Tannaka-Krein duality for compact groupoids II, Fourier transform

TL;DR

This paper extends Tannaka-Krein duality to compact groupoids by developing Fourier and Plancherel transforms, enabling reconstruction of the groupoid from its representation theory and analyzing central functions within this framework.

Contribution

It introduces Fourier and Plancherel transforms for compact groupoids and proves the Plancherel theorem, advancing the harmonic analysis of groupoids beyond prior work on their representation theory.

Findings

Established Fourier and Plancherel transforms for compact groupoids

Proved the Plancherel theorem in this context

Analyzed central functions on isotropy groups

Abstract

In a series of papers, we have shown that from the representation theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study the Fourier and Fourier-Plancherel transforms and prove the Plancherel theorem for compact groupoids. We also study the central functions in the algebra of square integrable functions on the isotropy groups.