Semisimple strata for p-adic classical groups

Shaun Stevens^{∗}

July 1, 2000

Abstract

Let F_{0} be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F_{0}. In this paper, we prove some fundamental results towards the classification of the representations of G via types ([8]). In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has been calculated in [19].

Résumé

Soit F_{0} un corps local non-archimédien, de caractéristique résiduelle différente de 2, et soit G un groupe unitaire, symplectique ou orthogonal défini sur F_{0}. Dans cet article, nous démontrons des résultats fondamentals pour la classification des représentations de G par les types ([8]). En particulier, nous démontrons que toute représentation supercuspidale de G de niveau strictement positif contienne une strate gauche semisimple, c’est-à-dire, un caractère particulier d’un certain sous-groupe ouvert compact de G. L’entrelacement d’une telle strate a été calculé dans [19].

1 Introduction

Let F be a non-archimedean local field of residual characteristic different from 2, equipped with a galois involution with fixed field F_{0} (here, we allow the possibility F_{0} = F). Let V be an N-dimensional vector space over F and let h be a nondegenerate ϵ-hermitian form on V. We put A = End_{F} V and let be the adjoint involution on A induced by h. Put G̃ = Aut_{F} V and let σ be the involution of G̃ given by g ↦ \overline{g}^{−1}, for g ∈ G̃; σ also acts on the Lie algebra A via the differential, x ↦ −\overline{x}. Finally, we put G = G̃^{σ}, the fixed points of σ in G̃, a unitary group defined over F_{0} (possibly symplectic or orthogonal) and A_{−} = A^{σ}.

∗The research for this paper was partially funded by the EU network TMR “Arithmetic Algebraic Geometry” and by the Sonderforschungsbereich 478 “Geometrische Strukturen in der Mathematik”, Münster.

We are seeking a classification of the representations of G via the theory of types ([8]). Let π be an irreducible smooth complex representation of G. The representations of level zero of any connected reductive group have been classified by Morris [15] and Moy & Prasad [17] so we will only consider positive level representations here.

A basic result of Moy & Prasad [16] states that π contains an unrefined minimal K-type, that is, a certain character of a compact open subgroup of G. In this paper we both refine and make explicit these constructions.

Let Λ be a self-dual lattice sequence in V (see §2.1). Associated to Λ, we have a parahoric subgroup P, equipped with a filtration by normal open subgroups P^{n+1}, n ≥ 0. The characters of P^{n} trivial on P^{n+1} are parametrized by skew strata [Λ, n, n−1, b], for certain b ∈ A_{−}, and associated to each skew stratum is a characteristic polynomial φ(X). We call the stratum fundamental if φ(X) ≠ X^{N}.

In [18], Pan & Yu show that an unrefined minimal K-type is precisely a fundamental skew stratum (cf. also Morris [13]). In fact we can also deduce that π contains some fundamental skew stratum from our results here , using the notion of “optimal points” from [16].

We prove the following two results:

1. If π contains a skew stratum whose characteristic polynomial has a factor which is not fixed (upto sign) by σ (we call such a stratum G-split) then π is not supercuspidal (cf. [11], [7], [3], [12]).

2. Otherwise π contains a “refined” fundamental stratum, called a semisimple skew stratum.

The notion of semisimple here was proposed by the author in [19], [20] and generalizes that of a simple stratum (for G̃) from [6]; it is an orthogonal direct sum of simple or null skew strata which have coprime characteristic polynomials.

In particular, these results imply that any positive level irreducible supercuspidal representation of G contains a semisimple skew stratum. Moreover the intertwining of such a stratum is computable and conforms to the general philosophy of reducing to a smaller reductive group (see [19]). It makes sense, therefore, to think of this as the first step of an iterative process leading, eventually, to a full classification of the irreducible representations of classical groups. Furthermore, all of this strictly parallels the constructions for G̃ in [6], both formally and by explicit transfer.

The first result is proved using the method of covers ([8]), following the techniques of [7] §3 (see also [3] §2). In particular, the idea of using covers here is due to Bushnell. The spirit of the proof is also the same as the very general result of [12]; however, the language used there is very different and a comparison of the notions of “split” has not been done.

Part of this work formed a section of my doctoral thesis, although the proofs have changed considerably since then. I would like to thank my supervisor, Colin Bushnell, for setting me on this project and for his support and encouragement. Thanks also to Gopal Prasad for some very useful discussions. Particular thanks are due to Paul Broussous, for many explanations and conversations; indeed, this paper owes a great debt to [3].

2 Preliminaries and statement of results

2.1 Lattice sequences

Let F be a non-archimedean local field equipped with a galois involution with fixed field F_{0}; we allow the possibility F = F_{0}. Let 𝔬_{F} be the ring of integers of F, 𝔭_{F} its maximal ideal and k_{F} = 𝔬_{F} /𝔭_{F} the residue field, of characteristic different from 2. We denote 𝔬_{0}, 𝔭_{0}, k_{0} the same objects in F_{0}, and will use similar notation for any non-archimedean local field. We fix a uniformizer ϖ_{F} of F such that \overline{ϖ_{F}} = −ϖ_{F} if F/F_{0} is ramified, \overline{ϖ_{F}} = ϖ_{F} otherwise. We put ϖ_{0} = ϖ_{F}^{2} if F/F_{0} is ramified, ϖ_{0} = ϖ_{F} otherwise; so ϖ_{0} is a uniformizer of F_{0}.

Let V be an N-dimensional vector space over F, equipped with a nondegenerate ϵ-hermitian form, with ϵ = ±1. We put A = End_{F} V and denote by the adjoint (anti-)involution on A induced by h. Set also G̃ = Aut_{F} V and let σ be the involution given by g ↦ \overline{g}^{−1}, for g ∈ G̃. We also have an action of σ on the Lie algebra A given by a ↦ −\overline{a}, for a ∈ A (this is the differential of the action on G̃). We put Σ = {1, σ}, where 1 acts as the identity on both G̃ and A.

