Enveloping Algebra of GL(3) and Orthogonal Polynomials

Abstract

Let A be an associative algebra over C and L an invariant linear functional on it (trace). Let ω be an involutive antiautomorphism of A such that L(ω(a)) = L(a) for any a ω A. Then A admits a symmetric invariant bilinear form 〈a,b〉= L(aw(b)). For $U(\mathfrak{s}\mathfrak{l}(2))/\mathfrak{m})$, where $\mathfrak{m}$ is any maximal ideal of $U\mathfrak{s}\mathfrak{l}((2))$, 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 $A = U\mathfrak{g}\mathfrak{l}((3))/\mathfrak{m}$ for the maximal ideals $\mathfrak{m}$ which annihilate irreducible highest weight $\mathfrak{g}\mathfrak{l}(3)$-modules of particular form (generalizations of symmetric powers of the identity representation). In whis way we obtain multivariable analogs of Hahn polynomials. Clearly, one can similarly consider $\mathfrak{g}\mathfrak{l}(n)$ and $\mathfrak{g}\mathfrak{l}(m|n)$ instead of $\mathfrak{g}\mathfrak{l}(3)$ but the amount of calculations is appalling.

I am thankful to D. Leites for encouragement and help and to ESI, Vienna, for hospitality and support.