The Infinitesimal Method

Abstract

Up to now, with exception of the Schrödinger representation of the Heisenberg group, all irreducible representations treated here were finite-dimensional. Before we go further into the construction of infinite-dimensional representations, we discuss a method which allows a linearization of our objects and hence simplifies the task to determine the structure of the representations and classify them. As we shall see, this is helpful for compact and noncompact groups as well. The idea is to associate to the linear topological group G a linear object, its Lie algebra g = LieG, and study representations $\hat \pi$ of these algebras, which are open to purely algebraic and combinatorial studies. One can associate to each representation π of G an infinitesimal representation dπ of g. Conversely, one can ask which representations $\hat \pi$ may be integrated to a unitary representation π of G, i.e., which are of the form $\hat \pi = d\hat \pi ,{\mathbf{ }}\pi \in \hat G$ . As we will see in several examples, this method allows us to classify the (unitary) irreducible representations and, hence, gives a parametrization of $\hat G$ . It will further prove to be helpful for the construction of explicit models for representations (πℌ)