B) is strongly transitive and type-preserving.

Details of the proof may be found in Brown [5, pp. 112–113]. Axiom 4 means that ∆(G,B)will always be a thick building. 2

We will choose the fundamental chamber to be the chamber corresponding to B and thefundamental apartment to be the apartment corresponding to the poset of special cosets of theform wP , where w ∈ W and P is a special subgroup. The map wP → wW ′ for P = BW ′Bcorresponds to an isomorphism of the fundamental apartment to the Coxeter complex associatedto (W,S) [5, pp. 112–113].

5.3.1 The Iwahori Hecke algebra

5.12. Definition. If G is a group with a BN -pair, and each chamber of ∆(G,B) is adjacent tofinitely many other chambers, (G,B) is a Hecke pair. Its Hecke algebra H(B\G/B) is called anIwahori Hecke algebra.

The Bruhat decomposition means that the Iwahori Hecke algebra is spanned by the char-acteristic functions of double cosets [BwB] for w ∈ W . Let ` : W → N be the word-lengthfunction on reduced words in W , with respect to its set of generators S (if w′ is a not a reducedword, then set `(w′) = `(w) where w = w′ and w is reduced). Multiplication in the IwahoriHecke algebra is determined as follows:

H1 [Bw1B] ∗ [Bw2B] = [Bw1w2B] for w1, w2 ∈W , if `(w1) + `(w2) = `(w1w2).

H2 [BsB] ∗ [BwB] = L(s)[BswB] + (L(s)− 1)[BwB] for s ∈ S and w ∈W , if `(sw) < `(w).

Iwahori [10] demonstrated that H(B\G/B) is generated as an algebra by the identity [B]together with the set {[BsB] : s ∈ S} and the relations

54

I1 [BsB]2 = L(s)[B] + (L(s)− 1)[BsB] for s ∈ S.

I2 If mij <∞, for all distinct si, sj in S

1. ([BsiB] ∗ [BsjB])r [BsiB] = [BsjB] ([BsiB] ∗ [BsjB])r if mij = 2r + 1.

2. ([BsiB] ∗ [BsjB])r = ([BsjB] ∗ [BsiB])r if mij = 2r.

5.3.2 The link of a simplex

We have already seen a that the link of a simplex in the building is a building which canbe associated to the left coset space of an (Iwahori) Hecke subalgebra of H(B\G/B). Here isan algebraic way to understand the link. The BN -pair axioms tell us that if W ′ is the groupgenerated by S′ ⊂ S, then BW ′B = ∪

w∈W ′BwB is a group. If S′ is a proper subset of S,

BW ′B is precisely the stabilizer of the simplex v corresponding to W ′. In this case, BW ′Bcontains a finite number of left cosets of B (since B is mapped to finitely many chambers byBW ′B) and if P = BW ′B, then H(B\P/B) is a Hecke subalgebra of H(B\G/B), and thelink of v is isomorphic to the building ∆(P,B). In general, if v is the simplex in the buildingstabilized by g−1Pg, where P = BW ′B, then H(g−1Bg\g−1Pg/g−1Bg) ∼= H(B\P/B) is theHecke subalgebra associated to the link of v.

Given any vertex v ∈ ∆, let the neighborhood of v, nbhd(v), be the set of chambers in∆ which have v as a vertex. Then nbhd(v) is a subcomplex of ∆ which contains link(v). Infact, functions supported on left cosets corresponding to the chambers of nbhd(v) form a Heckesubalgebra which is isomorphic to the subalgebra associated to link(v). The isomorphism onalgebras follows from associating each chamber in nbhd(v) to its set of faces which are disjointfrom v.

The following lemma is the analog of Lemma 5.4 for buildings.

5.13. Lemma. Let v be a vertex of the building ∆ and let C1, . . . , Cr be the elements of nbhd(v).Then the set of chambers of ∆ is the disjoint union of the sets C1, . . . ,Cr, where

Ci = {D ∈ chambers(∆) : d(D,Ci) < d(D,Cj) for i 6= j}.

Sketch of Proof. For each Ci ∈ nbhd(v), choose an apartment Σi which contains Ci andlet ρCi be the retraction of ∆ onto Σi as constructed in Section 5.2. Observe that ρCi(Cj) forj 6= i is a chamber in Σi which has v as a vertex but does not coincide with Ci. The fact thatΣi is a Coxeter complex means that we can construct the set

Ci = {D ∈ chambers(Σi) : d(D,Ci) < d(D, ρCi(Cj)) for i 6= j}.

