Weyl-Pedersen calculus for some semidirect products of nilpotent Lie groups

TL;DR

This paper investigates the Weyl-Pedersen calculus on specific nilpotent Lie groups formed as semidirect products, focusing on invariant differential operators and their role in understanding boundedness properties across all 3-step nilpotent groups.

Contribution

It introduces a unified approach to analyze the Weyl-Pedersen calculus for a class of semidirect product nilpotent Lie groups, extending results to all 3-step cases.

Findings

Established boundedness criteria for Weyl-Pedersen calculus on these groups

Demonstrated applicability to all unitary irreducible representations of 3-step nilpotent groups

Connected invariant differential operators to representation theory

Abstract

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl-Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.