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Keywords:
semidirect products; invariant symbolic calculus; coadjoint orbit; unitary representation; Berezin quantization; Weyl quantization; Poincaré group
Summary:
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 Poincaré group.
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