Equivalence of the Metaplectic Representation with Sums of Wavelet Representations for a Class of Subgroups of the Symplectic Group

Equivalence of the metaplectic representation with a sum of affine representations is discussed for a class of subgroups of the symplectic group. The paper begins with a study of sums of wavelet transforms, and it is shown that the usual admissibility conditions for the wavelet transform apply to transforms by sums of affine representations as well. A construction of admissible vectors, bandlimited admissible vectors and frames for sums of affine representations by means of transversals is given. A class of subgroups of the symplectic group which are compact extensions affine groups are identified, and it is shown how subrepresentations of the metaplectic representation can be equivalent to affine representations. This equivalence is illuminated by several examples.