Sheaf-Theoretic 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 SU(2), the group of 2x2 unitary complex matrices, and O(3,R), the group of orthogonal 3x3 matrices with real entries. 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. This research will use geometric methods to make advances in modular representation theory. Many of the anticipated results are motivated by known facts in complex representation theory, but new tools and techniques must be developed in the modular case. In connection with this research, the P.I. will also undertake research-training activities aimed at Ph.D. students and other early-career researchers. 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 research will build on these developments with projects on three different topics: (i) the topology of global Schubert varieties; (ii) Kazhdan-Lusztig cells, tensor ideals, and tilting modules; and (iii) "silting" complexes of coherent sheaves. Topic (i) has connections to number theory and to a potential "modular ramified Satake equivalence". Topic (ii) has the most direct links to classical questions in representation theory, while topic (iii) is expected to lead to new avenues of research in K-theory and categorification, for instance in the context of symmetric spaces. 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.