Level Zero Types and Hecke Algebras for Local Central Simple Algebras

Level Zero Types and Hecke Algebras for Local Central Simple Algebras

Martin Grabitz, Allan J. Silberger, and Ernst-Wilhelm Zink Abstract:

Let^Dbe a central division algebra and^A =^GLm(^D) the unit group of a central simple algebra over a p-adic eld^F. The purpose of this paper is to give types (in the sense of Bushnell and Kutzko) for all level zero Bernstein components of ^A and to establish that the Hecke algebras associated to these types are isomorphic to tensor products of Iwahori Hecke algebras. The types which we consider are lifted from cuspidal representations of ^M(^kD), where ^M is a standard Levi subgroup of

GLmand ^kD is the residual eld of^D. Two types are equivalent if and only if the corresponding pairs (^M(^kD) ) are conjugate with respect to

A . The results are basically the same as in the split case^A =^GLn(^F) due to Bushnell and Kutzko. In the non split case there are more equiv- alent types and the proofs are technically more complicated.

0. Introduction

Let F be a p-adic local eld, let D := Dd be a central F-division algebra of index d, and letA:=Mm(D) be a central simpleF-algebra of reduced degree n := dm. The purpose of this paper is to give a classication of types (see BK2]) for all level zero Bernstein components of the unit group A and to establish that the Hecke algebras associated to these types are isomorphic to tensor products of Iwahori Hecke algebras, as in the split case (see BK1] and BK3]).

In M1] Morris proved Hecke algebra isomorphism theorems which apply to the level zero representations of general reductive groups and in M2] he extended this earlier work to show that cuspidal level zero representations of the nite eld points of Levi factors of reductive groups inate to types for level zero Bernstein components. Our paper, in eect, presents a special case of Morris's general theory, an example which is at the same time more general and given in greater detail than in Bushnell and Kutzko's work ( BK1] and BK3]) for the level zero case ofMn(F).

We think that the present extension of the split case is interesting enough to merit being spelled out, as we have attempted in this paper. Like Howe/Moy and Bushnell/Kutzko, we construct level zero types by inating cuspidal rep- resentations of Levi factors with coecients in the residue eld ofD to repre- sentations of unit groups of hereditary orders. Our situation is also analogous to the split case in that representations of Levi factors which are conjugate under inner automorphisms ofA inate to types for the same Bernstein com- ponent. However, there are more inner automorphisms acting on the set of

cuspidal representations some of these can be interpreted as a Galois action which is trivial in the split case. Although the Hecke algebra of a simple type looks like the group algebra of a semi-direct product of an innite cyclic group normalizing a Coxeter group, the cyclic group object which serves as a part of the support of the Hecke algebra in the case of a simple type for A need not normalize a principal order or Iwahori subgroup-like object. The multipli- cation of double cosets is also more complicated in the case of general simple algebras. We prove our Hecke algebra isomorphism theorems for natural rep- resentatives of each Galois orbit of simple level zero types after arguing that all representatives of the same orbit have isomorphic Hecke algebras, that one representative is a type if and only if all are. As in other level zero situations (e.g. M2]), we obtain our results by reducing the proofs to general arguments due to Bushnell and Kutzko ( BK2]).

We begin the paper with some background information and give statements of our main theorems (Theorems 1 and 2) in ^xx0.6 and 0.7. Parts 1-4 are concerned with Hecke algebras, whereas Part 5 concludes the classication of types by applying Bushnell/Kutzko's theory of covers ( BK2] 8.).

We thank Peter Schneider for a helpful remark.

x0.1 The Bernstein Spectrum and Decomposition

Let G = G(F) denote the group of F-points of a connected reductive F- group. A cuspidal pair (M ) for G consists of a Levi subgroup M of G and an irreducible supercuspidal representation of M the Bernstein spectrum (G) is dened as the set of G-conjugacy classes of cuspidal pairs (M ). For any irreducible smooth representation of G its supercuspidal support is a unique element of (G).

