A Whittaker category for the Symplectic Lie algebra

TL;DR

This paper introduces a new category of Whittaker modules for the symplectic Lie algebra, showing their equivalence to modules over the even Weyl algebra and classifying simple modules in certain blocks.

Contribution

It establishes an equivalence between non-singular Whittaker modules for  ext{sp}_{2n} and modules over the even Weyl algebra, extending classical Whittaker module theory.

Findings

Equivalence of certain Whittaker module categories to modules over the even Weyl algebra

Classification of simple modules in specific blocks as Nilsson's modules

Characterization of algebra homomorphisms from  ext{U}( ext{sp}_{2n}) to the Weyl algebra

Abstract

For any $n\in \mathbb{Z}_{\geq 2}$, 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 $\mathcal{WH}_{\mathbf{a}}^{\mu}$ 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 $\mathcal{WH}_{\mathbf{i}}^{-\frac{1}{2}\omega_n}$ 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.