Quantum dimensions and their non-Archimedean degenerations

We derive explicit dimension formulas for irreducible M_F-spherical K_F-representations, where K_F is the maximal compact subgroup of the general linear group GL_d(F) over a local field F and M_F is a closed subgroup of K_F such that K_F/M_F realizes the Grassmannian of n-dimensional F-subspaces of F^d. We explore the fact that (K_F, K_F) is a Gelfand pair whose associated zonal spherical functions identify with various degenerations of the multivariable little q-Jacobi polynomials. As a result, we are led to consider generalized dimensions defined in terms of evaluations and quadratic norms of multivariable little q-Jacobi polynomials, which interpolate between the various classical dimensions. The generalized dimensions themselves are shown to have representation-theoretic interpretations as the quantum dimensions of irreducible spherical quantum representations associated to quantum complex Grassmannians.