Canonical Gelfand–Zeitlin Modules over Orthogonal Gelfand–Zeitlin Algebras. (22nd February 2019)

• Record Type: Journal Article
• Authors: Early, Nick; Mazorchuk, Volodymyr; Vishnyakova, Elizaveta
• Is Part Of: International mathematics research notices. Volume 2020:Number 20(2020)
• Page Start: 6947
• Page End: 6966
• Publication Date: 2019-02-22
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnz002
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We prove that every orthogonal Gelfand–Zeitlin algebra $U$ acts (faithfully) on its Gelfand–Zeitlin subalgebra $\Gamma $ . Considering the dual module, we show that every Gelfand–Zeitlin character of $\Gamma $ is realizable in a $U$ -module. We observe that the Gelfand–Zeitlin formulae can be rewritten using divided difference operators. It turns out that the action of the latter operators on $\Gamma $ gives rise to an explicit basis in a certain Gelfand–Zeitlin submodule of the dual module mentioned above. This gives, generically, both in the case of regular and singular Gelfand–Zeitlin characters, an explicit construction of simple modules, which realize the given Gelfand–Zeitlin characters.