Cohomological globalization and duality conjecture for Harish-Chandra modules

Let $G$ be a connected semisimple Lie group with finite center, let ${\bf g}$ be its Lie algebra, and let $K$ be a maximal compact subgroup. Let $(\pi,V_{\pi,K})$ be a Harish-Chandra module, with smooth, analytic, distribution, and hyperfunction globalizations $V_{\pi,\infty}$, $V_{\pi,\omega}$, $V_{\pi,-\infty}$, and $V_{\pi,-\omega}$, respectively. Let $\Gamma\subset G$ be a torsion-free discrete subgroup of finite covolume, and let $\widetilde{\pi}$ denote the contragredient representation. Then, if $\Gamma$ is cocompact and $p\geq 0$, the relevant cohomology groups satisfy the following isomorphisms:

$$H^p(\Gamma,V_{\pi,\omega})\cong H^p(\Gamma,V_{\pi,\infty}),\qquad H^p(\Gamma,V_{\pi,-\omega})\cong H^p(\Gamma,V_{\pi,-\infty}),$$ $$H^p(\Gamma,V_{\pi,*})^*\cong H^{\dim(G/K)-p}(\Gamma,V_{\widetilde{\pi},-*}),\qquad *=\omega,\infty,$$

and all these vector spaces are Hausdorff and finite-dimensional. If $\Gamma$ has finite covolume, then $H^*(\Gamma,V_{\pi,-\infty})$ is finite-dimensional and Hausdorff. Cohomological globalization and duality conjecture. Under the stated hypotheses, the above comparison, duality, Hausdorffness, and finite-dimensionality assertions hold. This result establishes that the cohomology of different globalizations agrees in the cocompact case and has the expected duality, while the finite-covolume distribution-globalization cohomology is also finite-dimensional and Hausdorff. The assertion is proved in Theorem of the paper.