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[Separated from another question]

Can you give an example of a non-semisimple module over $\mathbb C\mathrm{GL}_n(\mathbb C)$? (Preferably one without direct summands, i.e. an indecomposable module)

Are such modules always infinite dimensional over $\mathbb C$?

How do typical examples look like?

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Peter Patzt
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  • In modern representation theory, one does work with $C_c^\infty(G)$ for $G$ non-finite Lie group. – Marc Palm Mar 11 '14 at 11:02
  • Can you make that object more precise by defining it or giving it a name that I can google? – Peter Patzt Mar 11 '14 at 11:05
  • It is the algebra of smooth, compactly supported functions on $G$ with convolution wrt to the Haar measure. – Marc Palm Mar 11 '14 at 11:06
  • And that algebra is semisimple, or has only infinite nonsemisimple modules? What is the upside of working with it and what does it have to do with my question? – Peter Patzt Mar 11 '14 at 11:07
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    Irreducible modules correspond then to irreducible reps of $G$. Unitary reps are automatically semisimple, but smooth admissible ones need not be. For example, the trivial representation can be found in the parabolic induction of the modulus character of the Borel subgroup. But you seem to be intersted only in the fact that $GL_n(C)$ is a group, not a Lie group, because the group algebra adresses only $GL_n(C)$ as a discrete group? – Marc Palm Mar 11 '14 at 11:08
  • Do you have a reference where I can read up on this? I am especially interested in whether certain inductions of irreducible representations are again semisimple or possibly even fin-dim. (See this question: http://math.stackexchange.com/questions/707929/reciprocity-for-branching-rules-of-mathrmgl-n-mathbb-c ) – Peter Patzt Mar 11 '14 at 11:16
  • Are you interested in $GL(n,C)$ as a discrete group or a Lie group? For the Lie group part, there are references but I'd start with understanding the classification of smooth, admissible representations of $SL(2,C)$ and go on from there. Everything centers around parabolic induction. – Marc Palm Mar 11 '14 at 11:18
  • let us continue this discussion in chat – Peter Patzt Mar 11 '14 at 11:23

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