Caractères à valeurs dans le centre de Bernstein.
J. reine angew. Math. 508 ; 61-83 (1999). pdf dvi
We study some aspects of finitely generated projective smooth complex
G-modules for a p-adic group G. We consider such a module, say P,
as a module over the Bernstein center Z of
the category of all smooth G-modules. If we reduce P modulo any complex-valued character
of Z, we get a finite length smooth representation of G ; then, for a generic character,
we describe the image of this representation in the appropriate Grothendieck group.
Under some regularity assumption, we even describe this image for all characters,
by defining and studying a Z-valued character on the Z-admissible (but not admissible
!) representation P.
The case of Serre's universal module is an interesting example : we deduce from our approach
a compatibility property between the Satake
isomorphism and Bernstein's description of the center of the Hecke-Iwahori algebra.
This compatibility has been used by people working on Shimura varieties
with Iwahori level structures (see e.g. T. Haines, Manuscripta Math. 101 (2000),
no. 2, 167--174, who also quotes a proof (probably better) by Lusztig of this compatibility).
There should be an analogue (in a sense which is made precise in the text) of Serre's universal module for any Bernstein block
(not necessarily the Iwahori block). We verify this for GL(n) using Bushnell
and Kutzko's theory of types.
Also when the regularity assumption above fails, one
has to use Cohen-Macaulay techniques as in Bernstein, Braverman,
Gaitsgory, The Cohen-Macaulay property of the
category of (g,K)-modules. Selecta
Math. (N.S.) 3 (1997), no. 3, 303--314.
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On the K0 of a p-adic group. Invent. Math. 140 (1)
; 171-226 (2000). pdf dvi
This article deals with various topics related with Grothendieck groups, invariant
distributions, parabolic and compact inductions... for a p-adic group G. The main result
is a description of the K0 of the Hecke algebra H(G) of G in terms of discrete series of Levi subgroups, which has an interesting behavior with regard to parabolic restriction and
induction. A similar description - but no more compatible with
these parabolic functors - is obtained for the
cocenter H(G)/[H(G),H(G)] and the Hattori rank map
gets an easy description in this dictionary.
We follow a beautiful idea of J. Bernstein which consists in comparing
two natural filtrations on these objects, one of a combinatorial nature and
one of a topological nature. The combinatorial filtrations are related to the
structure of Levi subgroups in G and have natural counterparts on many classical
objects of interest, such as the Grothendieck group of finite length G-modules
R(G), the set of regular semi-simple conjugacy classes, and
the variety of infinitesimal
characters. These filtrations will turn out to be
"compatible", in a sense
to be specified, with regard to all the classical operations or morphisms between these objects.
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Une preuve courte du principe de Selberg abstrait pour un
groupe p-adique.
Proc. Amer. Math. Soc. 129 (4) ; 1213-1217 (2001) pdf dvi
The "abstract" Selberg principle for a p-adic group is the vanishing of orbital integrals
at non-compact elements of the "trace" of a projective finitely generated representation.
This is at least the third proof of this principle. The first one is due to Blanc and
Brylinski in Cyclic homology and the
Selberg principle. J. Funct. Anal. 109 (1992),
no. 2, 289--330,
the more conceptual is due to Peter Schneider in The cyclic
homology of p-adic reductive
groups. J. Reine Angew. Math. 475 (1996),
39--54, and ours appears as the shortest one...
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Quelques propriétés des idempotents centraux
des groupes p-adiques.
J. reine angew. Math. 504 ; 69-103 (2004). pdf dvi
For a finite group G, the central primitive idempotents of the group algebra C[G]
have been well known for a long time ; they are in 1-1 correspondence with the classes
of irreductible representations and one gets a formula by taking the character of such
a representation.
For a p-adic group G, Bernstein gave a spectral description of the central primitive
idempotents of the Hecke algebra of G (suitably completed), whereas Harish Chandra's
Plancherel theorem in principle yields a formula (an invariant distribution on G).
But the latter is not fully explicit since Harish Chandra's mu-functions are generally
not known. In this paper we derive from the two previous papers another formula for
these invariant distributions, in terms of the K-theory of G.
As an application, we seek a bound for the possible denominators occuring in such a
formula. For finite groups, such denominators are well known to divide the order of
the group. In the p-adic case, a similar statement is expected, and we prove here that
this similar statement is equivalent to a conjectural property of the K-theory of G, namely
that it should be generated by the K-theory of its open compact subgroups (after maybe
some mild localization of scalars).
For the group GL(n), we prove the latter conjecture using our description of K-theory
(second paper above) together with Bushnell-Kutzko's theory of types and developments by
Schneider-Zink.
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Types et inductions pour les représentations modulaires de groupes p-adiques.
