Induced representations of completely solvable Lie groups
We study the invariant symbolic calculi associated with the unitary irreducible representations of a compact Lie group.
Let be the semidirect product where is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space . Let be a unitary irreducible representation of which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of whose little group is a maximal compact subgroup of . We construct an invariant symbolic calculus for , under some technical hypothesis. We give some examples including the Poincaré group.