The Bernstein spectrum has the structure of a complex locally algebraic va- riety. Let Xnr(M) denote the group of unramied characters of M with its natural complex structure. Then the connected component of (G) which contains the G-orbit of a cuspidal pair (M ) is the image of the map

Xnr(M)^!(G) ^7!G^;orbit of (M ):

Let ^M(G) denote the category of smooth G-representations. For a con- nected component (G) let ^M() denote the full subcategory of G- representations, all irreducible subquotients of which have supercuspidal sup- port in . The Bernstein decomposition of^M(G) Be] is dened as the equiv- alence

M(G) = ^YM() (1)

where runs over the connected components of (G).

x0.2 Hecke Algebras and Intertwining Functions

LetKbe an open compact subgroup ofGand let (K W) be a representation of K in W. We call the convolution algebra consisting of all compactly supported functions f :G^!End^C(W) such that

f(k¹gk²) =(k¹)f(g)(k²) (2)

2

for allk¹ k^{2 2}K the Hecke algebra ofGwith respect to (K ) and we denote it^H(G K ).

The unit element is

e (x) =

( (K)^;1(x) forx²K

0 otherwise

where (K) = ^R_K1dy.

We shall make use of a generalized algebra of \intertwining functions" in this paper and we include here for reference purposes a brief discussion of these functions. For any pair (i Wi K) of irreducible representations of K act- ing in vector spaces Wi (i = 1 2) we call intertwining function any com- pactly supported function f := f²¹ such that f : G ^! Hom^C(W¹ W²) and f(k⁰gk) =²(k⁰)f(g)¹(k). For functions f³² f²¹ we have a natural convo- lution product

f ³² f^{2 1}(g) :=^Z_Gf³²(x)f²¹(x^;1g)dx which produces a functionf ³¹.

For'²HomK(W² V) and w²W¹ we dene

((f²¹)')(w) :=^Z_G(g^;1)'(f²¹(g)w)dg:

We obtain a mapping (f^{2 1}) : HomK(W² V) ^! HomK(W¹ V). We note that (f³² f²¹) = (f²¹)(f^{3 2})

which means that HomK(W¹ V) is a right module for the Hecke algebra

H(G K ¹).

x0.3 The Concept of a Type

Bushnell and Kutzko BK2] call a pair (K ), where K is an open compact subgroup and is an irreducible representation of K, a type for G if the cat- egory ^M (G) of allG-representations which are generated by their -isotypic components is closed under the formation of subquotients. In particular, they call (K ) a type for the connected component (G) if^M (G) =^M(), i.e. if for every irreducible representation ofG the restriction ^jK contains if and only if the supercuspidal support of belongs to .

For (K ) a type the category ^M (G) is equivalent to a category of modules over the Hecke algebra^H(G K ). More precisely, Bushnell and Kutzko show in BK2](4.3) that the mapping

M (G)³( V)^7!HomK(W V)²Mod(^H(G K )^opp)

is an equivalence of categories if and only if the pair (K ) is a type. (We have to take the opposite algebra because our denition of ^H(G K ) uses instead of the contragredient of as in Bushnell and Kutzko's work.)

3

Henceforth we consider only the special case G=A .

x0.4 The Connected Components of (A )

We recall the formal set-up of Bernstein and Zelevinsky which provides a parameterization for the connected components of (A ). Let ^C := ^C(D) be a set of representatives for the unramied twist classes of irreducible pre- unitary supercuspidal representations of GLs(D) for all s 1. For ^{2 C} a representation of GLs(D) we dene the degree of to be d() := s. Let Div⁺(^C) denote the set of eective divisors over ^C. To any eective divisor

D=^P

2C

m we associate the triple:

- its degree d(^D) = ^Pmd()

- the Levi subgroup M^D GLd^(D)(D), where M^D =^Q(GLd⁽⁾(D)) ^m (assuming some ordering of the factors), and

- the supercuspidal representation ^D of M^D such that ^D =(^m ): For each ^{D 2}Div⁺(^C) of degree m let ^D (A ) denote the connected component which contains the A -orbit of (M^{D D}).

