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.
Dimensional upper bounds for admissible subgroups for the metaplectic representation,
DE MARI CASARETO DAL VERME, F.;
2010-01-01
Abstract
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.
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