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Some simple examples in the representation theory of semisimple Lie groups. (English) Zbl 0833.22019
Starting out with calculations in \(\text{Sl}(2,\mathbb{R})\)-modules consisting of functions in two variables and different degrees of homogeneity the author discusses several families of representations of classical Lie groups. The guiding strategy for all these examples is: Consider a natural action of the group on some finite dimensional space and associate a family of infinite dimensional modules of functions on this space with it. Then consider a Cartan decomposition \(G = K\text{exp}({\mathfrak p})\) of the group, calculate the \(K\)-types and the way operators from \(\mathfrak p\) interchange them. Finally use these data to determine composition series and unitarizability. In this way an amazing amount of information is obtained using only very elementary methods.

22E46 Semisimple Lie groups and their representations