1. Fact

: The mapping ^{D 7!D} parameterizes the connected components of (A ) by degree m divisors over ^C.

x0.5 Standard Hereditary Orders

Let O denote the ring of integers of D and ^pthe maximal ideal of O. We x the maximal order^A1 =Mm(O), with Jacobson radical ^P1 =Mm(^p). We also x the minimal order ^Am ^A1 which consists of those elements of ^A1 which have all matrix elements below the main diagonal in ^p. The Jacobson radical

Pm ^Am has coecients on and below the main diagonal in ^p. A hereditary order ^A such that ^Am ^{A A1} will be called standard. Every hereditary order of Ais conjugate to a unique standard hereditary order. If the standard hereditary orders satisfy^{A0 A}, then the Jacobson radicals satisfy the reverse inclusions ^{P1 PA PA0 P}m. The mapping ^{A 7! A} = ^A=^P1 sends the set of standard hereditary orders bijectively to the set of upper block triangular matrix rings in Mm(kD). In particular, to any standard hereditary order ^A there corresponds a tuple of positive integers s¹ ::: sr with summ such that the quotient ring ^A=^PA is the semi-simple algebra

A=^P=Ms¹(kD)Msr(kD) (3)

each factorMsi(kD) being a complete matrix algebra over the residual eldkD

of D. The multiplicative group of (3) is

A := (^A=^P) =^A =(1 +^P)=GLs¹(kD)GLsr(kD): (4)

x0.6 The Cuspidal Support of a Level Zero Representation

An irreducible smooth representation ( V) ofA is called level zero ifV^{1+P1 6}= (0), i.e. if there exists a (1 +^P1)-xed vector.

For ( V) a level zero representation we interpret V^1+P as the Jacquet re- striction of V^1+P1 with respect to the parabolic subgroup ^A =(1 +^P1) ^A1.

4

It is natural to take ^Aminimal (^PA maximal) such thatV^{1+PA 6}= 0. It follows that, as a representation of (4), all irreducible constituents =¹r

occurring in V^1+PA are cuspidal, i.e. they are tensor products in which each tensor factor i of GLsi(kD) is a cuspidal representation. In this case, we call (^A ) a cuspidal level zero pair and we write supp() :=^f1 ::: r^g: As we have seen, every level zero representation has a cuspidal level zero pair as a component. We will prove that each of these pairs is a type and that the con- nected component (A ) corresponding to the type (^A ) is determined as follows. Consider supp() and introduce the following equivalence relation on the set of cuspidal representations of GLs(kD) for all s 1: ⁰ if and only if ⁰ = for some ² Gal(kD^jk) acting coecientwise on GLs(kD).

We write ] for the Gal(kD^jk)-equivalence class of,r^]() for the number of elements in supp() belonging to ], and d( ]) :=s for the degree of and

]. To we associate the eective divisor () := ^X

^] r^]() ]

where the sum ranges over Gal(kD^jk)-equivalence classes of cuspidal represen- tations occurring in supp().

In 5.1 and 5.2 we show that ] determines an unramied twist class of irre- ducible supercuspidal representations ofGLd^(])(D), hence a unique ^]2C. Therefore, () determines ^D() :=^P]r^]()^{] 2}Div⁺(^C) and we prove:

Theorem 1

^{: Let (A} ) be any cuspidal level zero pair. Then (^A ) is a type for the connected component ^D (A ), where ^D =^D() is determined by ().

We call two cuspidal level zero pairs (^A ) and (^{A0 0}) equivalent if () = (⁰) and we obtain a bijection between equivalence classes of such pairs and level zero connected components of (A ). In the case of Mn(F) there is no Galois action and this means that () = (⁰) if and only if supp()=supp(⁰) as multisets.

x0.7 The Hecke Algebra of a Level Zero Type

Let (^A ) be a cuspidal level zero pair. In verifying that (^A ) is a type we study the structure of the Hecke algebra^H(A ^A ). We prove:

Theorem 2

: If (^A ) has the divisor (), then

H(A ^A )=^O

^]

H(r^]() q^dd(]))

a tensor product of ane Hecke algebras (see Part 4 for the notation^H(r z)).

