  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 $(K_F,M_F)$ 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.
