Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of p-adic Simple Algebras

WEAK EXPLICIT MATCHING FOR LEVEL ZERO DISCRETE SERIES OF UNIT GROUPS OF

^p

-ADIC SIMPLE ALGEBRAS

Allan J. Silberger and Ernst-Wilhelm Zink

Abstract. Let ^F be a p-adic local eld and let ^Ai be the unit group of a cen- tral simple ^F-algebra ^Ai of reduced degree ^{n >} 1 (ⁱ = 12). Let ^R2(^Ai ) denote the set of irreducible discrete series representations of ^Ai . The \Abstract Match- ing Theorem" asserts the existence of a bijection, the \Jacquet-Langlands" map,

JL

A

2 A

1 :^R2(^A1)^!R2(^A2) which, up to known sign, preserves character values for regular elliptic elements. This paper addresses the question of explicitly describing the map^JL, but only for \level zero" representations. We prove that the restriction

JL

A

2 A

1 :^R20(^A1)^!R20(^A2) is a bijection of level zero discrete series (Proposition 3.2) and we give a parameterization of the set of unramied twist classes of level zero discrete series which does not depend upon the algebra ^Ai and is invariant under

JL

A

2 A

1 (Theorem 4.1).

x

0. Introduction.

This paper is the third of a series of papers (SZ], GSZ]) in which the authors are working toward an explicit description of the Jacquet-Langlands correspondence for the level zero case.

For the proofs of this paper we depend upon the Abstract Matching Theorem (AMT) (see ^x0.3), so our results are of the nature, \If AMT is true, then the correspondence has to be this correspondence."

x

0.1 Some Structure and Notation.

This section will be used throughout and is presented here for easy reference. Our notation will be consistent with that of GSZ].

LetF be a p-adic local eld and n >1 an integer. Ford 1 let D:=D_d denote a central F-division algebra of index d and let A := Mm(D) a central simple F- algebra of reduced degree n:=dm.

Let o:= o_F denote the ring of integers ofF, $_F denotes a prime element of o, and ^pF :=$Fo the maximal ideal of o. Let k := kF denote the residual eld of F with q :=^jk^j its order.

Let O := OD denote the ring of integers of D, $ := $D a prime element of O such that $^d = $F, and let ^p := ^pD = $O be the prime ideal of O. We x a maximal unramied eld extensionFd Dwhich is normalized by$. The residual eld kD := O=^p is of order q^d and may be identied with kd, the residual eld of Fd. More generally, for` 1 we writeF`for an unramied extension ofF of degree

`and k<sub>`</sub> for a nite eld extension ofk of degree `.

We write X(k<sub>`</sub> ) for the group of multiplicative characters of k` and Xt(F_{`</sub> ) for the group of tame multiplicative characters of F<sub>`}. Thus ² X_t(F_` ) has the

reduction ² X(k_` ) (see equation (2.3) equation numbers are always within parentheses).

For any ordered partition s1::: s_r of m we call the block diagonal subgroup M =GLs¹GLsr a standard Levi subgroupof GLm and we call the parabolic subgroup P = M ⁿU which contains the upper triangular subgroup of GL_m a standard parabolic subgroupof GLm. When necessary we writeG(R) to denote the group of R points of the algebraic group with respect to the ring R. Usually we abuse notation to identify algebraic subgroups of GLm with their D-points.

For a hereditary order ^A A = M_m(D) we write ^P := ^PA for its Jacobson radical. Fix the maximal hereditary order ^A1 = Mm(O) A and write ^P1 :=

P

A

1 =Mm(^p). We call^Astandardif ^AA1 (hence^P1 ^PA) and if (^A=^P1) is a standard parabolic subgroup of GLm(kD). Most of the time we consider standard principal orders ^A_r, with Jacobson radical^P_r, which are determined by the period r ^jmand which are such that (^Ar=^Pr) is the standard Levi subgroup GLs(kD)]^r of GLm(kD).

x

0.2 The Category of Level Zero Representations.

Consider the Bernstein spectrum (A ), the set ofA conjugacy classes M] of cuspidal pairs in whichM is a standard Levi subgroup ofA andis an irreducible supercuspidal representation of M. Each irreducible smooth representation of A has a well dened supercuspidal support ^CS()² (A ) and thus we have a surjective nite to one map

CS : Irr(A ) ^;!(A ):

The partition of (A ) into connected components pulls back to a partition Irr(A ) =^G

Irr() where

Irr() := ^f : ^CS()^2g: This partition gives rise to the Bernstein decomposition

(0.1) ^M(A ) =^Y

M()

of the category of smooth representations of A into subcategories ^M(). The set of objects of ^M() consists of those representations which have all irreducible subquotients in Irr().

