Representation theory and combinatorics related to symmetric superspaces

I start with presenting certain general results and conjectures about spherical supervarieties.
Then I will concentrate on the examples of symmetric superspaces G/K associated with Jordan superalgebras via TKK construction, compute the algebra of invariant differential operators in the ring of regular functions on G/K. Our results show that in all cases the spectra of invariant differential operators can be described by certain supersymmetric polynomials which are specializations of Macdonald polynomials. (joint work with S. Sahi and H. Salmasian).