# Pseudos on locally compact groups

The general framework for pseudo-differential operators on locally compact unimodular groups of type I has been developed in this paper:

M. Mantoiu, M. Ruzhansky, Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups, Doc. Math., 22 (2017), 1539-1592. offprint (open access), link, arxiv

Compact and Abelian groups are type I. So are the Euclidean and the Poincaré groups. Among the connected groups, real algebraic, exponential (in particular nilpotent) and semi-simple Lie groups are type I. Not all the solvable groups are type I but there is a criterion. A discrete group is type I if and only if it is the finite extension of an Abelian normal subgroup.

• We develop a rigorous framework for the analysis of pseudo-differential operators on locally compact groups of type I.

• We introduce notions of Wigner and Fourier-Wigner transforms, and of Weyl systems, adapted to this general setting. These notions are used to define and analyse τ-quantizations (or quantization by Weyl systems) of operators modelled on families of quantizations on ${\mathbb R}^n$ that include the Kohn-Nirenberg and Weyl quantizations.

• We develop the $C^\ast$-algebraic formalism to put τ-quantizations in a more general perspective, also allowing analysis of operators with ‘coefficients’ taking values in different $C^\ast$-algebras. The link with a left form of τ-quantization is given via a special covariant representation, the Schrödinger representation. This is further applied to investigate spectral properties of covariant families of operators.
• Although the initial analysis is set for operators bounded on $L^2(G),$ this can be extended further to include densely defined operators and, more generally, operators acting on ‘distributions’. Since $G$ does not have to be a Lie group (i.e. there may be no compatible smooth differential structure on $G$) we show how this can be done using the so-called Bruhat space $D(G)$, an analogue of the space of smooth compactly supported functions in the setting of general locally compact groups.
• The results are applied to a deeper analysis of τ-quantizations on nilpotent Lie groups. On one hand, this extends the setting of graded Lie groups to more general nilpotent Lie groups, also introducing a possibility for Weyl-type quantizations there. On the other hand, it extends the invariant Melin calculus on homogeneous groups to general non-invariant operators with the corresponding τ-versions of scalar-symbols on the dual of the Lie algebra.
• We give a criterion for the existence of Weyl-type quantizations in our framework, namely, to quantizations in which real-valued symbols correspond to self-adjoint operators. We show the existence of such quantizations in several settings, most interestingly in the setting of general groups of exponential type.