Hecke algebras and harmonic analysis on finite groups1-2)/27-52.pdf · [3] Hecke algebras and...

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Rendiconti di Matematica, Serie VII Volume 33, Roma (2013), 27 – 52 Hecke algebras and harmonic analysis on finite groups FABIO SCARABOTTI – FILIPPO TOLLI To Alessandro Fig` a Talamanca on the occasion of his retirement Abstract: Let G be a finite group, K a subgroup and (σ,V ) an irreducible represen- tation of K. Then the Hecke algebra associated with the triple (G, K, σ) is the commutant of the induced representation Ind G K σ. In [3] Curtis and Fossum derived several explicit ex- pressions for the characters of Hecke algebras. In the present paper we give an exposition of their results (see also [5], pp. 279-291) in the language of finite harmonic analysis. In particular, we show the connection with the theory of finite Gelfand pairs. 1 – Introduction Let K G be finite groups and denote by X = G/K the corresponding homoge- neous space. A function f : G -→ C is said to be bi-K -invariant if f (k 1 gk 2 )= f (g) for all g G, k 1 ,k 2 K . The bi-K -invariant functions form an algebra under con- volution that coincides with the commutant of the permutation representation of G on X . In other words, any G-invariant operator on X is given by the convolution with a suitable bi-K -invariant kernel. We recall that (G, K ) is a (finite) Gelfand pair when the permutation repre- sentation of G on the space G/K decomposes without multiplicity; equivalently, when the algebra of bi-K -invariant functions is commutative. In this setting it is possible to develop a harmonic analysis based on a particular basis of the space of bi-K -invariant functions constitued by the so called spherical functions. The the- ory of spherical functions has many applications; we refer to ([6, 13, 1]) for complete Key Words and Phrases: Hecke algebra – Irreducible character – Spherical func- tion A.M.S. Classification: Primary 20C08, Secondary 20C15, 20E22, 43A90.

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Page 1: Hecke algebras and harmonic analysis on finite groups1-2)/27-52.pdf · [3] Hecke algebras and harmonic analysis on finite groups 29 Lemma 2.1. Let E,E 1 and E 2 be idempotents and

Rendiconti di Matematica, Serie VIIVolume 33, Roma (2013), 27 – 52

Hecke algebras and harmonic analysison finite groups

FABIO SCARABOTTI – FILIPPO TOLLI

To Alessandro Figa Talamancaon the occasion of his retirement

Abstract: Let G be a finite group, K a subgroup and (!, V ) an irreducible represen-tation of K. Then the Hecke algebra associated with the triple (G,K,!) is the commutantof the induced representation IndG

K!. In [3] Curtis and Fossum derived several explicit ex-pressions for the characters of Hecke algebras. In the present paper we give an expositionof their results (see also [5], pp. 279-291) in the language of finite harmonic analysis. Inparticular, we show the connection with the theory of finite Gelfand pairs.

1 – Introduction

Let K ! G be finite groups and denote by X = G/K the corresponding homoge-neous space. A function f : G "# C is said to be bi-K-invariant if f(k1gk2) = f(g)for all g $ G, k1, k2 $ K. The bi-K-invariant functions form an algebra under con-volution that coincides with the commutant of the permutation representation of Gon X. In other words, any G-invariant operator on X is given by the convolutionwith a suitable bi-K-invariant kernel.

We recall that (G,K) is a (finite) Gelfand pair when the permutation repre-sentation of G on the space G/K decomposes without multiplicity; equivalently,when the algebra of bi-K-invariant functions is commutative. In this setting it ispossible to develop a harmonic analysis based on a particular basis of the space ofbi-K-invariant functions constitued by the so called spherical functions. The the-ory of spherical functions has many applications; we refer to ([6, 13, 1]) for complete

Key Words and Phrases: Hecke algebra – Irreducible character – Spherical func-tionA.M.S. Classification: Primary 20C08, Secondary 20C15, 20E22, 43A90.

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28 FABIO SCARABOTTI – FILIPPO TOLLI [2]

expositions. In [9, 10, 11] and [12] we extended the theory of spherical functions tohomogeneous spaces whose associated permutation representation decomposes withmultiplicity and gave several applications, mainly to probability and statistics. Oneof the goals of the present paper is to show how the results in [3] may be seen asgeneralizations of the theory of spherical functions associated with finite Gelfandpairs.

The plan of the paper is the following. In Section 2 we give some basic results onidempotents in a finite dimensional algebra focusing on the case of the commutantof a representation of a finite group. In Section 3 we show the connection betweenidempotents and ideals in the group algebra. In Section 4 we give the results in [3] forgeneral Hecke algebras, showing the connections between the irreducible charactersof the Hecke algebra and the irreducible characters of the group. In Section 4.1 wefocus to the case of Hecke algebras that are commutant of induced representationsof one-dimensional representations. In this case, the results of the general case maybe formulated in a form suitable for applications. The last two sections containnew material with respect to our original sources. In Section 4.2, we show how theresults in [3] are even more transparent when the Hecke algebra is the commutantof a permutation representation. In the final section we treat the case of a finiteGelfand pair and we show that the main formulas in [3] generalized the well knownformulas that express a character in terms of the corresponding spherical functionand the spherical function in terms of the character.

In the present paper we assume that all representations are complex and unitary.We use the notation in our books [1] and [2], where one can find introductions torepresentation theory of both finite groups and finite dimensional associative alge-bras and to finite harmonic analysis.

2 – Ideals and idempotents

We begin by recalling some basic facts on idempotents in unitary spaces (see [7]).Let V be a complex vector space endowed with a hermitian scalar product anddenote by End(V ) the vector space of all linear operators T : V # V . E $ End(V )is called an idempotent when E2 = E. If E is an idempotent then V = RanE %KerE (with direct sum) and E is the projection on RanE along KerE. Moreover,Ran(I "E) = KerE, Ker(I "E) = RanE, I "E is also an idempotent and it is theprojection on KerE along RanE. Conversely, if V = V1%V2 then the projection of V1

along V2 is an idempotent. An idempotent E $ End(V ) is an orthogonal projectionwhen RanE & KerE and this is equivalent to E being selfadjoint. If E1, E2 areidempotents we have E1E2 = 0 if and only if RanE2 ! KerE1; in particular, ifE1E2 = E2E1 = 0 then RanE1 ' RanE2 = {0}. From these observations weimmediately deduce the following fact.