Let ^C := ^C(^D) be a set of representatives for the set of unramied twist classes of irreducible unitary supercuspidal representations of GLs(D) for all s 1, and let Div⁺(^C) be the set of eective divisors on ^C. Then, with the notation of the second proof of Proposition 2.1,

D=^Xm ^7;!D = theconnectedcomponentof M^DD]²(A ) denes a one-one correspondence between the set of divisors of degree d(^D) =

Pmd() =m and the set of connected components of (A ).

Let ^A1^P1 as in ^x0.1. Then the unit representation 11+^P1 is a type for A and the subcategory^M0(A ) of level zero representations ofA has as its set of objects the set of representations (V) which are generated by the set of (1 +^P1)-xed vectors in V. It follows either from BD], Corollaire 3.9, or, more explicitly, from GSZ], Theorem 5.5 that^M0(A ) is closed under forming subquotients. Moreover,

(0.2) ^M0(A ) =level zero^M()

is a nite partial sum of (0.1). (For a general result see BK2], beginning of Section 4, especially Theorem 4.3.) In the decomposition (0.2) the level zero components may be further represented as = ^D for divisors ^D = ^Pm such that all from the support have (1 +^P1)-xed vectors (GSZ], Theorem 5.5).

More precisely, consider the residual eld kD and the set ^C(kD) of irreducible cuspidal representations of GLs(kD) for all s 1. With respect to the action of Gal(kD^jk) on matrix elements in GLs(kD), hence on the set of cuspidal represen- tations, let ^C(kD) denote the set of Gal(kD^jk) orbits in ^C(kD). Using GSZ], (27) and Theorem 5.5(ii), we have the bijection

(0.3) ^C(kD) ^!C0 ]^7!]

onto the subset of level zero unramied twist class representatives ^C0 ^C. From (0.3) we obtain a bijection of divisors

Div⁺(^C(kD)) ^!Div⁺(^C0) ^7!D:

We write ^D = if ^D corresponds to and note that (0.2) becomes

M0(A ) =^M()

where runs over the nite set of degree m divisors on ^C(kD).

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0.3 The Abstract Matching Theorem (AMT).

The Abstract Matching Theorem (AMT) of Deligne/Kazhdan/Vigneras DKV], Rogawski Ro], and Badulescu Ba] asserts that for any two central simple F- algebras A1 = Mm¹(Dd¹), A2 = Mm²(Dd²) of reduced degree n = d1m1 = d2m2

there exists a bijective mapping, the \Jacquet-Langlands correspondence", (0.4) ^JL_A²_A¹ :^R2(A₁)^;!R2(A₂) ^A2 :=^JL_A²_A¹(^A1)

where ^R2(A_i ) denotes the set of discrete series representations of the unit group A_i =GL_mi(D_di) (i= 12), such that the characters Ai satisfy

(0.5) (^;1)^m1;1A1(x) = (^;1)^m2;1A2(x)

for all regular ellipticx. (We assume an identication of the regular elliptic conju- gacy classes among all the unit groups of reduced degree n simpleF-algebras.)

x

0.4 The Weak Explicit Matching Theorem.

Let^R₂₀(A )^R2(A ) denote the set of level zero discrete series representations and let denote a level zero Bernstein component. Proposition 2.8 shows:

(i) Irr()^\R₂₀(A )⁶= ⁽⁾ = r] ² Div⁺(^C(kD)) is a simple degree m divisor, i.e. rd(]) =m.

(ii) For a simple divisor as in (i) let ^SA := Irr()^\R₂₀(A ). Then ^SA is comprised of precisely one unramied twist class of discrete series represen- tations.

We say that two representations ^{0 2 R2}(A ) are inertially equivalent and write ⁰ if there exists an unramied character of A such that = ⁰ . To prove (ii) as well as a certain multiplicity one statement for discrete series representations we reduce to the unramied split case where these facts are well known. For this we use our description of the Hecke algebras of level zero types GSZ] and establish a reformulation of BK1](7.7.5) for our context. The Appendix is devoted to these reformulations of Bushnell/Kutzko's work.

