Email: snshah at math dot columbia dot edu

Office: Math 505

Tuesday and Thursday lectures are from 2:40-3:55pm in Math 407.

Friday lectures will be announced here and are from 1-2:30pm in Math 622.

The structure and representation theory of p-adic reductive groups plays a central role in the theory of automorphic forms. Via conjectures of Langlands, these topics mirror aspects of the representation theory of Galois groups of local fields. This course aims to develop a clear picture of the category of representations as well as of the internal structure of a single representation. We will emphasize concrete results over abstract theory, and incorporate examples as often as possible.

Friday lectures will be occasionally added in order to cover background material. Topics covered on Friday lectures will include the theory of reductive groups, finite groups and their representations, discussions of the global Langlands program, and applications of the main topics of the course.

The schedule below will be updated over the course of the semester.

- Bushnell and Henniart, the Local Langlands Conjecture for GL(2)
- Casselman's notes
- Many papers that will be linked in the lectures below.

- Friday, September 8th: Reductive groups and their classification. Structure and representation theory of reductive groups. References include a short article of Springer as well as the books of Springer, Borel, and Humphreys.
- Tuesday, September 12th: Motivation. Smooth representations of p-adic reductive groups and their basic properties. Admissible representations. (From Chapter 1 of Bushnell and Henniart.)
- Thursday, September 14th: Induction/compact induction. (From Chapter 1 of Bushnell and Henniart.)
- Friday, September 15th: Application to p-adic deformation of automorphic representations. (See Section 3 of Urban's paper.) Weil-Deligne representations. (See Tate's article.) Local Langlands conjecture for GL_n. (See this survey article of Kudla.)
- Tuesday, September 19th: Hecke algebras. (From Chapter 1 of Bushnell and Henniart.) Definition of principal series and overview of main theorems concerning them.
- Thursday, September 21st: "Shallow" structure theory, with proofs for GL_2 and GL_n. Define presentations of spherical and Iwahori-level Hecke algebras. (See Gross's article for a statement/discussion of the Satake isomorphism and Rostami's paper for the Iwahori-Hecke algebra in full generality.)
- Tuesday, September 26th: Satake isomorphism, with emphasis on GL_n, following Bump. Alternate viewpoint using the "universal principal series" following Haines, Kottwitz, and A. Prasad. (Also see Cartier's article for an accessible account of a rather general case. The most general results are due to Haines and Rostami and Haines.)
- Thursday, September 28th: Explicit computation of Satake transform of minuscule cocharacters. For a discussion of computational aspects of the Satake transform, see this paper of Panchishkin and Vankov.
- Tuesday, October 3rd: Jacquet modules and admissibility. (From Chapter 3 of Casselman's notes.)
- Thursday, October 5th: Coxeter systems. Buildings. (See Garrett's self-contained introduction.) Iwahori-Matsumoro presentation of the Hecke algebra. (See Garrett's notes on representations with Iwahori-fixed vectors.)
- Tuesday, October 10th: Jacquet module of unramified principal series, I: projection of Iwahori invariants. (From Chapter 6 of Casselman's notes and Casselman's paper.)
- Thursday, October 12th: Jacquet module of unramified principal series, II: filtration of the unramified principal series via Bruhat cells. (From Chapter 6 of Casselman's notes.)
- Tuesday, October 17th: Jacquet module of unramified principal series, III: Jacquet module of the associated graded. (From Chapter 6 of Casselman's notes. An alternate reference is the proof of (VI.5.1.3) in Renard's book on pages 245-251.)
- Thursday, October 19th: Normalization of intertwining operators. (From Section 3 of Casselman's paper.)
- Tuesday, October 24th: Irreducibility of principal series. (From Section 3 of Casselman's paper.)
- Thursday, October 26th: Proof of Macdonald's formula. (From Section 4 of Casselman's paper.)
- Friday, October 27th: Example of explicit Satake: Spin L-function of GSp_4. (From Appendix of this paper.)
- Tuesday, October 31st: Whittaker models, I: Uniqueness. (From Section 1 of Casselman and Shalika.)
- Thursday, November 2nd: Whittaker models, II: Holomorphy and value on simple cells. (From Sections 2, 3, and 4 of Casselman and Shalika.)
- Friday, November 3rd: An application of explicit Satake: the method of Piatetski-Shapiro and Rallis. (From Section 3 of this paper.)
- Friday, November 10th: Whittaker models, III: the formula of Casselman and Shalika. (From Section 5 of Casselman and Shalika.)
- Tuesday, November 14th: Supercuspidals, I: Matrix coefficients. (From Chapter 5 of Casselman's notes.)
- Thursday, November 16th: Supercuspidals, II: Injectivity and Projectivity. (From Chapter 5 of Casselman's notes.)
- Tuesday, November 21st: The Steinberg representation. (From Chapter 8 of Casselman's notes and Section 10 of these notes of Savin.)
- Tuesday, November 28th: Supercuspidals, III: An example for SL_2(Q_2). (From Section 11 of these notes of Savin.)
- Thursday, November 30th: Theory of Bernstein, I: The Bernstein decomposition. (From Roche's article in these proceedings.)
- Friday, December 1st: Theory of Bernstein, II: The Bernstein center. (From Roche's article in these proceedings.)