Local tame lifting for GL(N) I: Simple characters

Abstract

L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 14 106 COLIN J. BUSHNELL AND GUY HENNIART Our aim here is to make the first substantial step towards defining l^y in explicit local terms when the extension K/F is tamely ramified, but not necessarily Galois. We start from the classification of the irreducible smooth representations of the group GL(N, F), in terms of explicit local data, given in [BK1]. Our approach is thus in the spirit of [K], [KM], [Pa], and very different from the globally-derived " base change " methods of, for example, [L], [AC]. This makes our task here a dual one. We must first give a rigorous (albeit partial) definition of local tame lift, and then connect it with the theory of base change. This is reflected in the structure of the paper. The central concept of [BK1] is that of simple type. This is constructed in three stages, starting with a simple stratum, which is basically a field-theoretic object. The second step is a simple character, which is an arithmetically defined abelian character of a certain compact open subgroup of GL(N, F) determined by the underlying simple stratum. The final step will not concern us in this paper. An irreducible supercuspidal representation TC ofGL(N, F) must contain a simple character 6p, say. Its lift ^/p^), whatever that may be, is not necessarily supercuspidal. However, it will be built, via a familiar process of parabolic induction [Ze], from a uniquely determined collection of irreducible supercuspidal representations p^ of groups GL(N,, K), with S,N,=N. Each of these p^ will contain a simple character 6^. We proceed on the tentative hypothesis that the collection { OK } is in some way determined by the original character 6p (and, we might add, conversely). We therefore seek a way of lifting simple characters. There are, however, a number of other factors which need to be taken into account. First, our original supercuspidal representation n will contain many different simple characters. Any two of these will, of necessity, intertwine in GL(N, F). A fundamental result of [BK1] then shows that they will be conjugate in GL(N, F). However, they may arise from quite different constructions. In particular, they can be attached to distinct simple strata, and it is not straightforward, given two explicitly defined simple characters, to determine whether or not they are conjugate. See [BK3], [BK2], [KP] for some discussion of this matter. Further, we have families of relations between simple characters in GL(N, F) and simple characters in GL(N', F) for any integer N'. These relat...

Semisimple characters for p-adic classical groups

“…One of the main ingredients in the description of the admissible dual of G by Bushnell and Kutzko ( [6], [7]) is the notion of simple characters: these are arithmetically defined characters of certain compact open subgroups of G. To obtain all the irreducible supercuspidal representations of G in [6], there are three main steps: first, to show that these simple characters have some rather remarkable properties of functoriality (and it turns out that they even have such properties when the dimension N is allowed to vary (see [6], [10]) and similarly for the base field F (see [4], [5] and sequels); second, to show that any irreducible supercuspidal representation of G contains a simple character θ of a group denoted H 1 ; and finally, to find the representations of the normalizer in G of θ which contain θ.…”

The essentially tame local Langlands correspondence, I

“…(Here, we regard ξ as a quasicharacter of W E via class field theory, and denote by Ind E/F the functor of induction from representations of W E to representations of W F : see (A.3) below for more details. ) We combine the structure theory for supercuspidal representations [9] with the technique of tame lifting of simple characters [3] to construct a canonical bijection…”

“…Under the Langlands correspondence, these correspond respectively to base change b K/F and automorphic induction A K/F . When K/F is also tamely ramified, the theory of tame lifting [3], [6] describes how the classification from [9] interacts with these operations.…”

“…At that time, the latter was only available in characteristic zero [25]. We, on the other hand, can proceed much more directly and generally: the construction of the map (4) is an instance of the theory of tame lifting [3], [6] and its bijectivity follows from the classification theorems of [9].…”

“…The operations of base change and automorphic induction are constructed, in [1] and [16] respectively, on the hypothesis that F has characteristic zero. The operation of tame lifting (of endo-classes of simple characters) [3] is defined in arbitrary characteristic. The main results of [3], which are basic to everything we do here, relate tame lifting to base change and automorphic induction.…”