Representations of the group SL(2,R), where R is a ring of functions (RMS 28:5 (1973) 87–132)

Summary

We obtain a construction of the irreducible unitary representations of the group of continuous transformations X → G, where X is a compact space with a measure m and G = PSL(2, R), that commute with transformations in X preserving m.

This construction is the starting point for a non-commutative theory of generalized functions (distributions). On the other hand, this approach makes it possible to treat the representations of the group of currents investigated by Streater, Araki, Parthasarathy, and Schmidt from a single point of view.

Introduction

One stimulus to the present work was the desire to extend the theory of generalized functions to the non-commutative case. Let us explain what we have in mind.

Let R be the real line, X a compact manifold, and f(x) an infinitely differentiable function on X with values in R, that is, a mapping X → R. A group structure arises naturally on the set of functions f(x), which we denote by R^X. Irreducible unitary representations of this group are defined by the formula f(x) → e^il(f) where l is a linear functional in the space of “test” functions f(x). Thus, to each generalized function (distribution) there corresponds an irreducible representation of R^x. If we replace R by any other Lie group G, then it is natural to ask for the construction of irreducible unitary representations of the group G^x, regarded as a natural non-commutative analogue to the theory of distributions. Such an attempt was made in [1], § 3.