In Part 1 we determine the support of^H(A ^A ) and, applying BK2](7.2)(ii), we show that ^H(A ^A ) is a tensor product of Hecke algebras such that

A=^Ar is a principal order and such that all tensor factors of are Gal(kD^jk)- equivalent representations. In (1.10), the nal result of the part, we show that in this particular case^H(A ^Ar ) is isomorphic to^H(A ^Ar ^r), where ²supp() and^r :=^r. Thus, we conclude that^H(A ^A ) depends only

5

upon the divisor (). In Part 4 we show that ^H(A ^A_r ^r)= ^H(r q^dd()) (see Theorem 4.2), i.e. is isomorphic to an ane Hecke algebra of type A (see BK1], Chapter 5 for the case Mn(F)). As a preparation in Parts 2 and 3 we study the multiplication of double cosets, the main results being Propositions 2.6, 2.7, and 3.1. Here dierences in the proofs between the general and the split case become visible. As we have noted, the nal results do not reect these dierences.

x0.8 Generalized Tits Systems for Unit Groups of Simple Algebras This section is included for reference purposes and to establish notation.

Let oF denote the ring of integers of the p-adic local eld F, ^pF the prime ideal of oF, and k =oF=pF the residual eld of F. In D we x a pair (Fd $) consisting of a maximal unramied extension of F and a prime element of D which normalizesFd in this case $^d =$F is a prime element ofF. We may also identify kD with the residual eld of Fd and note thatkD^jk is a degree d extension.

Let ⁰A denote the subgroup of A consisting of all elements x such that NrdA^jF(x)²oF. Note that all compact subgroups of A are in ⁰A.

Let ~WA denote the subgroup of A consisting of all monomial matrices with non-zero entries which are powers of $. Each element w ² W^~A has a unique product representation of the form

w=$^vp (5)

where v = (v¹ ::: vm) ^{2 Zm}, $^v = diag($^v1 ::: $^vm), and p = (i⁽j⁾)_mij⁼¹ is the matrix of a permutation ^{2 S}m, i.e. w =$^vp is the matrix obtained by permuting the columns of$^v by . The subgroupWA= ~WA^{\ 0}A consists of all w = $^v p such that v¹ ++vm = 0, and we have the semi-direct product

W~A =WA^ohhAⁱ:

Let^IA be the minimal standard hereditary order in A, and let NAA and

0NA=NA^{\ 0}A denote the subgroups of monomial matrices.

Set hA= Im^;1

$

:

For i= 1 ::: m^;1 letsiA denote the matrix of the transpositioni ^$i+ 1 and set s⁰A :=hAs¹Ah^;1A :

2. Fact

(see I]): The triple (A ^IA NA) is a generalized Tits system with W~A=NA=(^I_A^\NA) as generalized Weyl group. The setSA =^fs⁰A ::: sm^;1A^g

is a Coxeter system of type ~Am^;1 it generates the groupWA= ⁰NA=(⁰NA^\IA) and (^I_A ⁰NA) is an ane BN-pair of the group ⁰A.

We have the Bruhat decomposition

W~A^!IAⁿA =^I_A and, more generally:

6

3. Fact

: Let ^Ai be a standard hereditary order for i = 1 2. Then there is a natural bijective correspondence

A

1

nA =^{A2 !}(^{A1 \}W~A)ⁿW~A=(^{A2 \}W~A):

IfM =GLs¹GLsr, wheres¹++sr =m, and^A =M(O)(1+^PA), then

A \W~A =M(O)^\WA=^Ss^1Ssr:

Write lA(w) for the length function on WA corresponding to the system SA. Forw²WA and s²SA such that lA(ws)> lA(w)

IAw^IAs^IA =^I_Aws^IA

(6)

and (^I_A :^I_A ^\w^I_Aw^;1) = q^dlA(w) (7)where q^d =^jkD^j.

Notations:

F oF $F ^pF p-adic local eld, integers, prime element, maximal ideal k =oF=^pF, q residual eld ofF, ^jk^j

D^jF central divisionF-algebra of indexd O ^p valuation ring and valuation ideal ofD kD :=O=^p q^d residual eld ofD, ^jkD^j

Fd $ a maximal unramied extension ofF in D, a prime element of D which normalizesFd ($^d =$F) A:=Mm(D) central simple algebra over F

n:=dm reduced degree of A^jF

A P:=^PA standard hereditary order in A, its Jacobson radical

Ar ^Pr standard principal order of period r^jm, its Jacobson radical in (3) we haves¹ ==sr =s= ^m_r.

IA:=^Am minimal standard hereditary order inA.

