Abstract:

In this talk, we will discuss the categorical action on the representation category of finite classical groups and its applications in representation theory. We construct a categorical double quantum Heisenberg action on the representation category of finite classical groups. Over a field of characteristic zero or positive characteristic, we deduce a categorical action of a Kac-Moody algebra on it. Furthermore, the categorical double quantum Heisenberg action gives rise to some new invariants. We show that those new invariants and the uniform projection can distinguish all irreducible characters of finite classical groups. We also show that the theta correspondence can explicitly determine the Kac-Moody action on the Grothendieck group of the whole category. If time permits, I will also discuss its application in some problems of modular representations of finite classical groups. This is a joint work with Peng Shan and Jiping Zhang.