Cocenters and representations of $p$-adic groups

Working Seminar on Representation Theory
Topic:Cocenters and representations of $p$-adic groups
Speaker:Xuhua He
Affiliation:University of Maryland; von Neumann Fellow, School of Mathematics
Date:Wednesday, October 12
Time/Room:11:00am - 12:00pm/S-101

It is known that the number of conjugacy classes of a finite group equals the number of irreducible representations (over complex numbers). The conjugacy classes of a finite group give a natural basis of the cocenter of its group algebra. Thus the above equality can be reformulated as a duality between the cocenter (i.e. the group algebra modulo its commutator) and the finite dimensional representations. Now let us move from the finite groups to the $p$-adic groups. In this case, one needs to replace the group algebra by the Hecke algebra. The work of Bernstein, Deligne and Kazhdan in the 80's establish the duality between the cocenter of the Hecke algebra and the complex representations. It is an interesting, yet challenging problem to fully understand the structure of the cocenter of the Hecke algebra. We will discuss some recent progress in this direction and discuss how this can (conjecturally) help us to understand both complex representations and modular representations of $p$-adic groups.