Journal of Lie Theory Volume 5 (1995) 101–127 ◯^{C} 1995 Heldermann Verlag Version of July 14, 1995

Types for elliptic non-discrete series representations of SL_{N}(F), N prime and F a p-adic field.

Guillaume D. Sanje Mpacko

Communicated by Simon Gindikin

Abstract

In this paper we give a classification of the elliptic non-discrete series representations of G = SL_{N}(F), F p-adic and N prime, by the method of types, that is by restriction to compact open subgroups. We give an explicit construction of certain types and determine their Hecke algebras. We then prove that an irreducible smooth non-discrete series representation of G is elliptic if and only if it ‘contains’ one of our types.

1. Introduction

We recall that for any algebraically connected (group of F -rational points of a) reductive p-adic group, the Langlands quotient theorem (extended to p-adic groups by Silberger) gives a classification of irreducible smooth (hence admissible) representations in terms of the irreducible tempered representations [20]. These include the discrete series and consist in general of irreducible constituents of representations unitarily parabolically induced from discrete series. An element of the group is called elliptic if it is contained in a Cartan subgroup which is compact modulo the center. One says that an irreducible tempered representation is elliptic if its character has support (in the sense of distributions) on the subset of regular elliptic elements. The subset of elliptic representations also include the discrete series. It is known that for the groups SL, unlike the groups GL, this inclusion is strict. D. Goldberg has given in [12] criteria for the constituents of parabolically induced representations of SL_{k}(F), where k is arbitrary, to be elliptic. This is based on the theory of the Knapp-Stein R-group and a result of J. Arthur in [1] (which are both valid for general p-adic groups). To be more precise we introduce some notations below.

Let χ denote a continuous character of the multiplicative group of the field F whose restriction to the group of integral units has order N. It determines canonically a character σ of the subgroup D of diagonal matrices in SL_{N}(F); σ

ISSN 0940–2268 / $2.50 ◯^{C} Heldermann Verlag extends trivially across the subgroup U of upper unipotent matrices, to a character of the Borel subgroup B = DU of upper triangular matrices (still denoted σ). It is a well known result of Howe and Silberger [16] that the unitarily induced representation Ind(B, G, σ) decomposes simply. It follows from [12] that its N inequivalent constituents are elliptic, furthermore any non-discrete series elliptic representation of SL_{N}(F) is a constituent of such an induced representation, since N is prime. We then construct a type (J, λ) which depends on χ. Our main Theorem is as follows: Main Theorem. An irreducible smooth representation π of SL_{N}(F) is an irreducible constituent of the induced representation Ind(B, G, σ) if and only if it contains (J, λ) upon restriction to J.

Our methods are inspired from A. Borel [5] and W. Casselman [9], [10]. They require a quite explicit knowledge of the structure of the Hecke algebras associated to our types. To determine the support of these algebras we rely on orbit theory in the spirit of Howe and Moy [15]. This enables us to describe the Hecke algebras explicitly by generators and relations. In [10] Casselman uses the ‘Iwahori’ Hecke algebra to classify the irreducible smooth representations with an Iwahori fixed vector, the so called unramified principal series (see also Borel [5]). This algebra is known to have the nice structure of an affine Hecke algebra. In [15] Howe and Moy showed (in the tame and prime cases) that much of the representation theory of a given GL_{k}(F) is controlled by the Iwahori Hecke algebras of (usually) smaller GL′s. This observation was later confirmed by Bushnell and Kutzko in [6], with no restriction on k and the residual characteristic. The situation for other groups proved to be more complicated. Indeed the works of A. Moy [18] and L. Morris [17] provide us with examples of some ‘degenerate’ Hecke algebras.

For the group SL_{N}(F) with N prime, we show that for each of our types, the Hecke algebra is isomorphic to the group algebra of some special subgroup in the affine Weyl group. We proceed by restriction from GL_{N}(F), and as a consequence we construct a series of types for GL_{N}(F) as well. We show that these are isomorphic to a tensor product of affine Hecke algebras. It is a conjecture of Bushnell and Kutzko that this should be true of any GL-type in ‘general position’.

Numerous papers have been devoted to the representation theory of the groups SL_{k}(F). Let us just mention the two most recent ones which we know of. In [8] Bushnell and Kutzko classify the supercuspidals by types. They proceed by restriction from GL_{k}(F) exploiting their own earlier work [6] on GL_{N}(F). There they constructed the so called simple types and gave another proof after L. Corwin [11] that the supercuspidals were induced from open and compact mod center subgroups. They extend this result to SL_{k}(F) also. Marco Tadic gives in [21] a description of the tempered representations of SL_{k}(F) by certain GL-parameters. His method is based on the Bernstein and Zelevinsky classification [3] of these representations for GL_{k}(F).

We now discuss the organization of this paper. We introduce notations and review some general facts in the second section. The third section discusses Goldberg’s theorem on elliptic representations. In the fourth section we give a construction of types and we state the main result (Theorem 1). We determine the structure of the Hecke algebra of a given type in section five. Section six deals with a discussion of ‘closely related’ types for GL_{N}(F). The structure of the Hecke algebras of our types is determined in section seven. We prove our main theorem in section eight. In the last section, as a consequence of the methods used here, we describe the smooth representations of GL_{N}(F) which contain the types constructed in section six.

Acknowledgements: These results are contained in the author’s Ph.D. Thesis [19]. Professor Philip Kutzko suggested this problem, I want to thank him for his guidance during our preliminary investigations with GL_{2}(F) and SL_{2}(F). I thank Professor David Keys for some helpful discussions. I also want to thank Professors Simon Gindikin, Roe Goodman and Siddhartha Sahi for their help in improving this document. I am grateful and honored for the encouragements received from Professors Joseph Bernstein, Amy-Cohen Corwin, Israel M. Gelfand, Roger Howe, Jim Lepowsky, Allen Moy, Paul Sally and Robert Wilson.

2. General background

Notation. We fix a p-adic field F. O_{F} is the ring of integers in F. P_{F} is the maximal ideal in O_{F} with generator ϖ. F^{+} the additive group of the field is filtered by the ideals

P_{F}^{m} = ⟨ϖ^{m}⟩, m ∈ ℤ.

U_{F} is the group of units in O_{F}. U_{F} is filtered by the groups

U_{F}^{(i)} = 1 + P_{F}^{i} , i ≥ 1.

We let k_{F} = O_{F} /P_{F} denote the residue field with cardinality q = p^{r}, where r is some positive integer and p is the residual characteristic. N denotes a fixed arbitrary prime number. M_{N}(F) stands for the ring of N by N matrices with entries in F.

In this paper we let G̃= GL_{N}(F) be the group of units in M_{N}(F), the invertible N by N matrices with entries in F. We let G = SL_{N}(F), the subgroup of G̃ consisting of matrices with determinant equal to 1, except otherwise noted.

We let B̃ = D̃Ũ denote the standard Borel subgroup of upper triangular matrices with Levi component the subgroup D̃ of diagonal matrices and unipotent radical the subgroup Ũ of upper unipotent matrices. If H̃ is a subgroup of G̃ then we let H = H̃ ∩ G. In particular we get the standard Borel subgroup B = DU of G, where D and U have the obvious meaning. Observe that Ũ = U.

Particularly important for us are the parahoric subgroups of G̃ and G. Let G̃_{0} = GL_{N}(O_{F}); this is a maximal compact open subgroup of G̃. We note in passing the Iwasawa decomposition

G̃ = B̃G̃_{0}.

G̃_{0} is filtered by the compact open subgroups

C̃_{i} = 1 + ϖ_{F}^{i} M_{N}(O_{F}), i ≥ 1.

These form a fundamental system of neighborhoods of the identity in G̃. We see that the reduction mod P_{F} homomorphism

G̃_{0} → GL_{N}(k_{F})

yields an isomorphism:

G̃_{0}/C̃_{1} ≅ GL_{N}(k_{F}).

The inverse image in G̃_{0} of a parabolic subgroup of GL_{N}(k_{F}) is a parahoric subgroup of G̃. Among these is the standard Iwahori subgroup which is the inverse image of the standard Borel subgroup of GL_{N}(k_{F}). Restricting to SL_{N}(F) we obtain the parahoric subgroups of G.