We put G = G̃^{Σ} = {g ∈ G̃ : h(gv, gw) = h(v, w) for all v, w ∈ V }, a unitary, symplectic or orthogonal group over F_{0}, and A_{−} = A^{Σ} ≃ Lie G.

Recall from [7] §2, that an 𝔬_{F} -lattice sequence in V is a function Λ from ℤ to the set of 𝔬_{F} -lattices in V such that

(i) Λ(k) ⊂ Λ(j), for k ≥ j;

(ii) there exists a positive integer e = e(Λ|𝔬_{F}), called the 𝔬_{F} -period of Λ, such that ϖ_{F} Λ(k) = Λ(k + e), for all k ∈ ℤ.

An 𝔬_{F} -lattice sequence Λ is called strict if Λ(k) ≠ Λ(k +1), for all k ∈ ℤ. Note also that an 𝔬_{F} -lattice sequence is certainly an 𝔬_{0}-lattice sequence.

We also recall the definition of the direct sum of two lattice sequences: if V = V_{1} ⊕ V_{2} and, for i = 1, 2, Λ^{i} is an 𝔬_{F} -lattice sequence in V_{i} of 𝔬_{F} -period e, then the direct sum Λ = Λ^{1} ⊕ Λ^{2} is given by

Λ(k) = Λ^{1}(k) ⊕ Λ^{2}(k), for all k ∈ ℤ.

For L an 𝔬_{F} -lattice in V, we put L^{#} = {v ∈ V : h(v, L) ⊂ 𝔭_{F} }. Then we call an 𝔬_{F} -lattice sequence Λ self-dual if there exists d ∈ ℤ such that Λ(k)^{#} = Λ(d − k), for all k ∈ ℤ. If Λ = Λ^{1} ⊕ Λ^{2} is a direct sum of self-dual 𝔬_{F} -lattice sequences such that V = V_{1} ⊥ V_{2}, then we write Λ = Λ^{1} ⊥ Λ^{2}.

Given an 𝔬_{F} -lattice sequence Λ in V, there are two operations we can apply to it. First, we have “translation”: for k_{0} ∈ ℤ we define Λ′ by Λ′(k) = Λ(k+k_{0}), for k ∈ ℤ. Second, we have “normalization”: for m ∈ ℕ, we define Λ′ by Λ′(k) = Λ([\frac{k}{m}]), for k ∈ ℤ, where [x] denotes the greatest integer not greater than x. These two operations do not change the associated objects which we describe below, except upto a renormalization of the index. In particular, we may assume, where necessary, that a self-dual 𝔬_{F} -lattice sequence Λ in V is normalized such that Λ(k)^{#} = Λ(1 − k).

Associated to an 𝔬_{F} -lattice sequence Λ in V, we have a decreasing filtration {𝔞_{n}(Λ) : n ∈ ℤ} of A by 𝔬_{F} -lattices, given by

𝔞_{n} = 𝔞_{n}(Λ) = {x ∈ A : aΛ(k) ⊂ Λ(k + n) for all k ∈ ℤ}.

Moreover, 𝔞_{0} is a hereditary 𝔬_{F} -order in A and 𝔞_{1} is its Jacobson radical. If Λ is self-dual, then each 𝔞_{n}(Λ) is fixed by σ and we put 𝔞_{n}^{−} = 𝔞_{n}^{−}(Λ) = 𝔞_{n}(Λ)^{Σ} = 𝔞_{n}(Λ) ∩ A_{−}, which gives a filtration of A_{−} by 𝔬_{F} -lattices.

The filtration on A gives rise to a valuation ν_{Λ} on A, by

ν_{Λ}(x) = sup{k ∈ ℤ : x ∈ 𝔞_{n}(Λ)},

with the understanding that ν_{Λ}(0) = +∞. If Λ is self-dual then ν_{Λ} is fixed by σ.

Given an 𝔬_{F} -lattice sequence Λ, we also put

U = U(Λ) = 𝔞_{0}(Λ)^{×};
U_{n} = U_{n}(Λ) = 1 + 𝔞_{n}(Λ),  for n ≥ 1.

Then U is a compact open subgroup of G̃ and {U_{n} : n ≥ 1} is a filtration by normal subgroups. Moreover, for all n ≥ 1, we have an isomorphism

𝔞_{n}/𝔞_{n+1} →^{∼} U_{n}/U_{n+1}  (2.1)

induced by x ↦ 1 + x. If Λ is self-dual, then U, U_{n} are fixed by σ and we put

P = P (Λ) = U(Λ)^{Σ} = U(Λ) ∩ G;
P_{n} = P_{n}(Λ) = U_{n}(Λ)^{Σ} = U_{n}(Λ) ∩ G,  for n ≥ 1.

As before, P is a compact open subgroup of G, with a filtration by normal subgroups P_{n}, and the isomorphism (2.1) induces an isomorphism ([14] (2.1.4)(b))

𝔞_{n}^{−}/𝔞_{n+1}^{−} →^{∼} P_{n}/P_{n+1}.  (2.2)

We define the normalizer 𝔎(Λ) to be

𝔎(Λ) = ⋂_{n∈ℤ}NG̃(U_{n}(Λ)),

where NG̃ denotes the normalizer in G̃. An element x ∈ A is called Λinvertible if x ∈ 𝔎(Λ); equivalently, if xΛ(k) = Λ(k + ν_{Λ}(x)), for all k ∈ ℤ. Finally, note that if Λ is self-dual, then we have 𝔎(Λ) ∩ G = P (Λ).

Now we turn our attention to the characters of the groups U_{n}, P_{n}. We fix ψ_{0} an additive character of F_{0} with conductor 𝔭_{0} and put ψ_{F} = ψ_{0} ∘ tr_{F/F_{0}}, where tr denotes trace. Since F/F_{0} is at worst tamely ramified, ψ_{F} has conductor 𝔭_{F}. We also set ψ_{A} = ψ_{F} ∘ tr_{A/F}.

For S an 𝔬_{F} -lattice in A, we put

S^{∗} = {x ∈ A : ψ_{A}(xS) = 1}.

