Leonard pairs and the Askey-Wilson relations

Abstract

Let K denote a field and let $V$ denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^*:V\to V$ which satisfy the following two properties: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. Referring to the above Leonard pair, we show there exists a sequence of scalars $β,γ,γ^*, \varrho,\varrho^*,ω, η, η^*$ taken from K such that both (i) A^2 A^*-βA A^*A+A^*A^2-γ(AA^*+A^*A) -\varrho A^* =γ^*A^2+ωA+\etaI; (ii) A^{*2}A-βA^*AA^*+AA^{*2}-γ^*(A^*A+AA^*) -\varrho^*A =γA^{*2}+ωA^*+η^*I. The sequence is uniquely determined by the Leonard pair provided the dimension of $V$ is at least 4. The equations above are called the Askey-Wilson relations.