Whittaker categories of quasi-reductive Lie superalgebras and principal finite W-superalgebras

Abstract

We study the Whittaker category $\mathcal N(\zeta)$ of the Lie superalgebra $\mathfrak g$ for an arbitrary character $\zeta$ of the even subalgebra of the nilpotent radical associated with a triangular decomposition of $\mathfrak g$. We prove that the Backelin functor from either the integral subcategory or any strongly typical block of the BGG category to the Whittaker category sends irreducible modules to irreducible modules or zero. The category $\mathcal N(\zeta)$ provides a suitable framework for studying finite $W$-superalgebras associated with an even principal nilpotent element. For the periplectic Lie superalgebras $\mathfrak{p}(n)$, we formulate the principal finite $W$-superalgebras $W_\zeta$ and establish a Skryabin-type equivalence. For a basic classical and a strange Lie superalgebras, we prove that the category of finite-dimensional modules over a given principal finite $W$-superalgebra $W_\zeta$ is equivalent to $\mathcal N(\zeta)$ under the Skryabin equivalence, for a non-singular character $\zeta$. As a consequence, we give a super analogue of Soergel's Struktursatz for a certain Whittaker functor from the integral BGG category $\mathcal O$ to the category of finite-dimensional modules over $W_\zeta$.