Regular supercuspidal representations

Kaletha, Tasho¹

¹ Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Journal of the American Mathematical Society, Tome 32 (2019) no. 4, pp. 1071-1170

Résumé

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive $p$-adic group $G$ arise from pairs $(S,\theta )$, where $S$ is a tame elliptic maximal torus of $G$, and $\theta$ is a character of $S$ satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive $p$-adic groups.

Détail

DOI : 10.1090/jams/925

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