Lie groups and their representations occupy an important place in mathematics, with applications and repercussions over a wide front. The connections with various aspects of physics are of long-standing, as are the intimate relations with differential equations and differential geometry. More recently the global topology of Lie groups has provided a deep link with questions of number theory. Finally, when viewed as ‘noncommutative harmonic analysis’ the theory of representations is a branch of linear analysis.

The symposium held in Oxford in July 1977 was designed to provide an introduction to the representation theory of Lie groups on as wide a front as possible. The main lectures, which are reproduced in this volume, should give the reader some indication of the scope and results of the subject. Inevitably there are gaps in various directions, and some areas are treated in greater detail than others. This reflects the particular interests of the participants and is not to be taken as a measure of relevant importance. Broadly speaking the symposium centred on the classical case of real Lie groups and treated only briefly the p-adic and finite fields.

In Part I of these notes we have collected together the introductory material and in Part II, the more advanced lectures. The symposium was jointly sponsored and financed by the Science Research Council and the London Mathematical Society. The editorial work involved in turning lectures into manuscript was ably supervised by Glenys Luke and I am grateful to her, to the lecturers and to all others involved for their help in producing this volume.