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

TL;DR

This paper explores the Whittaker coinvariants functor for the general linear Lie superalgebra, establishing its properties, computing the algebra's center, and analyzing implications for category O block classifications.

Contribution

It introduces and studies the Whittaker coinvariants functor for $ ext{GL}(m|n)$, paralleling Soergel's functor, and computes the center of the associated $W$-algebra.

Findings

The functor has properties similar to Soergel's functor in Lie algebra category O.

The center of $W_{m|n}$ is explicitly computed.

Results impact the classification of blocks in category O.

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.