Patrick Lei

inFormal categOry O graD seminar (Fall 2021)

We plan to cover roughly the first half (7-8 chapters) of Humphreys’s book on category O ([H]) over the course of this semester. We will discuss various constructions in category O such as Vermas and contragredients, homological aspects such as relating to Exts, the BGG resolution, translation functors, and maybe some Kazhdan-Lusztig theory. This will probably take the majority of the semester, but if there is time left (and maybe next semester), we can discuss more geometric aspects, such as localization, Springer theory, the proof of the KL conjecture, and other topics that participants are interested in. We are inspired by the seminar run by Cailan, Henry, and Mrudul last year, but will (probably) remain much more basic than their seminar this semester.

Schedule
Sep 29
Kevin Chang
Review of semisimple Lie algebras and introduction to category O
I will begin by reviewing the structure theory and finite-dimensional representation theory of semisimple Lie algebras. I will finish by defining category O and describing some of its objects (Verma modules).
Reference: [H], Chapter 1
Notes
Oct 06
Fan Zhou
Beginnings in category O: Vermas, central characters, and blocks
We begin the study of category O by discussing some of its main characters (Vermas and their simple quotients) as well as central characters (Harish-Chandra) and “blocks” labeled by them.
Oct 13
Che Shen
Formal characters and application to finite dimensional modules
We will define formal characters for modules in category O. We will use them to derive the classical formulas of Weyl and Kostant on finite-dimensional modules, following the approach of Bernstein-Gelfand-Gelfand.
Oct 20
Kevin Chang
Duality and projectives in category O
In the first part of the talk, we will discuss how to take duals of representations in category O. In the second part, we will discuss the properties of projectives in category O. Along the way, we will talk about other important topics like dominance and standard filtrations.
Oct 27
Fan Zhou
Nov 3
TBD
Nov 10
TBD
Nov 17
TBD
Nov 24
No lecture (Thanksgiving)
Dec 01
TBD
Dec 08
TBD