Now let A1A2 be as in ^x0.3 in particular, assume that d1m1 =d2m2 =n. To give a natural identication of the simple degree m1 divisors on ^C(k_d¹) with the simple degreem2 divisors on ^C(kd²) we use Proposition 1.1, which constructs out of Green's parameterization of the cuspidal representations of the general linear group over a nite eld a natural bijection

(0.6) ^hinX(k_n) ^!thesetofsimpledegreemdivisorson ^C(k_d) ]^7!(]) for any factorization n = dm. The left side consists of the set of Gal(kn^jk) orbits of characters of the multiplicative group k_n. Using (0.6) we may set ^SA :=^S₍ ^A _]) and this is meaningful for any algebra A=M_m(D_d) of reduced degree n.

Our main result, the \Weak Explicit Matching Theorem" (Theorem 4.1), asserts that the Jacquet-Langlands correspondence (0.4) restricts to a bijection

JLA²A¹ :^R₂₀(A₁)^;!R₂₀(A₂)

of level zero discrete series sets such that^SA1 maps to ^SA2 for all ]^2hinX(k_n).

We actually prove a slightly ner result: Since AMT implies that^JLA²A¹ preserves central characters, we have a partition of ^SA1SA2 into nite subsets ^SA1SA2, respectively, of cardinalitye :=n=f,f :=^j]^j, where²Xt(F_n) has the reduction and F := ^j_F is the common central character of all the representations in

SA^{1 S}A². We prove that ^JL(^SA1) =^SA2.

Let us conclude this section by explaining why Theorem 4.1 is called explicit.

Let ^{2 hin}X(k_n) correspond to the simple degree m divisor r] on ^C(k_d) ( constructed via the correspondence (0.6) cf Proposition 1.1) and letA=Mm(Dd).

Then we give two explicit characterizations of the representations ^{2 SA}, which distinguish this class from all other discrete series representations ofA , each char- acterization being su cient to determine the inertial class of ^2R₂₀(A ):

(a) Let^A_r be the standard principal order of periodr inA thus^A_r=(1+^P_r)= GLs(kd)]^r (m=rs). Then ^2SA if and only if contains the lift of ^r to an irreducible representation of ^A_r.

(b) Let M be the standard block diagonal Levi subgroup GLs(D)]^r of A . Then ^{2 SA} if and only if the supercuspidal support of is the con- jugacy class of an unramied twist of the representation ^r_] of M. By GSZ], Proposition 5.1, the supercuspidal representation _] of GL_s(D) is

\explicitly" given by induction from a compact mod center subgroup ofA . Finally, the \weak explicit matching theorem" is \weak" in the sense that it is explicit only up to the inertial ambiguities which remain to be resolved.

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0.5 Concerning Some Commutative Diagrams.

So far we have considered algebras A^jF of xed reduced degree n. Now we compare A:=A⁽¹⁾ =Mm(D) with A^(r) :=M_m=r(D), where r ^jm.

In the second proof of Proposition 2.1 and in the subsequent Remarks we in- troduced for any ^{2 R2}(A ) the base representation ^b, which is a unitary su- percuspidal representation of A^(r) for some r^jm. The representation ^b has the property that the supercuspidal support^CS() has a standard representative of the form (^b)^r, where is a positive, real-valued character of the standard Levi sub- group GLs(D)]^r. The existence of^b is well known for D=F and can be deduced for other D from AMT (see DKV]). Because ^2R2(A ) is uniquely determined from ^b, which we prove in Proposition 2.4 only for the case of ^{2 R}₂₀(A ), we obtain for any r ^jman injection

'D :^R₂₀(A^(r) ),^!R₂₀(A )

which is dened uniquely such that ^(r) and 'D(^(r)) have the same base repre- sentation. For ] ^{2 hin}X(k_n) such that f = ^j]^j divides _nr we obtain a unique pull back ^(r)]^2hinX(k_n=r) and 'D restricts then to a bijection

'D :^SA((r^{r)) ;!S}A:

This implies that forr ^j(m1m2) and A_i =M_mi(D_di) withn=d_im_i (i= 12) the diagram

(0.7)

R20(A⁽₁^r) ) ^{;;J;L;! R}₂₀(A⁽₂^r) )

'_Dd1

?