The following realization of parahoric subgroups will be quite useful. By a lattice in F^{N}, we mean a free O_{F} -submodule of F^{N} of rank N. A periodic lattice flag L of period e is a sequence of lattices {L_{i} | i ∈ ℤ} such that

L_{i+1} ⊆ L_{i} and L_{i+e} = ϖL_{i} .

For a fixed lattice flag L let

A^{m} = A_{L}^{m} = {a ∈ M_{N}(F) | aL_{i} ⊆ L_{i+m}, all i}, m ∈ ℤ.

Then A = A^{0} is a subring of M_{N}(F) (in fact an hereditary O_{F} -order in M_{N}(F)) and A^{1} is an ideal in A (in fact the Jacobson radical of A). For a useful account (in this context) on the theory of hereditary orders we refer to [6]. The group K̃ = K̃_{L} of units of A is a parahoric subgroup of GL_{N}(F). The correspondence L → K̃_{L} gives a parametrization of the parahoric subgroups of G̃. In fact more is true, index shifting determines an equivalence relation on the set of periodic lattice flags; and the correspondence above carries in a one to one fashion, an equivalence class to a G-conjugacy class of parahoric subgroups. Again the restriction to G yields the parahoric subgroups of G. Now we set

K̃_{i} = 1 + A^{i}, i ≥ 1.

The K̃_{i} are compact open subgroups of GL_{N}(F) and normal in K̃.

Abelian characters and duality in M_{N}(F). F^{+} is a locally compact abelian group. O_{F} and the P_{F}^{m} are compact open subgroups of F^{+}. We fix once and for all a continuous character ψ of F^{+} with conductor P_{F}, that is ψ is trivial on P_{F} but not on O_{F} (such objects exist [22]). F^{+} is self-dual, that is isomorphic to its group of characters (F^{+})^{∧} (the Pontryagin dual) under the map a → ψ_{a} where

ψ_{a}(y) = ψ(ay).

In particular the Pontryagin dual of the finite abelian group P_{F}^{n}/P_{F}^{n+1} is given by

(P_{F}^{n}/P_{F}^{n+1})^{∧} ≅ P_{F}^{−n}/P_{F}^{1−n}.  (1)

Similar facts hold for M_{N}(F) which is also a locally compact abelian group under addition. With ψ as above we define ψ_{X}, X ∈ M_{N}(F) by

ψ_{X}(Y) = ψ(Tr(XY)), Y ∈ M_{N}(F)

where Tr denotes the usual trace function on M_{N}(F). Then ψ_{X} is a character of M_{N}(F) and the map X → ψ_{X} yields an isomorphism between M_{N}(F) and its Pontryagin dual M_{N}(F)^{∧}.

We note below some useful consequences of these duality results. We let K̃ = K̃_{L} be a parahoric subgroup of G̃ as in the previous section, with its natural filtration subgroups K̃_{i}, then we have an isomorphism of finite abelian groups

K̃_{j}/K̃_{i+1} ≃ A^{j}/A^{i+1} where 2j ≥ i + 1 ≥ j.

Thus the self duality of M_{N}(F) yields an isomorphism

(K̃_{j}/K̃_{i+1})^{∧} ≃ A^{−i}/A^{1−j} where 2j ≥ i + 1 ≥ j.  (2)
If we set K_{i} = K̃_{i} ∩ SL_{N}(F) then we (now) have
(K_{j}/K_{i+1})^{∧} ≃ A^{−i}/ (F ∩ A^{−i} + A^{1−j})  (3)

where we view F (abusing notation) as the scalar matrices in M_{N}(F). This is Proposition 1.1 of [7].

Smooth representations. In this section we will state some standard results on smooth representations which hold for an arbitrary (group of F -rational points of a) reductive algebraic group G defined over F.

By a representation of G we mean a pair (π, V) where V is a complex vector space and π a homomorphism G → End_{ℂ}(V). We will often simply write π or V depending on context. If H is a subgroup of G we let H^{g} = gHg^{−1} and if π is a representation of H we write π^{g} for the representation of H^{g} defined by

π^{g}(x) = π(g^{−1}xg), x ∈ H^{g}.

We say that a representation of G is smooth if for any vector v ∈ V its G−stabilizer

{g ∈ G | π(g)v = v}

is open.

We will call a one dimensional smooth representation of G a character of G. Examples of characters include the continuous homomorphisms from G to ℂ^{×}, where ℂ^{×} is given its usual topology. For GL_{N}(F) a standard construction is to form the composite γ∘ det, where γ is a continuous character of F^{×} [22] and det denotes the usual determinant homomorphism on GL_{N}(F). A character θ of F^{×} is called unramified if it is trivial on U_{F}. An unramified character of G is a character of the form θ ∘ η where η : G → F^{×} is a rational character of G (in the sense of algebraic groups) and θ is an unramified character of F^{×}.

We say that (π, V) is admissible if it is smooth and if for each compact open subgroup K ⊆ G, the subspace

V^{K} = {v ∈ V | π(k)v = v, all k ∈ K}

is finite dimensional. We say that (π, V) is irreducible if the only G−invariant subspaces are {0} and V . It is a basic fact that Theorem A. An irreducible smooth representation of G is admissible [9].

If P = LU denotes a parabolic subgroup of G with unipotent radical U then a (smooth) representation (σ, W) of the Levi factor L extends (trivially across U) to a (smooth) representation of P also denoted (σ, W). G acts on the space W (σ) of all functions f : G → W such that

f(hug) = σ(h)f(g), ∀h ∈ L, u ∈ U, g ∈ G.

by the formula

(π(g)f)(y) = f(yg), y, g ∈ G.

One usually refers to this as the action of π by right translations. The subspace of smooth vectors in W (σ), which one may refer to as smooth functions is Ginvariant so it affords a smooth representation, this is the parabolically induced representation denoted Ind_{P}^{G}σ.

Let δ_{P} denote the modular character of P, it is well known that if σ is unitarizable then Ind_{P}^{G}(δ_{P}^{1/2} ⊗ σ) is itself unitarizable. We will refer to it as the representation unitarily induced from σ. We will denote it ind_{P}^{G}σ.

We will also need the notion of compactly-induced representation denoted c-Ind_{H}^{G}σ for which G acts by right translations on the subspace of W (σ) consisting of compactly supported smooth functions. Clearly if G/H is compact then the two notions of induction coincide (this is the case for parabolic subgroups).

In this context, Frobenius reciprocity holds, that is

Hom_{H}(π|_{H}, σ) ≅ Hom_{G}(π, c − Ind_{H}^{G}σ).  (4)

We now state the following important property of induced representations

Theorem B. [9] A smooth representation of G induced from an irreducible smooth representation of a proper parabolic subgroup has finite composition length.

By constituents of such a representation we mean the irreducible composition factors (or subquotients). The so called supercuspidal representations are those smooth representations which do not occur as constituents of these ‘parabolically’ induced representations.

We now review the construction of Jacquet modules.

Let P = LU be a parabolic subgroup of G. The Jacquet module of a representation (π, V) of G with respect to P, denoted (π_{U}, V_{U}) is a representation of L on the quotient space

V_{U} = V/V (U)

where

V (U) = span{π(u)v − v | v ∈ V, u ∈ U}.

Write v̄ = v + V (U) then L acts by

π_{U}(h)v̄ = \overline{π(h)v}, h ∈ L, v ∈ V.

The subspace V (U) is also characterized as the subspace

{v ∈ V | for some compact open subgroup U′ ⊆ U, ∫_{U′} π(u′)vdu′ = 0},

where u′ ∈ U′ and du′ is a Haar measure on U′. It is a useful technical fact (see [9]) that for any compact open subgroups U_{1} ⊆ U_{2},

if ∫_{U_{1}} π(u)vdu = 0 then ∫_{U_{2}} π(u)vdu = 0 .  (5)

The correspondence π → π_{U} defines an exact functor from smooth Gmodules to smooth L-modules. Assuming now that π and σ (as above) are admissible; then the following second version of Frobenius reciprocity holds:

Hom_{L}(π_{U}, δ_{P}^{1/2} ⊗ σ) ≅ Hom_{G}(π, Ind_{P}^{G}σ).  (6)

Finally we state the following important result of Jacquet and Harish-Chandra [9]:

Theorem C. If π is an admissible representation of G then π_{U} is admissible.

