Representation theory of W-algebras

摘要

We study the representation theory of the W- algebra W-k((g) over bar) associated with a simple Lie algebra (g) over bar at level k. We show that the "-" reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k epsilon C. Moreover, we show that the character of each irreducible highest weight representation ofW(k)((g) over bar) is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra g of (g) over bar. As a consequence we complete ( for the "-" reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of W- algebras.