?

y

?

?

y'_Dd2 R20(A₁) ^{;;J;L;! R}₂₀(A₂)

is commutative up to unramied twist. From DKV], Theorem B.2.b we know that, in fact, (0.7) is commutative.

This suggests comparing the diagrams (0.7) against another set of obviously commutative diagrams. In a future paper, introducing \Langlands parameters", we will dene bijections

G

f^jn

hⁱⁿX_t(F_f )reg ³_f]^7;!_Af ^2R₂₀(A ) such that ^JL_A⁰_A(_Af) = ^A_f⁰. The diagrams

(0.8)

R20(A⁽₁^r) ) ^{;;J;L;! R}₂₀(A⁽₂^r) )

_Dd1

?

?

y

?

?

y_Dd2 R20(A₁) ^{;;J;L;! R}₂₀(A₂) in which

D_di : ^Af^{(ir) 7;!}_Af

are dened for f ^j _nr and are obviously commutative. When r^j(m1m2), these diagrams can be compared to the corresponding diagrams (0.7). When they can be compared, the diagrams (0.7) and (0.8) agree up to unramied character twist because _Af ^2SA, where =fN_Fn^j_Ff, but in general they are not the same. In fact, the mappings'_Ddi and _Ddi dier at most by twists by sign characters which depend upon nrf. In a future paper we shall make these twists explicit.

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1. Parameters for the Set of Level Zero Simple Components of

(A )

.

The purpose of the following Proposition is to prove (0.6), which gives a natural parameterization for the set of Bernstein components which correspond to level zero simple divisors. Only such components can have level zero discrete series subquotients associated to them.

1.1 Proposition.

Let kn^jk be a degree n extension of nite elds with generating automorphism and let n := dm. Then Green's parameterization of the cuspidal representations of nite general linear groups in terms of orbits of regular characters Gr] induces a natural bijection

(1.1) ^Gm :^hinX(k_n) ^{! G}

s^jmGLs(kd)^{^}_cusp=^hi

between the set of^hi-orbits in the group of charactersX(k_n)and the set of orbits of cuspidal representations of the groupsGLs(kd)for all s^jm. If]is a character orbit of lengthf and f ^jn, then the corresponding Galois orbit of cuspidal representations ] consists of representations of GLf⁰(kd) (f⁰ :=f=(df)) and the length of ] is (df). In particular,

j]^j=d()^j]^j=f⁰(df) d() =f⁰ being the degree of the divisor .

Remarks.

(i) For m= 1 and d=nthe correspondence ^G1 is the identity. For m=n and d= 1, since there is no Galois action on the right side of (1.1), we see that

Gn gives the usual Green's parameterization of the cuspidal representations of GLs(k) for s^jn.

(ii) On the right in (1.1) it would be enough to letbe a generator of the Galois group Gal(kd^jk).

Proof. Given a ^hi-orbit ]X(k_n) of lengthf we write lcm(df) =df⁰, and we use the following diagram of nite elds:

k_n

x

?

?

kf ^;;;;! kdf⁰

x

?

?

x

?

?

k(df) ^;;;;! kd

x

?

?

Since f ^j df⁰ and df^{0 j} n, for any k ² X(k_n) such that ] = f, we have a unique kd-regular character df^{0 2} X(k_df⁰) such that df⁰ N_kn^j_kdf⁰ = . Therefore, by Gr], we also have the unique cuspidal representation =(df⁰) of GLf⁰(kd) which has the character values

(x) = (^;1)^{f0;1 X}

²Gal(k_df ^jkd)_df (x)

for k_d-regular x²k_df⁰. Observing that the bijection of Green

h^dinX(k_df⁰)kd^;reg GLf⁰(kd)^{^}_cusp

which maps orbits of kd-regular characters in X(k_df⁰) to cuspidal representations of GLf⁰(kd), is compatible with the action of ^hi on both sides, we see that the

hⁱ-orbit (df⁰)] has length (df). Conversely, an orbit ] ² GLs(kd)^{^}_cusp=^hi determines an orbit of k_d-regular characters ] ^2hinX(k_ds) and a unique orbit ] X(k_n) such that = N_kn^j_kds. These orbits are of the same length specically, we have

j]^j=^j]^j=s^j]^j:

For ²X(k_n) we write e:=n=f ande⁰ := (em). From the equation dm=ef it follows that m = e⁰f⁰. Via ^G_m the character orbit ] determines the simple divisore⁰], which has the degree m.