7

1. The Support of the Hecke Algebra

Let ^A A be a standard hereditary order and let ( W) be an irreducible cuspidal representation of the group ^A . By ination we regard as a rep- resentation of ^A (see (4)). In this Part we want to construct a vector space basis for the Hecke algebra^H(A ^A ) (see ^x0.2).

For any x²A dene

^x(y) := (xyx^;1) (y^2A ):

1.1

Lemma

^{: For x 2 A} there exists f ^{2 H}(A ^A ) such that f(x) ⁶= 0 if and only if Hom^A\x^;1Ax( ^x) ⁶= (0), in which case setting f(x) :=

J ² Hom^A\x^;1Ax( ^x) uniquely determines f with support in ^A x^A . If ^jA\x^;1Ax is irreducible, then this space of functions is one-dimensional.

Proof: Fory^{2A \}x^;1A x we have

^x(y)f(x) = (xyx^;1)f(x) = f(xy) = f(x)(y) i:e: f(x) = J ²Hom^A\x^;1Ax( ^x):

Conversely, if 0⁶=J ²Hom^A\x^;1Ax( ^x) exists, thenf(y¹xy²) :=(y¹)J(y²) for y¹ y^{2 2 A} denes a function with support ^A x^A . Since y¹xy² = x im- plies that f(y¹xy²) = f(x), the function f is well dened. That the space of functions with support in ^A x^A is at most one-dimensional follows from Schur's Lemma, when ^jA\x^;1Ax is irreducible. ² We consider^A =U^;(^P)M(O)U⁺(O), the Iwahori factorization of^A , and write ^NA(M(O)) for the normalizer of M(O) in A . We shall also consider

A =M(kD) =_i^Qr

=1

GLsi(kD) as a block-diagonal group and =¹r, where i is a cuspidal representation of GLsi(kD).

1.2

Proposition:

The support of ^H(A ^A ) consists of the set of double cosets

A w^A , where w²W^~A and, in addition:

(i) w ^2NA(M(O))

(ii) conjugation by w xes the class of the representation of M(O).

Proof

: During the proof we write ^H:=^H(A ^A ). We proceed in several steps.

1.3

Lemma

^{: If} w=$^vp²W^~A and wis in the support of ^H, then $^v normalizes M(O).

Proof:

^{Consider w =$vp} and write the exponent vector v = (v¹ ::: vm) ²

Zm (see (^x0.8)) as a vector of vectorsv = (v⁽¹⁾ ::: v^(r)), where, for 1ir, the vector v⁽ⁱ⁾ is an si-vector. Replacing w by

p⁰w=p⁰$^vp^0;1p⁰p for p^{0 2A \}W^~A

we may assume that each of the subvectors v⁽ⁱ⁾ is a non-increasing sequences of integers. Assume that there is a j such that in some subvector v⁽ⁱ⁾ we have vj > vj⁺¹. Then

M⁰ = (GLjGLm^;j)^\M 8

is a proper Levi subgroup ofM we writeP⁰ =M⁰U⁰ M for the correspond- ing lower parabolic subgroup of M. Thus,

U⁰(D)^\A M(D)^\A =M(O)

$^;v(U⁰(D)^\A )$^v 1 +^P1 and, therefore,

f(u⁰$^vp) = f($^vp(p^;1$^;vu⁰$^vp)) =f($^vp)

for any f ^{2 H} and u^{0 2} U⁰(D) ^\A , since the conjugation by $^;v maps U⁰(D)^{\ A} into 1 + ^P1 and the permutation matrix p normalizes 1 +^P1. Noting that U⁰(D)^\A modulo 1 +^P is the unipotent radical of a proper parabolic subgroup of M(kD), we conclude that cuspidal implies that

0 = ^Z

U⁰⁽D^)\A (u⁰)f($^v p)du⁰ = ^Z

U⁰⁽D^)\A f($^vp)du⁰

so f($^vp) = 0. Therefore, all the subvectors v⁽ⁱ⁾ of v ^{2 Zm} have to be

\scalar" for$^vpto be in the support of^H in other words, it is necessary that

$^{v 2N}A(M(O)). ²

1.4

Lemma

: Assume that $^v normalizes M(O) but that the permutation matrix p does not normalize M. Then w=$^vpis not in the support of ^H.

