Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

Martijn Caspers

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

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 $SL(2,\mathbb C$) 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 $\mathbb Z_2$-grading, we obtain product formulae for little $q$-Jacobi functions.

Détail

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

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