Archimedean Bernstein-Zelevinsky Theory and Homological Branching Laws

Abstract: We develop the Bernstein-Zelevinsky theory for quasi-split real classical groups and employ this framework to establish an Euler-Poincar\'e characteristic formula for general linear groups. The key to our approach is establishing the Casselman-Wallach property for the homology of the Jacquet functor, which also provides an affirmative resolution to an open question proposed by A. Aizenbud, D. Gourevitch and S. Sahi. Furthermore, we prove the vanishing of higher extension groups for arbitrary pairs of generic representations, confirming a conjecture of Dipendra Prasad. We also utilize the Bernstein-Zelevinsky theory to establish two additional results: the Leibniz law for the highest derivative and a unitarity criterion for general linear groups. Lastly, we apply the Bernstein-Zelevinsky theory to prove the Hausdorffness and exactness of the twisted homology of split even orthogonal groups.