Then, for Λ a lattice sequence, we have 𝔞_{n}(Λ)^{∗} = 𝔞_{1−n}(Λ), by [7] (2.10). If S is fixed by σ, then, putting S_{−} = S ∩ A_{−}, we have

S^{∗} ∩ A_{−} = {x ∈ A_{−} : ψ_{A}(xS_{−}) = 1}.  (2.3)

Let denote the Pontrjagin dual. Then, for Λ an 𝔬_{F} -lattice sequence in V and n ≥ 1, we obtain a 𝔎(Λ)-equivariant isomorphism

𝔞_{−n}/𝔞_{1−n} →^{∼} (U_{n}/U_{n+1}) ,
b + 𝔞_{1−n} ↦ (ψ_{b} : x ↦ ψ_{A}(b(x − 1)), for x ∈ U_{n}).  (2.4)

Moreover, if Λ is self-dual then (by [16] (4.19)) this restricts to a P (Λ)- equivariant isomorphism

𝔞_{−n}^{−}/𝔞_{1−n}^{−} →^{∼} (P_{n}/P_{n+1}) ,
b + 𝔞_{1−n}^{−} ↦ (ψ_{b}^{−} : x ↦ ψ_{A}(b(x − 1)), for x ∈ P_{n}).  (2.5)

2.2 Strata

Definition 2.6 ([6] (1.5), [7] (3.1)) (i) A stratum in A is a 4-tuple [Λ, n, n − 1, b], where Λ is an 𝔬_{F} -lattice sequence, n ≥ 1 is an integer and b ∈ 𝔞_{−n}(Λ).

(ii) Two strata [Λ, n, n − 1, b_{i}], i = 1, 2, are called equivalent if b_{1} − b_{2} ∈ 𝔞_{1−n}(Λ).

(iii) A stratum [Λ, n, n − 1, b] is called skew if Λ is self-dual and b ∈ A_{−}.

Then, by (2.4), an equivalence class of strata corresponds to a character of U_{n}(Λ) and, by (2.5), an equivalence class of skew strata corresponds to a character of P_{n}(Λ).

Let [Λ, n, n − 1, b] be a stratum in A. Put y_{b} = ϖ_{F}^{n/g}b^{e/g} ∈ 𝔞_{0}(Λ), where e = e(Λ) and g = (n, e). Let Φ(X) ∈ 𝔬_{F} (X) be the characteristic polynomial of y_{b}. Then we define the characteristic polynomial φ_{b}(X) ∈ k_{F} [X] of the stratum to be the reduction modulo 𝔭_{F} of Φ(X). Note that this depends only on the equivalence class of the stratum and is, moreover, an intertwining invariant.

Definition 2.7 ([6] (2.3)) (i) A stratum [Λ, n, n − 1, b] in A is called fundamental if φ_{b}(X) ≠ X^{N}.

(ii) A stratum [Λ, n, n − 1, b] in A is called split if φ_{b}(X) has two coprime factors.

Let f(X) be a polynomial with coefficients in F or k_{F}, written f(X) = ∑_{i=0}^{n} a_{i}X^{i}. We define the polynomial \overline{f}(X) by \overline{f}(X) = ∑_{i=0}^{n} \overline{a}_{i}X^{i}.

Now suppose that [Λ, n, n − 1, b] is a skew stratum in A. Then we have y_{b} = η\overline{y}_{b}, for η = ± a sign (precisely, η = (−)^{e/g} if F/F_{0} is unramified, η = (−)^{n/g}(−)^{e/g} otherwise), and thus Φ(X) = \overline{Φ}(ηX) and φ_{b}(X) = \overline{φ}_{b}(ηX). Then, if we have a factorization Φ(X) = Φ_{1}(X)Φ_{2}(X), we have Φ(X) = \overline{Φ}(ηX) = \overline{Φ}_{1}(ηX)\overline{Φ}_{2}(ηX) so \overline{Φ}_{1}(ηX) is also a factor of Φ(X). The same applies to φ_{b}(X).

Definition 2.8 We say that the skew stratum [Λ, n, n − 1, b] is G-split if φ_{b}(X) has an irreducible factor ψ(X) such that (ψ(X), \overline{ψ}(ηX)) = 1.

Note that a G-split stratum is necessarily fundamental, since we have ψ(X) ≠ X. Further, a G-split stratum is split, since, by the argument above, \overline{ψ}(ηX) is also a factor of φ_{b}(X).

Definition 2.9 ([6] (1.5.5). [7] (5.1)) A stratum [Λ, n, n − 1, b] in A is called simple if

(i) the algebra E = F [b] is a field;

(ii) Λ is an 𝔬_{E}-lattice chain;

(iii) ν_{Λ}(b) = −n;

(iv) b is minimal, that is, writing e = e(E|F) for the ramification index and ν = ν_{E}(b) for the normalized valuation of b in E, we have (a) gcd(ν, e) = 1;

(b) ϖ_{F}^{−ν}b^{e} + 𝔭_{E} generates the residue field extension k_{E}/k_{F}.

Simple strata play an important role in the construction and classification of the representations of G̃ because the associated characters have a “nice” intertwining formula.

Definition 2.10 (cf. [19] (3.8)) A skew stratum [Λ, n, n − 1, b] in A is called semisimple if either it is simple or we have a non-trivial splitting V = V_{0} ⊥ ⋯ ⊥ V_{r} such that all the following hold:

(i) Λ = Λ^{0} ⊥ ⋯ ⊥ Λ^{r}, where Λ^{i}(k) = Λ(k) ∩ V_{i}, for i = 0, …, r;

(ii) b = b_{0} + ⋯ + b_{r}, where b_{i} = b|_{V_{i}}, for i = 0, …, r;

(iii) the polynomials φ_{b_{i}}(X) are pairwise coprime;

(iv) the strata [Λ^{i}, n, n − 1, b_{i}] in End_{F} (V_{i}) are simple, with the possible exception that b_{0} = 0.

2.3 The Theorems

Let π be a smooth representation of G. We say that π contains a skew stratum [Λ, n, n − 1, b] if it contains the associated character ψ_{b}^{−} of P_{n}(Λ).