3. On elliptic representations

Let P = LU be a (not necessarily proper) parabolic subgroup of G. Let L̃ ≅ ∏_{j=1}^{k} GL_{N_{j}} (for some N_{j} ’s) be the Levi subgroup of G̃ such that L = L̃ ∩ G. Now we have the following results of David Goldberg.

Lemma 1. [12] If for some i and j, N_{i} ≠ N_{j}, then a representation of the form mboxind_{P}^{G}σ can never have an elliptic constituent.

We now specialize to the case that N is prime and P = B. We then have the following criteria:

Proposition 1. [12] Suppose that all the N_{j} are equal (in fact they will then equal one, since N is prime). Let σ be a unitary character of D, then the following are equivalent:

a) ind_{B}^{G}σ has an elliptic constituent,

b) Every constituent of ind_{B}^{G}σ is elliptic,

c) σ is of the form ⊗_{j=1}^{N−1}χ̄^{j} where χ̄ is a character of F^{×} that has order N. The condition c) is formulated differently in [12]. The statement there is

that the R-group is isomorphic to the cyclic group ℤ_{N}. In particular ind_{B}^{G}σ decomposes into N inequivalent subrepresentations with multiplicity one. A simple verification shows that this condition is equivalent to the statement in c) above. This result motivates the construction given in the next section.

4. Construction of types

We consider a character \overline{χ} of F^{×} (the multiplicative group of F) whose restriction χ on U_{F} is nontrivial and satisfies χ^{N} = 1. Now we have:

Lemma 2. a) If p ≠ N then the powers χ^{j}, j ∈ {1, ⋯, N − 1}, all have conductor P_{F} (that is, they are trivial on U_{F}^{(1)} = 1 + P_{F} but not on U_{F}).

b) If p = N then the χ^{j} all have the same conductor P_{F}^{n+1} for some integer n ≥ 1.

Proof. a) If χ does not have conductor P_{F} then it has conductor P_{F}^{n+1} with n ≥ 1, thus χ may be identified with a character of the group

U_{F}^{(n)}/U_{F}^{(n+1)} ≅ k_{F}^{+}

which has order q = p^{r}. Hence N = p.

b) If N = p then χ must have conductor P_{F}^{n+1}, n ≥ 1; otherwise χ would identify with a character of the group

U_{F} /U_{F}^{(1)} ≅ k_{F}^{×}

which has order q − 1 = p^{r} − 1. Since p does not divide p^{r} − 1, this is impossible.

We now appeal to the self-duality of F^{+}. Composing the isomorphism (1) with the following isomorphism

U_{F}^{(n)}/U_{F}^{(n+1)} ≃ P_{F}^{n}/P_{F}^{n+1}
1 + y → y
we see that as a character of U_{F}^{(n)}/U_{F}^{(n+1)}, χ is given by
χ(1 + y) = ψ_{a}(y) where a ∈ P_{F}^{−n}/P_{F}^{1−n}.

It follows that

χ^{j} = ψ_{ja}.

Now each j ∈ {1, ⋯, N − 1} is a unit in O_{F} (since p = N); hence the ja all have the same valuation and the result follows.

Our goal is to construct the type corresponding to the pair (D, σ) where σ is a character of D determined by the χ^{j}, j ∈ {1, ⋯, N − 1}; σ is defined by:

σ ( \begin{matrix}d_{1} \\ & . & & & 0 \\ & & . \\ & & & . \\ & 0 & & & d_{N−1} \\ & & & & & ∏_{i=1}^{N−1} d_{i}^{−1} \end{matrix} ) = ∏_{j=1}^{N−1} χ^{j}(d_{j}).

The construction which follows goes by cases. Case N ≠ p:

Here the χ^{j}, j ∈ {1, ⋯, N − 1} have conductor P_{F}. We get a well defined character λ of the Iwahori subgroup as follows:

J = ( \begin{matrix}U_{F} \\ & . & & O_{F} \\ & & . \\ & P_{F} & & . \\ & & & & U_{F} \end{matrix} ) ∩ SL_{N}(F)

(O_{F} , U_{F} , P_{F} indicate where the entries belong)

λ ( \begin{matrix}u_{1} \\ & . & & ⋆ \\ & & . \\ & ⋆ & & . \\ & & & & u_{N} \end{matrix} ) = ∏_{j=1}^{N−1} χ^{j}(u_{j}),

(by ⋆ we mean some given entries which we do not need to show). We now state our main result:

Theorem 1. An irreducible smooth representation (π, V ) of G contains λ upon restriction to J if and only if there are unramified characters η_{j}, j ∈ {1, ⋯, N− 1} such that π embeds into Ind_{B}^{G}(⨂_{j=1}^{N−1} χ̄^{j}η_{j}) where ⨂_{j=1}^{N−1} χ̄^{j}η_{j} is a character of D defined by:

⨂_{j=1}^{N−1} χ̄^{j}η_{j} ( \begin{matrix}d_{1} \\ & . & & & 0 \\ & & . \\ & & & . \\ & 0 & & & d_{N−1} \\ & & & & & ∏_{i=1}^{N−1} d_{i}^{−1} \end{matrix} ) = ∏_{j=1}^{N−1} χ̄^{j}η_{j}(d_{j}).

Case N = p:

Here the χ^{j} have conductor P_{F}^{n+1}, n ≥ 1. We define a compact open subgroup J as follows: if n = 2m then we let

J = ( \begin{matrix}U_{F} \\ & . & & P_{F}^{m} \\ & & . \\ & P_{F}^{m+1} & & . \\ & & & & U_{F} \end{matrix} ) ∩ SL_{N}(F),

but if n = 2m − 1 then we let

J = ( \begin{matrix}U_{F} \\ & . & & P_{F}^{m} \\ & & . \\ & P_{F}^{m} & & . \\ & & & & U_{F} \end{matrix} ) ∩ SL_{N}(F) .

Again by setting

λ ( \begin{matrix}u_{1} \\ & . & & ⋆ \\ & & . \\ & ⋆ & & . \\ & & & & u_{N} \end{matrix} ) = ∏_{j=1}^{N−1} χ^{j}(u_{j}),

we get a well defined character λ of J. Again we need to prove Theorem 1 for this choice of J.

5. Support of the Hecke algebra of a type

Recall that given a pair (J, λ) where J is a compact open subgroup of G and λ is a character of J, the convolution Hecke algebra H(G∥J, λ) consists of all compactly supported functions f : G → ℂ such that

f(hgk) = λ^{−1}(h)f(g)λ^{−1}(k), g ∈ G, h, k ∈ J.

Note that this definition still makes sense when J is not open in G, but in that case many of the nice properties of Hecke algebras will no longer hold.

It is convenient to (and we will always) normalize Haar measure on G so that vol(J) = 1, when J is fixed.

Recall that the support of H(G∥J, λ) is the set

{g ∈ G | JgJ supports a nonzero function in H(G∥J, λ)}.

Since J is open it follows that there are only finitely many double cosets which support a given function in H(G∥J, λ). We will often denote f_{g} the unique function in H(G∥J, λ) (if any) such that f_{g}(g) = 1. An easy verification shows that the support of H(G∥J, λ) is the same as the G-intertwining of λ which is the set denoted I_{G}(λ) defined by

I_{G}(λ) = {g ∈ G | λ^{g} = λ, when restricted to J ∩ J^{g}}.

(Recall our notations J^{g} = gJg^{−1} and λ^{g}(x) = λ(g^{−1}xg) ).

It is also clear, that

g ∈ I_{G}(λ) if and only if JgJ ⊆ I_{G}(λ).

We now recall some standard facts on the structure of G which we shall need. If J is the Iwahori subgroup then we have the Bruhat decomposition:

G = JW J = ∐_{w∈W} JwJ (disjoint union),  (7)

where W is the affine Weyl group of G. Recall that if M is the normalizer of D in G and M_{0} = M ∩ SL_{N}(O_{F}) then W = M/M_{0} and the Bruhat decomposition amounts to:

G = JMJ,

and if w_{1}, w_{2} ∈ M then

Jw_{1}J = Jw_{2}J if and only if w_{1} = w_{2} (mod M_{0}).

