In this paper, using generating functions we study two categories E and C of modules for twisted affine Lie algebras ĝ[σ], which were firstly introduced and studied in [Li] for untwisted affine Lie algebras. We classify integrable irreducible ĝ[σ]-modules in categories E and C, where E is proved to contain the well known evaluation modules and C to unify highest weight modules, evaluation modul...

These are the notes of my talk at the conference “Double affine Hecke algebras and algebraic geometry” (MIT, May 18, 2010). The goal of this talk is to discuss some results and conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory for affine Lie algebras. These conjectures arose fr...

We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.

The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki, asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra g where g is gl ∞ or the affine Lie algebra A (1) l , and the irreducible representations correspond to the upper g...