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.