Algebra Seminar: Ram -- Combinatorics of representations of affine Lie algebras

This is the second lecture in a series of 3 by Arun Ram (University of Melbourne), the
other two are on June 14 and 17.  

Thursday 16 June, 12-1pm, Place: Carslaw 375 

Combinatorics of representations of affine Lie algebras 

This will be a survey of my current understanding of the combinatorial representation
theory of affine Lie algebras.  For category O at negative level, Verma modules have
finite composition series with decomposition numbers determined by Kazhdan-Lusztig
polynomials.  The structure of affine Weyl group orbits controls the pretty patterns.
For category O at positive level, Verma modules have infinite compositions with
decomposition numbers given by inverse Kazhdan-Lusztig polynomials, and at critical
level, the patterns correspond to the periodic Kazhdan-Lusztig polynomials. I’ll also
discuss parabolic category O.  Finite dimensional modules (which are level 0) are
indexed by Drinfeld polynomials and then there are various collections of smooth
representations where our combinatorial understanding has increased greatly in recent
years.