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

References

[1] AdamovićD, LüR, ZhaoK. Whittaker modules for the affine Lie algebra A₁⁽¹⁾. Adv. Math., 2016, 289: 438-479

[2] ArnalD, PinzconG. On algebraically irreducible representations of the Lie algebras sl2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak s}{\mathfrak l}_{2}$$\end{document}. J. Math. Phys., 1974, 15: 350-359

[3] BatraP, MazorchukV. Blocks and modules for Whittaker pairs. J. Pure Appl. Algebra, 2011, 215(7): 1552-1568

[4] BenkartG, OndrusM. Whittaker modules for generalized Weyl algebras. Represent. Theory, 2009, 13: 141-164

[5] ChristodoulopoulouK. Whittaker modules for Heisenberg algebras and imaginary Whittaker modules for affine Lie algebras. J. Algebra, 2008, 320(7): 2871-2890

[6] GuoX, LiuX. Whittaker modules over generalized Virasoro algebras. Comm. Algebra, 2011, 39(9): 3222-3231

[7] Guo X., Liu X., Whittaker modules over Virasoro-like algebra. J. Math. Phys., 2011, 52(9): Paper No. 093504, 9 pp.

[8] Guo X., Liu X., Xia L., Whittaker modules over Uq(sl3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U_{q}({\mathfrak s}{\mathfrak l}_{3})$$\end{document}. 2025, arXiv:2504.10262

[9] HartwigJT, YuN. Simple Whittaker modules over free bosonic orbifold vertex operator algebras. Proc. Amer. Math. Soc., 2019, 147(8): 3259-3272

[10] KostantB. On Whittaker vectors and representation theory. Invent. Math., 1978, 48(2): 101-184

[11] LiL, XiaL, ZhangY. On the centers of quantum groups of A_n-type. Sci. China Math., 2018, 61(2): 287-294

[12] Liu D., Pei Y., Xia L., Whittaker modules for the super-Virasoro algebras. J. Algebra Appl., 2019, 18(11): Paper No. 1950211, 13 pp.

[13] LiuX, GuoX. Whittaker modules over loop Virasoro algebra. Front. Math. China, 2013, 8(2): 393-410

[14] LuR, GuoX, ZhaoK. Irreducible modules over the Virasoro algebra. Doc. Math., 2011, 16: 709-721

[15] OndrusM. Whittaker modules for Uq(sl2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U_{q}({\mathfrak s}{\mathfrak l}_{2})$$\end{document}. J. Algebra, 2005, 289(1): 192-213

[16] OndrusM. Tensor products and Whittaker vectors for quantum groups. Comm. Algebra, 2007, 35(8): 2506-2523

[17] OndrusM, WiesnerE. Whittaker modules for the Virasoro algebra. J. Algebra Appl., 2009, 8(3): 363-377

[18] OndrusM, WiesnerE. Whittaker categories for the Virasoro algebra. Comm. Algebra, 2013, 41(10): 3910-3930

[19] SevostyanovA. Regular nilpotent elements and quantum groups. Comm. Math. Phys., 1999, 204(1): 1-16

[20] SevostyanovA. Quantum deformation of Whittaker modules and the Toda lattice. Duke Math. J., 2000, 105(2): 211-238

[21] XiaL, GuoX, ZhangJ. Classification on irreducible Whittaker modules over quantum group Uq(sl3,Λ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U_{q}({\mathfrak s}{\mathfrak l}_{3}, \Lambda)$$\end{document}. Front. Math. China, 2021, 16(4): 1089-1097

RIGHTS & PERMISSIONS

Peking University