Titlebook: Representations of Linear Groups; An Introduction Base Rolf Berndt Textbook 2007 Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH

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Basic Algebraic Concepts for Group Representations, concept of a ., which is completely adequate and sufficient for the study of representations of finite groups. In a later chapter we will enrich the discussion with elements from topology to refine the notion of a representation to that of a .. We start by stating the essential definitions and only

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Representations of Finite Groups,representation theory works. Thus we present the basic facts. We will not do it by using the abstract notions offered by modern algebra (as for instance in [FH]), where eventually the whole theory can be put into the nutshell of one page. Instead we follow very closely the classic presentation from

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Representations of Abelian Groups,at unitary irreducible representations are all one-dimensional (we proved this as a consequence of the “Unitary Schur” in 1.5). But in general the unitary dual is not denumerable. Since in this text we are mainly interested in groups like SL(2,R) and Heis(R), where unitary irreducible representation

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The Infinitesimal Method,nal. Before we go further into the construction of infinite-dimensional representations, we discuss a method which allows a linearization of our objects and hence simplifies the task to determine the structure of the representations and classify them. As we shall see, this is helpful for compact and

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Induced Representations,ialization one rediscovers Example 1.5 in our section 1.3, where we constructed representations in function spaces on .-homogeneous spaces. Already in 1898, Frobenius worked with this method for finite groups. Later on, Wigner, Bargman, and Gelfand-Neumark used it to construct representations of spe

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Geometric Quantization and the Orbit Method,Ki2] for the description of the unitary dual of nilpotent Lie groups. By fur- ther work of Kirillov and many others, in particular B. Kostant, M. Duflo, M. Vergne and D. Vogan, it grew into an important tool for explicit constructions of representations of various types of groups. The construction m

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The Infinitesimal Method,mples, this method allows us to classify the (unitary) irreducible representations and, hence, gives a parametrization of .. It will further prove to be helpful for the construction of explicit models for representations (.)