Jean-François DATProfesseur de MathÃ©matiques Ã Sorbonne Université.Chercheur à l'I.M.J. , projet Formes Automorphes. |
Institut de Mathématiques de Jussieu 4, place Jussieu 75252 Paris cedex 05 |
Tel : +33 1 44 27 54 28 | Fax : +33 1 44 27 78 18 | Bureau : 15-25 412 | e-mail : jean-francois.dat[at]imj-prg.fr |
I am interested in connections between Representation Theory, Algebraic Geometry and Number Theory (Galois representations), in the general framework of Langlands' Program. My first work was in the field of smooth representation theory of p-adic groups. More recently, I have focused on the cohomology of certain moduli spaces that arise naturally in this context as some p-adic avatars of Shimura varieties. I am particularly interested in those phenomena which cannot be dealt with Trace Formula arguments only : non semi-simplicity, extensions, finite or integral coefficients. One of my current projects is to obtain a cohomological realization of the mod l local Langlands correspondence, due to Vignéras, and in particular an interpretation of the nilpotent part of this correspondence.
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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.
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.
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...
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.
Bernstein blocks of complex representations of $p$-adic reductive groups have been computed in a large amount of examples, in part thanks to the theory of types a la Bushnell and Kutzko. The output of these purely representation-theoretic computations is that many of these blocks are equivalent. The motto of this paper is that most of these coincidences are explained, and many more can be predicted, by a functoriality principle involving dual groups. We prove a precise statement for groups related to $GL_{n}$, and then state conjectural generalizations in two directions : more general reductive groups and/or \emph{integral} $l$-adic representations.
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 . Moreover, Vincent Sécherre found a mistake. Here is an
erratum .
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.
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.
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...).
Let G be a general linear group over a p-adic field and let D* be an anisotropic inner form of G. The Jacquet-Langlands correspondence between irreducible complex representations of D* and discrete series of G does not behave well with respect to reduction modulo a prime l different from p. However we show that the Langlands-Jacquet transfer, from the Grothendieck group of admissible Q_l-representations of G to that of D* is compatible with congruences and reduces modulo l to a similar transfer for F_l-representations, which moreover can be characterized by some Brauer characters identities. Studying more carefully this transfer, we deduce a bijection between irreducible F_l-representations of D* and ``super-Speh'' F_l-representations of G. Via reduction mod l, this latter bijection is compatible with the classical Jacquet-Langlands correspondence composed with the Zelevinsky involution.
We prove that any level 0 block of Z_l representations of a p-adic general linear group is equivalent to the principal block of an appropriate product of general linear groups. A similar result, due to Bonnafé and Rouquier, is known for general linear groups over a finite field, where the equivalence is given by the cohomology of a suitable Deligne-Lusztig variety. Our strategy in the $p$-adic setting uses coefficient systems and consists roughly in "glueing" these equivalences along the building.
This note is motivated by the problem of ``uniqueness of supercuspidal support'' in the modular representation theory of $p$-adic groups. We show that any counterexample to the same property for a finite reductive group lifts to a counterexample for the corresponding unramified $p$-adic group. To this end, we need to prove the following natural property : any simple subquotient of a parabolically induced representation is isomorphic to a subquotient of the parabolic induction of some simple subquotient of the original representation. The point is that we put no finiteness assumption on the orginal representation.
