Introduction to geometric representation theory and local geometric Langlands

This is the website for the Fall 2024 iteration of the Algebra Participating Seminar (Math 290C).

Synopsis 

Geometric representation theory is a large subject, largely driven by applications, with many avenues of approach. In this course, we will take a largely historical route, by focusing on three discoveries from the early days of the subject which have been extremely influential on subsequent developments. These are (i) Kazhdan--Lusztig theory and Beilinson--Bernstein localization, (ii) Springer theory, and (iii) geometric Satake. Our goal will be to learn both the basic ideas of each of the three topics in some amount of detail, with an emphasis on examples, as well as the interrelations  between the three - for this, the language of categorical representation theory is  a convenient organization tool. 

Logistics

We meet 3:00-3:50pm on Wednesdays in MS 6201 - everyone is welcome, regardless of formal enrollment in Math 290C.   

The plan is to essentially cover the first four sections + the appendix of this survey, and scratch the surface of the remainder.  

Lectures (Preliminary schedule)

October 2nd - Overview of the seminar (Gurbir Dhillon).

October 16th - Representations of GL_n(F_q) and the finite Hecke algebra (Jacob Swenberg).

October 23rd - Function-sheaf dictionary and Springer theory (Zach Baugher). 

October 30th - Category O and translation functors (Colin Ni). 

November 6th and 8th - Kazhdan--Lusztig conjecture and Beilinson--Bernstein localization (Luna Gonzalez and Jon Hillery). 

November 13th - Categorical representations, character sheaves, Springer action as trace of translation functors (Gurkiran Dhaliwal). 

November 20th -Representations of GL_n(F_q((t))) and the affine Hecke algebra (John Zhou). 

December 4th - Geometric Satake equivalence and introduction to local geometric Langlands (Tom Han).