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 ∞,1 is an associative Banach algebra and the corresponding 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-Hörmander calculus is obtained for the Schrödinger representations of the finite-dimensional Heisenberg groups, and in this case we recover the results obtained by J. Sjöstrand (Math Res Lett 1(2): [185][186][187][188][189][190][191][192] 1994).

Analysis on singular spaces: Lie manifolds and operator algebras

“…There are many works dealing with pseudodifferential operators on groupoids, on singular spaces, or with the related C * -algebras, see for example [3,15,26,37,46,58,86,95,96,109,122,131,135,136,149,150,155].…”

Inverse-Closed Algebras of Integral Operators on Locally Compact Groups

“…In this section we also obtain our main inverse-closed algebras of integral operators on vector-valued functions on locally compact groups (Theorem 3.10). In Section 4 we provide a method for constructing larger inverse-closed algebras of integral operators, and the corresponding result (Theorem 4.3) applies in the situation that we encountered in our earlier paper [BB12] in connection with Weyl-Pedersen calculus for unitary irreducible representations of nilpotent Lie groups. Finally, certain symmetry groups of our inverse-closed algebras of integral operators are studied in Section 5, and some of their special features are established in the Lie group setting, with motivation coming fr om the recent results of [BG13] and [BB13].…”

Berezin Symbols on Lie Groups

“…This approach blends some techniques of reproducing kernels and some basic ideas of linear partial differential equations, in order to address a problem motivated by representation theory of Lie groups (see [C09], [C10], [C13], and [C14]). This problem is also related to some representations of infinite-dimensional Lie groups that occur in the study of magnetic fields (see [BB11a] and [BB12]). Let us also mention that linear differential operators associated to reproducing kernels have been earlier used in the literature (see for instance [BG14]).…”