## UW-PIMS Mathematics Colloquium: Jonathan Brundan

- Date: 03/09/2012
- Time: 14:30

University of Washington

Representations of General Linear Lie Superalgebras

Abstract

The general linear Lie algebra gln(C) -- endomorphisms of an

n-dimensional complex vector space V with operation [x,y] being the

commutator xy−yx -- is the most basic example of a Lie algebra. If we

assume instead that the vector space V is equipped with a Z/2-grading,

i.e. V=V0⊕V1 is the direct sum of an m-dimensional "even" and an

n-dimensional "odd" subspace, and replace commutator with the

"supercommutator" (which takes account of parity in the most natural

way), we get the general linear Lie superalgebra glm|n(C).

The representation theory of the general linear Lie algebra is

unbelievably rich and has a long history going back to Schur and Weyl,

who computed the characters of the finite dimensional irreducible

representations via the theory of symmetric functions. There is also an

important family of (mostly infinite dimensional) irreducible highest

weight representations for which an explicit character formula was

conjectured by Kazhdan and Lusztig in 1979, and dramatically proved by

Beilinson-Bernstein and Brylinski-Kashiwara in 1980, thereby giving

birth to a subject known as "geometric representation theory."

So what happens for the general linear Lie superalgebra? Even the finite

dimensional representations are quite difficult to understand, but now

we have a pretty good picture. There is also a version of the

Kazhdan-Lusztig conjecture which has just been proved (by Cheng, Lam and

Wang). In the talk I'll try to give you the flavor of these results,

starting with the base cases gl2(C) and gl1|1(C), which illustrate the

general picture perfectly despite being trivial from a combinatorial

perspective.