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[3] Hecke algebras and harmonic analysis on finite groups 29

Lemma 2.1. Let E,E1 and E2 be idempotents and suppose that E1E2 = E2E1 =0 and E = E1 + E2. Then RanE = RanE1 % RanE2.

Let G be a finite group and denote by (!, U) an irreducible representations ofG. The commutant of (!, U) is the algebra HomG(U,U) of all linear operators inEnd(U) that commute with the action of G:

HomG(U,U) = {T $ End(U) : !(g)T = T!(g), (g $ G}.

An idempotent E $ HomG(U,U) is primitive when the condition E = E1 + E2,with E1, E2 idempotents in HomG(U,U) and E1E2 = E2E1 = 0, always impliesthat either E1 = 0 or E2 = 0. If E is an idempotent in HomG(U,U) then

IE = {A $ HomG(U,U) : AE = A}

is a left ideal of HomG(U,U). It is easy to see that

IE = {A $ HomG(U,U) : KerA ) KerE}

because AE = A if and only if KerA ) KerE.We introduce the Hilbert-Schmidt hermitian scalar product on HomG(U,U) by

setting, for A,B $ HomG(U,U),

*A,B+HS = tr(B!A)

where B! is the adjoint of B and tr denotes the trace. Notice that if I is a left idealof HomG(U,U) then also I" = {B $ HomG(U,U) : *A,B+HS = 0, (A $ I} is a leftideal.

Lemma 2.2. Let I be a left ideal of HomG(U,U) and denote by E the orthogonalprojection from HomG(U,U) onto I. Then

E(AB) = AE(B) (2.1)

for all A,B $ HomG(U,U).

Proof. It follows immediately from the following two observations:

AE(B) $ I;AB "AE(B) = A[B " E(B)] $ I",

for all A,B $ HomG(U,U). !

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30 FABIO SCARABOTTI – FILIPPO TOLLI [4]

If W ! U we denote by EW the orthogonal projection onto W . Note thatEW belongs to HomG(U,U) if and only if W is a !-invariant subspace. In thefollowing propositions we will formulate certain results, that connect idempotentsE $ HomG(U,U) and !-invariant subspaces in U .

Proposition 2.3.

1. The map W ,# EW is a linear bijection between the !-invariant subspaces and theorthogonal projections in HomG(U,U). Moreover, W is irreducible if and onlyif EW is primitive. This in turn is equivalent to the fact that every idempotentE $ HomG(U,U), with RanE = W , is primitive.

2. The map E ,# IE is a linear bijection between the space of orthogonal idem-potents and the left ideals in HomG(U,U). Moreover, the ideal IE is minimal(i.e. it does not contain nontrivial ideals) if and only if E is primitive. This inturn is equivalent to the fact that every idempotent F $ HomG(U,U), such thatIF = IE, is primitive.

Proof. (1) The first part is obvious. Suppose that W is irreducible and thatE $ HomG(U,U) is an idempotent such that RanE = W . If E = E1 + E2 withE1, E2 $ HomG(U,U) idempotent and E1E2 = E2E1 = 0 then by Lemma 2.1 wehave W = RanE1 % RanE2 and RanE1,RanE2 are !-invariant. Therefore eitherRanE1 = {0} or RanE2 = {0}, and E is primitive. Conversely, assume that E $HomG(U,U) is a primitive idempotent and set W = RanE. Suppose that W1 ! Wis !-invariant and let E1 : U # W1 be the orthogonal projection onto W1. SetE2 = E"E1 so that E1, E2 $ HomG(U,U), they are idempotents E = E1+E2 andE1E2 = E2E1 = 0. Since E is primitive either E1 = 0 (and W1 = {0}) or E2 = 0and W1 = W . Therefore W is irreducible.

(2) The proof is similar to (1) and we limit ourselves to prove that for everyideal I there exists an idempotent E such that I = IE . Let E be as in Lemma 2.2.Setting E = E(IU ) we have

A $ I - A = E(A) = E(AIU ) =(!) AE(IU ) = AE

where =(!) follows from (2.1). Moreover

E2 = E(IU )E(IU ) = E(E(IU )) = E(IU ) = E

proving that E is an idempotent and that I = IE . !

Proposition 2.4. Suppose that W1,W2 ! U are two !-invariant subspaces andthat E1, E2 $ HomG(U,U) are two idempotents such that RanEi = Wi, i = 1, 2.Then, setting

V = {T $ HomG(U,U) : E2TE1 = T} . {E2TE1 : T $ HomG(U,U)},

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[5] Hecke algebras and harmonic analysis on finite groups 31

the mapV # HomG(W1,W2)T ,# T |W1

is a linear isomorphism. If W1 = W2 then it is also an isomorphism of algebras.

Proof. If T $ V then T |W1 $ HomG(W1,W2) and T |W1 = 0 if and only if T =0. Suppose that T $ HomG(W1,W2) and extend it to an operator T $ HomG(U,U)by setting T v = 0 for all v $ KerE1. Then we have

T v = TE1v = E2TE1v, (v $ U,

that is T $ V and certainly T |W1 = T . !

Proposition 2.5. Suppose that E $ HomG(U,U) is an idempotent. Then thefollowing facts are equivalent:

1. W = RanE is !-irreducible;2. ETE is a multiple of E for all T $ HomG(U,U);3. the algebra {T $ HomG(U,U) : ETE = T} is isomorphic to C.

Proof. By Schur’s Lemma we have RanE is irreducible if and only ifHomG(W,W ) /= C and by Proposition 2.4 HomG(W,W ) = {T $ HomG(W,W ) :ETE = T} . {ETE : T $ HomG(W,W )}. !

