Bun_G minicourse: Local Langlands

Abstract: This talk is the second part of a six-part series "$\mathrm{Bun}_G$, Shtukas, and the Local Langlands Program", held Tuesdays and Thursdays between 5 and 21 October, 2021.

Recordings and slides will appear here: sites.google.com/view/rampageseminar/home

Series abstract: The recent manuscript of Fargues-Scholze aims to "geometrize" the Langlands program for a p-adic group $G$, by relating the players in that story to the stack $\mathrm{Bun}_G$. Following a strategy of V. Lafforgue, the main result of [FS] is the construction of an L-parameter attached to a smooth irreducible representation of $G$.

The goal of this series is to review the main ideas of this work, and to discuss two related results: progress on the Kottwitz conjecture for local shtuka spaces by Hansen-Kaletha-Weinstein, and the construction of eigensheaves on $\mathrm{Bun}_G$ when $G=\mathrm{GL}_n$.

Talk abstract: We will review some representation-theoretic inputs to HKW. We’ll begin with reviewing the statements of the basic and refined local Langlands correspondence and the status of their proofs. We will then define the relative position of two members of a compound L-packet, which is an input to the Kottwitz conjecture, and the relative position of two regular semi-simple elements in inner forms. Based on the latter, we will define a Hecke transfer operator that transfers conjugation-invariant functions between inner forms, and discuss its effect on characters of supercuspidal representations.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

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Recent Advances in Modern p-Adic Geometry (RAMpAGe)

Series comments: The Zoom Meeting ID is: 995 3670 1681. The password for the series is: *The first three-digit prime*. Please visit the external homepage for notes and videos from past talks.

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