Final Report Summary - GLC (Langlands correspondence and its variants)
The main achievements are in three areas:
a) Representations of groups over local fields
b) The categorical problematic relevant for the representation theory
c) Representations and Hecke algebras for affine Kac-Moody groups over local fields
Part a) The main results are on the understanding of the structure of the endoscopic decomposition of the Schwartz on reductive groups over local fields, the geometric construction of the Bernstein's second adjointness morphism and a study of a relation between geometric and spectral aspects of the structure of the Bernstein's center.
Part b) The development etale analogue of the Bezrukavnikov-Nadler D-module construction of the Lusztig's category of character sheaves, a construction of a theory of constructible sheaves for a class of infinite-dimensional varieties and a development of some aspects of the theory of infinity categories needed for applications of these constructions to the representations theory.
c) The development of the notion of spherical and Iwahory Hecke algebras, a definition and a computation of the Tamagawa number and the development of basics of the theory of Eisentein series for affine Kac-Moody groups over local fields.
a) Representations of groups over local fields
b) The categorical problematic relevant for the representation theory
c) Representations and Hecke algebras for affine Kac-Moody groups over local fields
Part a) The main results are on the understanding of the structure of the endoscopic decomposition of the Schwartz on reductive groups over local fields, the geometric construction of the Bernstein's second adjointness morphism and a study of a relation between geometric and spectral aspects of the structure of the Bernstein's center.
Part b) The development etale analogue of the Bezrukavnikov-Nadler D-module construction of the Lusztig's category of character sheaves, a construction of a theory of constructible sheaves for a class of infinite-dimensional varieties and a development of some aspects of the theory of infinity categories needed for applications of these constructions to the representations theory.
c) The development of the notion of spherical and Iwahory Hecke algebras, a definition and a computation of the Tamagawa number and the development of basics of the theory of Eisentein series for affine Kac-Moody groups over local fields.