Harish Chandra showed how to associate to an admissible representation $(\pi,V)$ of a real semisimple Lie group $G$ the so-called Harish-Chandra module $V_K$ of $K$-finite vectors in $V$. This is a $(\mathfrak g, K)$-module which is essentially a $\mathfrak g$-module ($\mathfrak g$ being the complexification of the Lie algebra of $G$) with compatible action of $K$ (a maximal compact subgroup of $G$). Further he showed

a) that every such $(\mathfrak g, K)$-module arises as the Harish-Chandra module of an irreducible representation of $G$

b) any unitary representation of $G$ is admissible and that two unitary representations are infinitesimally equivalent (e.g. their Harish-Chandra modules are isomorphic) if and only if they are unitarily equivalent.

In various sources (Howe,Tan: Non-Abelian Harmonic Analysis, Varadarajan: Introduction to harmonic analysis on ss Lie groups, Vogan: Representations of real reductive Lie groups) one can find the classification of all irreducible admissible $(\mathfrak g, K)$-modules for $G=SL(2,\mathbb R)$. Here we have the finite-dimensional modules, the highest and lowest weight modules (also called discrete series) and the principal series representations which can all be organized by their infinitesimal character (the action of the Casimir element).

1) Is it feasible to work out in a similar elementary fashion such a classification of irreducible $(g,K)$-modules for examples of low rank? (e.g. for $G=Sp(2,\mathbb R)$ - symplectic group of rank $2$) Or do we need heavy machinery to do this in general (what are the obstacles?), and what would be catchphrases to search for here?

2) Where do I find (an exposition of) the proof of the general statement a) above?


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