Abstract:Let $G$ be the semidirect product $V\rtimes \,K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\pi$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$. We construct an invariant symbolic calculus for $\pi$, under some technical hypothesis. We give some examples including the Poincar\'e group.
Keywords: semidirect products; invariant symbolic calculus; coadjoint orbit; unitary representation; Berezin quantization; Weyl quantization; Poincar\'e group
DOI: DOI 10.14712/1213-7243.2015.244
AMS Subject Classification: 81S10 22E46 22E45 22D30 81R05