We study the Galois action on the equivariant cohomology complex of Drinfeld's p-adic symmetric spaces and show how it encodes Langlands' correspondence for the so-called "principal elliptic" representations of GL_d (i.e all G-stable subquotients of the space of smooth functions on the flag variety of GL_d). The use of some derived category formalism is fundamental here since the action of Galois on the cohomology is very poor (via characters) and that of GL_d is given by very specific elliptic principal series (by Schneider-Stuhler work). The main steps of the strategy are : 1- prove (observe) that the complex is cohomologically split, 2- compute all Ext spaces and cup-products between elliptic principal series, thus in particular get a descrition of the endomorphism algebra of the cohomology complex, 3- prove that the action of inertia is unipotent thus given by a nilpotent "monodromy" operator N, 4- prove that the order of nilpotence of N id d (this turns out to be sufficient to get N explicitly in our description of the endomorphism algebra). In order to prove the last estimation, we apply a version of Rapoport-Zink spectral sequence to some (non-algebraic) quotients of Drinfeld's symetric spaces. In the process we obtain a new proof of Deligne's weight-monodromy conjecture for the varieties which admit p-adic uniformization by these spaces. Also we give a new computation of the compactly supported cohomology of p-adic symmetric spaces.
We consider the Drinfeld and Lubin-Tate towers together with their action by the product
of GL(d), the division algebra of invariant 1/d and the Weil group. In the Lubin-Tate case,
Harris and Taylor proved that the supercuspidal part of the cohomology realizes both
the local Langlands and Jacquet-Langlands correspondences, as conjectured by Carayol.
Recently, Boyer computed the remaining part of the cohomology and exhibited two defects :
first, the representations of GL_d which appear are of a very particular and
restrictive form ; second, the Langlands correspondence is not realized anymore.
In this paper, we define and study the cohomology complex in a suitable equivariant derived category,
and show how it encodes Langlands correspondence for all elliptic representations.
Then we transfer this result to the Drinfeld tower via an enhancement of a theorem of
Faltings due to Fargues. We deduce that Deligne's weight-monodromy conjecture is true
for varieties uniformized by Drinfeld's coverings of his symmetric spaces. This
completes the computation of local L-factors of some unitary Shimura varieties.
To study the cohomology complex, we first split it as a direct sum of isotypic components
according to the action of the division algebra. For such a component, the strategy goes
the same way as in the previous article. The essential difference concerns step 4, i.e. the
estimation of the order of nilpotence of the monodromy operator. At this point we use
Boyer's description of the graded pieces of the monodromy filtration of vanishing cycles
on Harris-Taylor's Shimura varieties.
We define and study a Lefschetz operator on the equivariant cohomology complex of the Drinfeld and Lubin-Tate towers. For l-adic coefficients we show how this operator induces a geometric realization of the Zelevinski correspondence (the composition of the Langlands correspondence with the Zelevinski involution) for elliptic representations. Joint to our previous study of the monodromy operator, this suggests a possible extension of Arthur's philosophy for unitary representations occuring in the intersection cohomology of Shimura varieties to the (possibly) non-unitary representations occuring in the cohomology of Rapoport-Zink spaces. However, our motivation for studying the Lefschetz operator comes from the hope that its geometric nature will enable us to realize the mod-l Langlands correspondence due to Vignéras.
For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a way compatible with extension of scalars. When R is a field of characteristic l, this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals, and a new proof of its existence. When R runs over complete local algebras, this provides bijections between deformations of matching mod-l representations. Roughly speaking, we can decompose "the supercuspidal part" of the l-integral cohomology as a direct sum, indexed by irreducible supercuspidals \pi mod l, of tensor products of universal deformations of \pi and of its two mates. Besides, we also get a virtual realization of both the semi-simple Langlands-Vigneras correspondence and the l-modular Langlands-Jacquet transfer for all representations, by using the cohomology complex and working in a
This note is concerned with a cohomological consequence of a geometric construction due to Yoshida, which relates the tame level of the Lubin-Tate tower to some Deligne-Lusztig variety of Coxeter type. More precisely, we show that the equivariant morphism in cohomology which follows from Yoshida's construction is an isomorphism, whatever the coefficients are. In particular, this gives a conceptual explanation to the observation that l-adic cohomologies indeed were ``the same'', once computed independently on each side (by Boyer, resp. Lusztig). This also gives a ``simple'' proof of the absence of torsion in the integral cohomology of the tame Lubin-Tate space. Our main tool is a general result on vanishing cycles for schemes with semi-stable reduction which generalizes previous results of Zheng and Illusie. In rough terms, this states that the restriction of the nearby cycles complex to a closed stratum is the push-forward of its restriction to the corresponding open stratum.
