- organizers: Kevin Chang, Patrick Lei, Fan Zhou
- when: Wednesday 4-5:30 PM (Note: dinner offered!)
- where: 622 Mathematics
- notes from the seminar

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.

- 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