A Whittaker category for the Symplectic Lie algebra

Abstract

For any $n\in {\mathbb{Z}}_{\ge 2}$, let ${\mathfrak{m}}_n$ be the subalgebra of ${\mathfrak{sp}}_{2n}$ spanned by all long negative root vectors $X_{-2{\epsilon }_i}$, $i=1,\dots ,n$. An ${\mathfrak{sp}}_{2n}$-module $M$ is called a Whittaker module with respect to the Whittaker pair $({\mathfrak{sp}}_{2n},{\mathfrak{m}}_n)$ if the action of ${\mathfrak{m}}_n$ on $M$ is locally finite, according to a definition of Batra and Mazorchuk. This kind of modules are more general than the classical Whittaker modules defined by Kostant. In this paper, we show that each non-singular block ${\mathcal{W}\mathcal{H}}_{\mathbf{a}}^{\mu }$ with finite dimensional Whittaker vector subspaces is equivalent to a module category ${\mathcal{W}}^{\mathbf{a}}$ of the even Weyl algebra ${\mathcal{D}}_n^{ev}$ which is semi-simple. As a corollary, any simple module in the block ${\mathcal{W}\mathcal{H}}_{\mathbf{i}}^{-\frac{1}{2}{\omega }_n}$ for the fundamental weight ${\omega }_n$ is equivalent to the Nilsson's module $N_{\mathbf{i}}$ up to an automorphism of ${\mathfrak{sp}}_{2n}$. We also characterize all possible algebra homomorphisms from $U({\mathfrak{sp}}_{2n})$ to the Weyl algebra ${\mathcal{D}}_n$ under a natural condition.