Theorem 2.11 Let π be a smooth representation of G of positive level, that is, π has no fixed vector under P_{1}(Λ), for Λ any self-dual lattice sequence in V. Then π contains some fundamental skew stratum [Λ, n, n − 1, b]. Moreover, putting e = e(Λ|𝔬_{F}), g = (n, e), we have e/g ≤ N.

Proof The first assertion is given by [16] (5.2), where they call a fundamental stratum an “unrefined minimal K-type” (see [18], especially §5, for a translation into lattice-theoretic language). We remark that we could also deduce this from the results of §4 (see (4.3)).

For the second assertion, we put

S = {(Λ′, n′) : Λ′ is strict and (b + 𝔞_{1−n}) ∩ 𝔞′_{−n′} ≠ ∅} ,

where 𝔞′_{−n′} = 𝔞(Λ′)_{−n′}. This is clearly non-empty and, for (Λ′, n′) ∈ S, Λ′ is strict so e′ = e(Λ′|𝔬_{F}) ≤ N. Moreover, as in [9] (5.4) (or, more generally,

[16] (6.4)), we have n′/e′ ≥ n/e for all (Λ′, n′) ∈ S.

We choose (Λ′, n′) ∈ S with n′/e′ minimal and b′ ∈ (b + 𝔞_{1−n}) ∩ 𝔞′_{−n′}. If [Λ′, n′, n′ − 1, b′] is not fundamental then, by [5] Theorem 1, there exists (Λ′′, n′′) with Λ′′ strict, b′ + 𝔞′_{1−n′} ⊂ 𝔞′′_{−n′′} and n′′/e′′ < n′/e′. But then (Λ′′, n′′) ∈ S, contradicting the minimality of n′/e′. Hence [Λ′, n′, n′ − 1, b′] is fundamental so, again as in [9] (5.4), we have n′/e′ = n/e and the result follows. ■

Theorem 2.12 Let π be a smooth representation of G which contains a G-split skew stratum. Then π is not supercuspidal.

We will prove this in §3, where we construct a non-trivial Jacquet module for π.

Theorem 2.13 Let π be a smooth representation of G which contains a non-G-split fundamental skew stratum. Then π contains a semisimple skew stratum.

This is an easy consequence of the following proposition, which we prove in §4:

Proposition 2.14 Let [Λ, n, n − 1, b] be a non-G-split fundamental skew stratum in A. Then there exists a semisimple skew stratum [Λ′, n′, n′ − 1, β] in A such that

b + 𝔞_{1−n}^{−}(Λ) ⊂ β + 𝔞_{1−n′}^{−}(Λ′)

and n/e(Λ|𝔬_{F}) = n′/e(Λ′|𝔬_{F}).

In particular, these three theorems imply that any positive level supercuspidal representation of G contains a semisimple skew stratum [Λ, n, n − 1, β] such that n/e(Λ|𝔬_{F}) has denominator at most N (when written in its lowest terms). We also remark that these strata have a “nice” intertwining formula (see [19] (3.17)).

3 G-split strata

3.1 Intertwining

Let [Λ, n, n − 1, b] be a G-split skew stratum in A and put y_{b} = ϖ_{F}^{n/g}b^{e/g} ∈ 𝔞_{0}(Λ), where e = e(Λ) and g = (n, e). Let Φ(X) ∈ 𝔬_{F} (X) be the characteristic polynomial of y_{b} and φ_{b}(X) its reduction modulo 𝔭_{F}. Let ψ(X) be a monic irreducible factor of φ_{b}(X) such that ψ(X) ≠ \overline{ψ}(ηX) and write φ_{b}(X) = ψ(X)^{s}\overline{ψ}(ηX)^{s}θ(X), with θ(X) coprime to ψ(X) and \overline{ψ}(ηX), θ(X) = \overline{θ}(ηX). (Note that we may have θ(X) = 1 here.) By Hensel’s Lemma, there exist coprime polynomials Ψ(X), Θ(X) ∈ 𝔬_{F} [X], whose reductions modulo 𝔭_{F} are ψ(X)^{s}, θ(X) respectively, such that Φ(X) = Ψ(X)\overline{Ψ}(ηX)Θ(X).

We put V_{1} = ker Ψ(y_{b}), V_{−1} = ker \overline{Ψ}(ηy_{b}) and V_{0} = ker Θ(y_{b}). These spaces are preserved by b and we have

V = V_{0} ⊥ (V_{1} ⊕ V_{−1})

and V_{1}, V_{−1} are totally isotropic and in duality with respect to h. (Note that, if θ(X) = 1 then V_{0} = 0.) For i = −1, 0, 1, we define lattice sequences Λ^{i} in V_{i} by Λ^{i}(k) = Λ(k) ∩ V_{i}, for k ∈ ℤ. Then, as in [11] (3.5)(3.6), we have

Λ(k) = Λ^{1}(k) ⊕ Λ^{0}(k) ⊕ Λ^{−1}(k), for all k ∈ ℤ,

and, putting b_{i} = b|_{V_{i}} for −1 ≤ i ≤ 1, we have that, for i = 1, −1, b_{i} is Λ^{i}-invertible and ν_{Λ^{i}}(b_{i}) = −n. Indeed, in this situation, [Λ^{i}, n, n − 1, b_{i}] is non-split fundamental, for i = 1, −1.

Writing A^{ij} = Hom (V^{j}, V^{i}), by [6] (2.9) we have

𝔞_{k}(Λ) = ⨁_{−1≤i,j≤1} 𝔞_{k}(Λ) ∩ A^{ij},  for all k ∈ ℤ,
𝔞_{k}(Λ) ∩ A^{ii} = 𝔞_{k}(Λ^{i}),  for − 1 ≤ i ≤ 1, k ∈ ℤ.

We will abbreviate 𝔞_{k}^{ij} = 𝔞_{k}(Λ) ∩ A^{ij}, and in the block description

A = (\begin{matrix}A^{−1,−1} & A^{−1,0} & A^{−1,1} \\ A^{0,−1} & A^{0,0} & A^{0,1} \\ A^{1,−1} & A^{1,0} & A^{1,1} \end{matrix}) ,

we will usually omit the superscript ij.