Proof

^{: LetP =MU} be the upper block triangular parabolic subgroup ofA which has M as its Levi subgroup. If p² A is a permutation matrix which does not normalize M, then P⁰ =pPp^;1\M is a proper parabolic subgroup of M with the Levi decomposition

P⁰ =M⁰U⁰ = (pMp^;1\M)(pUp^;1\M):

In ^A we have P^0\A = (M^0\A )(U^0\A ) with U^0\A =pUp^;1\M(O).

Since is a cuspidal representation ofM(kD) and since the reduction ofU^0\A is the unipotent radical of a proper parabolic subgroup of M(kD), it follows that 0 = ^Z

U^0\A (u⁰)f($^vp)du⁰ = ^Z

U^0\A f($^vp(p^;1$^;vu⁰$^vp))du⁰

and our assertion follows from the fact thatp^;1$^;vu⁰$^vp²1+^P. To see this observe that the diagonal matrices lie in M ^\pMp^;1 hence $^v normalizes pUp^;1 and, by hypothesis, also M(O). Therefore $^v normalizes U^{0 \A} = pUp^;1\M(O) and we may conclude that$^;vu⁰$^{v 2}pUp^;1\M(O), i.e. that p^;1$^;vu⁰$^vp²U(O) 1 +^P. ² 1.5

Lemma

: If w²W^~A^\NA(M(O)), then

Hom^A\w^;1Aw( ^w) = HomM⁽kD⁾( ^w):

Proof

: Note that ^A =M(O)(1 +^P), where the normal subgroup 1 +^P is in the kernel of. The hypothesis w²W^~A^\NA(M(O)) implies that

A \w^;1A w=M(O)((1 +^P)^\w^;1(1 +^P)w) 9

and that and ^w are both trivial on (1 +^P)^\w^;1(1 +^P)w, i.e. ^w(x) = (wxw^;1) and conjugation by wmaps (1+^P)^\w^;1(1+^P)winto 1+^P. Thus the intertwining map factors through the projection of ^{A \}w^;1A w upon

A =M(kD). ²

Since is irreducible, 1.5 implies thatw²W^~A^\NA(M(O)) is in the support of ^H if and only if ^w =. This completes the proof of 1.2. ² From Part 0, Fact 3 and the observation that ~WA^\M(O) is a normal subgroup of ~WA^\NA(M(O)), we have the injective mapping

W~A^\NA(M(O))=( ~WA^\M(O)),^{!A n}A =^A :

Let Stab(M(O) ) denote the subgroup of ^NA(M(O)) consisting of those elements which x the class of and note that

W~A^\M(O) W~A^\Stab(M(O) ) W~A^\NA(M(O)):

Let ~WA() be a set of representatives for ( ~WA^\Stab(M(O) ))=( ~WA^\M(O)).

From 1.2 we obtain:

1.6

Corollary

: The mapping w^{7! A} w^A denes a bijection from ~WA() to the set of double cosets in the support of ^H(A ^A ).

In^x0.6 we introduced for any cuspidal level zero pair (^A ) the divisor ().

At this moment we do not yet know that this data determines a single type.

Lemma 1.7 and Proposition 1.10 show that any two interpretations (^{A1 1}) and (^{A2 2}) of this data lead to isomorphic Hecke algebras and representations ¹and²which occur as components of the same representations with the same multiplicities.

1.7

Lemma

: For i = 1 2 let ^Ai be a standard hereditary order and i a cuspidal representation of Mi(kD) = ^A_i such that (¹) = (²). Assume that the same representatives with the same multiplicities are chosen from each class ] so that¹ and² are tensor products in possibly permuted order of equivalent representations. Then the Hecke algebras^H(A ^{A1 1}) and^H(A ^{A2 2}) are isomorphic and, moreover, for any irreducible smooth representation of A the multiplicity of ¹ in ^jA1 equals the multiplicity of ² in ^jA2.