We take for generators of W the following (standard) representatives in M:

s_{1} = ( \begin{matrix}0 & 1 \\ −1 & 0 \\ & & I_{N−2} \end{matrix} ) , s_{2} = ( \begin{matrix}1 \\ & 0 & 1 \\ & −1 & 0 \\ & & & I_{N−3} \end{matrix} ) , ⋯

s_{N−1} = ( \begin{matrix}I_{N−2} \\ & 0 & 1 \\ & −1 & 0 \end{matrix} ) and s_{N} = ( \begin{matrix}& & ϖ^{−1} \\ & I_{N−2} \\ −ϖ \end{matrix} ) .

There is defined on W a ‘length’ function l with respect to J. The length of w ∈ W is the integer l(w) such that:

[J : J ∩ w^{−1}Jw] = q^{l(w)}.

In case J is the Iwahori subgroup (i.e N ≠ p) l(w) is equal to the minimum number of the s_{i} needed to express w. We have l(s_{i}) = 1 where i ∈ {1, ..., N}. We also note that

[J : J ∩ w^{−1}Jw] = vol(JwJ)/vol(J).

We list some standard properties:

Jw_{1}w_{2}J = Jw_{1}Jw_{2}J if l(w_{1}w_{2}) = l(w_{1}) + l(w_{2})
Js_{i}Js_{i}J = J ∪ Js_{i}J, where i ∈ {1, ⋯, N}.

We will denote [d_{1}, ⋯, d_{N}] the diagonal matrix

( \begin{matrix}d_{1} \\ & . & & & 0 \\ & & . \\ & & & . \\ & 0 \\ & & & & & d_{N} \end{matrix} )

We introduce the Weyl element w_{0} = s_{N−1}s_{N−2} ⋯ s_{2}s_{1}. We have

w_{0} = ( \begin{matrix}0 & 1 \\ & 0 & 1 & & 0 \\ & & . & . \\ & 0 & & . & . \\ & & & & 0 & 1 \\ (−1)^{N+1} & & & & & 0 \end{matrix} )

= (−1)^{N+1}e_{N1} + ∑_{i=1}^{N−1} e_{i,i+1}

where e_{ij} denotes the usual (δ_{ij}) matrix (δ_{ij} is the kronecker symbol). W acts by conjugation on the diagonal entries of elements in M_{N}(F), as the symmetric group S_{N}. For this action the diagonal matrices in W act trivially and w_{0} generates a subgroup of S_{N} consisiting of of cyclic permutations. We have w_{0}^{N} = (−1)^{N+1}I_{N}, where I_{N} denotes the N by N identity matrix. We also see by direct calculation that

w_{0}[d_{1}, ⋯, d_{N}]w_{0}^{−1} = [d_{2}, d_{3}, ⋯, d_{N}, d_{1}];

the action of w_{0}^{k}, k ∈ ℕ is clear from this. We form the semi-direct product:

W_{0} = ⟨w_{0}⟩ ⋉ D

We state:

Proposition 2. I_{G}(λ) = JW_{0}J, where in case p ≠ N, J is the Iwahori subgroup and in case p = N, J is the compact open subgroup defined in Section 4.

Proof. The cases p ≠ N and p = N require different techniques.

• Case p ≠ N: We first show that JW_{0}J ⊆ I_{G}(λ) or equivalently W_{0} ⊆ I_{G}(λ). It is enough to show that ⟨w_{0}⟩ ⊆ I_{G}(λ) (since the conjugation action of the diagonal matrices is trivial on diagonal entries). We show that w_{0} ∈ I_{G}(λ). We first observe the following simple fact. For u = (u_{ij}) in J, det(u)= 1 so we have:

∏_{j=1}^{N} u_{ii} = 1 − ∑_{s∈S_{N} −{1}} sgn(s)u_{1s(1)} ⋯ u_{Ns(N)},

where sgn(s) denotes the sign of the permutation s in the symmetric group S_{N}. We clearly have

∑_{s∈S_{N} −{1}} sgn(s)u_{1s(1)} ⋯ u_{Ns(N)} ∈ P_{F} .

This implies that χ(u_{s(N)}) = χ^{−1}(u_{s(1)} ⋯ u_{s(N−1)}) for any s ∈ S_{N}. Now we have

λ^{w_{0}} ( \begin{matrix}u_{1} \\ & . & & ⋆ \\ & & . \\ & ⋆ & & . \\ & & & & u_{N} \end{matrix} ) =  λ(w_{0}^{−1}[u_{1}, ⋯, u_{N}]w_{0}),

= χ(u_{N})χ^{2}(u_{1}) ⋯ χ^{N−1}(u_{N−2})
= χ(u_{1}) ⋯ χ^{N−2}(u_{N−2})χ^{−1}(u_{N}).

But χ^{−1} = χ^{N−1}, so that

λ^{w_{0}} = λ when restricted to J ∩ J^{w_{0}}.

We show in the same way that:

∏_{j=1}^{N−1} χ^{j}(u_{w_{0}^{k+1}(j)}) = ∏_{j=1}^{N−1} χ^{j}(u_{w_{0}^{k}(j)}), k ∈ ℤ,

where we abuse notation by letting w_{0}^{j} , j ∈ ℤ denote the permutation in S_{N} which implements the conjugation action of w_{0}^{j} on diagonal entries. So we have shown that JW_{0}J ⊆ I_{G}(λ).

To prove the converse we use again the fact that for w ∈ W, and any d ∈ D, wd ∈ I_{G}(λ) if and only if w ∈ I_{G}(λ). Now appealing to the Bruhat decomposition (7) we see that it is enough to show that if w is not in W_{0} then w is not in I_{G}(λ). Now let w ∈ W − W_{0} then (up to multiplication by an appropriate power of w_{0}) we may assume that w conjugates [u_{1}, u_{2}, ⋯, u_{N}] into [u′_{1}, u′_{2}, ⋯, u′_{N}], where u′_{1} = u_{1} and u′_{k} = u_{2} for some k ∈ {1, 3, ⋯, N} (the point here is that k ≠ 2).

Now we set u_{2} = u_{1}^{−1} and u_{j} = 1, for j ∈ {3, ⋯, N}. An easy calculation shows that λ^{w} and λ do not agree on [u_{1}, u_{1}^{−1}, 1, ⋯, 1]. This completes the proof in case p ≠ N.

• Case p = N: We see as above that JW_{0}J ⊆ I_{G}(λ). To prove the converse we can no longer invoke the Bruhat decomposition. We shall rely instead, on orbit theory in M_{N}(F) as developed by R. Howe (and exploited by Howe-Moy, Bushnell-Kutzko and others).

We consider the following standard lattice flags L_{0} = {L_{j}^{0}}, and L_{1} = {L_{j}^{1}} of period respectively 1 and N defined as follows, let e_{j} denote the standard basis of F^{N} then set:

L_{j}^{0} = ⨁_{i=1}^{N} P_{F}^{j} e_{i} , j ∈ ℤ.
L_{0}^{1} = O_{F} e_{1} ⊕ O_{F} e_{2} ⊕ ... ⊕ O_{F} e_{N−1} ⊕ P_{F} e_{N},
L_{1}^{1} = O_{F} e_{1} ⊕ ... ⊕ O_{F} e_{N−2} ⊕ P_{F} e_{N−1} ⊕ P_{F} e_{N},
⋮
L_{N−1}^{1} = O_{F} e_{1} ⊕ P_{F} e_{2} ⊕ ... ⊕ P_{F} e_{N−1} ⊕ P_{F} e_{N}

We will denote A_{0}^{j}, A_{1}^{j}, K_{0}^{(j)} and K_{1}^{(j)}, the corresponding filtrations of (compact open) ideals and subgroups, as in section 2.

In order to state a technical result on the geometry of conjugacy classes in A, we consider cases:

• Subcase n is odd: We let n = 2m − 1 and we set:

K_{0} = K_{0}^{(m)} ∩ G,

and

H_{0} = K_{0}^{(n+1)} ∩ G.

