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.
My work with Fargues on the geometrization of the local Langlands correspondence works with l-adic cohomology and hence the resulting L-parameters depend, a priori, on the auxiliary prime l. In this talk, I want to explain an approach for obtaining independence of l by upgrading our results to motivic coefficients, using the theory of rigid-analytic motives by Ayoub, Vezzani, ... .
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.
For any nice enough scheme with an action of the multiplicative group it is possible to define a t-structure on a certain category of monodromic D-modules using hyperbolic localization. I will discuss the construction of these t-structures as well as an application to geometric representation theory. Specifically, I will explain how the semi-infinite IC sheaf associated to a semisimple group arises as an intermediate extension in this t-structure for Drinfeld's Zastava spaces.
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.
In the original version of my work with Fargues, we constructed the spectral action with Z_\ell-coefficients, but omitting some small primes \ell. In the revised version, we removed this restriction on \ell, by giving an improved construction of the spectral action that also applies in the setting of Betti geometric Langlands.
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