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Is it true that each irreducible sl(2,$\mathbb{C}$)-module, $P(\lambda,\mu)$ with $\lambda \in \mathbb{Z}$ appears as the harish chandra module of some $(\pi_{\chi},V_{\chi})$ And given $\lambda\in\mathbb{Z}$ and $\mu\in\mathbb{C}$ such that neither of $\lambda\pm\sqrt{\mu+1}$ is an odd integer what is $\chi$

also any good references on this material? I'v only seen a cursory introduction in some course notes but can't find any good book on this.

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I am not an expert in this, but you might probably be able to find something similar in Daniel Bump's Automorphic Forms and Representations. There is one section in Chapter 2 on Harish-Chandra's $\mathfrak{g}-K$ module. I am not sure if this is what you are looking for.

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    I checked it out but it doesnt have what I need thanks though.2012-11-21