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Session S21 - Galois representations and automorphic forms

Wednesday, July 21, 16:00 ~ 16:40 UTC-3

Ramification of supercuspidal parameters

Michael Harris

Columbia University, United States   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 {\it pure} or {\it mixed}.

Assume $G$ is split semisimple. In work in progress with Gan and Sawin, 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 $G$. The last hypothesis allows us to use Fintzen's result that the representation is obtained by compact induction. 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, and we have some results on mixed supercuspidals as well. The method is specific to local fields of positive characteristic.

Joint work with Wee Teck Gan and Will Sawin.

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