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First, we establish the relation between the associated varieties of modules over Kac–Moody algebras   ${\hat {\mathfrak {g}}}$   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   ${{\mathrm {Ad}}}G$  -invariant subvarieties of the nullcone of   ${\mathfrak {g}}$  , 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 ].

Associated Varieties of Modules Over Kac–Moody Algebras and C 2 -Cofiniteness of W -Algebras