Tridiagonal pairs and the Askey-Wilson relations

The notion of a tridiagonal pair was introduced by Ito, Tanabe and Terwilliger. Let V denote a nonzero finite dimensional vector space over a field F. A tridiagonal pair on V is a pair (A, A*), where A : V → V and A* : V → V are linear transformations that satisfy some conditions. Assume (A, A*) is a tridiagonal pair on V. Recently Terwilliger and Vidunas showed that if A is multiplicity-free on V, then (A, A*) satisfy the following “Askey-Wilson relation” for some scalars β, γ, γ*, ϱ, ϱ*, ω, η, η*.A2A∗-βAA∗A+A∗A2-γ(AA∗+A∗A)-ϱA∗=γ∗A2+ωA+ηI,A∗2A-βA∗AA∗+AA∗2-γ∗(A∗A+AA∗)-ϱ∗A=γA∗2+ωA∗+η∗I.In the present paper, we show that, if a tridiagonal pair (A, A*) satisfy the Askey-Wilson relations, then the eigenspaces of A and the eigenspaces of A* have one common dimension, and moreover if F is algebraically closed then that common dimension is 1.