We claim that Ci consists of those chambers D such that ρCi(D) ∈ Ci, since retractions decreasedistances. 2

5.14. Definition. A subcomplex of ∆ of the form Ci will be called a cone of the vertex v.

5.4 Buildings associated to SLn(Qp)

5.4.1 Trees

Fix a prime p, and let G be the group SL2(Qp). Let B be the subgroup

B = {(a bc d

)∈ SL2(Zp) : c ≡ 0 (mod p)}.

55

Let N be the subgroup

N = {(a 00 a−1

): a ∈ Z×

p}⋃{(

0 −a−1

a 0

): a ∈ Z×

p},

and let S be the set

S = {s1, s2}, where s1 =(

0 −11 0

)and s2 =

(0 −p−1

p 0

).

Then one checks that we have a BN -pair [5, pp. 130–133].There are two parabolic subgroups of G which contain B: P = SL2(Zp) and xPx−1,

where x =(

1p 00 1

). Each vertex of the fundamental chamber is shared by p + 1 edges. A

chamber complex is (p+ 1)-regular if each chamber is adjacent to p other chambers. The groupG acts as symmetries of the (p+ 1)-regular tree which preserve the labelling of the vertices. Thesubgroup B is the stabilizer of the fundamental chamber.

The Iwahori Hecke algebra for SL2(Qp) In this case, the Iwahori relation I2 is vacuoussince m12 =∞. The Iwahori Hecke algebra is generated by

X = [Bs1B], Y = [Bs2B], and 1 = [B],

whereX2 = p+ (p− 1)X and Y = p+ (p− 1)Y.

There is a geometric connection between the action of G on the tree and the relationI1 on the generators of the Hecke algebra for the pair (G,B). The generator X of the Heckealgebra lies in the commutant of the action of G on the tree and can be thought of as the sumof p actions on the tree, shown in Figure 5.7 for p = 3. In fact, X and 1 together generate

B

Bs1B

+ +=X =

Fig. 5.7. The actions which form the generator X

H(B\SL2(Zp)/B). Applying X twice produces Figure 5.8. The identity operator appears ptimes and each component action of X appears p− 1 times. Therefore,

X2 = p+ (p− 1)X,

which is the relation I1 described by Iwahori.

56

X2 = + +

Fig. 5.8. The result of applying X twice

The Hecke algebra for the stabilizer of a vertex There is another Hecke algebra associatedto the action of SL2(Qp) on the (p+1)-regular tree. Its Hecke pair is (G,P ), where G is SL2(Qp)and P (for parabolic) is the stabilizer of a vertex. All Hecke algebras of this type are isomorphicto the algebra H(P\G/P ), where P = SL2(Zp), which we have studied in Sections 2.1 and 2.2.Left cosets of P correspond to the orbit of a white vertex—that is, the set of white vertices. Recallthat we showed in Section 2.3 that H(P\G/P ) may not be completed to a Hecke C∗-algebra,because the supremum norm ‖f‖ of any f ∈ H(P\G/P ) which is not a multiple of the identityis infinite.

5.4.2 Affine buildings

Let G be SLn(Qp). The construction of the building associated to G, which we will call∆n, is explained in detail in Brown [5, pp. 131–133]. The groups SLn(Qp) for n > 2 act on a classof (p+1)-regular buildings called An−1 buildings which may be thought of as higher-dimensionalversions of a tree. The apartments of the building ∆n are Euclidean (n−1)-spaces tessellated by(n− 1)-simplices. These buildings are called affine or Euclidean. The group SL3(Qp) acts on abuilding whose apartments are planes tessellated by equilateral triangles, as in Figure 5.9. Thesolid triangles are the chambers in this example. The tessellation above produces a so-called A2building.

In general, the link of a vertex in the building ∆n is isomorphic to the building associatedto the vector space kn, where k is the field Z/pZ. Figure 5.10 (left) shows the link of a vertex(shown in gray) for the building ∆3. In this case the link is the flag complex of the subspaces of(Z/2Z)3, as pictured on the right of Figure 5.10.

Let s1, s2, and s3 be the generators of the Weyl group ofH(B\G/B), whereG is SL3(Qp).Then the link of the vertex fixed by s1 and s2 is the building associated to the finite Hecke algebragenerated by the self-adjoint elements [B], [Bs1B], and [Bs2B]. Setting [B] = 1, [Bs1B] = Xand [Bs2B] = Y , the Iwahori relations become

I1 X2 = p+ (p− 1)X ; Y 2 = p+ (p− 1)Y

I2 XYX = Y XY .

Again, the Iwahori Hecke algebra relations correspond nicely to the geometry of the link. Weinterpreted the first relation earlier in this section for the tree. The second relation comes fromthe Weyl group relation (s1s2)3 = 1 which corresponds to symmetries of the apartments of thelink, which are hexagons.

