Associated Varieties of Modules Over Kac-Moody Algebras and C2-Cofiniteness of W-Algebras

First, we establish the relation between the associated varieties of modules over Kac-Moody algebras g and those over affine W-algebras. Secondly, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact, we show that the associated varieties of G-integrable admissible representations are irreducible Ad G-invariant subvarieties of the nullcone of g, by determining them explicitly. Thirdly, we prove the C-2-cofiniteness of a large number of simple W-algebras, including all the minimal series principal W-algebras [22] and the exceptional W-algebras recently discovered by Kac-Wakimoto [31].