Let p and \ell be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod \ell cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vigneras' local Langlands correspondence modulo \ell
Let K be a finite extension of Q_p with residue field F_q , and let l be a prime such that q = 1(mod l). We investigate the cohomology of the Lubin-Tate towers of K with coefficients in F_l, and we show how it encodes Vigneras' Langlands correspondence for unipotent F_l-representations.
The first part is about an analog on finite fields. It is readable by a student with a basic background in algebraic geometry. The second part requires the theory of reductive groups. The next parts are more advanced and use Berkovich's theory of analytic spaces on p-adic fields.
This paper is a continuation and a completion of~\cite{BR}. We extend the Jordan decomposition of blocks: we show that blocks of finite groups of Lie type in non-describing characteristic are Morita equivalent to blocks of subgroups associated to isolated elements of the dual group --- this is the modular version of a fundamental result of Lusztig, and the best approximation of the character-theoretic Jordan decomposition that can be obtained via Deligne-Lusztig varieties. The key new result is the invariance of the part of the cohomology in a given modular series of Deligne-Lusztig varieties associated to a given Levi subgroup, under certain variations of parabolic subgroups. We also bring in local block theory methods: we show that the equivalence arises from a splendid Rickard equivalence. Even in the setting of \cite{BR}, the finer homotopy equivalence was unknown. As a consequence, the equivalences preserve defect groups and categories of subpairs. We finally determine when Deligne-Lusztig induced representations of tori generate the derived category of representations. An additional new feature is an extension of the results to disconnected reductive algebraic groups, which is required to handle local subgroups.
Suppose you have a (non split) Levi subgroup L of a reductive group G over a finite field. Attached to any parabolic subgroup P with component L, there is a Deligne-Lusztig variety Y_P, whose cohomology makes a bridge between the representation theory of L and G. The variety Y_P highly depends on P, in fact even its dimension does. In general, also its cohomology does. However we give a criterion, depending on P, P' and a semisimple conjugacy class s in the dual group of L, that insures that the cohomology of Y_P and Y_P' coincide (in the derived category) after a suitable shift, and when cut out by the idempotent associated to s by Lusztig (for rational cohomology) or by Bonnafé and Rouquier (for integral cohomology).
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Haoran a notamment prouvé de manière purement locale la conjecture de Harris sur la forme des espaces de cohomologie rationnelle, ainsi que l'absence de torsion dans la cohomologie entière. Au passage il obtient des résultats sur la cohomologie d'une compactification de certaines variétés de Deligne-Lusztig apparaissant dans ce contexte, et sur les systèmes de coefficients sur l'immeuble.
Motivé par la quête d'une bonne notion de constructibilité pour les faisceaux étales sur les espaces analytiques non-archimédiens, il s'est intéressé aux différentes classes d'ensembles semi-algébriques, semi-analytiques, sous-analytiques surconvergents et sous-analytiques, et a notamment obtenu des résultats de finitude cohomologique pour les sous-analytiques surconvergents (basés sur une jolie interprétation géométrique de leur définition originale plutôt rébarbative) et une bonne notion de dimension pour les sous-analytiques.
Il a travaillé sur les cohomologies p-adiques et modulo p de variétés de Shimura ou d'espaces de Lubin-Tate, et sur les correspondances de Jacquet-Langlands associées. Il a notamment montré que la cohomologie mod p à support compact de la tour de Lubin-Tate ne contient aucune représentation supersingulière, ce qui change radicalement de la cohomologie mod l ou l-adique. Il a aussi isolé un formalisme issu des travaux d'Emerton pour prouver la modularité de représentations Galoisiennes pro-modulaires, et obtenu un résultat de compatibilité local-global d'une construction de Breuil-Herzig pour U(3).
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Credits. My research has been supported by the following agencies. | [Up] |