Irreducible Unitary Representations of the Group of Maps with Values in a Free Product Group

Abstract

This chapter explains the irreducible unitary representations of the group of maps with values in a free product group. Vershik, Gelfand, and Graev studied the construction of irreducible unitary representations of the group C∞ (X, G) of smooth maps of a compact manifold X with values in a Lie group G. Following the physical terminology, such a group is called a current group. In the case G= SL (2, R), they afforded factorizable irreducible unitary representations of the current group that depend upon measures on X. Their method reveals that the structure of a measure space is important rather than the structure of a manifold. In fact, they started with the construction of those representations of the weak current group G(X). A weak current group is the group of maps of a measurable space X with only finitely many values in a topological group G. Furthermore their method relies deeply on the structure of the neighborhood of the trivial representation of G= SL (2, R). The chapter also highilights that apart from the representation theory of current groups there has been a remarkable progress in harmonic analysis on free groups.