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Let $G$ be a real reductive group, $P=MN$ a parabolic subgroup with its Levi decomposition, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Now given a smooth representation $(\pi,V)$ of $G$, I wanna ask when the functor $H_0(\mathfrak{n},V)$ is exact? We may assume the representation $\pi$ has good properties, e.g., it is of moderate growth, $V$ is a nuclear Frechet space.

In general, the $\mathfrak{n}$ homology functor is only right exact on category of $\mathfrak{n}$ modules. So we may ask when this functor is exact if restricted to a subcategory.

Thanks.

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  • $\begingroup$ You can take the full subcategory of all representations that satisfy $H_1({\mathfrak n},V)=0$. The long exact homology sequence then tells you that the functor $H_0({\mathfrak n},.)$ is exact on that category. $\endgroup$
    – user1688
    Mar 1, 2011 at 16:20
  • $\begingroup$ My first reference for such questions is the paper of Casselman, Hecht, and Milicic (and Taylor, for the appendix?), "Bruhat filtrations and Whittaker vectors for real groups", in Proc. of Symp. in Pure Math., 2000. While I think your question is too vague for a real answer, I think this reference might provide answers in cases you care about, or at least the applicable methods and a good bibliography to follow. $\endgroup$
    – Marty
    Mar 2, 2011 at 5:03

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