## Category O Learning Seminar

This seminar focuses on learning about the Bernstein-Gelfand-Gelfand category O of representations of a finite-dimensional semisimple Lie algebra, primarily following J.E. Humphreys's book "Representations of Semisimple Lie Algebras in the BGG Category O".

### Organizational Meeting

by Brad Hannigan-Daley | University of Toronto
Time: 13:00  (Wednesday, May. 04, 2011)
Location: BA6183, Bahen Center, 40 St. George St.
Abstract:
This is the organizational meeting for a learning seminar on the BGG category O.

Suppose you are interested in the representations of a finite-dimensional complex semisimple Lie algebra. The category of all of its representations (finite-dimensional and otherwise) is too large to be understood algebraically. However, there is a nice subcategory, cut out by some finiteness conditions, which is small enough to be well-understood but large enough to contain most of the representations that we usually care about -- in particular, all of the finite-dimensional representations and all of the Verma modules (universal highest-weight modules).

This is the Bernstein-Gelfand-Gelfand (BGG) category O. Among the various reasons for studying this category, one can use it to realize the Weyl character formula, which computes the character of a finite-dimensional irrep V($\lambda$), as the Euler characteristic of the so-called BGG resolution of V($\lambda$). One of the goals in mind for this learning group is to understand this resolution. Another goal could be to understand the proof of the BGG theorem, which characterizes which Verma modules embed into a given Verma module. Yet another goal could be to understand at least the statement of the Kazhdan-Lusztig conjecture in this context; this may serve as some advertisement for Prof. Arkhipov's course next year on D-modules, which are used in the proof of this result.

The main references will probably be lecture notes by D. Gaitsgory (http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf) and/or the book "Representations of Semisimple Lie Algebras in the BGG Category O" by J.E. Humphreys (http://www.math.umass.edu/~jeh/bgg/main.pdf).

All are welcome.

#### Dates in this series

· Wednesday, May. 04, 2011: Organizational Meeting (Brad Hannigan-Daley)
· Wednesday, May. 11, 2011: Preliminaries and the PBW Theorem (Brad Hannigan-Daley)