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Induced Representations

Abstract

Induction is in this context a method to construct representations of a group starting by a representation of a subgroup. As we shall see, in its ultimate specialization one rediscovers Example 1.5 in our section 1.3, where we constructed representations in function spaces on G-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 special groups, in particular the Poincaré group. But it was Mackey, who, from 1950 on, developed a systematic treatment using essentially elements of functional analysis. Because it is beyond the scope of this book, we do not describe these here as carefully as we should. In most places we simply give recipes so that we can construct the representations for the groups we are striving for. We refer to Mackey [Ma], Kirillov [Ki] p.157ff, Barut-Raczka [BR] p.473ff, or Warner [Wa] p.365ff for the necessary background.

Keywords

Vector Bundle Line Bundle Unitary Representation Heisenberg Group Semidirect Product 
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Copyright information

© Friedr. Vieweg & Sohn Verlag | GWV Fachverlage GmbH, Wiesbaden 2007

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