Quantum dimensions and their non-Archimedean degenerations

We derive explicit dimension formulas for irreducible ℳF-spherical KF-representations, where KF is the maximal compact subgroup of the general linear group GLd(F) over a local field F and ℳF is a closed subgroup of KF such that KF/ℳF realizes the Grassmannian of n-dimensional F-subspaces of Fd. We explore the fact that (KF,ℳF) 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.