# Università degli Studi di Genova

## Dimensional upper bounds for admissible subgroups for the metaplectic representation

Abstract:
We prove dimensional upper bounds for admissible Lie subgroups H of G = {\Bbb H}_d\rtimes Sp(d,R), d\geq 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\leq d^2+ 2d, whereas if H=Sp(d, R), then dim H\leq d^2+1. Both bounds are shown to be optimal

Keywords:
metaplectic representation, reproducing formula, wavelets, Wigner distribution, semidirect product

MSC:
22, 43

Pubblicato su: Mathematische Nachrichten Vol. 283 (2010) N. 7 Pag. 000-000