We define 𝔬_{F} -lattices in A by

q𝔥_{1} = (\begin{matrix}𝔞_{n} & 𝔞_{q+1} & 𝔞_{q+1} \\ 𝔞_{n} & 𝔞_{n} & 𝔞_{q+1} \\ 𝔞_{n} & 𝔞_{n} & 𝔞_{n} \end{matrix}) , _{q}𝔥_{2} = (\begin{matrix}𝔞_{n+1} & 𝔞_{q+1} & 𝔞_{q+1} \\ 𝔞_{n} & 𝔞_{n+1} & 𝔞_{q+1} \\ 𝔞_{n} & 𝔞_{n} & 𝔞_{n+1} \end{matrix}) ,

for 0 ≤ q ≤ n. We put _{q}H_{j} = 1 +_{q}𝔥_{j}, for j = 1, 2, 0 ≤ q ≤ n; we abbreviate 0H_{j} = H_{j} and _{0}𝔥_{j} = 𝔥_{j}, for j = 1, 2. The sets _{q}H_{j} are compact open subgroups of G̃, for j = 1, 2, 0 ≤ q ≤ n, and the map x ↦ 1 + x induces isomorphisms of groups

q𝔥_{1}/_{q}𝔥_{2} → _{q}H_{1}/_{q}H_{2}.

For each q = 0, …, n, we define a character ψ_{b} of _{q}H_{1}, trivial on _{q}H_{2}, by ψ_{b}(1 + x) = ψ_{A}(bx), for x ∈ _{q}𝔥_{1}.

We write M = (A^{−1,−1})^{×} × (A^{0,0})^{×} × (A^{1,1})^{×}; this is a Levi subgroup of G̃. We also put A_{u} = A^{−1,0} ⊕ A^{−1,1} ⊕ A^{0,1}, A_{l} = A^{0,−1} ⊕ A^{1,−1} ⊕ A^{1,0} and N_{u} = 1 + A_{u}, N_{l} = 1 + A_{l}.

Let K be a compact open subgroup of G̃ and let ψ be a character of K. Then the G̃-intertwining of ψ is defined to be

IG̃(ψ|K) = {g ∈ G̃ : ψ(gkg^{−1}) = ψ(k), for all k ∈ K ∩ g^{−1}Kg} .

Proposition 3.1 (cf. [7] Theorem (3.7), [3] (2.3.2)) The G̃-intertwin-ing of the character ψ_{b}|H_{1} satisfies

IG̃(ψ_{b}|H_{1}) ⊂ H_{1} · M · H_{1}.

Proof This follows by iterating [7] (3.7), having observed (see [3] (2.3.2)), that we may indeed apply it in this, slightly more general, situation. Explicitly, we put

H′_{1} = (\begin{matrix}1 + 𝔞_{n} & 𝔞_{1} & 𝔞_{1} \\ 𝔞_{n} & 1 + 𝔞_{n} & 𝔞_{n} \\ 𝔞_{n} & 𝔞_{n} & 1 + 𝔞_{n} \end{matrix}) , and M′ = (\begin{matrix}A^{−1,−1} & 0 & 0 \\ 0 & A^{0,0} & A^{0,1} \\ 0 & A^{1,0} & A^{1,1} \end{matrix})^{×} .

By [7] (3.7), we have IG̃(ψ_{b}|H_{1}) ⊂ IG̃(ψ_{b}|H′_{1}) ⊂ H′_{1}M′H′_{1}. Since H′_{1} ⊂ H_{1} normalizes ψ_{b}|H_{1}, we in fact have IG̃(ψ_{b}|H_{1}) ⊂ H′_{1}I_{M′}(ψ_{b}|H_{1} ∩M′)H′_{1}. But, again by [7] (3.7), we have I_{M′}(ψ_{b}|H_{1} ∩ M′) ⊂ (H_{1} ∩ M′)M(H_{1} ∩ M′) and the result follows. ■

We observe now that all the groups _{q}H_{j} are fixed by σ. We put _{q}H_{j}^{−} = qH_{j} ∩ G and _{q}𝔥_{j}^{−} = _{q}𝔥_{j} ∩ A_{−}, for j = 1, 2, 0 ≤ q ≤ n, and M^{−} = M ∩ G, N_{u}^{−} = N_{u}∩G, N_{l}^{−} = N_{l}∩G. Note that P_{u}^{−} = M^{−}N_{u}^{−} is a maximal parabolic subgroup of G, with Levi component M^{−} and unipotent radical N_{u}^{−}, and P_{l}^{−} = M^{−}N_{l}^{−} is the opposite parabolic.

We write ψ_{b}^{−} for the restriction ψ_{b}|_{q}H_{1}^{−}.

Proposition 3.2 We have I_{G}(ψ_{b}^{−}|H_{1}^{−}) ⊂ H_{1}^{−} · M^{−} · H_{1}^{−}.

Proof By [20] (2.5), we have I_{G}(ψ_{b}^{−}|H_{1}^{−}) = IG̃(ψ_{b}|H_{1}) ∩ G and, by [19] (2.3) (see also ibid.(4.15)), we have H_{1}MH_{1} ∩ G = H_{1}^{−}M^{−}H_{1}^{−} so the result follows from (3.1). ■

3.2 Covers

We continue in the situation of §3.1. For −1 ≤ i ≤ 1, we put G_{i} = (A^{ii})^{×} and suppose that we are given:

(i) a subgroup K_{1} of U(Λ^{1}) containing H_{1}^{−} ∩ G_{1} and an irreducible representation ρ_{1} of K_{1} whose restriction to H_{1}^{−} ∩ G_{1} is a multiple of ψ_{b_{1}};

(ii) a subgroup K_{0}^{−} of P (Λ^{0}) containing H_{1}^{−} ∩ G_{0} and an irreducible representation ρ_{0}^{−} of K_{0}^{−} whose restriction to H_{1}^{−} ∩ G_{0} is a multiple of ψ_{b_{0}}^{−}.