Now consider the central simple algebraA=Mm(D),D :=Dd a central division algebra of index d over F. From GSZ](27) we know how to associate to ] a unitary supercuspidal representation ] ^2C(D) such that d(]) = d(]). Thus ] determines both a degree m divisor ^D_m(]) = e⁰_] ² Div(^C) and a simple level zero component ] := ^D_m( ])(A ).

1.2 Proposition.

The mapping ^hinX(k_n) ³ ] ^7! _] = ^D_{m( ])} gives a bi- jection from ^hinX(k_n) to the set of simple level zero connected components of (A ). If ] corresponds to the degree m divisor e⁰], then (^A_e^0e0) is a type for the component _]. If ²Irr(_]) is an irreducible representation with cusp- idal support in ], then its central character ! is a tame character of F with

! = ^j_k ²X(k ).

Proof. The proof follows from GSZ](5.4, 5.5). In particular, the central char- acter ! is tame because is a level zero representation and the reduction !

is constructed by restricting to the units of F and factoring mod the principal units. It is obviously determined by the type. From Proposition 1.1 we obtain

! = (df⁰)^e0j_k = ^j_k .

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2. A Parameterization of the Set of Inertial Classes

^R₂₀(A )=

.

We shall continue to use a \bar" over a Greek letter to denote a multiplicative character of a nite eld e.g. if is a tame multiplicative character of a p-adic eld, then usually will denote its \reduction", i.e. the reduction of its restriction to the group of units.

2.1 Proposition.

Let = (^D) be a level zero Bernstein component and assume that ^D is not a simple divisor. Then ^R2(A )^\Irr() =.

Proof. We give two proofs for this Proposition, the rst in the context of Bush- nell/Kutzko theory and the second based on a theorem of Harish-Chandra. The second proof will allow us to associate to any irreducible discrete series of A a unitary supercuspidal \base representation".

(i) Let (A ) be a level zero connected component. Assume that = (^D), where the degree m divisor ^D is not simple. By GSZ], Proposition 5.3 and The- orem 5.4 the types (^A ) associated to all have the same divisor () =

P

]r_]()] with r_] > 0 for at least two distinct orbits ]. Then we con- sider the proper Levi subgroupM^f=^Q_]GLr ^]( )d()(D)A . Using the Hecke algebra isomorphism GSZ]1.9 we obtain from BK2](8.4) that the parabolic induc- tion from M^fto G is an equivalence between the categories ^M_M^f() and ^MG(), where = (^D) is interpreted as a connected component of the Bernstein spec- trum (M^f) in the obvious way. This implies that ²Irr() is always parabolically induced from ~M and therefore cannot be discrete series. Indeed either by AMT (or, to give a local argument, by Harish-Chandra's Selberg principle) the character of a discrete series representation cannot completely vanish on the regular elliptic set, whereas the character of a parabolically induced representation does vanish for regular elliptic elements.

(ii) For ^D=^P2Cm ²Div⁺(^C) consider the cuspidal pair (M^D = ^Y

^2CGL_d()(D) ^mD =^2Cm)

and assume that the connected component (^D) is generated by the A orbit of (M^DD). Assume, moreover, that the supercuspidal support of ^2R2(A ) lies in (^D), hence that it is the A orbit of (M^D0), where 0 is an unramied twist of ^D. Let0 =1be the decomposition of 0, where 1 is unitary supercuspidal and is a positive, real-valued character of M :=M^D. LetT be the maximal split torus in the center of M and let W(T) :=^N_A (M)=M be the corresponding Weyl group. Let W(1) W(T) be the subgroup consisting of elements which x the class of 1. Since ^{2 R2}(A ), it follows from Si], Corollary 5.4.5.8 that W(1) contains re"ections with respect tor^;1 linearly independent T-roots ofA , where r is the number of diagonal blocks of M (r = ^P2Cm). It is easy to see that this implies that W(T) must permute the r blocks of M, so W(T) = ^Sr is the symmetric group on r letters and, moreover, W(1) = W(T). This implies that 1 = #^r, where the supercuspidal representation # ofGL_m=r(D) does not depend upon the choice of 0, because 0 and 1 are unique up to the action of W(T) and

#^r is W(T)-invariant. Since #^r = 1 is an unramied twist of ^D, it follows that ^D = m =r for a single ^{2 C}. In particular, it follows that ^D must be a simple divisor and # an unramied twist of . From (0.3) it follows that, when is level zero, then ^D=r, where =] for some orbit ]^2C(kD).