Clearly λ determines a character \overline{λ} of K_{0}/H_{0}. It follows that \overline{λ} = ψ_{a_{0}} , where

a_{0} ∈ A_{0}^{−n}/ (A_{0}^{1−m} + F ∩ A_{0}^{−n}) .

• Subcase n is even: Here we set n = 2m and we let

K_{1} = K_{1}^{(Nm+1)} ∩ G

and

H_{1} = K_{1}^{(Nn+1)} ∩ G.

Again we see that λ determines a character \overline{λ} = ψ_{a_{1}} of K_{1}/H_{1} where now

a_{1} ∈ A_{1}^{−Nn}/ (A_{1}^{−Nm} + F ∩ A_{1}^{−Nn}) .

In any case, we are in the ‘separated’ situation of Howe-Moy (see [15]). A direct adaptation of the proof of [15, Lemma 3.2], now yields:

Lemma 3. i) For any b ∈ A_{0}^{1−m}, K_{0} conjugates a_{0} + b to a diagonal matrix. ii) For any b ∈ A_{1}^{−Nm}, K_{1}^{(Nm)} ∩ G conjugates a_{1} + b to a diagonal matrix.

Remark 1. i) We note that K_{0} and K_{1}^{(Nm)} ∩ G are subgroups of J. ii) Observe that a_{0} and a_{1} have the form: [α, 2α, …, (N −1)α, 0], where v_{F} (α) = −n.

We now proceed with the proof of Proposition 2, as follows: We assume n is odd (the proof goes the same - with easy modifications - in case n is even).

It is enough to prove that I_{G}(ψ_{a_{0}}) ⊆ JW_{0}J. Calculations which are now standard (thanks to R. Howe [13]) show that

I_{G}(ψ_{a_{0}}) = {x ∈ G | x(a_{0} + A_{0}^{1−m} + F ∩ A_{0}^{−n})x^{−1} ∩ (a_{0} + A_{0}^{1−m}) ≠ Ø}. (8)

If x ∈ I_{G}(ψ_{a_{0}}) then there exists y, y′ ∈ A_{0}^{1−m}, and c ∈ F ∩ A_{0}^{−n} such that

x(a + y)x^{−1} = a + c + y′.  (9)
By Lemma 3, there exists z_{1}, z_{2} ∈ K_{0}, such that
z_{1}(a + y)z_{1}^{−1} = d and z_{2}(a + y′)z_{2}^{−1} = d′,

for some diagonal matrices d and d′. Noting also that K_{0} normalizes A_{0}^{m} and A_{0}^{1−m}, and K_{0} conjugates a_{0} to itself (mod A_{0}^{1−m}), it follows that d and d′ are congruent to a_{0} (mod A_{0}^{1−m}). In particular the entries of d (resp. d′) are all distinct. Set x_{0} = z_{2}xz_{1}^{−1}; then

x_{0}dx_{0}^{−1} = d′ + c.

This shows that there exists w ∈ W such that

d′ + c = wdw^{−1}.

In particular

c ∈ {α, 2α, …, (N − 1)α, 0} (mod A_{0}^{1−m}) (α is taken as in Remark 1).

(We trust the reader will distinguish between our use of c either as a scalar matrix or an element of the field). We also derive from this that

x_{0}^{−1}wdw^{−1}x_{0} = d,

hence w^{−1}x_{0} must be diagonal.

We claim that in fact w ∈ W_{0}. Indeed, if w is not in W_{0}, then (as noted before) we may assume that w^{−1} conjugates d = [d_{11}, …, d_{NN}] to [d′_{11}, …, d′_{NN}], where d′_{11} = d_{11} but d′_{1k} = d_{12} for some k ≠ 2.

Now writing d′ = [d_{21}, …, d_{2N}], we see that on one hand d_{21} + c = d_{11} and hence c ∈ P_{F}^{1−n}.

On the other hand we see that

c = d_{1k} − d_{12} ≡ (k − 2)α (mod P_{F}^{1−n}).

This is a contradiction. The proof is now complete in case p = N also.

6. Types for GL_{N}(F)

Implicit in our approach is the fact that our types may be constructed by restriction from similar types in G̃ = GL_{N}(F). We now discuss this explicitly and will later exploit the structure of two related GL_{N}(F)-algebras. In case N ≠ p we let J̃ denote the Iwahori subgroup of G̃. But in case N = p we define it to be the following compact open subgroup:

J̃ = ( \begin{matrix}U_{F} \\ & . & & P_{F}^{m} \\ & & . \\ & P_{F}^{m+1} & & . \\ & & & & U_{F} \end{matrix} ) if n = 2m;

J̃ = ( \begin{matrix}U_{F} \\ & . & & P_{F}^{m} \\ & & . \\ & P_{F}^{m} & & . \\ & & & & U_{F} \end{matrix} ) if n = 2m − 1.

(Recall that n + 1 is the conductoral exponent of our fixed character χ̄ of F^{×}). Clearly J = J̃∩G. We define a character λ̃ of J̃ in a manner completely analogous to the definition of λ (see section 4). It is then clear that λ is just the restriction of λ̃ to J. We review some basic facts about the affine Weyl group and the Bruhat decomposition for G̃. For this account we refer to the excellent survey [14] and the bibliography there.

The Weyl group W̃ of G̃ is isomorphic to the symmetric group S_{N} and may be realized as the subgroup of permutation matrices. W̃ normalizes D̃. We denote by D̃(ϖ) the subgroup consisting of diagonal matrices of the form

d_{i} = [ϖ^{i_{1}}, …, ϖ^{i_{N}}],

for i = (i_{1}, …, i_{N}) ∈ ℤ^{N}. Clearly if ∑_{k=1}^{N} i_{k} = 0 then d_{i} ∈ G. We will always assume this condition whenever we view d_{i} as an element of G. The affine Weyl group W̃^{a} may be realized as the semi-direct product

W̃^{a} = W̃ ⋉ D̃(ϖ).

In case N ≠ p the group J̃ yields the Bruhat decomposition

G̃ = J̃W̃^{a}J̃.

The length function l on W̃^{a} with respect to J̃ is defined as before by the formula

vol(J̃wJ̃) = q^{l(w)}vol(J̃), for w ∈ W̃^{a}.

Again in case N ≠ p, l coincides with the ‘Coxeter’ length function. In any case a simple calculation gives

l_{i} = l(d_{i}) = ∑_{1≤k<j≤N} |i_{k} − i_{j}|.

We next describe the structure of two algebras which we will need, namely H̃ = H(G̃∥J̃, λ̃) and \overline{H} = H(G̃∥J, λ). We note however that J is not open in G̃ so that \overline{H} is not an Hecke algebra in the usual sense. Next we prove the following

Proposition 3. The support of H(G̃∥J̃, λ̃) is equal to J̃D̃J̃ = J̃D̃(ϖ)J̃.

Proof. The technique of the proof is similar to that of Proposition 2. We will freely use some of the notations there. We indicate here only the changes that apply in this case. Again we consider cases. Case N ≠ p: One shows directly that D̃ (hence J̃D̃J̃) is in the support of H̃. We prove the converse. By the decomposition of W^{a} as a semi-direct product and the Bruhat decomposition we see as before that it suffices to prove that no w ∈ H̃ (except the identity) intertwines λ̃. For any w ∈ H̃ − {1} there exists j ∈ {1, …, N} such that w(j) ≠ j. Write k = w(j); then there exists u ∈ U_{F} such that

χ^{j}(u) ≠ χ^{k}(u).

But this means that λ^{w} and λ do not agree on the diagonal matrix ue_{jj} +∑_{i≠j} e_{ii}. This proves the result in this case. Case N = p: We proceed as in the proof of Proposition 2. The important modification is that by (2) and (8), the scalar matrix c there is now zero. We complete the argument below.

We consider the same lattice flags L_{0} and L_{1} as before. These yield compact open subgroups K_{0}^{(j)} and K_{1}^{(j)}, j ≥ 1. Assuming that the conductoral exponent n+1 of χ is even (argument is the same otherwise), we set n+1 = 2m; then λ determines a character ψ_{a_{0}} of

K_{0}^{(m)}/K_{0}^{(n+1)}.

