Representation Theory of W-Algebras

Abstract

This paper is the detailed version of math.QA/0403477 (T. Arakawa, Quantized Reductions and Irreducible Representations of W-Algebras) with extended results; We study the representation theory of the W-algebra $W_k(g)$ associated with a simple Lie algebra $g$ (and its principle nilpotent element) 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. Moreover, we show that the character of each irreducible highest weight representation of $W_k(g)$ is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of $g$.