Let !G be a complete system of pairwise inequivalent irreducible representationsof G. We recall [1, 2], that if U =

"!#J m!V! is the decomposition of U into

irreducible G-representations, where J 0 !G and m! = dimHomG(V!, U) is themultiplicity of V! in U , then

HomG(U,U) /=#

!#J

HomG(m!V!,m!V!) /=#

!#J

Mm!,m! (C), (2.2)

where Mm!,m! (C) is the full matrix algebra of (m! 1 m!)-complex matrices. Inparticular, if E! : U # m!V! is the orthogonal projection from U onto m!V!, then{E! : " $ J} is a basis for the center of HomG(U,U).

An idempotent E $ HomG(U,U) is central when it belongs to the center ofHomG(U,U). A central idempotent E $ HomG(U,U) is primitive when every de-composition E = E1+E2, with E1, E2 central idempotents and E1E2 = E2E1 = 0,is trivial, that is E1 = 0 or E2 = 0. Note that a central primitive idempotent isnot necessarily a primitive idempotent. The following proposition is an immediateconsequence of the above considerations.

Proposition 2.6. If E! is as above, then {E! : " $ J} are precisely the primitiveidempotents in HomG(U,U).

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32 FABIO SCARABOTTI – FILIPPO TOLLI [6]

For the representation theory of HomG(U,U) we refer to the first chapter of [2].Here we limit ourselves to point out that the irreducible character associated with" $ J is given by:

#!(T ) =m!$

i=1

t!i,i,

for all T $ HomG(U,U), where (t!i,j)i,j=1,...,m! $ Mm!,m! (C) is given by the iso-morphism (2.2).

3 – Ideals and idempotents in a group algebra

In this section we apply the previous analysis to the particular case of the regularrepresentation of a finite group G. The group algebra L(G) is the vector space{f : G # C} endowed with the convolution product:

f 2 $(g) =$

h#G

f(gh)$(h$1)

for all f,$ $ L(G) and g $ G. For g $ G, the Dirac function %g centered at g is

defined by setting %g(h) =

%0 if h 3= g

1 if h = g.Observe that the %1G is the unit element

of L(G).We denote by & the left regular representation of G on L(G), that is,

[&(g)f ](t) = f(g$1t)

for all g, t $ G and f $ L(G). It is easy to see that a subspace V 0 L(G) is&-invariant if and only if it is a left ideal of L(G); if it is a minimal ideal if and onlyif &-irreducible. In what follows, we set

f(g) = f(g$1)

for all f $ L(G) and g $ G; moreover for $ $ L(G) we denote by T" the corre-sponding convolution operator, that is

T"f = f 2 $

for all f $ L(G). We recall the following fundamental isomorphism ([1]):

Theorem 3.1. The map

L(G) # HomG(L(G), L(G))$ ,# T"

(3.1)

is an antiisomorphism of algebras.

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[7] Hecke algebras and harmonic analysis on finite groups 33

A function ' $ L(G) is called idempotent if '2' = '. By the above theorem wehave that ' is an idempotent if and only if the corresponding convolution operatorE = T# is an idempotent; moreover, it is easy to check that E is an orthogonalprojection if and only if ' = '. An idempotent is primitive if ' = '1 + '2 with'1,'2 idempotents such that '1 2 '2 = '2 2 '1implies that '1 = 0 or '2 = 0.

In the next proposition we collect the main properties of left ideals in the groupalgebra L(G). They are just a reformulation of the results of the Section 2, takinginto account Theorem 3.1 (and other observations made in this section).

Proposition 3.2.

1. If ' $ L(G) is an idempotent then V = L(G) 2 ' is a left ideal in L(G) andevery left ideal of L(G) may be represented in this way;

2. V = L(G)2' is minimal (that is, it is &-irreducible) if and only if ' is primitive;3. V = L(G) 2 ' is minimal if and only if the algebra ' 2 L(G) 2 ' is isomorphic

to C;4. if '1, '2 are two idempotents and V1, V2 the respective ideals, then the map

'2 2 L(G) 2 '1 # HomG(V1, V2)$ ,# T"

(3.2)

is a linear bijection and it is an antiisomorphism when V1 = V2.

Definition 3.3. Let (!, U) be a representation of G and f $ L(G). The Fouriertransform of f at (!, U) is the linear operator on U defined by setting

!(f) =$

g#G

f(g)!(g).

We can reconstruct the function f from its Fourier transform by the Fourier inver-sion formula ([1]):

f(g) =1

|G|$

$# !G

d$tr[!(g$1)!(f)]

for all g $ G, where d$ denotes the dimension of ! $ !G.

Note that ' $ L(G) is an idempotent (resp. orthogonal projector) if and only if!(') is an idempotent (resp. orthogonal projector). Moreover the Fourier transformis multiplicative: !(f1 2 f2) = !(f1)!(f2) for all f1, f2 $ L(G).

We recall the following fundamental results (see [2]).

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34 FABIO SCARABOTTI – FILIPPO TOLLI [8]

Lemma 3.4. Let L(G) ="

$# !G d$U$ be the isotypic decomposition of the regularrepresentation. Then we have:

1. The orthogonal projection E$ : L(G) # d$U$ is given by

E$f(g) =d$|G| tr[!(g

$1)!(f)]

for all f $ L(G) and g $ G;2. The map

d$U$ # Hom(U$, U$)f ,# !(f)

(3.3)

is an (explicit) isomorphism of vector spaces;3. The restriction of the map in Theorem 3.1 to the isotypic component d$U$ has

range isomorphic to HomG(d$U$, d$U$).

We end this section by introducing the following notation: given a G-representation! and a function f $ L(G) we set

#$(f) =$

g#G

#$(g)f(g) = tr[!(f)] (3.4)

where #$ denotes the character of the representation !.

4 – Hecke algebras

In this section we introduce the notion of the Hecke algebra associated to the inducedrepresentation from a subgroup.