Now Lemma 3 and Remark 1 apply. As a consequence there exists z_{2} , z_{1} ∈ J̃ and w ∈ W̃ such that

w^{−1}x_{0} is diagonal

where x_{0} = z_{2}xz_{1}. Now the fact that c is zero forces w to be the identity. The proof is now clear.

For i ∈ ℤ^{N} we let f̃_{i} denote the unique function in H̃ that takes value 1 on d_{i}. To state our version of the Hecke algebra isomorphism of Howe-Moy [15] we need another Hecke algebra, namely

H′ = H(D̃∥D̃ ∩ J̃, λ̃|_{D̃∩J̃}).

We let fi^{′} denote the unique function in H′ that takes value 1 on d_{i}. Set J′ = D̃ ∩ J̃. We now have:

Theorem 2. Normalize Haar measures on G̃ and D̃ so that J̃ and J′ have volume 1. The map Φ : H′ → H̃, given by Φ(fi^{′}) = q^{−l_{i}/2}f̃_{i}, is an algebra isomorphism. Proof. See [15, Theorem 2.1]. Note also that under our normalizations of Haar measures we have vol(J′d_{i}J′) = 1 and vol(J̃d_{i}J̃) = q^{l_{i}}.

Corollary 1. H̃ is a commutative algebra and the following relation holds: For i, j ∈ ℤ^{N}, f̃_{i} ∗ f̃_{j} = q^{1/2(l_{i}+l_{j}−l_{i+j})}f̃_{i+j} . Proof. H′ is clearly abelian (since D̃ is). As a consequence we have fi^{′} ∗fj^{′} = fi+j^{′}. Now apply Φ to get the desired relation.

Corollary 2. H̃ is isomorphic to the N -fold tensor product of the ‘one-dimens-ional’ affine Hecke algebras H(F^{×}∥U_{F} , χ^{j}), j ∈ {1, …, N} . Proof. This result holds for H′. Indeed we verify that the isomorphism is explicitly given by the map fi^{′} → ϕ_{i_{1}} ⊗…⊗ϕ_{i_{N}}, where ϕ_{i_{j}} is the unique function in H(F^{×}∥U_{F} , χ^{j}) such that ϕ_{i_{j}}(1) = 1.

In order to determine some useful relations in \overline{H} we now introduce the following Weyl element

t̃ = ( \begin{matrix}0 & 1 & & & 0 \\ & 0 & 1 \\ & & ... & ... \\ 0 & & & 0 & 1 \\ ϖ & & & & 0 \end{matrix} ) .

We let \overline{W} = ⟨t̃⟩ ⋉ D̃. Then a calculation as in Proposition 2 shows that t̃ as well as the diagonals in G̃ intertwine λ so that we obtain following:

Proposition 4. J\overline{W} J is contained in the support of \overline{H}. Let I_{k} denote the k by k identity matrix. Every x ∈ G̃ decomposes uniquely as zg where

z = ( \begin{matrix}det(x) & 0 \\ 0 & I_{N−1} \end{matrix} )

and g ∈ G. If we let Z be the subgroup consisting of diagonal matrices of the form

( \begin{matrix}α & 0 \\ 0 & I_{N−1} \end{matrix} )

where α ∈ F^{×}; then we have the decompositions

G̃ = Z ⋉ G

and

J̃ = (Z ∩ J̃) ⋉ J .

We note that Z ≅ F^{×}. Furthermore the first decomposition allows for a decomposition of a Haar measure dx on G̃ as

dx = dzdg

where dz (resp. dg) is Haar measure on Z (resp. G). The following simple result will be useful:

Lemma 4. a) For any d ∈ D we have J̃dJ̃ ∩ G = JdJ. In particular the restriction of f̃_{d} to G is equal to f_{d}.

b) Normalize Haar measures on G̃ , Z, and G such that J̃, Z ∩ J̃ and J all have volume 1. Then for d, d′ ∈ D, the restriction of f̃_{d} ∗ f̃_{d′} to G is equal to f_{d} ∗ f_{d′} .

Proof. Let g ∈ J̃dJ̃∩ G; then g = xdy for some x, y ∈ J̃ such that det(xy)=1. Now decomposing x = z_{1}g_{1} and y = z_{2}g_{2} as above we see that g = g_{1}dg_{2}. This proves a). To prove b) we compute

(f̃_{d} ∗ f̃_{d′})(g) = ∫G̃ f̃_{d}(gx^{−1})f̃_{d′}(x)dx
= ∫_{Z⋉G} f̃_{d}(gy^{−1}z^{−1})f̃_{d′}(zy)dzdy.

The integrand is nonzero only if gy^{−1}z^{−1} ∈ J̃dJ̃ and zy ∈ J̃d′J̃. In particular det(z) ∈ U_{F} hence z ∈ Z ∩ J̃. This implies that

f̃_{d}(gy^{−1}z^{−1})f̃_{d′}(zy) = f̃_{d}(gy^{−1})λ̃(z^{−1})λ̃(z)f̃_{d′}(y)
= f̃_{d}(gy^{−1})f̃_{d′}(y)
= f_{d}(gy^{−1})f_{d′}(y).

This shows that the integral is equal to (f_{d} ∗ f_{d′})(g), the result is now clear.

We observe that t̃ normalizes J. In particular l(t̃) = 0. Let ē_{λ} denote the identity in \overline{H}. Let f̄_{i} (resp. f̄t̃) denote the unique function in \overline{H} that takes value 1 on d_{i} (resp. t̃).

Remark 2. We note that H(G∥J, λ) embeds into \overline{H}, by extending functions to G̃ as zero off G. We will use the same notation for a function and its extension. Furthermore the restriction of f̃_{d} to G may be regarded as an element of \overline{H} by Lemma 4.

Next we will prove Proposition 5. a) For i, j ∈ ℤ^{N}, f̄_{i} ∗ f̄_{j} = q^{1/2(l_{i}+l_{j}−l_{i+j})}f̄_{i+j}.

b) f̄t̃ is invertible and for any d ∈ D̃, f̄ −1 ; t̃ ∗ f̄_{d} ∗ f̄t̃ = f̄_{t̃^{−1}dt̃}.

c) f̄ k ; t̃ = f̄t̃^{k}.

Proof. The relation in a) follows from Corollary 1, Lemma 4 and Remark 1. We prove the statements in b). We clearly have

Jt̃Jt̃^{−1}J = J.

This implies that

f̄t̃ ∗ f̄t̃^{−1} = β_{1}ē_{λ}.

Evaluating f̄t̃ ∗ f̄t̃^{−1} at 1 we get β_{1} = 1. This means that f̄t̃^{−1} is the inverse of ft̃. For any w ∈ \overline{W} we have

Jt̃^{−1}JwJ = Jt̃^{−1}wJ .

This shows that

f̄t̃^{−1} ∗ f̄_{w} = β_{2}f̄_{t̃^{−1}w}.

Fix Haar measures on G and J normalized by the condition that vol(J)=1. We compute

β_{2} = (f̄t̃^{−1} ∗ f̄_{w})(t̃^{−1}w)
= ∫_{G} f̄t̃^{−1}(t̃^{−1}wg^{−1})f̄_{w}(g)dg
= vol(JwJ) ∫_{J×J} f̄t̃^{−1}(t̃^{−1}wk^{−1}w^{−1}h^{−1})f̄_{w}(hwk)dhdk
= vol(JwJ) ∫_{J} f̄t̃^{−1}(t̃^{−1}wk^{−1}w^{−1})λ(k)dk.

The integrand is nonzero only if k ∈ w^{−1}Jw. Under this condition we have

f̄t̃^{−1}(t̃^{−1}wk^{−1}w^{−1}) = λ(wk^{−1}w^{−1})
= λ(k^{−1}) (since w intertwines λ).

It follows that the integral is equal to vol(JwJ)vol(J ∩ w^{−1}Jw) = 1. This shows the relation

f̄t̃^{−1} ∗ f̄_{w} = f̄_{t̃^{−1}w}.

We prove similarly that

f̄_{w} ∗ f̄t̃ = f̄_{wt̃}.

