Dimensional upper bounds for admissible subgroups for the metaplectic representation

We prove dimensional upper bounds for admissible Lie subgroups H of G = ℍd ⋊ Sp (d, ℝ), d ≥ 2. The notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d2 + 2d, whereas if H ⊂ Sp (d, R), then dim H ≤ d2 + 1. Both bounds are shown to be optimal (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)