Proof

: Since the symmetric group ^Sr is generated by transpositionsri which switch the pair (i i+ 1) for 1i < r, it is sucient to consider pairs (^{A1 1}) and (^{A2 2}) in which i and i⁺¹ are inequivalent and such that ¹ and ² dier by a transposition of the i-th and i+ 1-th tensor factors. This means that M¹ and M² can dier as block diagonal groups in at most theiri-th and i+ 1-th blocks. LetM⁰ denote the Levi factor which contains M¹ and M², in which the two blocksGLsiGLsi⁺¹ are replaced by the single blockGLsi⁺si⁺¹. Let^A0 denote the standard hereditary order such that ^A0 =M⁰(kD). Let

⁰ := Ind^AA0

1

¹ = Ind^AA02

² 10

⁰ being an irreducible representation since i ⁶ i⁺¹. It is well known both that ⁰ is irreducible and that the class of ⁰ does not depend upon the order of the tensor factors i i⁺¹. Therefore, if ^A0 contains ⁰ if and only if ^A_i contains i for i= 1 2, and the multiplicities are the same.

To see that the commuting algebras ^H(A ^{A1 1}) and ^H(A ^{A2 2}) are iso- morphic we use BK1](4.1.3) to deduce that for each i = 1 2 the algebras

H(A ^Ai i) and ^H(A ^A0 Ind^AA0_i(i)) are isomorphic. ² Using 1.7 we may assume without loss of generality that the tensor factors of are ordered such that the set^f1 ::: r^gis partitioned into subsets such that if i and i⁰ belong to the same Gal(kD^jk)-orbit, then the same is true for all i⁰⁰ between i and i⁰. In other respects we may assume that the representative of the divisor is arbitrary.

Next we write M^f for the smallest Levi subgroup of A such that M M^f and such that ~WA^\Stab(M(O) )M^f(D). To specifyM^f we rst use (4) to dene the partition of ^f1 ::: m^gsuch that

P =^P1Pr ^Pi =^fhi^;1+ 1 ::: hi^g hi =^Xi

v⁼¹sv: (8)

We set ^Pi ^Pj if i and j are Gal(kD^jk)-equivalent. The union of the ^Pi

over Gal(kD^jk)-equivalent i gives a coarser partition ^Pe = ^{Pe1 Pe}t of

f1 ::: m^g. Let l ^~l denote the functions on ^f1 ::: m^g such that l(x) = i if x^2Pi and ~l(x) =iifx^2Pei. The Levi subgroupM A corresponds to^P l and has the representation

M =^f(aij)²A ^jaij = 0 l(i)⁶=l(j)^g

the larger Levi subgroup M^f corresponds to ^Pe ~l and has the similar represen- tation

Mf =^f(aij)²A ^jaij = 0 ~l(i)⁶= ~l(j)^g:

As usual, we writed() :=sif the cuspidal representation is a representation of GLs(kD). Associated to the divisor (), we have

Mf=^Y

^]GLr^{]( )}d⁽⁾:

Our assumption that i i⁰⁰ i⁰ and i i⁰ implies i i⁰⁰ implies that Mf is block diagonal. We may therefore represent the upper and lower block triangular parabolic subgroups which have M^f as their Levi factors in the form

Pe =M^fnU^e =^f(aij)²A ^j aij = 0 ~l(i)>^~l(j)^g Pe^;=M^fnU^e; =^f(aij)²A ^j aij = 0 ~l(i)<^~l(j)^g:

11

The following Lemma is now obvious:

1.8

Lemma

^:

(i) ^A = (^{A \}U^e;)(^{A \}M^f)(^{A \}U^e)

(ii) ^{A \}U^e; and ^{A \}U^e are contained in1 +^P.

We now have the following important consequence of BK2](7.2):

1.9

Proposition:

^{Let (A Mf}) as before. There is a canonical isomorphism t_P^e :^H(M^f M^f\A )^;!H(A ^A )

such that supp(t_P^ef) = ^A supp(f)^A for all f ^2H(M^f M^f\A ).