Lemma 4.1. Let G be a group and K ! G a subgroup. Suppose that V is a leftideal in L(K) and ' $ L(K) an idempotent that generates V . Then IndGKV . {f $L(G) : f 2 ' = f}. In other words, ' (which is clearly an idempotents in L(G))generates IndGKV as a left ideal in L(G).

Proof. Recall thatL(G) # IndGKL(K)f ,# F

(4.1)

where F (g, k) = f(gk), is an explicit linear isomorphism. Indeed, IndGKL(K) is theset of all functions F : G 1K # C such that F (gk1, k) = F (g, k1k) and (4.1) is a

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[9] Hecke algebras and harmonic analysis on finite groups 35

particular case of transitivity of induction (see [2, Proposition 1.6.6]). Then, in thecorrespondence f ,# F , we have

f $ IndGKV - F (g, k) =$

t#K

F (g, kt$1)'(t)

- f(gk) =$

t#K

f(gkt$1)'(t)

- f = f 2 '. !

Lemma 4.2. Keeping the same assumptions of the previous lemma, suppose that(!, U) is a representation of G. Then the map

HomK(V,ResGKU) # !(')UT ,# T (')

is a linear isomorphism of vector spaces.

Proof. Denote by &K the left regular representation of K. Note that we haveT&K(k) = !(k)T , for every T $ HomK(V,ResGKU), and therefore

T (') = T

&$

k#K

'(k)%k

'

=$

k#K

'(k)T (%k)

=$

k#K

'(k)T [&K(k)%1K ]

=$

k#K

'(k)!(k)T (%1K )

= !(')T (%1K ) $ !(')U.

Moreover, for every f $ V we have:

T (f) = T (f 2 ')

= T

&$

k#K

f(k)&K(k)'

'

=$

k#K

f(k)!(k)T (')

= !(f)T ('),

and therefore T is determined by T (') and the map T ,# T (') is injective.

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36 FABIO SCARABOTTI – FILIPPO TOLLI [10]

To prove that this map is also surjective, we show that, for u $ U , the mapT : V # U , T (f) = !(f)!(')u for all f $ V , belongs to HomK(V,ResGKU). Indeed,we have

T (f) = !(f 2 ')u = !(f)u

and therefore, for k $ K, we have

T [&K(k)f ] = ![&K(k)f ]u

= !(%k 2 f)u= !(k)!(f)u

= !(k)T (f). !

Corollary 4.3. If (!, U) is irreducible, then the multiplicity of U in IndGKV isequal to

1. dimHomG(U, IndGKV );

2. dim[!(')U ];3. #$(').

Proof. (1) it is well known.(2) it follows from (1), Frobenius reciprocity and the preceding lemma.Since !(') : U # !(')U is the projection of U onto !(')U , we have also

dim[!(')U ] = tr[!(')] = #$(').

This proves (3). !

Definition 4.4. Let K ! G and ' be an idempotent in L(K). Then theHecke algebra H(G,K,') is the subalgebra ' 2 L(G) 2 ' of L(G). Clearly theantiisomorphism (3.2), the decomposition in (2.2) and Lemma 4.1 ensure us that

H(G,K,') /= HomG(IndGKV, IndGKV ).

Remark 4.5. Often, it is used the notation H(G,K,#), where # is the characterof the representation of K on L(K) 2 '; see [4].

In the same notation of Lemma 4.1, suppose that IndGKV ="

$#J m$U$ is the

decomposition of IndGKV into irreducible G-representations (i.e. J 0 !G and m$ isthe multiplicity of ! in IndGKV ). Let d$U$ be the !-isotypic component in L(G).

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[11] Hecke algebras and harmonic analysis on finite groups 37

Lemma 4.6.

1. m$U$ = {f 2 ' : f $ d$U$};2. the map

{$ $ d$U$ : ' 2 $ 2 ' = $} # HomG(m$U$,m$U$)$ ,# T"|m"U"

is an antiisomorphism.

Proof. 1) Clearly, m$U$ 0 d$U$ and therefore from Lemma 4.1 we deducethat

m$U$ = {f $ d$U$ : f 2 ' = f} = {f 2 ' : f $ d$U$}.

2) d$U$ is a two sided ideals of L(G) (isomorphic to End(U$)) and therefore byvirtue of Proposition 3.4

d$U$ # HomG(d$U$, d$U$)$ ,# T"

is an antiisomorphism of algebras. Then we can apply (3.2) (m$U$ is a left idealin d$U$) noting also that we can replace ' with the orthogonal projection ontod$U$. !

Lemma 4.7. An idempotent ( $ H := H(G,K,') is primitive in H if and onlyif it is primitive in L(G).

Proof. Since ( = ' 2 ( 2 ' = ( 2 ' = ' 2 (, we have

( 2 L(G) 2 ( = ( 2 ' 2 L(G) 2 ' 2 (= ( 2H 2 (.

Then the lemma is a consequence of Proposition 3.2 and Proposition 2.5. !

Lemma 4.8. Let $ $ H(G,K,') and consider the convolution operator T" as-sociated with $. If the restriction of T" to the isotypic component m$U$ is the nulloperator, then !($) = 0.

Proof. Indeed, for all f $ d$U$ we have that f 2 ' $ m$U$ and thus

!(f)!($) = !(f)!(' 2 $)= !(f 2 ' 2 $)= 0.

Since {!(f) :f $d$U$}/=Hom(U$, U$) (see (3.3)) we necessarily have !($)=0. !

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38 FABIO SCARABOTTI – FILIPPO TOLLI [12]

Corollary 4.9.

HomG(m$U$,m$U$) /= {$ $ d$U$ : ' 2 $ 2 ' = $}.

In other words, if $ $ H(G,K,') and

T"|m!U! = 0 (4.2)

for all " $ J \ {!}, then $ $ d$U$.

Proof. Using Lemma 4.8 we deduce from (4.2) that "($) = 0 if " $ J \ {!}.If " 3$ J then we have that f 2 ' = 0 for all f $ m!U! which immediately impliesthat "($) = 0. These facts force $ $ d$U$ (see (1) in Lemma 3.4). !

