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