This book is based on several courses given by the authors since 1966.
It introduces the reader to the representation theory of compact Lie
groups. We have chosen a geometrical and analytical approach since we
feel that this is the easiest way to motivate and establish the theory
and to indicate relations to other branches of mathematics. Lie
algebras, though mentioned occasionally, are not used in an essential
way. The material as well as its presentation are classical; one might
say that the foundations were known to Hermann Weyl at least 50 years
ago. Prerequisites to the book are standard linear algebra and analysis,
including Stokes' theorem for manifolds. The book can be read by German
students in their third year, or by first-year graduate students in the
United States. Generally speaking the book should be useful for
mathematicians with geometric interests and, we hope, for physicists. At
the end of each section the reader will find a set of exercises. These
vary in character: Some ask the reader to verify statements used in the
text, some contain additional information, and some present examples and
counter- examples. We advise the reader at least to read through the
exercises.