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 can we conclude about $\chi$
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