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International Mathematics Research Papers (2006) Vol. 2006 : article ID 54701, 53 pages, doi:10.1155/IMRP/2006/54701
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Quantum dimensions and their non-Archimedean degenerations

Uri Onn and Jasper V. Stokman

Einstein Institute of Mathematics, The Hebrew University of Jerusalem Edmond Safra Campus, Givat Ram, Jerusalem 91904, Israel E-mail address: urion{at}math.huji.ac.il
Korteweg-de Vries Institute for Mathematics, Faculty of Science, Universiteit van Amsterdam Plantage Muidergracht 22-24, TV Amsterdam 1018, The Netherlands E-mail address: jstokman{at}science.uva.nl

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.


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