Triangular subgroups of Sp(d, R) and reproducing formulae

We consider the (extended) metaplectic representation of the semidirect product G = H-d (sic) Sp(d, R) between the Heisenberg group and the symplectic group. Subgroups H = Sigma (sic) D, with Sigma being a d x d symmetric matrix and D a closed subgroup of GL(d, R), are our main concern. We shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in Cordero et al. (2006) [4]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrodinger representation of R-2d or the wavelet representation of R-d (sic) D, with D closed subgroup of GL(d, R). Finally, we shall provide new examples of reproducing groups of the type H = Sigma (sic) D, in dimension d = 2. (c) 2013 Elsevier Inc. All rights reserved.