Abstracts for Seminar "Arithmetic Geometry and Representation Theory"

I will explain my recent work with Masao Oi on parametrizing the local Langlands correspondence using the Kottwitz set and try to justify the assertion that our construction is "natural". I will discuss some instructive examples as well as our partial progress on a generalized endoscopic character identity.

Central sheaves were introduced by Gaitsgory in order to geometrize the identification of the center of the Iwahori-Hecke algebra and the spherical Hecke algebra. This concept is an important step towards Bezrukavnikov's equivalence. After a wrap-up concerning the motivic Satake equivalence, I will talk about a motivic approach to a central sheaves functor. This is joint work with Robert Cass and Thibaud van den Hove.

Supercuspidal representations are the elementary particles in the representation theory of reductive p-adic groups and arise in number theory as local factors of cuspidal automorphic representations. Constructing such representations explicitly, via (compact) induction, is a longstanding open problem. Although the problem has been solved for large p, a solution remains out of reach in general. I'll discuss work in progress joint with Jessica Fintzen towards constructing some of the missing "tamely ramified" supercuspidals when p is (very!) small.

In the study of special values of L-functions and p-adic L-functions, it is often necessary to have a good theory of p-adic families of nearly holomorphic automorphic forms (nearly overconvergent forms) and the p-adic iteration of Maass--Shimura differential operators. There are several candidates for this theory in the literature, however there is usually a restriction, e.g., on the slopes of the nearly overconvergent forms or the p-adic variation of the differential operators.