Categorical local Langlands seminar
Spring 2024

(Organized by Wee Teck and Dave)

The local Langlands correspondence predicts a precise parametrization of smooth representations of p-adic groups by Galois-theoretic data. In recent years it has become clear that this set-theoretic bijection should upgrade to an equivalence of categories. In this seminar we will explore the precise structure and consequences of this hoped-for equivalence, mostly following Dave's "Beijing notes" and building on the remarkable recent work of Fargues-Scholze. In the last two lectures we will go beyond the Beijing notes and outline a full proof of the categorical local Langlands conjecture for GL2, following work in progress of Hansen-Mann.

The seminar will meet on Thursdays from 12:15 to 2 in room S17-05-11 (except for the first meeting, see below).

Schedule

Lecture 1 (January 25): Overview
Speaker: Dave
This meeting ONLY in room S17-05-12.

Lecture 2 (February 1): Bun_G crash course
Material: As much of [FS2] as possible, and some of [H] Section 1.1.
Speaker: Jhan-Cyuan

Lecture 3 (February 8): Finiteness conditions and t-structures
Material: Finish [H] Section 1.1, and then cover [H] Section 1.2.
Speaker: Chung-Ru

Lecture 4 (February 15): Moduli stack of L-parameters; rough form of categorical conjecture
Material: [H] Section 1.3, filling in details via [Z] and Chapter 8 of [FS]; then Section 1.4 of [H].
Speaker: Bryan

Lecture 5 (February 22): Further properties of the categorical conjecture
Material: Most of sections 1.5-1.7 of [H].
Speaker: Dave

(no lecture February 29)

Lecture 6 (March 7): The B(G) local Langlands correspondence
Material: First four sections of [BMO], and the motivational discussion at the beginning of Section 2 of [H].
Speaker: Wee Teck

Lecture 7 (March 14): Generous L-parameters
Material: Section 2.1 of [H].
Speaker: Jhan-Cyuan

Lecture 8 (March 21): Trivial L-parameter
Material: Section 2.2 of [H]; if time permits, state the main result of Section 2.4 (with a brief indication of the proof).
Speaker: Nhat Hoang

Lecture 9 (March 28): Admissible sheaves and generic L-parameters
Material: Section 3.1 of [H].
Speaker: Alex

Lecture 10 (April 4): Generalized coherent Springer sheaves
Material: Section 3.2 of [H].
Speaker: Alex

Lecture 11 (April 11): Outline of the proof of categorical local Langlands for GL2
Material: [HM].
Speaker: Dave

Lecture 12 (April 18): Proof outline for GL2 cont'd; some comments on GLn
Material: [HM].
Speaker: Dave

References

[BMO] A. Bertoloni Meli and M. Oi, The B(G)-parametrization of the local Langlands correspondence
[FS] L. Fargues and P. Scholze, Geometrization of the local Langlands correspondence
[FS2] L. Fargues and P. Scholze, The Langlands program and the moduli of bundles on the curve
[H] D. Hansen, Beijing notes on the categorical local Langlands conjecture
[HM] D. Hansen and L. Mann, in preparation
[Z] X. Zhu, Coherent sheaves on the stack of Langlands parameters