Remarks.

(i) The second argument applies to general discrete series representations and proves that a discrete series representation must lie in a Bernstein compo- nent corresponding to a simple divisor.

(ii) If ^{2 R2}(A ), we write ^b := #, where # is as constructed in Proof (2) above. We call ^b the supercuspidal base representation of .

2.2 Corollary.

If ^R₂₀(A ) ^\Irr() ⁶= , then = ] for some Galois orbit ]²X(k_n).

Proof. Proposition 2.1 implies that, if ^R₂₀(A )^{\ 6}= , then = (^D) for some simple divisor ^D. On the other hand, Proposition 1.2 gives a parameterization of the set of simple level zero components such that each is of the form for some ²X(k_n).

2.3 Denition.

We write ^SA Irr(_]) for the set of irreducible discrete series representations which have supercuspidal support in ].

From Proposition 2.1, Proof (ii), it follows that the base representation mapping ^{7! b} is dened for any ^{2 R2}(A ). We want to give some of the properties of this mapping but we shall be able here to give proofs only in the level zero case, to which we now return.

Let ^R₀₀(A ) ^R₂₀(A ) denote the subset consisting of unitary supercuspidal level zero representations.

2.4 Proposition.

The base representation map denes a bijection

R20(A )^{37;! b2 G}

s^jm

R00(GLs(D)):

Proof. Injectivity: Let # := ^b1 = ^b2 ^{2 R}00(GLs(D)), let r = m=s, and let M = GLs(D)]^r be a block diagonal Levi subgroup of A . Since the supercuspidal supports of 1 and 2 are inertially equivalent to #^r, it follows that 12 ²

Irr() for the same connected component of the Bernstein spectrum. From the classication of these components given in Proposition 1.2, we know that = ], i.e. 12 ^{2 S}A for some ] ² X(k_n)=^hi. Applying Proposition 2.8, we see that 1 = 2~for some unramied character ~ =Nrd_A^j_F of A . We also write ~⁰ =Nrd_A^0j_F forA⁰ =Ms(D). LetP =MⁿU be the standard (upper triangular) parabolic subgroup of A with M as Levi factor. Then 2 i_{A P}(#^r), where is a positive, real-valued unramied character of P=U = M and i_{A P} denotes normalized (unitary) parabolic induction. Therefore,

1= 2~~i_{A P}(#^r) =i_{A P}((~⁰#)^r)

which implies that ^b1 = ~⁰#. Our hypothesis further implies that # = ~⁰#, i.e.

that 1 and 2 have the same supercuspidal support. We may conclude from this that 1 = 2, since it is known that the composition series of i_{A P}(#^r) con- tains only one discrete series component.

Surjectivity: Assume that rs = m and that ^{2 R}₀₀(GLs(D)). From GSZ], Proposition 5.5(ii) we know that is an unramied twist of _] for some ] ² GLs(kD)^{^}_cusp=^hi. By Proposition 1.1 there is a unique ]^2hinX(k_n) such that

Gm(]) = ]. This means that ^Dm(]) = r]. Again applying Proposition 2.8 we see that ^{SA 6}= . The supercuspidal support of ^{2 SA} is an unramied twist of ^r_] and ^b is an unramied twist of _]. But for an unramied character ~ as above we have seen that ^b(~) = ^b~⁰. Thus we can realize any un- ramied unitary twist of ], in particular , as the base representation of some ^2R₂₀(A ).

Remark.

Let (A ) be a connected component of the Bernstein spectrum.

The proof of Proposition 2.4 depended upon the following two basic facts:

(i) The set Irr() contains at most one unramied twist class of discrete series representations. We used this fact to prove injectivity.

(ii) There exists an unramied twist class of discrete series representations in Irr() when = (^D) for a simple divisor^D. We needed this fact to prove surjectivity.

