Whittaker Modules of Singular Type over Lie Algebra of Type A₂

Abstract

The concept of Whittaker modules for finite-dimensional Lie algebras was introduced by Kostant [Invent. Math., 1978, 48(2): 101–184], where he studied the Whittaker modules with respect to nonsingular Whittaker functions in full detail. However, the Whittaker modules are not considered for the singular case, which turns out to be more complicated. In this paper, we study the Whittaker modules for the Lie algebra

\mathfrak{s}{\mathfrak{l}}_3

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak s}{\mathfrak l}_{3}$$\end{document}

, the first Lie algebra that admits singular Whittaker functions. We investigate the submodule structure of the singular universal Whittaker modules and obtain a full description of all Whittaker vectors. Consequently, we determine all maximal submodules and provide a complete classification of all simple Whittaker modules of singular type in terms of quotient modules of the universal Whittaker modules with two parameters.

Keywords

Whittaker module / Whittaker function / Whittaker vector / simple module

RIGHTS & PERMISSIONS

Peking University