Combining these two facts we get the desired relation. The proof of c) follows from the simple fact that the k-fold coset Jt̃Jt̃… t̃J is equal to Jt̃^{k}J, and an obvious inductive argument.

This proposition clearly establishes the following

Theorem 3. The subalgebra of \overline{H} generated by the f̄_{i}, (i ∈ ℤ^{N}) and f̄t̃ is isomorphic to the group algebra ℂ[\overline{W}].

7. Structure of the Hecke Algebra

We now take up the task of proving the following:

Theorem 4. As a ℂ-algebra H(G∥J, λ) is isomorphic to the group algebra ℂ[W_{0}]. First we make some remarks. By Proposition 2, we know that the support of H(G∥J, λ) is JW_{0}J. So the functions f_{w}, w ∈ W_{0}, generate H(G∥J, λ) as a ℂ-vector space. We now make use of the results of the previous section. We will freely use the notations there.

To determine relations in H(G∥J, λ) we may - by Remark 1 - compute inside \overline{H}. We may also restrict relations from H̃ by restricting functions to G. The Theorem will follow clearly from

Proposition 6. a) For any i, j ∈ ℤ^{N} we have f_{i} ∗ f_{j} = q^{1/2(l_{i}+l_{j}−l_{i+j})}f_{i+j}.

b) f_{w_{0}} is invertible and f_{w_{0}}^{−1} ∗ f_{d} ∗ f_{w_{0}} = f_{w_{0}^{−1}dw_{0}} , d ∈ D.

c) f_{w_{0}}^{k} = q^{\frac{k(k−1)}{2}} f_{w_{0}}_{k}, where k ∈ ℕ. In particular f_{w_{0}}^{N} = q^{\frac{N(N−1)}{2}} χ((−1)^{N+1})e_{λ}. Proof. The relation in a) follows from Corollary 1 and Lemma 4. To prove the relations in b) and c) we consider the following diagonal matrix

γ_{k} = ( \begin{matrix}(−1)^{N+1}ϖ^{−1}I_{k} & 0 \\ 0 & I_{N−k} \end{matrix} ) , k ∈ ℤ.

A calculation shows that

w_{0}^{k} = t̃^{k}γ_{k}.

In particular l(w_{0}^{k}) = l(γ_{k}) = k(N − k). We now compute in \overline{H}. Computing directly as in the proof of Proposition 5 we get the following decompositions:

f_{w_{0}} = f̄t̃ ∗ f̄_{γ_{1}} and f_{w_{0}^{−1}} = f̄_{γ_{1}^{−1}} ∗ f̄ −1 ; t̃.

Hence

f_{w_{0}^{−1}} ∗ f_{w_{0}} = f̄_{γ_{1}^{−1}} ∗ f̄_{γ_{1}}
= q^{l(γ_{1})}ē_{λ} (by Proposition 5 a))
= q^{N−1}ē_{λ}.

This shows in particular that f_{w_{0}} is invertible. We make use of Proposition 5 and the simple fact that the conjugation action of t̃ does not change the length of a Weyl element; we compute:

f_{w_{0}^{−1}} ∗ f_{i} ∗ f_{w_{0}} = f̄_{γ_{1}^{−1}} ∗ f̄ −1 ; t̃ ∗ f_{i} ∗ f̄t̃ ∗ f̄_{γ_{1}}
= f̄_{γ_{1}^{−1}} ∗ f̄_{t̃^{−1}dt̃} ∗ f̄_{γ_{1}}
= q^{1/2(l_{i}+l(γ_{1})−l(t̃^{−1}d_{i}t̃γ_{1}))}f̄_{t̃^{−1}d_{i}t̃γ_{1}}
= q^{1/2(l_{i}+l(γ_{1})−l(t̃^{−1}d_{i}t̃γ_{1}))}q^{1/2(l(γ_{1})+l(t̃^{−1}d_{i}t̃γ_{1})−l_{i})}f̄_{γ_{1}^{−1}t̃^{−1}d_{i}t̃γ_{1}}
= q^{N−1}f_{w_{0}^{−1}d_{i}w_{0}}.

We have thus shown the required relation in b), recalling also from above that

f_{w_{0}^{−1}} = q^{N−1}f_{w_{0}}^{−1}.

To prove the relations in c), we proceed by induction.

Using again the decomposition f_{w_{0}} = f̄t̃ ∗ f̄_{γ_{1}} we see that f_{w_{0}}^{2} = qf_{w_{0}^{2}} . We now proceed with the induction as follows

f_{w_{0}}^{k+1} = f_{w_{0}}^{k} ∗ f_{w_{0}}
= q^{\frac{k(k−1)}{2}} f_{w_{0}}_{k} ∗ f_{w_{0}}
= q^{\frac{k(k−1)}{2}} f̄t̃^{k} ∗ f̄_{γ_{k}} ∗ f̄t̃ ∗ f̄_{γ_{1}}
= q^{\frac{k(k−1)}{2}} f̄t̃^{k+1} ∗ f̄_{t̃^{−1}γ_{k}t̃} ∗ f̄_{γ_{1}}
= q^{\frac{k(k−1)}{2}} q^{\frac{l(γ_{k})+l(γ_{1})−l(w{k+1 \atop 0})}{2}} f_{w_{0}^{k+1}}
= q^{\frac{(k+1)k}{2}} f_{w_{0}}_{k+1}

This proves the first relation in c). The second relation is an obvious consequence of this relation.

8. Classification

Our goal here is to prove Theorem 1 (see section 4). We let F: V → V_{U} denote the canonical Jacquet map with respect to the Borel subgroup B = DU. For simplicity of notations we will write v̄ = F(v) where v ∈ V . We note that J admits an Iwahori decomposition with respect to B:

J = (J ∩ U)(J ∩ D)(J ∩ U^{−}),

a product which can be taken in any order. We let λ_{D} denote the restriction of λ on J ∩ D and λ_{DU^{−}} the restriction of λ to (J ∩ D)(J ∩ U^{−}). We then prove the following result (known as Casselman’s lemma):

Theorem 5. The canonical map F: V → V_{U} determines an isomorphism (of ℂ-vector spaces)

V^{λ} ≅ (V_{U})^{λ_{D}}.

To prove this theorem we need the following results:

Proposition 7. (Casselman [9]) Let G be a reductive connected p-adic group. If P = LU is a proper parabolic subgroup and if U_{1}, U_{2} are two compact open subgroups of U then there exists a ∈ T, where T is the maximal split torus in the center of L, such that a^{−1}U_{2}a ⊆ U_{1}.

Remark 3. For G = SL_{N}(F) we let P = B = DU then T = L = D ≅ (F^{×})^{N−1}.

Lemma 5. For any smooth representation (π, V) of G and any d ∈ D the operator π(e_{λ}) ∘ π(d) ∘ π(e_{λ}) is an automorphism of V^{λ}.

Lemma 6. V^{λ} and V^{λ_{DU^{−}}} have the same image in V_{U}. We will give the proofs of these Lemmas later; now we use them to prove Theorem 5. Proof of Injectivity: The canonical map F: V → V_{U} yields by restriction the map V^{λ} → V_{U} whose kernel is V^{λ} ∩ V (U). By way of contradiction, let v ∈ V^{λ} ∩ V (U), v ≠ 0; there exists a compact open subgroup U_{1} such that

∫_{U_{1}} π(u)vdu = 0.

Proposition 7 implies that there exists d ∈ D such that d^{−1}U_{1}d ⊆ J ∩ U. Now Lemma 5 says that: (π(e_{λ}) ∘ π(d) ∘ π(e_{λ}))(v) = π(e_{λ})π(d)v is a non-zero vector in V^{λ}. Observe that for any nonzero vector w in V^{λ} we have:

∫_{J∩U} π(u)wdu ≠ 0,

since λ is trivial on J ∩ U; but we have that

∫_{J∩U} π(u) ∘ π(e_{λ}) ∘ π(d)vdu = π(e_{λ}) ∫_{J∩U} π(u)π(d)vdu
= π(e_{λ}) ∘ π(d) ∫_{J∩U} π(d^{−1}ud)vdu
= π(e_{λ}) ∘ π(d) ∫_{d(J∩U)d^{−1}} π(u)vdu
= 0 (since U_{1} ⊆ d(J ∩ U)d^{−1}, see (1.6)).