We proved our assertion only for level zero representations and depended upon Proposition 2.8 for the two facts. We remark here that both facts are known in complete generality. In particular, they follow from AMT (DKV], Theorem B.2.b), which reduces these assertions to the split case.

Applying (0.3), which denes a bijection of ^F_s^j_mGLs(kd)^{^}_cusp to the set of in- ertial classes ^F_s^j_m^R₀₀(GLs(D))=, we may then use (1.1) and Proposition 2.4 to parameterize the set of level zero inertial classes ^R₂₀(A )= by the set of Galois orbits X(k_n)=^hi. We thus obtain a map ^7!SA (See (2.5)).

In the following we use notation explained in the paragraph prior to Proposition 1.2.

2.5 Proposition.

Assume that ^Dm(]) = e⁰], i.e. ] (A ) is the con- nected component which contains the cuspidal pair(M^e_]⁰)with respect to the Levi subgroup M = GLf⁰(D)] ^e0 A . For any ^{2 SA} the normalized Jacquet func- tor r_MA () is irreducible specically, r_MA () = ^e_]⁰ for some unramied character of M which is not unitary.

Proof. The type (^A_e⁰) for ] is a cover of (M ^\A_e^0e0), which is a type for the supercuspidal component _](M) (M) (see GSZ] 5.3 and its proof).

Therefore, by BK2](7.9)(ii), we have a natural isomorphism of complex vector spaces

(2.1) Hom^A_e0()= Hom^A_e0

\M(rMA ()):

Let!denote the central character of . Then we also have the natural isomorphism

(2.2) Hom^A_e0

h$Fⁱ(!)= Hom^A_e0(): Since is irreducible, Hom^A_e0

h$Fⁱ(!) is a nite-dimensional simple module over the Hecke algebra ^H(A ^A_e^0h$Fⁱ!). Since is a discrete series representation, we may apply Proposition A9 with (K) = (^A_e^0h$Fⁱ!). This implies that the modules in (2.2) are complex vector spaces of dimension one.

Considering now the right side of (2.1), we see thatr_MA () contains the type (^A_e⁰) with multiplicity one and therefore r_MA () contains an irreducible com- ponent ^e_]⁰ and indeed only one. The character cannot be unitary, since if it were unitary, then would be a subrepresentation of a unitarily induced repre- sentation and therefore not a discrete series representation. Moreover, any other component ofr_MA () has to be supercuspidal and conjugate to^e_]⁰under the normalizer of M, i.e. it has to be of the form ^e_]⁰⁰ (see e.g. BR], III.2 Theorem 18). Any such component contains ^j_M^\A_e0. By BK2](7.9) this is impossible since occurs as a component of ^jA_e0 only once. Thus r_MA () = ^e_]⁰ and is irreducible.

Remark.

Since r_MA () is irreducible, it belongs to the supercuspidal support of . Therefore, we have

r_MA () = ^be01 = (]~⁰)^e01

where1is a positive, real-valued character and ~⁰is a unitary unramied character.

As before, (^A_e⁰) is the type such that =^e0. We write ^1:::e0 :=^{1 e0}:

The set of tuples (1::: e⁰)²(^Z=((df)))^e0 parameterizes the (df)^e0 conjugates of .

2.6 Corollary.

Let (V)^2SA and let V^1+Pe0 denote the subspace of V consist- ing of all (1 +^P_e⁰)-xed vectors, where ^P_e⁰ denotes the Jacobson radical of ^A_e⁰. As a representation of ^A_e⁰ _{the space} V^1+Pe0 decomposes as ^1:::e0, where, for 01::: e⁰ <(df), each representation occurs with multiplicity one.

Proof. From GSZ](5.5)(i) we know that each of the above types occurs and with the same multiplicity. As we have seen in the proof of 2.4, occurs simply in ^jA_e0. Therefore, the same is true for each of the types^1:::e0.

