Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

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 $SL(2,\mathbb{C}$) 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.

References in corpus (4)

• A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries
• The Fourier transform in quantum group theory
• On a correspondence between quantum SU(2), quantum E(2) and extended quantum SU(1,1)
• Modular properties of matrix coefficients of corepresentations of a locally compact quantum group

Cited by in corpus (2)

• Matrix-valued orthogonal polynomials related to the quantum analogue of $(SU(2)\times SU(2),\text{diag})$
• Weak amenability of locally compact quantum groups and approximation properties of extended quantum SU(1,1)