Searcharxiv⌕ Search

arXiv subjects

Anne-Marie Aubert

Publications and source records attributed to Anne-Marie Aubert.

46 records · Page 3Linked to original sources
arXiv2008

Localisation de faisceaux caractères

Nous obtenons une formule pour les valeurs de la fonction caractéristique d'un faisceau caractère en fonction de la théorie des représentations de certains groupes finis, liés au groupe de Weyl. Cette formule, qui généralise des résultats antérieurs de Mœglin et de Waldspurger, dépend de la connaissance de certains sous-groupes réductifs admettant un faisceau caractère cuspidal. Dans un second temps, afin de rendre la formule plus explicite dans le cas d'un groupe quasi-simple, nous déterminons ces sous-groupes à conjugaison près. We obtain a formula for the values of the characteristic function of a character sheaf, in terms of the representation theory of certain finite groups related to the Weyl group. This formula, a generalization of previous results due to Mœglin and Waldspurger, depends on knowledge of certain reductive subgroups that admit cuspidal character sheaves. For quasi-simple groups, we make the formula truly explicit by determining all such subgroups upto conjugation.

math.RT↗
arXiv2008

On Cuspidal Representations of General Linear Groups over Discrete Valuation Rings

We define a new notion of cuspidality for representations of $\GL_n$ over a finite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedean local field $F$ using geometric and infinitesimal induction functors, which involve automorphism groups $G_λ$ of torsion $\Oh$\nobreakdash-modules. When $n$ is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of $\GL_n(F)$. We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of $\GL_n(\Oh_k)$ for $k\geq 2$ for all $n$ is equivalent to the construction of the representations of all the groups $G_λ$. A functional equation for zeta functions for representations of $\GL_n(\Oh_k)$ is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for $\GL_4(\Oh_2)$ are constructed. Not all these representations are strongly cuspidal.

math.RT↗
arXiv2007

Springer correspondences for dihedral groups

Recent work by a number of people has shown that complex reflection groups give rise to many representation-theoretic structures (e.g., generic degrees and families of characters), as though they were Weyl groups of algebraic groups. Conjecturally, these structures are actually describing the representation theory of as-yet undescribed objects called ''spetses'', of which reductive algebraic groups ought to be a special case. In this paper, we carry out the Lusztig--Shoji algorithm for calculating Green functions for the dihedral groups. With a suitable set-up, the output of this algorithm turns out to satisfy all the integrality and positivity conditions that hold in the Weyl group case, so we may think of it as describing the geometry of the ''unipotent variety'' associated to a spets. From this, we determine the possible ''Springer correspondences'', and we show that, as is true for algebraic groups, each special piece is rationally smooth, as is the full unipotent variety.

math.RT↗
arXiv2007

Représentations de Springer pour les groupes de réflexions complexes imprimitifs

To a spetsial complex reflection group, equipped with a root lattice in the sense of Nebe, we attach a certain finite set playing a role which is analogous to the role of the set of unipotent classes of an algebraic group. In the case of imprimitive groups, we give a combinatoric parametrization of it in terms of Malle-Shoji generalized symbols. This result provides a link between the works of Shoji on Green functions for complex reflection groups and of Broue, Kim, Malle, Rouquier, et. al. on the cyclotomic Hecke algebras and their families of characters. ----- A un groupe de reflexions complexe spetsial, muni d'un reseau radiciel au sens de Nebe, nous associons un certain ensemble fini qui doit jouer un role analogue a celui de l'ensemble des classes unipotentes d'un groupe algebrique. Dans le cas des groupes imprimitifs, nous en donnons un parametrage combinatoire en termes des symboles generalises de Malle et Shoji. Ce resultat fournit un lien entre les travaux de Shoji sur les fonctions de Green pour les groupes de reflexions complexes et ceux de Broue, Kim, Malle, Rouquier, et al. sur les algebres de Hecke cyclotomiques et leurs familles de caracteres.

math.RT↗
arXiv2006

Character Sheaves and Depth-zero representations

In this paper we provide a geometric framework for the study of characters of depth-zero representations of unramified groups over local fields with finite residue fields which is built directly on Lusztig's theory of character sheaves for groups over finite fields and uses ideas due to Schneider-Stuhler. Specifically, we introduce a class of coefficient systems on Bruhat-Tits buildings of perverse sheaves sheaves on affine algebraic groups over an algebraic closure of a finite field and to each supercuspidal depth-zero representation of an unramified $p$-adic group we associate a formal sum of these coefficient systems, called a model for the representation. Then, using a character formula due to Schneider-Stuhler and a fixed-point formula in etale cohomology we show that each model defines a distribution which coincides with the Harish-Chandra character of the corresponding representation, on the set of regular elliptic elements. The paper includes a detailed treatment of SL(2), Sp(4) and GL(n) as examples of the theory.

math.RT↗
arXiv2006

Cycles in the chamber homology of GL(3)

Let F be a nonarchimedean local field and let GL(N) = GL(N,F). We prove the existence of parahoric types for GL(N). We construct representative cycles in all the homology classes of the chamber homology of GL(3).

math.KT↗
arXiv2005

The Hecke algebra of a reductive p-adic group: a geometric conjecture

Let H(G) be the Hecke algebra of a reductive p-adic group G. We formulate a conjecture for the ideals in the Bernstein decomposition of H(G). The conjecture says that each ideal is geometrically equivalent to an algebraic variety. Our conjecture is closely related to Lusztig's conjecture on the asymptotic Hecke algebra. We prove our conjecture for SL(2) and GL(n). We also prove part (1) of our conjecture for the Iwahori ideals of the groups PGL(n) and SO(5).

math.RT↗
arXiv2004

Plancherel measure for GL(n,F) and GL(m,D): explicit formulas and Bernstein decomposition

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n) = GL(n,F). Let νdenote Plancherel measure for GL(n). Each component Ωin the Bernstein variety Ω(GL(n)) has several numerical invariants attached to it. We provide explicit formulas for the Bernstein component ν_Ω of Plancherel measure in terms of these invariants. We also prove some new formal degree formulas, a transfer-of-measure formula for GL(n), and a transfer-of-measure formula from GL(n,F) to GL(m,D).

math.RT↗
arXiv2003

Supports unipotents de faisceaux caracteres

Soit G un groupe algebrique reductif sur la cloture algebrique d'un corps fini F_q et defini sur ce dernier. L'existence du support unipotent d'un caractere irreductible du groupe fini G(F_q), ou d'un faisceau caractere de G, a ete etablie dans differents cas par Lusztig, Geck et Malle, et le second auteur. Dans cet article, nous demontrons que toute classe unipotente sur laquelle la restriction du faisceau caractere ou du caractere donne est non nulle est contenue dans l'adherence de Zariski de son support unipotent. Pour etablir ce resultat, nous etudions certaines representations des groupes de Weyl, dites "bien supportees". Let G be a reductive algebraic group over the algebraic closure of a finite field F_q that is defined over F_q. Under various conditions, Lusztig, Geck and Malle, and the second author have shown that an irreducible character of the finite group G(F_q), or a character sheaf on G, has a unipotent support. In this paper, we show that every unipotent class on which a given character or character sheaf has nonzero restriction is contained in the Zariski closure of its unipotent support. This result is obtained by studying the class of so-called "well-supported" representations of Weyl groups.

math.RT↗