Whittaker coinvariants for $\mathrm{GL}(m|n)$

Abstract

Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra ${\mathfrak{gl}}_{m|n}(\mathbb{C})$. In this paper we study the {\em Whittaker coinvariants functor}, which is an exact functor from category $\mathcal{O}$ for ${\mathfrak{gl}}_{m|n}(\mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergel's functor $\mathbb{V}$ in the setting of category $\mathcal{O}$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $\mathcal{O}$ up to Morita/derived equivalence.

58 pages, minor changes

References in corpus (2)

• 2-Kac-Moody algebras
• Homological invariants in category O for the general linear superalgebra

Cited by in corpus (2)

• Minimal W-superalgebras and modular representations of basic Lie superalgebras
• Gorenstein homological algebra for rngs and Lie superalgebras