Geometric Methods in the Representation Theory of Affine Hecke Algebras, Finite Reductive Groups, and Character Sheaves

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures, using groups of invertible matrices. Representation theory of finite groups has numerous applications to other areas, including number theory and mathematical physics. In this project the linear structures are themselves finite matrix groups, or more generally matrix groups whose entries satisfy divisibility properties with respect to a fixed prime number. Geometric and combinatorial techniques will be brought to bear to study representations of these groups, especially in the important case when the representing matrices themselves have entries in a finite field. A central aim of this project is to continue the new approach to the representation theory of reductive groups over a finite field and the theory of character sheaves on reductive groups in which the notion of categorical center plays a key role. This will lead to a better understanding of the classification of irreducible representations and of character sheaves, and in particular will help to remove some non-canonical features from earlier work in the area. The project is also concerned with the study of characters of irreducible or tilting modules of a semisimple group in positive characteristic, and with the study of the canonical basis of Hecke algebras with unequal parameters using the theory of parabolic character sheaves. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.