A Duality Approach to Representations of Baumslag-solitar Groups

We give an operator theoretic approach to the constructions of multiresolutions as they are used in a number of basis constructions with wavelets, and in Hilbert spaces on fractals. Our approach starts with the following version of the classical Baumslag-Solitar relations ut = tu where t is a unitary operator in a Hilbert space H and u is an isometry in H. There are isometric dilations of this system into a bigger Hilbert space, relevant for wavelets. For a variety of carefully selected dilations, the “bigger” Hilbert space may be L(R), and the dilated operators may be the unitary operators which define a dyadic wavelet multiresolutions of L(R) with the subspace H serving as the corresponding resolution subspace. That is, the initialized resolution which is generated by the wavelet scaling function(s). In the dilated Hilbert space, the Baumslag-Solitar relations then take the more familiar form utu = t. We offer an operator theoretic framework including the standard construction; and we show how the representations of certain discrete semidirect group products serve to classify the possibilities. For this we analyze and compare several types of unitary representations of these semidirect products: the induced representations in Mackey’s theory, the wavelet representations on L(R), the irreducible representation on the dual, the finite dimensional representations, and the the regular representation.