Introduction

Summary

This book is about the representation theory of analytic groups (an analytic group is a connected complex Lie group and a representation of it as matrices of size n × n is a holomorphic homomorphism from the analytic group to the group GL_nℂ of all invertible n × n complex matrices). As it is usually viewed, the main problem of representation theory is: given the group, determine, in terms of some ‘parameters’, the representations of the given group. For example, the classification of representations of a simply-connected simple Lie group in terms of the high weights of irreducible components, and the description of the possible high weights from the root system of the Lie algebra of the group, is a profound and inspiring solution to the problem for the groups to which it applies. (One can get an idea of how inspiring this solution was by consulting, for example, the bibliography of [1].)

There is also, however, the converse problem, which will be our major concern: given its representations, determine the group. The problem, as stated, is not very well-posed (what does it mean to be ‘given the representations’?) although, as is often the case in the development of mathematics, that was not a serious deterrent historically (the historical development is summarized below), and currently the concepts of category theory allow a precise statement, as we shall see. It turns out, however, that the problem does not always have a solution.