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First, we establish the relation between the associated varieties of modulesover Kac-Moody algebras \hat{g} and those over affine W-algebras. Second, weprove the Feigin-Frenkel conjecture on the singular supports of G-integrableadmissible representations. In fact we show that the associated variates ofG-integrable admissible representations are irreducible G-invariantsubvarieties of the nullcone of g, by determining them explicitly. Third, weprove the C_2-cofiniteness of a large number of simple W-algebras, includingall minimal series principal W-algebras and the exceptional W-algebras recentlydiscovered by Kac-Wakimoto.

Associated varieties of modules over Kac-Moody algebras and $C_2$-cofiniteness of W-algebras