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.

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
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
more on the structure, simplicity criternion, and embedding of verma modules
We move on to Chapter 4 of Humphreys. We will further discuss the module structure of Vermas, embeddings between them, and a criterion for when they are simple. We will also take a field trip to Hunan, where much of this theory was first developed. (Little known fact: the name U(n) comes from hUNan.)
Nov 3
Patrick Lei
cats, bondage, and why you can’t do representation theory without geometry
We define strong linkage, and then discuss the Jantzen filtration and use it to prove a relationship between strong linkage and multiplicity in the composition series of a Verma.
Reference: [H], Chapter 5
Nov 10
Fan Zhou
why you should care about the bgg resolution, and more on homological considerations in category o
We move on to roughly Chapter 6 of [H]. We will prove the BGG resolution (modulo some combo things that nobody wants to see), possibly from more than one perspective. We will discuss the relationship between BGG and Lie algebra cohomology (namely Kostant/Bott), and on that topic turn to more homological considerations such as Ext groups. We will also take a field trip to the land South of the Clouds, where much of this theory was second developed. Little known fact: the name U(n) actually comes from yUNnan. That’s a rock fact!
Nov 17
Fan Zhou
Vermas and simples under the translation functor, facets, chambers, and walls
Perhaps after some last remarks on homological matters, we move on to Chapter 7 of Humphreys, which discusses linguistics, parkour, and chamber music. Linguistics will appear because we will study translation functors and their actions on familiar friends such as Vermas, simples, and projectives, parkour will appear because we will do some wall-crossing, and chamber music appears because chambers and musical notation (the so-called natural sign) appear. A warning: many rock facts will appear due to facets, chambers, and walls.
Nov 24
Fan Zhou
THANKS to our work so far, we will be GIVING another proof of bgg
In this bonus talk, we will discuss a filtration on category O, more homological magic in that setting, and how this all pertains to another proof of the BGG resolution whose key input is Kostant’s theorem on (co)homology. Recall that BGG implies Kostant, so maybe worth noting is then the equivalence between Kostant and BGG. For those also following the DWIC seminar, this gives an example of how derived categories are used in representation theory.
Dec 01
Kevin Chang
Kazhdan-Lusztig theory
In the first part of this talk, I will construct the Kazhdan-Lusztig polynomials (with proofs). In the second part of this talk, I will discuss their geometric origins (without proofs).
Dec 08
Patrick Lei
Koszul duality for people who aren’t Peter May
I will tell you what a Koszul algebra is and what Koszul duality is, and then I will tell you how this is related to category O. Note: no operads were harmed during the formation of this lecture.