Unitary representations of a loop ax +b group, Wiener measure and Γ -function

We construct a family of irreducible unitary representations of the loop affine group of a line (ax + b group) with central extension on the Hilbert space of square-integrable functions with respect to theWiener measure. We relate the matrix coefficients of the elements of the loop ax +b group to the loop analogue of the Γ -function. © 2012 Elsevier Inc. All rights reserved