Enveloping superalgebra U(osp(1|2)) and orthogonal polynomials in discrete indeterminate

Abstract

Let $A$ be an associative simple (central) superalgebra over ${\mathbb C}$ and $L$ an invariant linear functional on it (trace). Let $a\mapsto a^t$ be an antiautomorphism of $A$ such that $(a^t)^ t=(-1)^{p(a)}a$, where $p(a)$ is the parity of $a$, and let $L(a^t)=L(a)$. Then $A$ admits a nondegenerate supersymmetric invariant bilinear form $\langle a, b\rangle=L(ab^t)$. For $A=U({\mathfrak{sl}}(2))/{\mathfrak{m}}$, where ${\mathfrak{m}}$ is any maximal ideal of $U({\mathfrak{sl}}(2))$, Leites and I have constructed orthogonal basis in $A$ whose elements turned out to be, essentially, Chebyshev (Hahn) polynomials in one discrete variable. Here I take $A=U({\mathfrak{osp}}(1|2))/{\mathfrak{m}}$ for any maximal ideal ${\mathfrak{m}}$ and apply a similar procedure. As a result we obtain either Hahn polynomials over ${\mathbb C}[\tau]$, where $\tau^2\in{\mathbb C}$, or a particular case of Meixner polynomials, or --- when $A=\mbox{Mat}(n+1|n)$ --- dual Hahn polynomials of even degree, or their (hopefully, new) analogs of odd degree. Observe that the nondegenerate bilinear forms we consider for orthogonality are, as a rule, not sign definite.