Ann. Scient. Éc. Norm. Sup, 32 (1) ; 1-38 (1999). pdf dvi
For complex smooth representations of a p-adic group G, Bushnell and Kutzko introduced in
Smooth representations of reductive $p$-adic groups: structure theory via types.
Proc. London Math. Soc. (3) 77 (1998) a notion of type for a Bernstein block
(not necessarily supercuspidal). Roughly speaking, it is a pair (J,t) consisting of an
open compact subgroup and an irreducible representation of this subgroup whose compactly
supported induced representation to G is a progenerator of the Bernstein block we start with.
If the block is parabolically induced from a Levi subgroup M, they explained how to construct
a type (J,t) from a type (J(M),t(M)) for the corresponding block relative to M. This is the
notion of a G-cover. Among the beautiful consequences of such a construction, one gets an
isomorphism between the compactly supported induced representation from t and the parabolic
induction of the compactly supported induced representation from t(M).
Consider now smooth representations of G with coefficients in a positive characteristic
field. The notion of Bernstein block in general doesn't exist anymore, but the notion of
G-cover still makes sense. In this paper, we gave sufficient conditions to insure the
existence of an isomorphim between the two induced representations, as above.
The main motivation was the feeling that it should be a crucial ingredient towards the
solution of important open questions on finiteness properties of modular representations, as
in M.-F. Vignéras' appendix to this paper.
As a matter of fact, it is an elaboration of this feeling which lead to a solution in
the eighth paper below.
Let us mention that many results of this paper have been improved and simplified by
Corinne Blondel in Quelques propriétés des paires couvrantes,
pdf .
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Nu-tempered representations of p-adic groups I : l-adic case.
Duke Math. J. 126 (3) ; 397-469 (2005). pdf dvi
The so-called tempered
complex smooth representations of p-adic groups have been much
studied and used, in connection with automorphic forms.
Nevertheless, the smooth representations which are realized
geometrically often have l-adic coefficients, so that archimedean
estimates of their matrix coefficients hardly make sense.
We investigate here a
notion of tempered representation
with coefficients in any normed field of characteristic different from
p. The theory turns out to be different according to the norm
being Archimedean, non-Archimedean with $|p|\neq 1$ or non-Archimedean
with $|p|= 1$.
In this paper, we concentrate on the last case.
The main applications
concern modular representation theory (i.e. on a positive
characteristic field), and
in particular the study of reducibility properties of the parabolic
induction functors ; one of the main results is the generic
irreducibility for induced families. Once a
suitable theory of rational
intertwining operators developed, this allows us to define Harish
Chandra's $\mu$-functions and show in some special cases how they
track down
the cuspidal constituents of parabolically induced representations.
Besides, we discuss
the admissibility of parabolic restriction functors and derive some
lifting properties for supercuspidal modular representations.
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Représentations lisses p-tempérées des groupes p-adiques. (2006, 21 pages).
pdf dvi
As in the former paper we consider aymptotic properties of matrix coefficients called
"temperedness" and "discreteness", but this time for a p-adic norm on the field of
coefficients. We concentrate on two topics :
1- integrality questions : there is a notion of Langlands classification of all irreducible
representations in terms of "p-adic
Langlands quotients of induced-from-twisted-p-tempered". Then we classify all "locally"
integral representations according to their p-adic Langlands parameter.
Here "locally integral" means
that for any open compact subgroup H the H-invariants admit a Hecke-invariant lattice. For classical groups we show that
an irreducible representation is locally integral if and only if its infinitesimal character lyies in an explicit affinoid
subdomain of the spectrum of Bernstein's center.
We conjecture that such representations are actually integral, i.e. admit a "global" G-invariant
lattice.
2- as in the complex coefficients case, one can define, in a non-trivial way,
a p-adic Schwartz-Harish Chandra algebra. We show in some special cases (GL(n) should be OK)
that a p-adic Plancherel-type formula holds for p-adic Schwartz functions.
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Finitude pour les représentations lisses des groupes p-adiques.
(2006, 44 pages).
pdf dvi
We study basic properties of the category of smooth representations of some p-adic group
G with coefficients in any commutative ring R in which p is invertible. The main purpose
is to establish Bernstein's second adjunction property and the noetheriannity of Hecke
algebras in this context. The first step is to prove that the first property implies the
second one ; this uses aspects of our paper "nu-tempered representations" above and holds
without any condition on G.
In order to study the second adjunction property, we introduce new tools, called
``parahoric functors" which relate representations of G_x
with those of M_x, where x is a point of the building of the Levi subgroup M.
For classical and linear groups, this tool together with Stevens and
Bushnell-Kutzko theories of (semi)simple characters, allows us to conclude.
The same strategy should also work in a ``tame"
context as in Yu's papers.
For general groups it provides partial results (level 0, principal series...).
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Integral structures in Bernstein's center. Needs corrections. dvi