Representations of Linear Groups (PDF)

Introduction (S. VIII-IX)

In this book, the groups enumerated in the Preface are introduced and treated as matrix groups to avoid as much as possible the machinery of manifolds, Lie groups and bundles (though some of it soon creeps in through the backdoor as the theory is further developed). Parallel to information about the structure of our groups we shall introduce and develop elements of the representation theory necessary to classify the unitary representations and to construct concrete models for these representations. As the main tool for the classi.cation we use the in.nitesimal method linearizing the representations of a group by studying those of the Lie algebra of the group.

And as the main tools for the construction of models for the representations we use – tensor products of the natural representation, – representations given by smooth functions (in particular polynomials) living on a space provided with an action of the group, and – the machinery of induced representations. Moreover, because of the growing importance in physics and the success in deriving branching relations, the procedure of geometric quantization and the orbit method, developed and propagated by Kirillov, Kostant, Du.o and many others shall be explained via its application to some of the examples above.

Besides the sources already mentioned, the author was largely in.uenced by the now classical book of Kirillov: "Elements of the Theory of Representations" [Ki] and the more recent "Introduction to the Orbit Method" [Ki1]. Other sources were the books by Barut and Raczka: "Theory of Group Representations and Applications" [BR], S. Lang: "SL(2,R)" [La], and, certainly, Serre: "Linear Representations of Finite Groups" [Se]. There is also the book by Hein: "Einfuhrung in die Struktur- und Darstellungstheorie der klassischen Gruppen" [Hei], which follows the same principle as our text, namely to do as much as possible for matrix groups, but does not go into the

Prof. Dr. Rolf Berndt, Department of Mathematics, University of Hamburg, Germany.

"This book is a very elementary introduction to some of the fundamental ideas in representation theory [...] Its emphasis on examples from physics and number theory makes it appealing to a broad audience."

Mathematical Reviews, 2008g