Enveloping algebra U(gl(3)) and orthogonal polynomials in several discrete indeterminates

Abstract

Let A be an associative complex algebra and L an invariant linear functional on it (trace). Let i be an involutive antiautomorphism of A such that L(i(a))=L(a) for any a in A. Then A admits a symmetric invariant bilinear form (a, b)=L(a i(b)). For A=U(sl(2))/m, where m is any maximal ideal of U(sl(2)), Leites and I have constructed orthogonal basis whose elements turned out to be, essentially, Chebyshev and Hahn polynomials in one discrete variable. Here I take A=U(gl(3))/m for the maximal ideals m which annihilate irreducible highest weight gl(3)-modules of particular form (generalizations of symmetric powers of the identity representation). In this way we obtain multivariable analogs of Hahn polynomials.

12p., Latex

References in corpus (3)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Shifted Schur Functions
• Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices

Cited by in corpus (5)

• Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras
• Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices
• Twisted generalized Weyl algebras and primitive quotients of enveloping algebras
• Cylindrical Dyck paths and the Mazorchuk-Turowska equation
• Classification of twisted generalized Weyl algebras over polynomial rings