We think of K_{1} embedded in G as \{(\begin{matrix}\overline{k} & −1 \\ & & 1 \\ & & & k \end{matrix}) : k ∈ K_{1}\} .

Corollary 3.3 (cf. [7] (3.9)) (i) The set K^{−} = (K_{1} × K_{0}^{−}).H_{1}^{−} is a group.

(ii) There is a unique irreducible representation ρ_{−} of K^{−} which is trivial on K^{−} ∩ N_{u}^{−}, K^{−} ∩ N_{l}^{−} and whose restriction to K_{1} × K_{0}^{−} is ρ_{1} ⊗ ρ_{0}^{−}.

(iii) The pair (K^{−}, ρ_{−}) is a G-cover of (K_{1} × K_{0}^{−}, ρ_{1} ⊗ ρ_{0}^{−}).

Proof This is identical to [7] (3.9), except we take the element ζ to be

ζ = (\begin{matrix}ϖ_{F} \\ & 1 \\ & & \overline{ϖ_{F}} & −1 \end{matrix}) .

■

3.3 Jacquet modules

We again continue with the notation of §3.1.

Lemma 3.4 (cf. [3] (2.3.9)) Let 1 ≤ q < n and put N_{l,q}^{−} = N_{l}^{−} ∩P_{n−q}(Λ). Then the group N_{l,q}^{−} normalizes _{q}H_{1}^{−} and acts transitively by conjugation on the set of characters of _{q−1}H_{1}^{−} agreeing with ψ_{b}^{−} on the subgroup _{q}H_{1}^{−}.

Proof By [7] (3.7) Lemma 4, the map y ↦ yb − by, y ∈ A, induces an isomorphism 𝔞_{n−q} ∩ A_{l} → 𝔞_{−q} ∩ A_{l}. Moreover, this map preserves A_{−} and hence restricts to an isomorphism, 𝔞_{n−q}^{−} ∩ A_{l} → 𝔞_{−q}^{−} ∩ A_{l}, and the result follows. ■

Proposition 3.5 (cf. [3] (2.4.4)) Let (π, V) be a smooth representation of G containing the G-split skew stratum [Λ, n, n − 1, b]. Then it contains the character ψ_{b}^{−}|H_{1}^{−} also.

Proof Given the previous lemma, this is identical to [3] (2.4.4). ■

Theorem 3.6 (cf. [3] (2.4.2)) Let (π, V) be a smooth representation of G containing the G-split skew stratum [Λ, n, n − 1, b]. Then π is not supercuspidal.

Proof By the previous proposition, (π, V) contains the character ψ_{b}^{−}|H_{1}^{−} and, by the corollary above, (H_{1}^{−}, ψ_{b}^{−}) is a G-cover of the pair (H_{1}^{−} ∩ M, ψ_{b}^{−}|H_{1}^{−} ∩ M). Then, by [8] (7.9), we have an isomorphism of ℂ-spaces

V^{ψ_{b}^{−}} → V_{u}^{ψ_{b}^{−}|H_{1}^{−}∩M},

where V_{u} is the Jacquet module of V attached to P_{u}^{−} and V^{ψ_{b}^{−}} is the ψ_{b}^{−}- isotypic component. In particular, V_{u} ≠ 0 so π is not supercuspidal. ■

Remarks 3.7 (i) To obtain this, we could have appealed to the very general result of [12]. However, the language used there is very different and a translation into the lattice-theoretic language used here has not yet been done.

(ii) Thanks to [8] (8.3), Corollary 3.3 above gives a method for constructing types for certain (non-supercuspidal) components of the Bernstein spectrum, assuming we have all the supercuspidal types (cf. [1], [2]).

4 Semisimple strata

We now turn to the proof of Proposition 2.14. We adapt the notation of [10] to our situation.

Let Λ be a self-dual 𝔬_{F} -lattice chain, normalized so that Λ(k)^{#} = Λ(1−k) for k ∈ ℤ, and we put e_{0} = e(Λ|𝔬_{0}). We consider the quotients

Λ̃(k) = Λ(k)/Λ(k + 1), k ∈ ℤ.

Multiplication by ϖ_{0} allows us to identify Λ̃(k) with Λ̃(k + e_{0}), for k ∈ ℤ. Given k ∈ ℤ, let k̃ denote its image in ℤ/e_{0}ℤ. We put

Λ̃ = ∑_{k̃∈ℤ/e_{0}ℤ} Λ̃(k̃),

a vector space over k_{F}, hence over k_{0}. We consider End_{k_{F}} (Λ̃); we have

End_{k_{F}} (Λ̃) = ∑_{j̃∈ℤ/e_{0}ℤ}End(Λ̃)j̃,  where

End(Λ̃)j̃ = ∑_{k̃∈ℤ/e_{0}ℤ}Hom_{k_{F}} (Λ̃(k̃), Λ̃(k̃ + j̃)).

Further, we have End(Λ̃)ĩEnd(Λ̃)j̃ ⊂ End(Λ̃)_{ĩ+j̃}, for ĩ, j̃ ∈ ℤ/e_{0}ℤ. Altogether, we have the structure of a ℤ/e_{0}ℤ-graded algebra on End_{k_{F}} (Λ̃).

Given an element b ∈ 𝔞_{−n}(Λ), we obtain, by reduction, maps b̃ĩ: Λ̃(ĩ) → Λ̃(ĩ− ñ), for each ĩ ∈ ℤ/e_{0}ℤ, and hence a map

b̃ = ∑_{ĩ∈ℤ/e_{0}ℤ} b̃ĩ ∈ End(Λ̃) ˜−n.

We now describe the duality on Λ̃ and End_{k_{F}} (Λ̃) induced by h. We have, for k ∈ ℤ, a well-defined pairing

h̃_{k} : Λ̃(k) × Λ̃(−k) → k_{F} ,
(v + Λ(k + 1), v′ + Λ(1 − k)) ↦ h(v, v′) + 𝔭_{F} .

