Verma modules and finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

2026/01/01 by Hau-Wen Huang

paper · doi:10.1017/nmj.2025.10096

crossref issued 2026/01/01 · crossref published 2026/01/01 · crossref published-print 2026/01/01 · crossref published-online 2026/02/03 · crossref created 2026/02/03 · crossref deposited 2026/02/03 · crossref indexed 2026/07/30

Abstract

Abstract The Askey–Wilson algebras illustrate the bispectral property of orthogonal polynomials in the Askey scheme. The universal Askey–Wilson algebra \triangle _q is a central extension of the Askey–Wilson algebras associated with the most general orthogonal polynomials in the Askey scheme. The Verma \triangle _q -modules are a family of infinite-dimensional \triangle _q -modules with marginal weights. Under the condition that q is not a root of unity, it was shown that every finite-dimensional irreducible \triangle _q -module has a marginal weight and is isomorphic to a quotient of a Verma \triangle _q -module. Assume that q is a root of unity. We prove that every finite-dimensional irreducible \triangle _q -module with a marginal weight is isomorphic to a quotient of a Verma \triangle _q -module. More precisely, two natural families of finite-dimensional quotients of Verma \triangle _q -modules contain all finite-dimensional irreducible \triangle _q -modules with marginal weights up to isomorphism. Furthermore, we classify the finite-dimensional irreducible \triangle _q -modules with marginal weights up to isomorphism.

Citations

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