Let E be an idempotent on a vector space U of dimension d. Suppose thatT $ Hom(U,U) satisfies T = ETE. Then KerT ) KerE and RanT 0 RanE.Let {u1, u2, . . . , ud} be a basis of U such that {u1, u2, . . . , um} is a basis of RanEand {um+1, um+2, . . . , ud} is a basis of KerE. Let (ti,j)i,j=1,2,...,d be the matrixrepresenting T in the basis {u1, u2, . . . , ud}, that is

Tui =d$

j=1

ti,juj .

Observe that ti,j = 0 if i > m or j > m and therefore

{T $ Hom(U,U) : T = ETE} /= Mm,m(C).

These considerations applied to the case E = !(') and T = !($) lead to the proofof the following lemma.

Lemma 4.10. Let {u1, u2, . . . , ud"} be a basis of U$ such that {u1, u2, . . . , um"}is a basis of !(')U$ and {um"+1, um"+2, . . . , ud"} is a basis of Ker!('). Then themap

{$ $ d$U$ : ' 2 $ 2 ' = $} # Mm",m" (C)$ ,# (t"i,j)i,j=1,2,...,m"

where t"i,j are the coe!cients of !($) in the basis {u1, u2, . . . , ud"}, is an explicitisomorphism.

Corollary 4.11. The irreducible characters of H /="

$#J Mm",m" (C) aregiven by the restrictions of the characters of the representations in J .

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[13] Hecke algebras and harmonic analysis on finite groups 39

For each ! $ J , we define )$ by setting (recall (3.4))

)$($) = #$($) =$

g#G

$(g)#$(g) = tr[!($)].

By virtue of Corollary 4.11 {)$ : ! $ J} are the irreducible characters of H. Werecall (see [1]) that an alternative expression of the orthogonal projection of L(G)onto the isotypic component d$U$

/= U$ 4 U$! is given by

E$f =d$|G|f 2 #$ =

d$|G|f 2 #$!

,

where !% is the adjoint of !. Equivalently

E$f =d$|G|

$

g#G

#$(g)&(g$1)f.

Finally we observe that ' 2 #$ = #$ 2 ' $ IndGKV (see Lemma 4.1).

Lemma 4.12. The orthogonal projection from IndGKV onto m$U$ is given by

E#$ f =

d$|G|f 2 ' 2 #$.

Proof. If f $ IndGKV then f = f 2 ' and therefore

E$f = E#$ f.

If f $ m$U$ 0 d$U$ then f = E$f = E#$ f , while if " 3/ ! and f $ m!U! then

0 = E$f = E#$ f . !

Corollary 4.13 ([8], Janusz). If the multiplicity of U$ in IndGKV is equal to 1,then E#

$ is a primitive idempotent in L(G) whose range is a subspace isomorphicto U$.

An idempotent ( in H or in L(G) is called central when (2$ = $2( for all $ $ H.A central idempotent ( is called primitive when every decomposition ( = (1 + (2,with (1, (2 central idempotents and (1 2 (2 = (2 2 (1 = 0, is trivial, i.e. (1 = 0 or(2 = 0.

Corollary 4.14. The central primitive idempotents of H are given by(

d$|G|' 2 #$ : ! $ J

).

Proof. It is an immediate consequence of Proposition 2.6 and Lemma 4.12. !

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40 FABIO SCARABOTTI – FILIPPO TOLLI [14]

In the following theorem the characters of the representations of G contained inIndGKV are expressed in terms of the characters of H.

Theorem 4.15. For g $ G, let C(g) be the conjugacy class of g and denote by1C(g) the characteristic function of C(g). Then we have, (! $ J ,

#$(g) =|G|

|C(g)|)$(' 2 1C(g) 2 ')*$

h#G

)$(' 2 %h"1 2 '))$(' 2 %h 2 ')+$1

.

Proof. Since the characteristic function 1C(g) is a central function, we have' 2 1C(g) 2 ' = 1C(g) 2 '. Therefore

!(' 2 1C(g) 2 ') = !(1C(g))!('). (4.3)

Moreover, using again the centrality of 1C(g), we have !(1C(g)) = &I$ and taking

the trace of both sides we get that & = |C(g)|%"(g)d"

and therefore

!(1C(g)) =|C(g)|#$(g)

d$I$. (4.4)

Plugging (4.4) in (4.3) and taking the trace of both sides, we get

#$(' 2 1C(g) 2 ') =|C(g)|#$(g)

d$#$('). (4.5)

From the orthogonality relations for matrix coe!cients it follows that !(#$) = I$|G|d"

and therefore

!(#$ 2 ') = !(#$)!(') = !(')|G|d$

.

Taking the trace of the first and the last term we get

#$(#$ 2 ') = #$(')|G|d$

. (4.6)

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[15] Hecke algebras and harmonic analysis on finite groups 41

Being ' = ' 2 ' and recalling the notation %g for the Dirac function we have

#$ 2 ' = ' 2 #$ 2 '

=$

g#G

#$(g$1)' 2 %g 2 '

=$

g#G

#$(g$1)' 2 ' 2 %g 2 ' 2 '

=$

g#G

#$(g$1)$

s,t#G

(' 2 %sgt 2 ')'(s)'(t)

(setting h = sgt) =$

h#G

#$

,

-$

s,t#G

'(s)'(t)%th"1s

.

/' 2 %h 2 '

=$

h#G

#$(' 2 %h"1 2 ')' 2 %h 2 '.

(4.7)

Taking into account (4.6), we have

#$(') =d$|G|

$

h#G

#$(' 2 %h"1 2 ')#$(' 2 %h 2 ')

(by Corollary 4.11) =d$|G|

$

h#G

)$(' 2 %h"1 2 '))$(' 2 %h 2 ').(4.8)

Then the theorem follows from (4.5) and (4.8). !

