Local parameters of supercuspidal representations

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 $\CL^{ss}(π)$ 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 $\CL(π)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(π)$ 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 $\CL^{ss}(π)$ 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 Poincaré series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II.

Appendix by Raphaël Beuzart-Plessis added to version 3. The result on tempered Weil-Deligne parameters has been extended to discrete series, using the results of the Appendix. The result on ramification of pure supercuspidal parameters is now stated for general unramified reductive groups

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• On the generalized Ramanujan and Arthur conjectures over function fields