Ramification of supercuspidal parameters

Abstract: Let $G$ be a reductive group over a local field $F$ of characteristic $p$. Genestier and V. Lafforgue have constructed a semi-simple local Langlands parametrization for irreducible admissible representations of $G$, with values in the $\ell$-adic points of the $L$-group of $G$; the local parametrization is compatible with Lafforgue's global parametrization of cuspidal automorphic representations. Using this parametrization and the theory of Frobenius weights, we can define what it means for a representation of $G$ to be "pure".

Assume $G$ is split semisimple. In work in progress with three collaborators, whose names will be revealed on October 18, we have shown that a pure supercuspidal representation has ramified local parameter, provided the field of constants in $F$ has at least $3$ elements and has order prime to the order of the Weyl group of $G$. In particular, if the parameter of a pure representation $\pi$ is unramified then $\pi$ is a constituent of an unramified principal series. We are also able to prove in some cases that the ramification is wild.

number theoryrepresentation theory

Audience: researchers in the discipline

The 2020 Paul J. Sally, Jr. Midwest Representation Theory Conference

Series comments: The 44th Midwest Representation Theory Conference will address recent progress in the theory of representations for groups over non-archimedean local fields, and connections of this theory to other areas within mathematics, notably number theory and geometry.

In order to receive information on how to participate (to be sent out closer to the conference), please register by October 14 here: forms.gle/zFAnQBnuPGRnKzMr7