Whittaker vectors in singular Whittaker modules

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra with Borel subalgebra $\mathfrak{b}$ and corresponding nilradical $\mathfrak{n}$. We show that singular Whittaker modules M are simple if and only if the space $\text{Wh} M$ of Whittaker vectors is 1-dimensional. For arbitrary locally $\mathfrak{n}$-finite $\mathfrak{g}$-modules V, an immediate corollary is that the dimension of $\text{Wh} V$ is bounded by the composition length of V.

KEYWORDS:

• Composition length
• Whittaker modules
• Whittaker vectors

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17B10
• Secondary: 17B20

Acknowledgments

This paper is the result of a MITACS Globalink undergraduate research internship held by Dulam, Ghate, and Pathak in Summer 2023 under the direction of Lau at Université Laval. The authors also thank Charlotte Lavoie-Bel for many useful discussions throughout the summer.

Disclosure statement

The authors report that there are no competing interests to declare.

Additional information

Funding

Financial and logistical support of MITACS and Université Laval are gratefully acknowledged.