Invariant symbolic calculus for semidirect products. (English) Zbl 1463.81016

Cahen, Benjamin

Commentat. Math. Univ. Carol. 59, No. 2, 253-269 (2018).

The interesting paper under review presents the construction of an invariant symbolic calculus for unitary irreducible representations of Lie groups of the form \(G=V\rtimes K\), where \(V\) is a finite-dimensional real vector space that is acted on by a connected noncompact semisimple Lie group \(K\). The focus is on unitary irreducible representations \(\pi\colon G\to\mathcal{B}(\mathcal{H})\) associated with coadjoint orbits \(\mathcal{O}\) of \(G\) whose corresponding so-called little groups are maximal compact subgroups of \(G\). In this framework, an invariant symbolic calculus is a one-to-one linear mapping \(\mathcal{S}\) from a vector space of linear operators on \(\mathcal{H}\) to a vector space of generalized functions on \(\mathcal{O}\) satisfying the conditions that \(\mathcal{S}(A^*)\) is the complex conjugate of \(\mathcal{S}(A)\) and \(\mathcal{S}(\pi(g)A\pi(g)^{-1})=\mathcal{S}(A)(g^{-1}x)\) for all operators \(A\) in the domain of definition of \(\mathcal{S}\) and all \(g\in G\) and \(x\in\mathcal{O}\). The general construction is illustrated by the examples of the Poincaré group and the tangent group of \(\mathrm{SU}(n,1)\).

Reviewer: Daniel Beltiţă (Bucureşti)

MSC:

81S10	Geometry and quantization, symplectic methods
22E46	Semisimple Lie groups and their representations
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods
22D30	Induced representations for locally compact groups
81R05	Finite-dimensional groups and algebras motivated by physics and their representations

Keywords:

coadjoint orbit; unitary representation; Berezin quantization; Weyl quantization; Poincaré group; semidirect products; invariant symbolic calculus

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