57

Fig. 5.9. A plane tessellated by equilateral triangles

Fig. 5.10. The link of a vertex in ∆3

58

Chapter 6

SL2(Qp), SL3(Qp), and Iwahori Hecke algebras

The Iwahori Hecke algebra associated to SLn(Qp) can be realized as the commutant ofthe action of SLn(Qp) on the affine building ∆n. In this chapter, we will use this interplaybetween Hecke algebras and the geometry of buildings to prove that the product 〈〈, 〉〉 introducedin Section 3.4 is positive for SL2(Qp) with its Iwahori subgroup, and we will present some resultsfor SL3(Qp).

6.1 SL2(Qp)

We have already observed that SL2(Qp) acts as type-preserving automorphisms of the(p+ 1)-regular tree. Moreover, the subgroup B given by

B = {(a bc d

)∈ SL2(Zp) : c ≡ 0 (mod p)},

is the stabilizer of a chamber in the tree, and (G,B) is an Iwahori Hecke algebra. We will referto the edge stabilized by B as the fundamental chamber and the line containing the images ofthis edge under the generators of the Weyl group s1 and s2 as the fundamental apartment, aspictured in Figure 6.1.

s1s2B s1B B s2B s2s1B s2s1s2Bs1s2s1B

Fig. 6.1. The fundamental apartment of ∆2

If 1 = [B], X = [Bs1B], and Y = [Bs2B], then X and Y are self-adjoint and the Heckealgebra is generated by X and Y with the relations

X2 = p+ (p− 1)X and Y 2 = p+ (p− 1)Y.

The finite subalgebra corresponding to the link of each vertex is made up of p+ 1 vertices and isisomorphic to the algebra generated by X . Recall that the inner product 〈〈, 〉〉 is positive for allfinite Hecke algebras.

59

6.1.1 Properties of the product 〈〈, 〉〉

We will show that the product 〈〈, 〉〉 is positive for the Hecke algebra H(B\G/B), whereG is SL2(Qp) and B is its Iwahori subgroup. The proof involves both the geometry of the treeand the fact that the link of each vertex is a building corresponding to a finite Hecke algebra.

6.1. Lemma. Let G be SL2(Qp), B its Iwahori subgroup, W its Weyl group, and ` the lengthfunction on W . Then

1. Ξ(g) = 1 for all g ∈ G.

2. d(B,wB) = `(w) for w ∈W .

3. L(w) = p`(w)

Proof. Item (1) follows from the compactness of B.If w = si1

. . . sir is a reduced word, then

B, si1B, (si1si2)B . . . , (si1 . . . sir )B = wB

is a gallery of length r = `(w) from B to wB. With a little work, one can show that if w is areduced word, this is a minimal gallery [6, p. 38]. Item (3) follows from (2) and the fact that thetree is (p+ 1)-regular. 2

6.2. Corollary. Let 〈〈, 〉〉 : C[G/B]×C[G/B]→ H(B\G/B) be the product defined by

〈〈f1, f2〉〉(g) =1

L(g)

∑γ∈G/Bg

f1(γ) f2(γg) Ξ(γ) (6.1)

for f1, f2 ∈ C[G/B] and g ∈ G. Then

〈〈[g1B], [g2B]〉〉 =1

pd(g1B,g2B)[Bg−1

1g2B],

for g1, g2 ∈ G.

Proof. If g1, g2 ∈ W , then Equation 6.1 follows immediately from Lemma 6.1. If g1B andg2B correspond to chambers in some apartment Σ′, then there exists k ∈ G so that the actionof k maps Σ′ isomorphically to the fundamental apartment, and we choose w1, w2 ∈ W so thatw1 ∈ g1B and w2 ∈ g2B. 2

The product 〈〈, 〉〉 is closely related to the “W -valued distance function” δ discussed inBrown [6, pp. 38–41]. The function δ : W ×W →W is defined on the Weyl group by

δ(w1, w2) = w−11w2.

If C is the set of chambers in the fundamental apartment Σ, then the natural identification ofW with C allows us to define δ : C × C → W . This function extends to a W -valued function onthe chambers in the building in the usual way (that is, by mapping the apartment containingchambers C and D isomorphically to the fundamental apartment, and computing δ(C,D) in thefundamental apartment). The relationship between Brown’s function δ, the building distancefunction d, and our product 〈〈, 〉〉 is

〈〈[g1B], [g2B]〉〉 =1

pd(g1B,g2B)[B(δ(g1B, g2B))B].

60

The following lemma gives a useful fact about the geometry of the product.

6.3. Lemma. If the chamber xB is contained in a gallery of minimal length between the cham-bers gB and hB, then

〈〈[gB], [xB]〉〉 ∗ 〈〈[xB], [hB]〉〉 = 〈〈[gB], [hB]〉〉.

