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Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup with Langlands decomposition $P=MAN$.

Associated to a irreducible representation $V$ of $M$ and $\nu\in \mathfrak{a}^*\otimes \mathbf{}C$, we can construct the principle series of $G$ by induced representation $C^\infty(G,V)^P$.

Question: Under what condition, the $\Gamma$ cohomology $H(\Gamma, C^\infty(G,V)^P)$ is finite dimension. If not, for other globalization, e.g. hyperfunction $C^{-\omega}(G,V)^P$, will the $\Gamma$-cohomology be finite dimension?

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