This is a contradiction. Proof of Surjectivity: We note that (by Theorem C) (π_{U}, V_{U}) is admissible since (π, V) is; hence (V_{U})^{λ_{D}} is finite dimensional. We also note the following facts:

• (f1) There exists a finite dimensional subspace E ⊆ V^{λ_{D}} mapping onto (V_{U})^{λ_{D}} under the canonical map F: V → V_{U}.

• (f2) There exists a compact open subgroup U_{0}^{−} ⊆ U^{−} that fixes E.

• (f3) For any d ∈ D, π_{U}(d) is an automorphism of (V_{U})^{λ_{D}}.

Proof of (f1): Define an operator P on V by:

P (v) = ∫_{J∩D}(π ⊗ λ_{D}^{−1})(x)vdx

where dx is normalized by the condition vol(J ∩D) = 1. Choose a basis f̄_{1}, …, f̄_{r} of (V_{U})^{λ_{D}}; then set

e_{i} = P (f_{i}) ,

where f_{i} is any representative in f̄_{i}. We show that ē_{i} = f̄_{i}. If K is a compact open subgroup of G that fixes f_{i} then:

e_{i} = vol(J ∩ D ∩ K) × ∑_{x∈J∩D/J∩D∩K}(π ⊗ λ_{D}^{−1})(x)f_{i},

therefore

\overline{e}_{i} = vol(J ∩ D ∩ K) × ∑_{x∈J∩D/J∩D∩K} λ_{D}^{−1}(x)π_{U}(x)f̄_{i} = f̄_{i}.

Also a direct computation shows that P (v) ∈ V^{λ_{D}} for any v ∈ V , in particular e_{i} ∈ V^{λ_{D}}. Now we define E = span{e_{1}, …, e_{r}}. □

Proof of (f2): Let E = span{e_{1}, …, e_{r}} as above, let U_{i}^{−} ⊆ U^{−} be a compact open subgroup that fixes e_{i}, now set U_{0}^{−} = ⋂_{i=1}^{r} U_{i}. □

Proof of (f3): It is enough to show that for all d ∈ D, π_{U}(d)(V_{U})^{λ_{D}} ⊆ (V_{U})^{λ_{D}}. Take \overline{v} ∈ (V_{U})^{λ_{D}}, d′ ∈ D then:

π_{U}(d′)π_{U}(d)\overline{v} = π_{U}(d)π_{U}(d′)\overline{v}
= π_{U}(d)λ_{D}(d′)\overline{v}
= λ_{D}(d′)π_{U}(d)\overline{v}.

We now proceed with the proof of surjectivity. We choose d ∈ D such that

d^{−1}(J ∩ U)d ⊆ U_{0}^{−}

then we claim that π(d)E ⊆ V^{λ_{DU^{−}}}. To see this let v ∈ E, d′ ∈ J ∩D, u ∈ J ∩U^{−}; then

π(d′u)π(d)v = π(d′)π(d)π(d^{−1}ud)v
= π(d′)π(d)v
= π(d)π(d′)v
= π(d)λ(d′)v
= λ(d′)π(d)v.

We have

π_{U}(d)(V_{U})^{λ_{D}} = F(π(d)E) ⊆ F(V^{λ_{DU^{−}}}) = F(V^{λ})

(the last equality follows from Lemma 6). We also have

(V_{U})^{λ_{D}} = π_{U}(d)(V_{U})^{λ_{D}} ⊆ F(V^{λ})

(the first equality follows from (f3)). This proves surjectivity. □

Proof of Lemma 5: We define, for φ ∈ H(G, λ) and x ∈ G, ^{x}φ by

xφ(y) = φ(x^{−1}y).

A direct computation shows that

de_{λ} = δ_{d} ∗ e_{λ},

where δ_{d} is the Dirac function at d. We also have:

π(d) ∘ π(e_{λ}) = π(^{d}e_{λ}),

hence

π(e_{λ}) ∘ π(d) ∘ π(e_{λ}) = π(e_{λ} ∗^{d}e_{λ}).

It is then enough to show that e_{λ} ∗ ^{d}e_{λ} is an invertible element of the algebra e_{λ} ∗ H(G) ∗ e_{λ}. Clearly

e_{λ} ∗^{d}e_{λ} = e_{λ} ∗ δ_{d} ∗ e_{λ}

thus we see that e_{λ} ∗^{d}e_{λ} has support JdJ. We have JdJ = Jd_{i}J for some i ∈ ℤ^{N}. It follows that e_{λ} ∗^{d}e_{λ} is a nonzero multiple of f_{i}, hence (by Proposition 4) it is invertible as required. □

Proof of Lemma 6: It is enough to show that F(V^{λ_{DU^{−}}}) ⊆ F(V^{λ}) (since V^{λ} ⊆ V^{λ_{DU^{−}}}). We need to show that given v ∈ V^{λ_{DU^{−}}} there exists v′ ∈ V^{λ} such that v′ − v ∈ V (U). We let dx be the Haar measure on J normalized by the condition vol(J) = 1. We define

v′ = ∫_{J}(π ⊗ λ^{−1})(x)vdx

then v′ ∈ V^{λ}. A direct calculation and the fact that v ∈ V^{λ_{DU^{−}}} show that v′ − v ∈ V (U). □

Proof of Theorem 1: For convenience we write σ = ⨂_{j=1}^{N−1} χ^{j}η_{j}, and π = Ind_{B}^{G}σ. The restriction of π to J decomposes (by Mackey’s Theorem) as

⨁_{x∈(J|G|B)} c-Ind_{J∩B^{x}}^{J}σ^{x},

where (J|G|B) denotes the set of all J − B double cosets in G. The restrictions of λ and σ to J ∩ B coincide, therefore by Frobenius reciprocity (4) c-Ind_{J∩B}^{J} σ contains λ, hence π contains λ.

We prove the converse. By assumption we have V^{λ} ≠ 0, Theorem 5 then implies that (V_{U})^{λ_{D}} ≠ 0. But (V_{U})^{λ_{D}} is finite dimensional (by Theorem C, see section 2). Thus

V_{U} = (V_{U})^{λ_{D}} ⊕ Ṽ

as J ∩ D-spaces but also as D-spaces, where Ṽ is some subspace in V_{U}, since D centralises J ∩D. D is abelian hence there exists a one dimensional D-quotient in (V_{U})^{λ_{D}}. It follows by Frobenius reciprocity (6) that π embeds into Ind_{B}^{G}γ where γ is a character of D. We consider the restriction of Ind_{B}^{G}γ to J. It follows (by Mackey’s theorem and Frobenius reciprocity again) that for some x ∈ (J|G|B), the restrictions of λ and γ^{x} on J ∩ D^{x} coincide. We deduce from this that γ is of the form ⨂_{j=1}^{N−1} χ^{j}η_{j}, where the η_{j} are unramified characters of F^{×}. This concludes the proof of Theorem 1. □

9. Concluding remarks

The analogue of Theorem 1 for G̃ is stated below. We denote ⊗_{j=1}^{N}χ^{j}η_{j} the character of D̃ defined by

⊗_{j=1}^{N}χ^{j}η_{j}([d_{1}, …, d_{N}]) = ∏_{j=1}^{N} χ^{j}η_{j}(d_{j}).

We then introduce the (Bernstein-Zelevinsky) notation

×_{j=1}^{N}χ^{j}η_{j}

for the induced representation

IndG̃ ; B̃(⊗_{j=1}^{N}χ^{j}η_{j}) .

The Hecke algebras corresponding to the GL_{N}(F)-types introduced in section 6, are described by Corollary 2. The same methods used here to prove Theorem 1 yield now the following

Theorem 6. An irreducible smooth representation (π, V ) of G̃ contains λ̃ upon restriction to J̃ if and only if there are unramified characters η_{j}, j ∈ {1, ⋯, N} such that π is equivalent to ×_{j=1}^{N}χ^{j}η_{j}.

Observe that by results of Bernstein and Zelevinsky [3], ×_{j=1}^{N}χ^{j}η_{j} is irreducible.

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Rutgers State University New Brunswick, NJ 08854 USA

Received March 28, 1995