On finite dimensional representations of finite W-superalgebras

Abstract

Let $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}+{\mathfrak{g}}_{\bar{1}}$ be a basic Lie superalgebra, ${\mathcal{W}}_0$ (resp.$\mathcal{W}$) be the finite W-(resp.super-) algebras constructed from a fixed nilpotent element in ${\mathfrak{g}}_{\bar{0}}$. Based on a relation between finite W-algebra ${\mathcal{W}}_0$ and W-superalgebra $\mathcal{W}$ found recently by the author and Shu, we study the finite dimensional representations of finite W-superalgebras in this paper. We first formulate and prove a version of Premet's conjecture for the finite W-superalgebras from basic simple Lie superalgebras. As in the W-algebra case, the Premet's conjecture is very close to give a classification to the finite dimensional simple $\mathcal{W}$-modules. In the case of $⋙$ is Lie superalgebras of basic type \Rmnum{1}, we prove the set of simple $\mathcal{W}$-supermodules is bijective with that of simple ${\mathcal{W}}_0$-modules; presenting a triangular decomposition to the tensor product of $\mathcal{W}$ with a Clifford algebra, we also give an algorithm to compute the character of the finite dimensional simple $\mathcal{W}$-supermodules with integral central character.

arXiv commentV2, typos corrected,16 pages, a part overlap with arxiv:2002.10604. V3 a bad but not serious error is corrected( see Proposition 4.1) V4 accepted version

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