Proof

: In the terminology of BK2](6.1), Lemma 1.8 implies that the pair (^A ) is decomposed with respect to (M^f P^e). Moreover, in the terminology of BK2](6.2), we have

H(A ^A ) =^H(A ^A )_M^e

since ~WA^\Stab(M(O) ) M^f, i.e. the support of our Hecke algebra is in

A

Mf^A . It follows that BK2](7.2)(ii) implies the present Proposition. ² Since M^f is the direct product of subgroups

Mfv =^fa²M^f aij =ij if ~l(i) = ~l(j)⁶=v^g

and M^fv ^\A supports all constituents of which are in a single Gal(kD^jk)- equivalence class ] = ]v, we have the isomorphism

O

v ^H(M^fv M^fv^{\A (v)})^;!H(M^f M^f\A ): (9)

As a consequence 1.9 reduces the study of the structure of level zero Hecke algebras to the case in which the cuspidal pair (^A ) corresponds to a prin- cipal order^A and with only Gal(kD^jk)-equivalent tensor factors. The Hecke algebra isomorphism assertion of the next Proposition could have been given a simpler proof by using 1.9. However, the following Proposition also completes the proof that one representative of a divisor is a type if and only if the same is true for all representatives of . Thus, after 1.10, it will be sucient to prove that is a type when all Gal(kD^jk)-equivalent tensor factors of are equivalent.

1.10

Proposition

: The Hecke algebra ^H(A ^A ) depends, up to isomorphism, only on the divisor (). Moreover, if for i = 1 2 the pair (^A_i i) consists of a standard hereditary order and cuspidal representation of ^A_i such that (¹) = (²), then, for any irreducible smooth representation of A , the representations ¹ and ² occur in restrictions of with the same multiplici- ties. Thus ¹ is a type if and only if ² is a type and for the same Bernstein components.

12

Proof

: In view of 1.7 it is enough to prove 1.10 for ¹ =¹ r, where ¹ ::: ` are Gal(kD^jk)-conjugate and no other tensor factor of ¹ belongs to the Gal(kD^jk)-orbit of ¹, and ² = ^{1 2} r. In this case,

A:=^A1 =^A2 clearly, s:=s¹ ==s`. In fact, 1.7 implies that the order of the tensor factors in ² can be arbitrary, so we redene

² :=²` <sup>1</sup>`⁺¹r:

With these denitions of ¹ and ² we want to construct two compactly sup- ported intertwining functions f²¹ and f^{1 2} (see ^x0.2) such that

f ²¹(a⁰xa) =²(a⁰)f²¹(x)¹(a) andf¹²(a⁰xa) =¹(a⁰)f ¹²(x)²(a) (10)

for alla a^{0 2A} and x²A and

f^{1 2} f ²¹ =e¹ and f^{2 1} f¹² =e ²: (11)

For functions with the properties (10) and (11) it is easy to see that we have the isomorphisms

f^{1 2 H}(A ^{A 2})f²¹ =^H(A ^{A 1}) and f^{21 H}(A ^{A 1})f ¹² =^H(A ^{A 2}):

Moreover, let be a smooth representation of A and consider the space Hom^A(¹ ). As remarked in^x0.3 this space is a module for the Hecke alge- bra ^H(A ^{A 1}). For any function ' ² Hom^A(¹ ) we have (f^{1 2})' ² Hom^A(² ). If '⁶= 0, then (11) implies that (f¹²)'⁶= 0 too. Similarly one sees that (f ²¹) denes a monomorphism of the ^H(A ^{A 2})-module Hom^A(² ) to Hom^A(¹ ). Since these convolutions are inverses of each other, the morphisms are surjective as well, so isomorphisms, allowing each module to be regarded as a module over either Hecke algebra. The asser- tions in 1.10 to the eect that in any irreducible smooth representation the multiplicities of ¹ and ² as components are the same follow from these ob- servations. Thus ¹ is a type if and only if ² is and for the same Bernstein components.

It remains to construct the functionsf ²¹ and f^{1 2}. We consider the matrix h⁰ =

0

B

@

I⁽`^;1)s

$Is

Im^;`s

1

C

A

and claim that the double cosets ^A h^0A and^A h^{;10 A} support the respective functions f²¹ and f¹². First, we want to check (10) for the functionf²¹ we shall leave the analogous verication for f¹² to the reader.

To check that a function satisfying (10) can be dened with support ^A h^0A we must verify, as in 1.1, that

f ²¹(h⁰)¹(y) =f²¹(h⁰yh^;10 h⁰) = ²(^h0y)f²¹(h⁰) (12)

13