Quantum dimensions and their non-Archimedean degenerations

Uri Onn, Jasper V. Stokman

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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, KF) 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.

Original languageEnglish
Article number54701
JournalInternational Mathematics Research Papers
StatePublished - 11 Dec 2006


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