We consider the unramied extension Fn^jF of degree n and we write Xt(F_n ) for the set of tamely ramied characters of F_n . Then we have a reduction map (2.3) X_t(F_n)^37;!2X(k_n)

where denotes ^j_on regarded as a character of k_n. Here we identify k_n with the residual eld of Fn and we use the assumption that is a tame character. Under these assumptions the is well dened we call the reduction of . Noting that F_nhas a prime element inF, we see that (2.3) is compatible with the Galois action of Gal(Fn^jF) = Gal(kn^jk) = ^hi, the lengths of orbits being preserved. Moreover, the character ²X_t(F_n) is uniquely determined by its reduction ²X(k_n) and its restriction F ² Xt(F ). For ^{2 SA} it follows from Denition 2.3 and from the last statement of Proposition 1.2 that the central character ! ² Xt(F ) has the reduction ! = ^j_k . From this we obtain a partition ^SA = ^S_A, where ² Xt(F_n ) runs over all tame characters which have the prescribed reduction and where:

2.7 Denition.

For ²Xt(F_n) set ^S_A :=^f2SA : ! =F^g:

Like ^SA (cf. Def. 2.3) the denition of ^S_A depends only upon the ^hi-orbit of .

2.8 Proposition.

If ²Xt(F_n ) generates a ^hi-orbit ] of length f, then ^S_A is a set of order e = n=f. In particular, the group of unramied characters of order dividing n acts transitively on ^S_A. The set ^SA is a single unramied twist class.

Proof. We note that^S_Aconsists of all ^2SA which have a xed central character.

By Proposition 1.2 the representations of ^S_A admit the type (^A_e⁰), where = ^e0. Therefore, we may apply Corollary A10 with r = e⁰ and l = (df), which according to Proposition 1.1 is ^j]^j. This implies that dr=l = de⁰=(df) = e, in which the second equality follows fromef =dmande⁰ = (em). If²X(F ) and ~=Nrd_A^j_F, then we know that!~ =!ⁿ, which implies that the group of unramied characters of order dividing n acts on each set ^S_A. Therefore, it su ces to show that, for ^2SA, we have ~F = if and only if^f_F = 1. In this

case, since n=f = e, we see that all e inequivalent discrete series representations with the same central character lie in a single unramied twist class. Since the group of unramied characters of A acts transitively on the set of all^S for which the reduction of is , it follows that ^SA is a single unramied twist class.

Let us proceed with the proof that ~ = if and only if ^f = 1. From Proposition 2.4 and the comments immediately prior to Proposition 1.2, which explain how to construct a degree m simple divisor from a cuspidal divisor e⁰], where e⁰d() = e⁰f⁰ = m, it follows that the supercuspidal support of ^{2 SA} has the form ^e_]⁰ , where is some non-unitary unramied character of M = (GLf⁰(D)) ^e0. By Jacquet's subrepresentation theorem Ind_AP (^e_]⁰), where P =MⁿUP is a parabolic subgroup ofA and the induction is normalized. Thus, ~ is contained in Ind_AP (^e_]⁰ ~^j_M). From the irreducibility of the Jacquet module we see that ~= implies that

(2.4) ^e_]⁰ =^e_]⁰~^j_M:

Furthermore Nrd_A^j_F^j_M = (Nrd_f⁰)^e0, where Nrd_f⁰ denotes the reduced norm map- ping Mf⁰(D)^jF. Therefore from (2.4) we obtain

] =](Nrdf⁰)

for the supercuspidal representation ] of GLf⁰(D). With this justication we reduce to the supercuspidal case = _], f⁰ = m, and e⁰ = 1. In this case from GSZ]5.1 we know that = cInd(~), where ~ is an extension of to ^h$^liA₁. Here and in GSZ] l =^j]^j. Thus l = (df), since corresponds to ] of length f (see Proposition 1.1). The reduced norm Nrd induces an injection

Nrd_A^j_F :^h$^liA₁=^A₁ ,^!F =^o_F with image of index ml =f⁰(df) =f. Therefore,

~cInd(~) = cInd(~~) = cInd(~) which is true if and only if ~~ = ~, if and only if ^f = 1.

Writing^R₂₀(A )=for the set of inertial equivalence classes of level zero discrete series representations, we now have the bijection

(2.5) ^hinX(k_n) ^37;!SA2R₂₀(A )=: In concluding this section, we lift (2.5) to a bijection

Gal(F_n^jF)ⁿX_t(F_n)³]^7;!S_A ^2R₂₀(A )=

where we set 1 2 if 1 = 2 for some unramied character of A which is trivial on the center of A . To formulate our result, which summarizes what has been proved to this point, we also extend (^A_e⁰) to (^A_e⁰F F), F being the extension of =^e0 to ^A_e^0h$Fⁱ=^A_e⁰F such thatF1 is the restriction of F

to F .