Algebras of symbols associated with the Weyl calculus for Lie group representations

TL;DR

This paper extends the Weyl calculus for infinite-dimensional Lie group representations by analyzing the algebraic and boundedness properties of symbols in modulation spaces, with applications to nilpotent and semidirect product groups.

Contribution

It establishes continuity and algebraic properties of the Moyal product for symbols in modulation spaces, generalizing classical Weyl calculus to new Lie group contexts.

Findings

Modulation space $M^{ abla,1}$ forms an associative Banach algebra.

Operators associated with symbols in these spaces are bounded.

Classical Weyl-H"ormander calculus is recovered for Schr"odinger representations.

Abstract

We develop our earlier approach to the Weyl calculus for representations of infinite-dimensional Lie groups by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. For instance, we prove that the modulation space of symbols $M^{\infty,1}$ is an associative Banach algebra and the corrresponding operators are bounded. We then apply the abstract results to two classes of representations, namely the unitary irreducible representations of nilpotent Lie groups, and the natural representations of the semidirect product groups that govern the magnetic Weyl calculus. The classical Weyl-H\"ormander calculus is obtained for the Schr\"odinger representations of the finite-dimensional Heisenberg groups, and in this case we recover the results obtained by J.~Sj\"ostrand in 1994.