The big q-Jacobi function transform

Abstract

We give a detailed description of the resolution of the identity of a second order $q$-difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The $q$-difference operator and the two choices of Hilbert spaces naturally arise from harmonic analysis on the quantum group $SU_q(1,1)$ and $SU_q(2)$. The spectral analysis associated to $SU_q(1,1)$ leads to the big $q$-Jacobi function transform together with its Plancherel measure and inversion formula. The dual orthogonality relations give a one-parameter family of non-extremal orthogonality measures for the continuous dual $q^{-1}$-Hahn polynomials with $q^{-1}>1$, and explicit sets of functions which complement these polynomials to orthogonal bases of the associated Hilbert spaces. The spectral analysis associated to $SU_q(2)$ leads to a functional analytic proof of the orthogonality relations and quadratic norm evaluations for the big $q$-Jacobi polynomials.