Analytic features of reproducing groups for the metaplectic representation

We introduce the notion of admissible subgroup H of G = H^d \rtimes Sp(d,R) relative to the (extended) metaplectic representation μ_e via the Wigner distribution. Under mild additional assumptions, it is shown to be equivalent to the fact that the identity f= \int_H \la f,μ_e(h) φ \ra μ_e(h) φ dh holds (weakly) for all f ∈ L^2(R^d ). We use this equivalence to exhibit classes of admissible subgroups of Sp(2,R). We also establish some connections with wavelet theory, i.e., with curvelet and contourlet frames.