Representations of finite reductive groups, character sheaves and theory of total positivity

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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.More precisely, the central aims of this project are to (1) investigate a new approach to representations of Weyl groups and unipotent representations of finite reductive groupsin terms of a new basis of the Grothendieck group; (2) investigate a new formulation of the character formula for semisimple groups in positive characteristic; (3) study Hecke algebras with unequal parameters in the framework of the theory of parabolic character sheaves; (4) study strata of a reductive group; and finally (5) investigate new W-graphs associated to involutions in two-sided cells of a Coxeter groups.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.