Explicit categorical mod l local Langlands correspondence for depth-zero supercuspidal part of GL_2

Abstract: Let $F$ be a non-archimedean local field. I will explicitly describe:

(1) (The category of quasicoherent sheaves on) The connected component of the moduli space of Langlands parameters over $\overline{\mathbb{Z}_l}$ containing an irreducible tame L-parameter with $\overline{\mathbb{F}_l}$ coefficients; (2) the block of the category of smooth representations of $G(F)$ with $\overline{\mathbb{Z}_l}$ coefficients containing a depth-zero supercuspidal representation with $\overline{\mathbb{F}_l}$ coefficients.

The argument works at least for (simply connected) split reductive group $G$, but I will focus on the example of $\mathrm{GL}_2$ for simplicity. The two sides turn out to match abstractly. If time permits, I will explain how to get the categorical local Langlands correspondence for depth-zero supercuspidal part of $\mathrm{GL}_2$ with $\overline{\mathbb{Z}_l}$ coefficients in Fargues-Scholze's form.

Mathematics

Audience: researchers in the topic

Comments: Hybrid talk

Zoom livestream: ID 743 736 2326 / Password 013049

PKU/BICMR Number Theory Seminar