Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras

Abstract

We extend Kostant's result on annihilator ideals of non-singular simple Whittaker modules over Lie algebras to (possibly singular) simple Whittaker modules over Lie superalgebras. We describe these annihilator ideals in terms of certain primitive ideals coming from the category $\mathcal O$ for quasireductive Lie superalgebras. To determine these annihilator ideals, we develop annihilator-preserving equivalences between certain full subcategories of the Whittaker category $\tilde{\mathcal N}$ and the categories of certain projectively presentable modules in the category $\mathcal O$. These equivalences lead to a classification of simple Whittaker modules that lie in the integral central blocks when restricted to the even subalgebra. We make a connection between the linkage classes of integral blocks of $\mathcal O$ and of $\tilde{\mathcal N}$. In particular, they can be computed via Kazhdan-Lusztig combinatorics for Lie superalgebras of type $\mathfrak{gl}$ and $\mathfrak{osp}$. We then give a description of the integral blocks of the category $\tilde{\mathcal N}$ of Whittaker modules for Lie superalgebras $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2|2n)$ and $\mathfrak{pe}(n)$.