Character Sheaves and Depth-zero representations

This paper proves the existence of global rational structures on spaces of cusp forms of general reductive groups. We identify cases where the constructed rational structures are optimal, which includes the case of GL($n$). As an application, we deduce the existence of a natural set of periods attached to cuspidal automorphic representations of GL($n$). This has consequences for the arithmetic of special values of $L$-functions that we discuss in subsequent articles. In the c...

In this paper, we give a new realization of the local Langlands correspondence for PGL(2,F), where F is a p-adic field of odd residual characteristic. In this case, supercuspidal representations of PGL(2,F) are parameterized by characters of elliptic tori. Taking a cue from real groups, we propose that supercuspidal representations are naturally parameterized by characters of covers of tori. Over the reals, Harish-Chandra defined the discrete series representations by specify...

This is a report on joint work with T. Hausel and L. Migliorini, where we prove, for each of the groups GL(2,C), PGL(2,C), SL(2,C), that the non-Abelian Hodge theorem identifies the weight filtration on the cohomology of the character variety with the perverse Leray filtration on the cohomology of the domain of the Hitchin map. We review the decomposition theorem, N\^go's support theorem, the geometric description of the perverse filtration and the sub-additivity of the Leray...

This article provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed by W.Schmid in [Sch]. A corresponding problem in the compact group setting was solved by N.Berline, E.Getzler and M.Vergne in [BGV] by an application of the theory of equivariant forms and particularly the fixed point integral localization formula. This article (besides its representation-theoretical significance) provides a whol...

In the first chapters, this paper contains a survey on the theory of ordinary characters of finite reductive groups with non-connected centre. The last chapters are devoted to the proof of Lusztig's conjecture on characteristic functions of character sheaves for all finite reductive groups of type A, split or not.

This work is intended as an introduction to the statement and the construction of the local Langlands correspondence for GL(n) over p-adic fields. The emphasis lies on the statement and the explanation of the correspondence.

These are notes for my Takagi lecture at the University of Tokyo in November, 2016. I survey what is known about simple modules for reductive algebraic groups. The emphasis is on characteristic p>0 and Lusztig's character formula. I explain ideas connecting representations and constructible sheaves (Finkelberg-Mirkovic conjecture) in the spirit of the Kazhdan-Lusztig conjecture. I also discuss a conjecture with S. Riche (a theorem for GL_n) which should eventually make comput...

Ces notes en fran\c{c}ais sont un r\'esum\'e de la pr\'epublication arXiv:1402.2501, o\`u en collaboration avec Peter Schneider, nous \'etablissons des formules de caract\`ere pour la s\'erie discr\`ete de GL(N) d'un corps local non archim\'edien. Elles s'adressent \`a un public de sp\'ecialistes suffisamment \`a l'aise avec les notations et les concepts de la Th\'eorie des Types de Bushnell et Kutzko. These notes, written in french, are a summary of the preprint arXiv:1402...

This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $p$-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which Harish-Chandra local character expansion holds). We construct a parameter space $B$ (that depe...

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in \cite{L3}. We also prove some properties of the character sheaves on the group compactification.