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.

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.

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 â€œPerverse sheaves on certain infinite-dimensional stacks and affine Springer theoryâ€ (a joint paper with A. Bouthier and Y. Varshavsky).

b) In the paper â€œIwahoriHecke algebras for p-adic loop groupsâ€ (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 Cherednikâ€™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 â€œRemarks on the asymptotic Hecke algebraâ€ (a joint work with A. Braverman).

d) In the paper â€œCharacter values and Hochschild homologyâ€ (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 â€œA spectral decomposition of orbital integrals for PGL(2, F)â€ 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).

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