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We construct a family of irreducible unitary representations of the loopaffine group of a line (ax+b group) with central extension on the Hilbert spaceof square integrable functions with respect to the Wiener measure. We relatethe matrix coefficients of the elements of the loop ax+b group to the loopanalogue of the Gamma-function.

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