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

We consider the (extended) metaplectic representation of the semidirect product G=Hd⋊Sp(d,R) between the Heisenberg group and the symplectic group. Subgroups H=Σ⋊D, with Σ being a d×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 Schrödinger representation of R2d or the wavelet representation of Rd⋊D, with D closed subgroup of GL(d,R). Finally, we shall provide new examples of reproducing groups of the type H=Σ⋊D, in dimension d=2.