A family of tridiagonal pairs

AbstractLet F denote a field, and let V denote a vector space of finite positive dimension over F. Let A, A∗ denote a tridiagonal pair on V of diameter d⩾3. Assume the eigenvalue and dual eigenvalue sequences of A, A∗ satisfy θi=q2iθ, θi*=q2d-2iθ* for some nonzero scalars θ, θ∗, q∈F, where q is not a root of unity. Assume that not all eigenvalues of A and A∗ have multiplicity one. Let M and M∗ denote the subalgebras of End(V) generated by A and A∗, respectively, and assume that V=Mv∗+M∗v for some eigenvectors v∗ of A∗ associated with θ0* and v of A associated with θd. We find a nice basis for V and describe the action of A, A∗ on this basis in terms of six parameters