Sheaves on adic groups for p-adic representation theory
by
Anne-Marie Aubert
DMA, Ecole Normale Supérieure, Paris, France
Coauthors: Clifton Cunningham (University of Calgary, Canada)

The beginnnig of the course will be devoted to the structure of algebraic groups over a field or a discrete valuation ring (we will adopt the point of view of group schemes). We will explain that algebraic groups over finite fields occur as reductive quotients of some bounded subgroups of algebraic groups over non Archimedean local fields (called parahoric subgroups). Then we will define Bruhat-Tits buildings and show how to associate an adic algebraic group G_F^ad to any algebraic group G over the unramified closure F of a p-adic field, and an adic affinoid group G_{x, r}^ad to each Moy-Prasad filtration subgroup G_{x, r} of a parahoric subgroup of G (here x is a point in the Bruhat-Tits building of G(F) and r a real). We will also describe the corresponding adic affinoid Lie algebras and give several examples of parahoroid groups and Lie algebras.

Then we will define a continuous morphism p_{x, r} from G_{x, r}^ad to M_{x, r} in the category of ringed group objects, which we call the reduction morphism, where M_{x, r} is a reductive group scheme over the residue field k of F. We will use the reduction morphism p_{x, 0} to define a depth-zero stratification of G_{x, 0}^ad. We will recall the notion of étale sheaves and use them to define fundamental strata for G_F^ad. In the sequel of the course, we will describe Deligne-Lusztig theory of complex characters of finite groups of Lie type and partition the set of irreducible characters of any finite group of Lie type G(q) into Lusztig series, each series being attached to a semisimple element in the Langlands dual of G(q) (that is, the group with root system dual to those of G(q)). In general, Deligne-Lusztig characters do not provide a basis of the space of class functions on G(q). In order to obtain such a basis, we will describe Lusztig theory of character sheaves (they are certain perverse sheaves on an algebraic group over an algebraic closure of a finite field F_q and define over F_q). In particular, we will describe induction and restriction of character sheaves.

Finally, we will explain how the theory of l-adic cohomology on adic spaces should provide a connection between characteristic functions of character sheaves and characters of certain smooth representations of p-adic groups.

Date received: March 2, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagt-04.