This pairing is ϵ-hermitian (or ϵ-bilinear if k_{F} = k_{0}) and is, moreover, nondegenerate: for v ∈ Λ(k), h̃_{k}(v+Λ(k+1), Λ̃(−k)) = 0 implies v ∈ Λ(−k)^{#} = Λ(k + 1). Multiplication by ϖ_{0} transforms h̃_{k} into h̃_{k+e_{0}}, for all k ∈ ℤ, and hence we obtain nondegenerate ϵ-hermitian pairings

h̃k̃ : Λ̃(k̃) × Λ̃( ˜−k) → k_{F} .

Putting these all together, we have a nondegenerate pairing

h̃ : Λ̃ × Λ̃ → k_{F} .

The pairing h̃ induces an adjoint involution on End_{k_{F}} (Λ̃), which we denote . Then, for b ∈ A we have \overline{b̃} = ˜ \overline{b}. In particular, if we put End_{k_{F}}^{−} (Λ̃) = \{ϱ ∈ End_{k_{F}} (Λ̃) : ϱ + \overline{ϱ} = 0\}, then the reduction map sends A_{−} onto End_{k_{F}}^{−} (Λ̃).

Definition 4.1 Let Λ be a self-dual 𝔬_{F} -lattice sequence. We call a self-dual 𝔬_{F} -lattice sequence Λ′ a refinement of Λ if there exists m ∈ ℕ odd such that Λ(k) = Λ′(mk) for all k ∈ ℤ.

Note that a k_{F} -subspace V of Λ̃(k̃) corresponds to a unique 𝔬_{F} -lattice L such that Λ(k) ⊃ L ⊃ Λ(k + 1) (more precisely, to the set of lattices {ϖ_{0}^{i}L : i ∈ ℤ}). Moreover, if we put V^{⊥} = {ṽ ∈ Λ̃(−k̃) : h̃(V, ṽ) = 0}, then V^{⊥} corresponds to the lattice L^{#}. In particular, a refinement Λ′ of Λ corresponds to flags of k_{F} -subspaces

Λ̃(k̃) = W0 ; k̃ ⊃ W1 ; k̃ ⊃ ⋯ Wm ; k̃ = 0, for k̃ ∈ ℤ/e_{0}ℤ,

with (Wi ; k̃)^{⊥} = W_{−k̃}^{m−i} and ϖ̃_{F}Wi ; k̃ = W_{k̃+ẽ}^{i}, for k̃ ∈ ℤ/e_{0}ℤ, 0 ≤ i ≤ m, where

e = e(Λ|𝔬_{F}).

Proposition 4.2 With notation as above, let b ∈ 𝔞_{−n}^{−}(Λ) \ 𝔞_{1−n}^{−}(Λ). Then there exists a refinement Λ′ of Λ and n′ ∈ ℤ such that

(i) n′/e(Λ′|𝔬_{F}) = n/e(Λ|𝔬_{F});

(ii) 𝔞_{1−n}(Λ) ⊂ 𝔞_{1−n′}(Λ′);

(iii) b ∈ 𝔞_{−n′}^{−}(Λ′) and the reduced map b̃′ ∈ End_{k_{F}} (Λ̃′) is semisimple.

Proof Consider the reduced map b̃ ∈ End_{k_{F}} (Λ̃) and let b̃ = b̃_{ss} + b̃_{np} be its Jordan decomposition. We have b̃_{np}^{m} = ˜0, for some m ∈ ℤ, which we may (and do) assume odd, m = 2s − 1. Then, following [18] (5.5), we put

Vi ; k̃ = (b̃_{np})^{i}Λ̃(k̃ + ĩñ) ⊂ Λ̃(k̃)  for k̃ ∈ ℤ/e_{0}ℤ, 0 ≤ i ≤ m;
Wi ; k̃ = ⋂_{q−p=2(k−1−i)}(Vp ; k̃ + (Vq ; ˜−k)^{⊥}) for k̃ ∈ ℤ/e_{0}ℤ, 0 ≤ i ≤ s − 1;
Wi ; k̃ = (Wm−i ; ˜−k)^{⊥}  for k̃ ∈ ℤ/e_{0}ℤ, s ≤ i ≤ 2s − 1.

Thus we have flags of k_{F} -subspaces Λ̃(k̃) = W0 ; k̃ ⊃ ⋯ Wm ; k̃ = 0, which give rise to a refinement Λ′ of Λ, with

Λ′(km + i)/Λ(k + 1) = Wi ; k̃, for k ∈ ℤ, 0 ≤ i ≤ m − 1.

We put n′ = nm and (i) is clear since e(Λ′|𝔬_{F}) = me(Λ|𝔬_{F}).

For a ∈ 𝔞_{1−n}(Λ), we have aΛ′(km + i) ⊂ aΛ(k) ⊂ Λ(k − n + 1) = Λ′(km + (m − mn)); but m − i − mn ≥ 1 − n′ so a ∈ 𝔞_{1−n′}(Λ′) as required. Also, since b̃Wi ; k̃ ⊂ W_{k̃−ñ}^{i}, we have that bΛ′(km + i) ⊂ Λ′((k − n)m + i), that is b ∈ 𝔞_{−n′}(Λ′).

Finally, we have Λ̃′(k̃′m′ + ĩ′) ≃ Wi ; k̃/Wi+1 ; k̃ (where k̃′ denotes the image

of k in ℤ/e_{0}mℤ) so we may think of b̃′ as a further reduction of b̃. Now b̃_{np} reduces to ˜0′ so b̃′ is semisimple as required. ■

Remark 4.3 If, in the situation of (4.2), b̃ is nilpotent, we have

b + 𝔞_{1−n}(Λ) ⊂ 𝔞_{1−n′}(Λ′)

and (n′ − 1)/e(Λ′) < n/e(Λ). However, we cannot use this directly to deduce, as in [5], [10], [13], that a smooth representation π of G contains a fundamental skew stratum, since e(Λ′) is not bounded. For this, we must use the notion of an “optimal point” from [16] §6.

