Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

TL;DR

This paper extends the theory of Gelfand pairs to the setting of locally compact quantum groups, introducing a quantum spherical Fourier transform and analyzing specific examples with new product formulas for special functions.

Contribution

It provides an operator algebraic framework for quantum Gelfand pairs and develops a quantum spherical Fourier transform, with explicit examples and product formulas for special functions.

Findings

Quantum Plancherel transformation restricts to a spherical transform.

Example of quantum Gelfand pair with at most two invariant vectors per corepresentation.

Derived product formulas for little q-Jacobi functions.

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.