Proof. Without loss of generality, suppose gB, hB, and xB correspond to chambers inthe fundamental apartment, and so g, h, x ∈W . Since

d(gB, xB) + d(xB, hB) = d(gB, hB),

we have`(g−1x) + `(x−1h) = `(g−1h).

Applying the Hecke algebra relation H1,

[Bg−1xB] ∗ [Bx−1hB] = [Bg−1hB].

In conjunction with the Equation 6.1 above, this shows that

〈〈[gB], [xB]〉〉 ∗ 〈〈[xB], [hB]〉〉 = 〈〈[gB], [hB]〉〉.

2

6.1.2 Positivity of 〈〈, 〉〉 for the Iwahori Hecke algebra

The proof that 〈〈, 〉〉 is positive is by induction on larger and larger subtrees (correspondingto larger and larger submodules of the domain of definition of 〈〈, 〉〉). We will first prove a usefulfact relating the positivity of 〈〈, 〉〉 on a finite submodule of its domain to the positivity of acertain matrix with entries in the Hecke algebra. This matrix approach yields an alternate proofthat 〈〈, 〉〉 is positive for finite Hecke algebras, which furnishes the first step in the inductiveargument. Moreover, the matrix approach allows us to use the geometry of the tree and resultsfrom Section 6.1.1 to show that we can enlarge the domain of 〈〈, 〉〉 while preserving positivity.

Matrix notation In general, if A is a ∗-algebra, let Mn(A) be the n×n matrices with elementsin A. We say that a matrix M ∈ Mn(A) is positive if there exist matrices Mi ∈ Mn(A) suchthat M can be written as a finite sum M =

∑iM∗iMi. If A is a C∗-algebra, then Mn(A) is a

C∗-algebra.Let (G,S) be a Hecke pair.

6.4. Definition. Let Γ = {g1S, . . . , gnS} be a finite subset of the set of left cosets of S in G.Define MΓ ∈ Mn(H(S\G/S)) to be the self-adjoint matrix indexed by the elements of Γ wherethe entry of MΓ indexed by the pair gS, hS is given by

Mg,h = 〈〈[gS], [hS]〉〉.

6.5. Lemma. Let Γ be a finite subset of the set of left cosets of S in G and f ∈ C[G/S] bea function supported on Γ. If MΓ is a positive matrix then 〈〈f, f〉〉 is a positive element inH(S\G/S).

61

Proof. Let n = |Γ|. If MΓ is positive, it may be written MΓ =∑iM∗iMi for some finite

collection of matrices Mi ∈Mn(H(S\G/S)). Let v be the vector (f(g1), . . . , f(gn)). Then

〈〈f, f〉〉 = v∗MΓv =∑i

(Miv)∗Miv

which is a positive element of H(S\G/S). 2

Finite Hecke algebras Recall that 〈〈, 〉〉 is positive for finite Hecke algebras. Indeed, we canshow that the matrix MG/S is a positive multiple of its own square if H(S\G/S) is a finite Heckealgebra.

6.6. Lemma. Let H(S\G/S) be a finite Hecke algebra, and let P = MG/S . Then P 2 = ΛP ,where

Λ =∑

γ∈G/S〈〈[S], [γS]〉〉 ∗ 〈〈[γS], [S]〉〉.

Proof. The element of P 2 indexed by the pair gS, hS is

(P 2)g,h =∑

γ∈G/S〈〈[gS], [γS]〉〉 ∗ 〈〈[γS], [hS]〉〉.

In the course of the proof of Lemma 4.9, we showed that∑γ∈G/S

〈〈[gS], [γS]〉〉 ∗ 〈〈[γS], [hS]〉〉 = Λ ∗ 〈〈[gS], [hS]〉〉,

where Λ ∈ H(S\G/S) is defined by

Λ =∑

γ∈G/S〈〈[S], [γS]〉〉 ∗ 〈〈[γS], [S]〉〉.

Therefore,P 2 = ΛP.

2

This gives an alternate proof that 〈〈, 〉〉 is positive for finite Hecke algebras.

6.7. Corollary. The product 〈〈, 〉〉 is positive for finite Hecke algebras.

Proof. Recall that H(S\G/S) = C∗(S\G/S). Since Λ is a central, positive, invertibleelement in a C∗-algebra (Lemma 4.11), its inverse is central and positive. Since Λ−1 and P 2

are commuting positive elements of the C∗-algebra Mn(C∗(S\G/S)), where n = [G : S], thespectrum of Λ−1P 2 is positive. Therefore, P is positive. 2

The following result will be needed for our inductive argument.

