Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 087, 39 pages      arXiv:1104.2459      https://doi.org/10.3842/SIGMA.2011.087
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

Martijn Caspers
Radboud Universiteit Nijmegen, IMAPP, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Received April 14, 2011, in final form August 30, 2011; Published online September 06, 2011

Abstract
We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in SL(2,C) together with its diagonal subgroup into a pair for which every irreducible corepresentation admits at most two vectors that are invariant with respect to the quantum subgroup. Using a Z2-grading, we obtain product formulae for little q-Jacobi functions.

Key words: locally compact quantum groups; Plancherel theorem; Fourier transform; spherical functions.

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