Categorical local Langlands seminar

Spring 2024

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

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