6.8. Lemma. Let H(S\G/S) be a finite Hecke algebra. If P = MG/S , x ∈ G, and P is thematrix indexed by G/S where

P g,h = 〈〈[gS], [xS]〉〉 ∗ 〈〈[xS], [hS]〉〉,

then P − P is a positive matrix.

62

Proof. Lemma 6.6 shows that P 2 = ΛP . We will prove that (P − P )2 = Λ(P − P ). Nowif r is the row of P indexed by xS, the matrix P is equal to r∗r. We have

(P − P )2 = (P − r∗r)2 = P 2 − Pr∗r − r∗rP + r∗r r∗r.

Since r∗ is an eigenvector for P , Pr∗ = Λr∗. In addition, rr∗ = 〈〈xS, xS〉〉Λ = Λ. Therefore,

(P − P )2 = Λ(P − r∗r) = Λ(P − P ),

and hence P − P is positive. 2

Now we will return to the case where G is SL2(Qp) and B is its Iwahori subgroup. We willuse the notation Γ to refer both to a finite subtree and to the set of left cosets gB correspondingto the edges in Γ. The use of the notation will be clear from the context. In Figure 6.2, Γ isa finite subtree of the 3-regular tree, and v is a vertex of Γ. Let Γv be the subtree formed byadding the neighborhood of the vertex v to Γ.

Γ xB

v

Γv

Fig. 6.2. The subtrees Γ and Γv

6.9. Lemma. Let Γ be a finite subtree of the (p + 1)-regular tree, and suppose M = MΓ ispositive. Let v be a vertex of Γ so that Γ is contained in a cone of v. If Γv is the subtree formedby extending Γ by nbhd(v), then Mv = MΓv is positive.

Proof. Let xB = Γ∩nbhd(v). We will assume that the indices of M and Mv correspondingto edges in Γ agree.

Let Q be the matrix whose rows are indexed by Γ and whose columns are indexed by Γv

given byQg,h =

{ 〈〈[gB], [hB]〉〉 if gB is the nearest edge in Γ to hB0 otherwise.

Observe that Qg,h = 1 if gB = hB. Then Q has the form

Q =

←− Γv −→↑Γ↓xB

1 0 · · · 0 · · · 0 · · ·0 1 · · · 0 · · · 0 · · ·...

.... . .

......

0 0 · · · 1 · · · 〈〈[xB], [gB]〉〉 · · ·

.

63

since xB is the closest element in Γ to every edge in Γv\Γ.We assumed the matrix M is positive, and therefore Q∗MQ is positive. Now Q∗MQ has

the form←− Γ −→

↑Γ↓

... ∗ · · · ∗· · · 〈〈[gB], [hB]〉〉 · · ·

...∗ ∗...

. . .∗ ∗

,

where “∗” indicates an entry of the form

〈〈[gB], [xB]〉〉 ∗ 〈〈[xB], [hB]〉〉.

If only one of gB, hB is in Γ, then xB lies in the minimal gallery between gB and hB, so

〈〈[gB], [xB]〉〉 ∗ 〈〈[xB], [hB]〉〉 = 〈〈[gB], [hB]〉〉.

Therefore, Mv and Q∗MQ agree except on those entries which have neither index in Γ.We are done if we can show Mv −Q∗MQ is positive. The matrix Mv −Q∗MQ has the

form←− nbhd(v) −→

↑nbhd(v)↓

0 0 0

. . ....

...0 0 0

0 · · · 0 ∗ · · · ∗...

...0 · · · 0 ∗ · · · ∗

,

where “∗” indicates the entry

〈〈[gB], [hB]〉〉 − 〈〈[gB], [xB]〉〉〈〈[xB], [hB]〉〉.

Let N be the submatrix of Mv−Q∗MQ indexed by nbhd(v)×nbhd(v). The entries of Nlie in a finite Hecke subalgebra. The matrix N is precisely a matrix of the form P −P describedin Lemma 6.8, which we showed was positive. We can now write Mv explicitly as the sum ofpositive matrices:

Mv = Q∗MQ+ (Mv −Q∗MQ).

Therefore, Mv is positive. 2

Lemma 6.9 provides the induction step in our proof that the product 〈〈, 〉〉 is positive.

6.10. Theorem. Let p be a prime, G be the group SL2(Qp), and B the Iwahori subgroup of Ggiven by

B = {(a bc d

)∈ SL2(Zp) : c ≡ 0 (mod p)}.

Then the product 〈〈, 〉〉 : C[G/B]×C[G/B] −→ H(B\G/B) is positive, in the sense of ∗-algebras.

