Local parameters of supercuspidal representations

TL;DR

This paper refines the local Langlands correspondence for supercuspidal representations of reductive groups over local fields of positive characteristic, establishing ramification properties of associated parameters using global methods.

Contribution

It uniquely refines the Genestier-Lafforgue parameter to a tempered L-parameter and analyzes ramification properties of these parameters for supercuspidal representations.

Findings

Refinement of Genestier-Lafforgue parameters to tempered L-parameters.

Ramification properties of parameters for unramified groups and supercuspidal representations.

Conditions under which parameters are ramified or wildly ramified.

Abstract

For a connected reductive group $G$ over a non-archime\-dean local field $F$ of positive characteristic, Genestier and Lafforgue have attached a semisimple parameter $\CL^{ss}(\pi)$ to each irreducible representation $\pi$. Our first result shows that the Genestier-Lafforgue parameter of a tempered $\pi$ can be uniquely refined to a tempered L-parameter $\CL(\pi)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(\pi)$ for unramfied $G$ and supercuspidal $\pi$ constructed by induction from an open compact (modulo center) subgroup. If $L^{ss}(\pi)$ is pure in an appropriate sense, we show that $\CL^{ss}(\pi)$ is ramified (unless $G$ is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal{L}^{ss}(\pi)$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincar\'e series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II.