REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

Ben Webster

ABSTRACT: Contrary what some undergraduates think, not all vector spaces come into our lives equipped with bases. In particular, an irreducible representation of your favorite group or Lie algebra doesn't seem to have any one basis which is better than all the others.  Certainly none is part of the obvious structure of the representation.

But sometimes appearances are deceiving: any finite dimensional representation of a semi-simple Lie group or Lie algebra (like the special linear group) really does have a distinguished basis, which is "best" for certain purposes, the "canonical basis" of Lusztig.

This basis is actually a hint of a much more deep and surprising structure; the representations in question actually have categories which upgrade their structure (one can think of this as defining maps between certain of the vectors of the representation).  This is a big and complicated picture, but I'll explain what I can of it in 1 hour.