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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11464_2023_160_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathfrak s}{\mathfrak l}_{3}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="fraktur">s</mi> </mrow> </mrow> <msub> <mrow> <mrow> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow> <mn>3</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>, 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.</p>

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Whittaker Modules of Singular Type over Lie Algebra of Type A2

  • 期刊论文

  • 发表日期：

    2025年6月5日

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  • Limeng Xia,
  • Xiangqian Guo

摘要

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 \({\mathfrak s}{\mathfrak l}_{3}\) s l 3 , 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.