Geometric Methods in Rep Theory Seminar – Thomas Haines (UMaryland)

Cellular pavings of convolution fibers and applications

Abstract. A convolution morphism is the geometric analogue of a convolution of functions in a Hecke algebra. The properties of fibers of convolution morphisms are used in a variety of ways in the geometric Langlands program and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibers of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, rationality of the BBD Decomposition Theorem over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence. If time permits, I will describe a new combinatorial model for generalized Mirkovic-Vilonen intersections and the branching to Levi subgroups.

Geometric Methods in Representation Theory