Proof. Let ∆2 be the building associated to the pair (G,B). That is, ∆2 is the (p + 1)-regular tree. Let Γ0 be the subtree consisting of the neighborhood of one vertex in ∆2. Then thespace of functions supported on the double coset space B\G/B restricted to Γ0 is a finite Heckealgebra, and hence the matrix MΓ0

is positive, by Corollary 6.7. Since any subtree of ∆2 of finitediameter can be covered a tree formed by extending Γ0 a finite number of times in the mannerof Lemma 6.9, the matrix MΓ is positive for all finite subtrees Γ. For any f ∈ H(B\G/B),

64

there exists a subtree Γ containing the support of f . Since MΓ is positive, 〈〈f, f〉〉 is positive, byLemma 6.5. 2

6.2 SL3(Qp)

Let G be SL3(Qp) and B its Iwahori subgroup—that is, the stabilizer of a chamber inthe A2 building ∆3 on which G acts. We claim that 〈〈, 〉〉 is positive for functions whose supportcorresponds to a certain class of subcomplex of ∆3.

Let Γ be a subcomplex of ∆3 and let v be a vertex of Γ. Let Γv be the subcomplex formedby adjoining nbhd(v) to Γ.

6.11. Definition. The subcomplex Γv is called a tame extension of Γ if Γ is contained in conesof the vertex v which share a common wall W .

Recall that Lemma 5.13 shows that every chamber in ∆3 has a unique nearest chamberin nbhd(v).

Figure 6.3 shows a tame extension.

Γ

v

nbhd(v)

W

Γv

Fig. 6.3. A tame extension

We need one more technical lemma before proceeding to our main result, Proposition 6.13.

6.12. Lemma. Let Γv be a tame extension of Γ. Then for every chamber C ∈ nbhd(v),

1. There is a unique chamber C in nbhd(v) ∩ Γ closest to C.

2. If D is any chamber in nbhd(v) ∩ Γ, then

d(C,D) = d(C,C) + d(C,D).

Proof. Item (1) follows from the fact that the wall W partitions the building into disjointsets of chambers which are closer to one of the chambers in nbhd(v) ∩ Γ than the others.

65

Suppose Item (2) were false. That is, suppose C 6= D and let the gallery

C = C1, . . . , C, . . . Cr = D

be the composite of minimal galleries from C to C and C to D, and suppose this gallery is notminimal. Then there exists a wall in the building which is crossed twice—once between C and C,and once between C and D. But the only wall between C and D is the wall W , and no minimalgallery from C to C crosses W . Therefore, the gallery above is minimal. 2

Returning to our consideration of the Hecke algebra H(B\G/B), observe that Lemma 6.1and Lemma 6.3 hold for (p+ 1)-regular buildings of the form ∆n(G,B), since we did not use thefact that the building was a tree in their proofs.

6.13. Proposition. Let G be SL3(Qp), B its Iwahori subgroup, and ∆3 the (p + 1)-regularaffine building ∆3(G,B). Let Γ be a finite subcomplex of ∆3 and let Γv be a tame extension ofΓ. If M = MΓ is positive, then Mv = MΓv is positive.

Proof. Let M = MΓ and choose Mv = MΓv so that the indices agree on the submatrixindexed by Γ. Let the wall W be as in the definition of a tame extension. For gB ∈ ∆3, letgB ∈ nbhd(v) ∩ Γ be the chamber in nbhd(v) ∩ Γ which is closest to gB. Let Q be the matrixwhose rows are indexed by Γ and whose columns are indexed by Γv given by

Qg,h ={ 〈〈[gB], [hB]〉〉 if gB is the nearest edge in Γ to hB

0 otherwise.

We assumed the matrix M is positive, and therefore Q∗MQ is positive. The entries inQ∗MQ are given by

(Q∗MQ)g,h =

〈〈[gB], [hB]〉〉 if gB, hB ∈ Γ

〈〈[gB], [gB]〉〉 ∗ 〈〈[gB], [hB]〉〉 if gB /∈ Γ and hB ∈ Γ〈〈[gB], [hB]〉〉 ∗ 〈〈[hB], [hB]〉〉 if gB ∈ Γ and hB /∈ Γ

〈〈[gB], [gB]〉〉 ∗ 〈〈[gB], [hB]〉〉 ∗ 〈〈[hB], [hB]〉〉 if gB, hB /∈ Γ.

(6.2)

If gB /∈ Γ but hB ∈ Γ, then gB lies in a minimal gallery from gB to hB by Lemma 6.12, andsimilarly for the case gB ∈ Γ but hB /∈ Γ. So the matrix Q∗MQ has the form

Q∗MQ =

· · · 〈〈[gB], [hB]〉〉 · · · · · ·

...

nbhd(v) ∗

nbhd(v)

where ∗ indicates an entry of the form

〈〈[gB], [gB]〉〉 ∗ 〈〈[gB], [hB]〉〉 ∗ 〈〈[hB], [hB]〉〉

So Mv and Q∗MQ agree except on those entries indexed by nbhd(v).We are done if we can show Mv − Q∗MQ is positive. The matrix Mv − Q∗MQ is zero

outside its submatrix indexed nbhd(v). Therefore, the entries of Mv − Q∗MQ lie in a finiteHecke algebra isomorphic to the algebra generated by X and Y .

