Geometric Methods in Modular Representation Theory

Louisiana State University, Baton Rouge LA

Investigators

Abstract

A matrix group is a set of invertible square matrices that contains all products and inverses of its members. Typical examples include O(3,R), the group of orthogonal 3x3 matrices with real entries, and SU(2), the group of 2x2 unitary complex matrices. Broadly speaking, the subject of representation theory deals with how such groups can act on a (complex) vector space via linear transformations. One can then ask what happens if we replace the complex numbers by a finite field (or the algebraic closure of a finite field). Modular representation theory is concerned with matrix groups with entries in such a field, acting on vector spaces over the same field. The proposed research will make advances in modular representation theory by geometric methods. Many of the anticipated results are analogous to known facts in complex representation theory, but new tools and techniques must be developed in the modular case. The past few years have seen the emergence of powerful new tools for applying geometric methods to the representation theory of algebraic groups in positive characteristic, including parity sheaves and the mixed modular derived category. This project will build on these developments with projects on three different topics: (i) monodromy operators in positive characteristic; (ii) generalized Springer theory for coherent sheaves; and (iii) Kazhdan-Lusztig cells, tensor ideals, and tilting modules. Topic (i) is essentially geometric in nature, while topics (ii) and (iii) are expected to have concrete consequences for representation theory. 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.