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

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

2010-04-09

arxiv.org/abs/1004.1554v1

Mathematics

Quantum Algebra

Scientific paper

First, we establish the relation between the associated varieties of modules over Kac-Moody algebras $\hat{g}$ and those over affine W-algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of $G$-integrable admissible representations for the degenerate cases in the sense of Frenkel-Kac-Wakimoto. In fact we show that the associated variates of $G$-integrable degenerate admissible representations are irreducible $\Ad G$-invariant subvarieties of the nullcone of $g$, by determining them explicitly. Third, we prove the $C_2$-cofiniteness of a large number of simple W-algebras, including all the (non-principal) exceptional W-algebras recently discovered by Kac-Wakimoto.

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