We adopt the notation of [16], noting that the lattice sequence Λ′ corresponds to a rational point x_{Λ′} in the building of G̃ (see [4]) which is fixed by Σ. Then 𝔞_{m}(Λ′) corresponds to 𝔤_{x_{Λ′},m/e(Λ′)}^{∗}, when we have identified the Lie algebra A = 𝔤 with its dual.

We choose a self-dual basis for Λ′ as in [14] (1.7); this gives rise to a maximal torus in G̃ fixed by Σ and a simplex S in the apartment determined by this torus which contains x_{Λ′} and is also fixed by Σ (and maximal for this property – it is of codimension at most 2). Let C be a chamber containing S in its closure and let Φ be the set of affine roots which take values (strictly) between 0 and 1 on C. Then Φ ∩ ^{σ}Φ is the set of affine roots which take values between 0 and 1 on S.

For Ξ ⊂ Φ∩^{σ}Φ fixed by Σ, the optimal point x_{Ξ} (which may, and will, be taken as a barycentre of a face of C – that is, a point corresponding to a strict lattice sequence, period at most N) is in the closure of S and, moreover, ^{σ}x_{Ξ} is also an optimal point for Ξ in the closure of S. Hence y_{Ξ} = \frac{1}{2}(x_{Ξ} +^{σ}x_{Ξ}) is also an optimal point, fixed by Σ, and the corresponding lattice sequence Λ_{Ξ} is self-dual and of period at most 2N.

Now let Ξ ⊂ Φ ∩^{σ}Φ be the set of affine roots ψ such that 0 < ψ(x_{Λ′}) ≤ \frac{1−n′}{e(Λ′)} − [\frac{1−n′}{e(Λ′)}]. This is fixed by Σ and, by the definition of optimal point, there exists n_{Ξ} ∈ ℤ such that 𝔞_{1−n′}(Λ′) ⊂ 𝔞_{−n_{Ξ}}(Λ_{Ξ}) and n_{Ξ}/e(Λ_{Ξ}) < (n′ − 1)/e(Λ′).

We now complete the proof of Proposition 2.14.

Theorem 4.4 Let [Λ, n, n−1, b] be a non-G-split fundamental skew stratum in A. Then there exists a semisimple skew stratum [Λ′, n′, n′ − 1, β] in A, with Λ′ a refinement of Λ, such that

(i) n′/e(Λ′|𝔬_{F}) = n/e(Λ|𝔬_{F});

(ii) 𝔞_{1−n}(Λ) ⊂ 𝔞_{1−n′}(Λ′);

(iii) b ∈ β + 𝔞_{1−n′}^{−}(Λ′).

Proof From (4.2) we obtain Λ′, n′ such that (i) and (ii) are satisfied and, since the stratum is fundamental, the reduction b̃′ ∈ End_{k_{F}} (Λ̃′) is non-zero semisimple.

Put y = b^{e′/g′}ϖ_{F}^{n′/g′}, where e′ = e(Λ′|𝔬_{F}) and g′ = (e′, n′); so y = η\overline{y}, for η = ± a sign. Let Φ(X) ∈ 𝔬_{F} [X] be the characteristic polynomial of y and let φ_{b}(X) ∈ k_{F} [X] be the characteristic polynomial of the stratum [Λ′, n′, n′ − 1, b]. Since the stratum is non-G-split, we have

φ_{b}(X) = ∏_{i=1}^{r} ϕ_{i}(X)^{s_{i}} · X^{M},

where the ϕ_{i}(X) are monic, irreducible, pairwise coprime, ϕ_{i}(X) = \overline{ϕ}_{i}(ηX) and ∑_{i=1}^{r} s_{i} + M = N. (Note that we may have M = 0 here.) By Hensel’s Lemma, we may lift this to

Φ(X) = ∏_{i=1}^{r} Φ_{i}(X) · Θ(X),

where the Φ_{i}(X) are monic, pairwise coprime, Φ_{i}(X) = \overline{Φ}_{i}(ηX) and reduce modulo 𝔭_{F} to ϕ_{i}(X)^{s_{i}}, and Θ(X) reduces modulo 𝔭_{F} to X^{M}.

Put

V_{i} = ker Φ_{i}(y),  for i = 1, …, r;
V_{0} = ker Θ(y).

Then, as in [11] (3.4), (3.5), we have V = V_{0} ⊥ ⋯ ⊥ V_{r}, this decomposition is fixed by b and, putting Λ^{i}(k) = Λ(k) ∩ V_{i} for i = 0, …, r, we have

Λ(k) = ⨁_{i=0}^{r} Λ^{i}(k), for k ∈ ℤ.

Hence we obtain skew strata [Λ^{i}, n′, n′−1, b_{i}] in A^{i} = EndV^{i}, where b_{i} = b|_{V^{i}}, and the stratum [Λ′, n′, n′ − 1, b] is the sum of these strata. Moreover, for i = 1, ..., r, the stratum is non-split fundamental, while for i = 0 it is nonfundamental.

We treat first the case i = 0. The reduction of b_{0} in End_{k_{F}} (Λ̃^{0}) is semisimple but also nilpotent, since the stratum is non-fundamental. Hence it is 0 and we put β_{0} = 0.

Now let 1 ≤ i ≤ r. The skew stratum [Λ^{i}, n′, n′ − 1, b_{i}] is non-split fundamental and the reduction of b_{i} in End_{k_{F}} (Λ̃^{i}) is semisimple; hence, by [6] (2.5.8), the stratum is equivalent to a simple stratum [Λ^{i}, n′, n′ − 1, α_{i}]. Now α_{i} + \overline{α}_{i} ∈ 𝔞_{1−n′}(Λ^{i}) so, by [20] (1.10), the stratum is equivalent to a simple skew stratum [Λ^{i}, n′, n′ − 1, β_{i}].

We now put β = ∑_{i=0}^{r} β_{i}; then [Λ′, n′, n′ − 1, β] is semisimple skew and is equivalent to [Λ′, n′, n′ − 1, b], as required. ■

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Shaun Stevens Mathematical Institute 24-29 St. Giles’ Oxford OX1 3LB United Kingdom email: ginnyshaun@bigfoot.com