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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - HAS (Harmonic Analysis and l-adic sheaves)

Teaser

The main goal of this project is the study of the category C(G) of smooth representations of reductive group G over a local non-archimedian field F.

Summary

The main goal of this project is the study of the category C(G) of smooth representations of reductive group G over a local non-archimedian field F.

Work performed

There are five different directions of my research which are stated in the order of their importance in my eyes.

a) The first is the development of the affine analogue of Lusztig\'s theory of character sheaves. Lusztig used successfully the algebra-geometric theory of perverse sheaves to study representations of reductive group over finite fields. Reductive groups over local non-archimedian fields can be considered as infinite dimensional variety over a finite field.

My goal is to extend the theory of perverse sheaves to such varieties and to use this new theory for the study of representations of reductive groups over local non-archimedian fields. An important step of this program is achieved in the preprint “Perverse sheaves on certain infinite-dimensional stacks and affine Springer theory” (a joint paper with A. Bouthier and Y. Varshavsky).

b) In the paper “IwahoriHecke algebras for p-adic loop groups” (a joint paper with A. Braverman and M. Patnaik) we defined and study the Iwahori-Hecke algebras for loop groups over non-archimedian local fields. We show that this algebra is closely related to Cherednik’s double affine Hecke algebra and gave an explicit description of the affine Satake isomorphism. This work could lead to a construction of the KL-basis in these algebras.

c) The natural map from the center of C(G) to its cocenter fails to be an isomorphism and this failure is a source of many difficulties.

The second part of the project is on a definition of a variant C\'(G) of the category C(G) such that the map from the center of C\'(G) to its cocenter is an isomorphism and on an application of this construction to the study of C(G). The first construction of C\'(G) and study of some of its properties is in the paper “Remarks on the asymptotic Hecke algebra” (a joint work with A. Braverman).

d) In the paper “Character values and Hochschild homology” (a joint work with R. Bezrukavnikov) we present a conjecture (and a proof for G = SL(2)) generalizing a result of J. Arthur which expresses a character value of a cuspidal representation of a group over a non-archimedian local field as a weighted orbital integral of its matrix coefficient.

e) In the paper “A spectral decomposition of orbital integrals for PGL(2, F)” I described the spectral decomposition of functionals of elliptic orbital for the group PGL(2, F).

Now I am finishing (in a joint work with S. Debacker) an extension of this result to the case of G=PGL(3, F).

Final results

A construction of the theory of character sheaves of depth zero.