Global and Concrete Quantizations on General Type I Groups

TL;DR

This paper extends the framework of global quantization and pseudo-differential calculus from specific groups to all locally compact groups of type I, using Fourier analysis and concrete parametrizations.

Contribution

It generalizes the operator-valued symbol calculus to arbitrary locally compact groups of type I, utilizing Fourier theory and explicit parametrizations.

Findings

Developed a manageable formalism for non-unimodular groups

Applied the theory to examples of solvable groups

Provided concrete forms for the abstract dual and Plancherel measure

Abstract

In recent papers and books, a global quantization has been developed for unimodular groups of type I. It involves operator-valued symbols defined on the product between the group $\mathsf{G}$ and its unitary dual $\widehat{\mathsf{G}}$, composed of equivalence classes of irreducible representations. For compact or for graded Lie groups, this has already been developed into a powerful pseudo-differential calculus. In the present article we extend the formalism to arbitrary locally compact groups of type I, making use of the Fourier theory of non-unimodular second countable groups. The unitary dual and its Plancherel measure being quite abstract in general, we put into evidence situations in which concrete forms are available. Kirillov theory and parametrizations of large parts of $\widehat{\mathsf{G}}$ allow rewriting the basic formulae in a manageable form. Some examples of completely solvable groups are worked out.