Arkiv för Matematik

Volume 61 (2023)

Number 1

A Whittaker category for the symplectic Lie algebra and differential operators

Pages: 123 – 140

DOI: https://dx.doi.org/10.4310/ARKIV.2023.v61.n1.a7

Authors

Yang Li (School of Mathematics and Statistics, Henan University, Kaifeng, Henan, China)

Jun Zhao (School of Mathematics and Statistics, Henan University, Kaifeng, Henan, China)

Yuanyuan Zhang (School of Mathematics and Statistics, Henan University, Kaifeng, Henan, China)

Genqiang Liu (School of Mathematics and Statistics & Institute of Contemporary Mathematics, Henan University, Kaifeng, Henan, China)

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 $X_{-2 \varepsilon_i} , i=1, \dotsc , n$. Then ($\mathfrak{sp}_{2n}$, $\mathfrak{m}_n$) is a Whittaker pair in the sense of a definition given by Batra and Mazorchuk. In this paper, we use differential operators to study the category of $\mathfrak{sp}_{2n}$-modules that are locally finite over $\mathfrak{m}_n$. We show that when $\mathbf{a} \in (\mathbb{C}^\ast)^n$, each non-empty block $\mathcal{WH}^{\chi \mu}_{\mathbf{a}}$ with the central character $\chi \mu$ is equivalent to the Whittaker category $\mathcal{W}^{\mathbf{a}}$ of the even Weyl algebra $\mathcal{D}^{ev}_n$ which is semi-simple. Any module in $\mathcal{WH}^{\chi \mu}_{\mathbf{a}}$ has the minimal Gelfand–Kirillov dimension $n$. We also characterize all possible algebra homomorphisms from $U(\mathfrak{sp}_{2n}$) to the Weyl algebra $\mathcal{D}_n$ under a natural condition.

Keywords

even Weyl algebra, Whittaker pair, Whittaker module, semi-simple

2010 Mathematics Subject Classification

17B05, 17B10, 17B30, 17B35

Received 30 May 2022

Received revised 2 August 2022

Accepted 15 August 2022

Published 26 April 2023