2006/06/09 by Uri Onn, Onn, Uri, Jasper V. Stokman +1 · 1 citation
Mathematics · #33D52 #33D80 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.math/0606222
openalex publication_date 2006/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive explicit dimension formulas for irreducible MF-spherical KF-representations where KF is the maximal compact subgroup of the general linear group GL(d,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.