Lecture on Springer theory

Announcements

No session on Friday, June 30.

General information

Dr. Konstantin Jakob, summer term 2023

Lecture: Mondays, 15:45-17:15 (changed), S2|15, Room 234, starting April 17, 

Exercise: Fridays, 11:40-13:20, S2|15, Room 51, starting April 21

Zoom-ID: 662 0229 8324, Password: Largest palindromic 9 digit prime 

The course will be held in English, and in hybrid format. If you are interested in attending, please email me directly. 

Contents

This lecture is intended to serve as an introduction to geometric representation theory. 

The classical Springer correspondence for a simple algebraic group G is a bijection between irreducible representations of the Weyl group W of G, and a refinement of nilpotent orbits of Lie(G). In its most basic form, this correspondence gives a bijection between irreducible representations of the symmetric group and conjugacy classes of nilpotent matrices. 

The object that serves as a bridge is a certain subvariety (called Springer fiber) of the variety of full flags in a vector space, related to a nilpotent endomorphism N. To obtain the correspondence, one has to construct an action of the Weyl group on the cohomology of Springer fibers. 

I will cover one of several approaches to the Springer correspondence, and some of the required background. Time permitting, I hope to discuss affine and global analogues of Springer theory, involving actions of the affine Weyl group on the cohomology of affine Springer fibers and parabolic Hitchin fibers. 

Prerequisites

Algebraic geometry, linear algebraic groups, representation theory, semisimple Lie algebras. If you have any questions, please contact me directly. 

Literature

N. Chriss, V. Ginzburg: Representation Theory and Complex Geometry

Z. Yun: Lectures on Springer theories and orbital integrals, available here