Harmonic Analysis Associated with the Jacobi-Dunkl Operator on (-π,π): Exotic Cases

We investigate some spectral properties of a second order differential-difference operator J(a,b) on L-2((-pi,pi), d mu(alpha,beta)), alpha, beta epsilon R, called the Jacobi-Dunkl operator of compact type. Using an idea of Hajmirzaahmad, in exotic cases, e.g. when at least one of the two parameters a, beta is <= -1, we construct exotic orthonormal bases that consist of eigenfunctions of J(alpha,beta). This allows one to consider natural self-adjoint exotic extensions of J(alpha,beta) and the corresponding exotic Jacobi-Dunkl and Jacobi-Dunkl-Poisson semigroups.