66

Let P = Mnbhd(v), R = Mnbhd(v)∩Γ, and let S be the matrix whose rows are indexedby nbhd(v) ∩ Γ and whose columns are indexed by nbhd(v) given by

Sg,h ={〈〈[gB], [hB]〉〉 if gB = hB

0 otherwise.

Now P is positive, R is positive (since the link of any simplex—in this case an edge—correspondsto a finite Hecke algebra), and S∗RS is positive. The elements of S∗RS are indexed by nbhd(v)and given by

(S∗RS)g,h = 〈〈[gB], [gB]〉〉 ∗ 〈〈[gB], [hB]〉〉 ∗ 〈〈[hB], [hB]〉〉

= 〈〈[gB], [hB]〉〉 ∗ 〈〈[hB], [hB]〉〉

= 〈〈[gB], [gB]〉〉 ∗ 〈〈[gB], [hB]〉〉

by Lemma 6.12.We claim that P − S∗RS is positive. Consider the matrix

(P − S∗RS)2 = P 2 − PS∗RS − S∗RSP + S∗RSS∗RS.

IfΛ =

∑γB∈nbhd(v)

〈〈1, [γB]〉〉 ∗ 〈〈[γB], 1〉〉,

then P 2 = ΛP . The entries of PS∗RS are given by

PS∗RSg,h =

∑γB∈nbhd(v)

〈〈[gB], [γB]〉〉 ∗ 〈〈[γB], [hB]〉〉 ∗ 〈〈[hB], [hB]〉〉

= Λ〈〈[gB], [hB]〉〉 ∗ 〈〈[hB], [hB]〉〉 = Λ(S∗RS)g,h

(applying Lemma 4.9). So PS∗RS = ΛS∗RS, and we can show in a similar manner thatS∗RSP = ΛS∗RS and (S∗RS)2 = ΛS∗RS. Therefore, P − S∗RS is positive, since

P − S∗RS = Λ−1(P − S∗RS)2.

But Mv−Q∗MQ is zero off the submatrix indexed by nbhd(v), on which it agrees with P−S∗RS.Therefore, Mv −Q∗MQ, and hence Mv, is positive. 2

Using tame extensions, it follows that 〈〈, 〉〉 is positive on any subcomplex whose projectionto an apartment is shown in Figure 6.4.

If we could show that positivity holds for so-called nontame extensions, as in Figure 6.5,that would prove that 〈〈, 〉〉 is positive on C[G/B]×C[G/B].

This suggests the following conjectures:

6.14. Conjecture. Let G be SL3(Qp) and B its Iwahori subgroup. Then 〈〈f, f〉〉 is positive forall f ∈ C[G/B].

6.15. Conjecture. Let G be SLn(Qp) and B its Iwahori subgroup. Then 〈〈f, f〉〉 is positive forall f ∈ C[G/B].

67

Fig. 6.4. The projection of a subcomplex formed by tame extensions

Γ

v

Γvnbhd(v)

Fig. 6.5. A nontame extension

68

6.3 The mysterious Λ

For finite Hecke algebras, the element Λ ∈ H(S\G/S) given by

Λ =∑

g∈G/S〈〈1, [gS]〉〉 ∗ 〈〈[gS], 1〉〉

proved extremely useful. We showed that Λ is positive, central, invertible, and, for P = MG/S ,

that P = Λ−1P 2. The fact that H(S\G/S) ∼= C∗(S\G/S) in the finite case allowed us toconclude that P is a positive matrix and hence that 〈〈, 〉〉 is positive.

What if H(S\G/S) is not a finite Hecke algebra? Then Λ is not an element in H(S\G/S).However, it does have some interesting properties. It is central with respect to elements inH(S\G/S). Moreover, if the infinite matrix P is given by P = MG/S , then we still have

P 2 = ΛP , and Λ is “positive” in the sense that it is the (infinite) sum of elements of the forma∗a, for a ∈ H(S\G/S). We might hope to construct some sort of formal inverse for Λ whichwould allow us to interpret P = Λ−1P 2 for infinite Hecke algebras. However, we have shownthat 〈〈, 〉〉 is not positive for the Hecke algebra formed from SL2(Qp) with its maximal compactsubgroup, so P cannot be positive for all Hecke algebras.

