Projective Schur Functions as a Bispherical Functions on Certain Homogeneous Superspaces

I show that the projective Schur functions may be interpreted as bispherical functions of either the triple (q(n), q(n)⊕q(n), q(n)), where q(n) is the “odd” (queer) analog of the general linear Lie algebra, or the triple (pe(n), gl(n|n), pe(n)), where pe(n) is the periplectic Lie superalgebra which preserves the nondegenerate odd bilinear form (either symmetric or skew-symmetric). Making use of this interpretation I characterize projective Schur functions as common eigenfunctions of an algebra of differential operators.