4.1 – The Hecke algebra of a representation of G induced by a one-dimensional repre-sentation of K

In this section we suppose that the K-representation V is one-dimensional. There-fore, if we denote by # its character, the idempotent generator is necessarily ' =1

|K|#. In the following lemma we give in this particular setting an explicit descrip-tion of the corresponding Hecke algebra H.

Theorem 4.16. In the above hypothesis, the Hecke algebra is given by

H = {f $ L(G) : f(k1gk2) = #(k1)#(k2)f(g), (k1, k2 $ K, g $ G}.

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42 FABIO SCARABOTTI – FILIPPO TOLLI [16]

Proof. Let f $ H, then f = 1|K|2# 2 f 2 # and therefore for k1, k2 $ K, g $ G,

we have

f(k1gk2) =1

|K|2 [# 2 f 2 #](k1gk2)

=1

|K|2$

t#k"12 g"1Kk#K

#(k1gk2t)f(t$1k)#(k$1)

(setting u = k2t) =1

|K|2$

u#g"1Kk#K

#(k1gu)f(u$1k2k)#(k

$1)

(setting h = k2k) =1

|K|2$

u#g"1Kh#K

#(k1gu)f(u$1h)#(h$1k2)

=1

|K|2$

u#g"1Kh#K

#(k1)#(gu)f(u$1h)#(h$1)#(k2)

=1

|K|2#(k1) · [# 2 f 2 #](g) · #(k2)

= #(k1)f(g)#(k2).

Vice versa, if f $ L(G) and f(k1gk2) = #(k1)f(g)#(k2) for all g $ (G), andk1, k2 $ K, then

[# 2 f 2 #](g) =$

t#g"1Kk2#K

#(gt)f(t$1k2)#(k$12 )

(setting gt = k1) =$

k1,k2#K

#(k1)f(k$11 gk2)#(k

$12 )

=$

k1,k2#K

#(k1)#(k$11 )f(g)#(k2)#(k

$12 )

= |K|2f(g). !Let G =

0s#S KsK be the decomposition of G into double K-cosets, with respect

a system of representatives S, with 1G $ S. Then the theorem just proved ensuresthat the values of f $ H on a double coset KsK is determined by the value of fon s: f(k1sk2) = #(k1)f(s)#(k2). We should note that on certain double cosets, fmust necessarily vanish, as shown in the next lemma.

Lemma 4.17. For s $ S and x $ sKs$1 ' K, we set #s(x) = #(s$1xs). LetT = {s $ S : #s(x) = #(x), (x $ sKs$1'K}. Then a function f $ H is determined

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[17] Hecke algebras and harmonic analysis on finite groups 43

by its values on T by the relation:

f(k1sk2) =

(#(k1)f(s)#(k2) if s $ T0 if s $ S \ T

for all k1, k2 $ T .

Proof. Let s $ S, x = sks$1 $ sKs$1 'K and f $ H. We have

f(s)#(x) = f(s)#(x$1) = f(x$1s)

= f(sk$1) = #(k$1)f(s) = #(k)f(s)

= f(s)#s(x).

Therefore, considering f $ H such that f(s) 3= 0, we deduce that necessarily #(x) =#s(s) for all x $ sKs$1 ' K and thus s $ T . Vice versa, let {*s : s $ T} be anarbitrary set of complex numbers and define f by setting:

f(k1sk2) =

(#(k1)*s#(k2) if s $ T0 if s $ S \ T.

Let us show that f is well defined, i.e. if k1sk2 = h1sh2 with k1k2, h1, h2 $ K ands $ T , then

#(k1)#(k2)f(s) = #(h1)#(h2)f(s). (4.9)

Since k1sk2 = h1sh2 implies that sk2h$12 s$1 = k$1

1 h1 $ sKs$1 'K and therefore

#(k$11 h1) = #s(k

$11 h1) = #(s$1k$1

1 h1s) = #(k2h$12 )

which gives#(h1)#(h2) = #(k1)#(k2)

which in turn immediately gives (4.9). Since it is obvious that the function f belongsto H, the proof is complete. !

Definition 4.18. The Curtis and Fossum basis of H is given by the elements{as : s $ T} defined by setting

as(g) =

(#(k1)#(k2)

1|K| if g = k1sk2 (k1, k2 $ K)

0 if g /$ KsK.(4.10)

Note that changing the double coset representatives will multiply each basis elementby some root of 1 (in the case V is the trivial representation of K such a root isjust 1). Note also that a1G . ' and, more generally, as(k1sk2) = |K|'(k1)'(k2).

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44 FABIO SCARABOTTI – FILIPPO TOLLI [18]

Lemma 4.19. For all s $ T we have

as =1

|sKs$1 'K| · |K|# 2 %s 2 #.

Proof. First of all we observe that

[# 2 %s 2 #](g) =$

t,u#G

#(gt)%s(t$1u)#(u$1) (4.11)

with the conditions that gt $ K, t$1u = s and u $ K. In particular,

g = gt · t$1 = gt · s · u$1 $ KsK

(in other words, if g /$ KsK the above convolution is 0). Let g = k1sk2 withk1, k2 $ K. Then (4.11) becomes (setting t = us$1)

[# 2 %s 2 #](k1sk2) =$

u#K

#(k1sk2us$1)#(u$1)

(x = sk2us$1) =$

x#sKs"1&K

#(k1)#(x)#(s$1x$1sk2)

(#(x) = #s(x)) = #(k1)#(k2)$

x#sKs"1&K

#(x)#(x)

= #(k1)#(k2)|sKs$1 'K|. !Clearly, there exist complex numbers µrst, r, s, t $ T , such that

ar 2 as =$

t#T

µrstat (4.12)

for all r, s $ T . These numbers are called the structure constants of the Heckealgebra H relative to the basis {as : s $ T}.

Lemma 4.20. The structure constants µrst are given by the following formula:

µrst = |K|$

g#(KrK)&(tKs"1K)

ar(g)as(g$1t).