69

References

[1] Dan Barbasch and Allen Moy. A unitary criterion for p-adic groups. Inventiones Mathe-maticae, 98:19–37, 1989.

[2] Armand Borel. Admissible representations of a semi-simple group over a local field withvectors fixed under an Iwahori subgroup. Inventiones Mathematicae, 35:233–259, 1976.

[3] Jean-Benoıt Bost and Alain Connes. Hecke algebras, type III factors and phase transitionswith spontaneous symmetry breaking in number theory. Selecta Mathematica (New Series),1(3):411–457, 1995.

[4] Berndt Brenken. Hecke algebras and semigroup crossed product C∗-algebras. Pacific Journalof Mathematics, 187(2):241–262, 1999.

[5] Kenneth S. Brown. Buildings. Springer-Verlag, New York, 1989.

[6] Kenneth S. Brown. Five lectures on buildings. In E. Ghys, A. Haefliger, and A. Verjovsky,editors, Group Theory from a Geometrical Viewpoint, pages 254–295, River Edge, NJ, 1991.World Scientific Publishing Co., Inc.

[7] Colin J. Bushnell and Philip C. Kutzko. Smooth representations of reductive p-adic groups:Structure theory via types. Proceedings of the London Mathematical Society, 77(3):582–634,1998.

[8] Kenneth R. Davidson. C∗-Algebras by Example. Number 6 in Fields Institute Monographs.American Mathematical Society, Providence, 1996.

[9] Fernando Q. Gouvea. p-adic Numbers. Springer-Verlag, New York, 1993.

[10] N. Iwahori and H. Matsumoto. On some Bruhat decompositions and the structure of theHecke rings of p-adic Chevalley groups. Publ. Math., 25:5–48, 1965.

[11] Richard V. Kadison and John R. Ringrose. Fundamentals of the Theory of Operator Algebras:Elementary Theory. Number 100-I in Pure and Applied Mathematics. Academic Press, NewYork, 1983.

[12] Aloys Krieg. Hecke Algebras. Number 435 in Memoirs of the American Mathematical Society.American Mathematical Society, Providence, Sept. 1990.

[13] Jean Pierre Serre. Trees. Springer-Verlag, New York, 1980.

[14] Jacques Tits. Structures et groupes de Weyl. Seminaire Bourbaki 1964-5, 288, Feb. 1965.

Vita

Rachel W. Hall

Rachel W. Hall was born on June 12, 1969 in Cincinnati, Ohio. She received a Bachelor ofArts magna cum laude from Haverford College in May, 1991, with a major in Ancient Greek atBryn Mawr College. She attended graduate school and taught at the Pennsylvania State Univer-sity from 1993 to 1999 and received a doctorate in Mathematics in December, 1999. Her thesis,Hecke C∗-Algebras, was completed under the direction of Nigel Higson. Her research interestsinclude Operator Algebras, Non-Commutative Algebras, and Group Rings. She is currently anAssistant Professor at St. Joseph’s University in Philadelphia.

AbstractHECKE C*-ALGEBRASRachel W. HallDoctor of Philosophy; December 1999The Pennsylvania State UniversityNigel D. Higson, Thesis Adviser

The purpose of this thesis is to introduce C∗-algebra techniques to the theory of Heckealgebras, especially to the representation theory of Hecke algebras. We will investigate when aHecke algebra may be completed to a Hecke C∗-algebra in a manner analogous to the completionof the group algebra to the group C∗-algebra, and we will explore the connections between theexistence of such a completion and the structure of the category of ∗-representations of the Heckealgebra.

The construction of the Hecke C∗-algebra first arose in Bost and Connes’ study of theHecke algebra H(P+

Z\P+Q/P+Z

) and the Riemann zeta function. We will focus our attentionon algebras quite closely related to their example. Our original motivation was to prove thatthe category of nondegenerate ∗-representations of the Hecke algebra H(S\G/S) is equivalentto the category of unitary representations of the group G generated by their S-fixed vectors,implying the existence of the Hecke C∗-algebra. However, the Hecke algebra formed from thegroup SL2(Qp) with the maximal compact subgroup SL2(Zp) shows that the Hecke C∗-algebradoes not exist in general. On the other hand, Borel proved an algebraic counterpart of thiscategory equivalence for the Hecke algebra H(B\G/B), where B is the Iwahori subgroup of asemi-simple group G over a non-Archimedean local field. In addition, Barbasch and Moy settledthe unitary case of the category equivalence by invoking the proof of the Deligne-Langlandsconjecture and examining all the representations of G. We will prove, in special cases, a similar,and in some ways stronger, result using C∗-algebra theory and the geometry of buildings. Wehave also considered cases relevant to Bost and Connes’ example.