This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students. I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover. One approach to this problem is to emphasize the beautiful representation theory of compact groups, and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III. But I did not want to omit important topics such as the Bruhat decomposition and the theory of symmetric spaces. For these subjects, compact groups are not sufficient. Part I covers standard general properties of representations of compact groups (including Lie groups and other compact groups, such as finite or p- adic ones). These include Schur orthogonality, properties of matrix coefficients and the Peter-Weyl Theorem.
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