Proof. On the one hand, from (4.10) and (4.12) we have

ar 2 as(t) =1

|K|µrst (4.13)

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[19] Hecke algebras and harmonic analysis on finite groups 45

for all r, s, t $ T . On the other hand, just computing the convolution, we get:

ar 2 as(t) =$

g#G

ar(g)as(g$1t)

=$

g#(KrK)&(tKs"1K)

ar(g) 2 as(g$1t).(4.14)

Comparing (4.13) and (4.14), the lemma follows. !

Lemma 4.21. For all ! $ J and s $ T , we have:

#$[# 2 %s"1 2 #] = #$[# 2 %s 2 #].

Proof. Since

[# 2 %s"1 2 #](g)$

k1,k2#K

#(k1)%s"1(k$11 gk$1

2 )#(k2)

and $

g#G

#$(g)%s"1(k$11 gk$1

2 ) = #$(k1s$1k2)

we have

#$[# 2 %s"1 2 #] =$

k1,k2#K

#(k1)#(k2)#$(k1s

$1k2)

= |K|$

k#K

#(k)#$(s$1k)

(replacing k with k$1) = |K|$

k#K

#(k)#$(ks).

Similarly,

#$[# 2 %s 2 #] = |K|$

k#K

#(k)#$(ks)

and the lemma follows. !In the following we will use this technical result.

Lemma 4.22. For every s $ S, we have:

|KsK| = |K|2

|K ' sKs$1| .

Proof. First note that the map k1sk2 ,# k1sk2s$1 is a bijection between KsKand KsKs$1. Since sKs$1 is a group, then

|KsK| = |K · sKs$1| = |K||sKs$1||K ' sKs$1| =

|K|2

|K ' sKs$1| . !

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46 FABIO SCARABOTTI – FILIPPO TOLLI [20]

Remark 4.23. Note also that, for every g $ KsK, we have

|K ' sKs$1| = |{(k1, k2) $ K 1K : g = k1sk2}|.

Theorem 4.24.

1. For each ! $ J , the central primitive idempotent of H associated with ! is

d$|G|

$

s#T

|sKs$1 'K|)$(as)as. (4.15)

2. The irreducible characters )$, ! $ J , satisfy the following orthogonality rela-tions: $

s#T

|sKs$1 'K|)$1(as))$2(as) = %$1,$2

|G|d$1

)$1(').

3. The dimension d$ of ! is also given by

d$ = |G|m$ ·*$

s#T

|sKs$1 'K| · |)$(as)|2+$1

,

where m$ is the multiplicity of ! $ IndGK#.

Proof. From Corollary 4.14 and (4.7), we get the following expression for thecentral primitive idempotent associated with ! $ J :

d$|G|

$

g#G

#$(' 2 %g"1 2 ')' 2 %g 2 '. (4.16)

Since, for k $ K, u $ G,

' 2 %k(u) = '(uk$1)

=1

|K|#(uk$1)

= '(u)#(k),

if g = k1sk2, s $ S, we have

#$(' 2%g"1 2')' 2%g2'=#(k$11 )#(k1)#(k

$12 )#(k2)#

$(' 2%s"1 2')' 2%s 2 '

=

12

3)$(as)as

|sKs$1 'K|2

|K|2 if s $ T

0 if s $ S \ T

(4.17)

where the last equality follows from Lemma 4.19 (and (4.11)) and Lemma 4.21.

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[21] Hecke algebras and harmonic analysis on finite groups 47

From (4.16) and (4.17), we get the following expression for the central primitiveidempotent associated with !:

d$|G|

$

s#T

|KsK| |sKs$1 'K|2

|K|2 )$(as)as =d$|G|

$

s#T

|sKs$1 'K|)$(as)as,

where the last equality follows from Lemma 4.22. This ends the proof of (1).(2) On the one hand, from Corollary 4.14 and (1) we have (for !1,!2 $ J):

)$1(' 2 #$2) =$

s#T

|sKs$1 'K|)$2(as))$1(as). (4.18)

On the other hand

)$1(' 2 #$2) = #$1(' 2 #$2)

=$

h#G

#$1(h)$

g#G

'(g)#$2(h$1g)

=$

g#G

(#$1 2 #$2)(g)'(g)

=|G|d$1

%$1,$2

$

g#G

#$1(g)'(g)

=|G|d$1

%$1,$2#$1(').

(4.19)

Comparing (4.18) and (4.19) (2) follows immediately.(3) It follows immediately from (2) and Corollary 4.3. !

4.2 – The Hecke algebra associated with the trivial character

In this section we study the Hecke algebra associated to a permutation representa-tion, that is we consider the case # = +K , where +K is the trivial character of K.We summarize the analysis of the previous section applied to this particular case inthe following theorem. Note that now T . S and that, by virtue of Theorem 4.16,H coincides with the algebra of bi-K-invariant functions.

Theorem 4.25. With the notation of the previous theorem suppose that # = +K .Then:

1. The Curtis-Fossum basis for H is simply given by(as =

1

|K|1KsK : s $ S

);

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48 FABIO SCARABOTTI – FILIPPO TOLLI [22]

2. The structure constants are given by:

µrst =1

|K| |(KrK) ' (tKs$1K)|;

3. Recalling that C(g) is the G-conjugacy class containing g, the character formulain Theorem 4.15 now becomes:

#$(g) =|G|

|C(g)| · |K|

*$

s#S

|sKs$1 'K|)$(as)|C(g) 'KsK|+

·*$

s#S

|sKs$1 'K||)$(as)|2+$1

.

(4.20)

Proof. (1) and (2) are left to the reader.(3) Since ' = 1

|K|1K , if u $ KsK we find that

' 2 %u 2 ' = ' 2 %s 2 ' =|sKs$1 'K|

|K| as,

because, for g $ KsK we have (taking into account Remark 4.23)

[' 2 %s 2 '](g) =1

|K|2$

k1,k2#K

%s(k$11 gk$1

2 ) =|sks$1 'K|

|K|2 .

