Asian Journal of Mathematics

Volume 25 (2021)

Number 3

Harish–Chandra modules over invariant subalgebras in a skew-group ring

Pages: 431 – 454

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n3.a6

Authors

Volodymyr Mazorchuk (Department of Mathematics, Uppsala University, Uppsala, Sweden)

Elizaveta Vishnyakova (Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Brazil; and Laboratory of Theoretical and Mathematical Physics, Tomsk State University, Tomsk, Russia)

Abstract

We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand–Zeitlin algebras and rational Galois algebras studied by [EMV, FGRZ, RZ, Har]. The algebras are defined via a geometric realization in terms of sheaves of functions invariant under an action of a finite group. A natural class of modules over these algebra can be constructed via a similar geometric realization. In the special case of a local reflection group, these modules are shown to have an explicit basis, generalizing similar results for orthogonal Gelfand–Zeitlin algebras from [EMV] and for rational Galois algebras from [FGRZ]. We also construct a family of canonical simple Harish–Chandra modules and give sufficient conditions for simplicity of some modules.

Keywords

Gelfand–Zeitlin modules, invariant polynomial, Gelfand–Zeitlin algebras, rational Galois algebras, Harish–Chandra modules

2010 Mathematics Subject Classification

14M10, 16W70, 17B35

Received 21 June 2019

Accepted 6 October 2020

Published 14 March 2022