Therefore,

#$(' 2 1C(g) 2 ') =$

s#S

|sKs$1 'K||K| |C(g) 'KsK|)$(as) (4.21)

and

$

h#G

#$(' 2 %h"1 2 ')#$(' 2 %h 2 ') =$

s#S

|KsK| |sKs$1 'K|2

|K|2 |)$(as)|2

=$

s#S

|sKs$1 'K| · |)$(as)|2(4.22)

where the first equality follows from Lemma 4.21 and the second equality fromLemma 4.22. By applying (4.21) and (4.22) to the formula in Theorem 4.15 theproof is complete. !

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[23] Hecke algebras and harmonic analysis on finite groups 49

4.3 – Relation with Gelfand pairs

We recall that (G,K) is a Gelfand pair when IndGK+K decomposes without multi-plicity; equivalently when the Hecke algebra H(G,K, 1

|K| +K) (which is isomorphic

to the algebra of bi-K-invariant functions) is commutative. If this is the case, let(!, V$) be an irreducible G-representation contained in IndGK+k and let u$ $ V$ bea K-invariant vector with 5u$5 = 1. Then the spherical function associated with !is the bi-K-invariant function defined by

$$(g) = *u$,!(g)u$+V" .

We recall that, by Frobenious reciprocity, u$ is unique up to a constant * $ C with|*| = 1. Moreover, {$$,! $ J} , where J 0 !G are the representations contained inIndGK+K , is a vector space basis of H. In [1], Exercise 9.5.8 the following formulasrelating $$ and #$ are presented:

$$(g) =1

|K|$

k#K

#$(gk), #$(g) =d$|G|

$

h#G

$$(h$1gh), (4.23)

for all g $ G. We want to show that these are particular cases of (4.15) and (4.20).To show the equivalence, decompose the double coset KsK into disjoint union

of left cosets: KsK = k1sK0

k2sK0

· · · knsK, with k1 = 1G. By Lemma 4.22

we have that n = |K||K&sKs"1| . Using the G-conjugacy invariance of the character #$

we have that $

g#kisK

#$(g) =$

k#K

#$(kisk) =$

k#K

#$(sk).

Therefore the central primitive idempotents of Theorem 4.24 evaluated at KsKequals

d$|G|

|K ' sKs$1||K| )$(as) =

d$|G|

|K ' sKs$1||K|2 )$(1KsK)

=d$|G|

|K ' sKs$1||K|2

$

g#KsK

#$(g)

=d$|G|

|K ' sKs$1||K|2

$

g#"n

i=1 kisK

#$(g)

=d$|G|

|K ' sKs$1||K|2

n$

i=1

$

k#K

#$(kisk)

=d$|G|

1

|K|$

k#K

#$(sk)

which agrees with the formula of the spherical function $$ in (4.23).

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50 FABIO SCARABOTTI – FILIPPO TOLLI [24]

Indeed, the central idempotent is equal to d"|G|$$, because $$ 2 $$ = |G|

d"(see [2,

Lemma 1.5.7]). In particular,

$$(h) =|K ' sKs$1|

|K| )$(as) (4.24)

for all h $ KsK, s $ S and ! $ J .Let now show that (4.20) reduces, in the setting of Gelfand pairs, to the second

formula in (4.23). We first observe that

$$(1K) =$

k#K

#$(k) = |K|

because the multiplicity of +K in ResGK! is equal to 1, and thus by Theorem 4.24

$

s#S

|sKs$1 'K||)$(as)|2 =|G|d$

.

Therefore

#$(g) =d$

|C(g)|

*$

s#S

|K ' sKs$1||K| )$(as) · |C(g) 'KsK|

+

( by (4.24)) =d$

|C(g)|$

s#S

$$(s)|C(g) 'KsK|

=d$

|C(g)|$

s#S

$

h#C(g)&KsK

$$(h)

=d$

|C(g)|$

h#C(g)

$$(h)

=d$|G|

$

h#G

$$(hgh$1).

REFERENCES

[1] T. Ceccherini-Silberstein – F. Scarabotti – F. Tolli: Harmonic analysis onfinite groups: representation theory, Gelfand pairs and Markov chains, CambridgeStudies in Advanced Mathematics 108, Cambridge University Press 2008.

[2] T. Ceccherini-Silberstein – F. Scarabotti – F. Tolli: Representation theoryof the symmetric groups: the Okounkov-Vershik approach, character formulas, andpartition algebras, Cambridge Studies in Advanced Mathematics 121, CambridgeUniversity Press 2010.

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[25] Hecke algebras and harmonic analysis on finite groups 51

[3] C. W. Curtis – T. V Fossum: On centralizer rings and characters of representationsof finite groups, Math. Z., 107 (1968), 402–406.

[4] Ch. W. Curtis – I. Reiner: Representation theory of finite groups and associativealgebras, Reprint of the 1962 original, Wiley Classics Library. A Wiley-IntersciencePublication. John Wiley & Sons, Inc., New York, 1988.

[5] Ch. W. Curtis – I. Reiner: Methods of Representation Theory. With Applicationsto Finite Groups and Orders, Vols. I and II, Pure Appl. Math., John Wiley & Sons,New York 1981, 1987.

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[7] H. Dym: Linear algebra in action, Graduate Studies in Mathematics 78, AmericanMathematical Society, Providence, RI, 2007.

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[10] F. Scarabotti – F. Tolli: Harmonic analysis on a finite homogeneous space, Proc.Lond. Math. Soc., (3) 100 (2010), 348–376.

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Lavoro pervenuto alla redazione il 5 settembre 2012ed accettato per la pubblicazione il 20 settembre 2012

Indirizzi degli Autori:Fabio Scarabotti – Dipartimento SBAI – Universita degli Studi di Roma “La Sapienza” – via A.Scarpa 8 – 00161 Roma – Email address: [email protected]

Filippo Tolli – Dipartimento di Matematica – Universita Roma TRE – L. San Leonardo Murialdo1 – 00146 Roma – Italy – Email address: [email protected]