Cohomologically induced distinguished representations and cohomological test vectors

Let $G$ be a real reductive group, and let $\chi$ be a character of a reductive subgroup $H$ of $G$. We construct $\chi$-invariant linear functionals on certain cohomologically induced representations of $G$, and show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two archimedean non-vanishing assumptions, which are crucial in the study of special values of L-functions via modular symbols.Comment: We still do not have a proof of "Theorem 4.3" of Version 1. The following correction is made in this version: the invariant bilinear form in the proof of Lemma A.4, which is incorrectly used in the last version, is now changed to an invariant inner produc

Notes on MVW-extensions

We review certain basic geometric and analytic results concerning MVW-extensions of classical groups, following Moeglin-Vigneras-Waldspurger. The related results for Jacobi groups, metaplectic groups, and special orthogonal groups are also included

Uniqueness of Rankin-Selberg periods

Let $k$ be a local field of characteristic zero. Rankin-Selberg's local zeta integrals produce linear functionals on generic irreducible admissible smooth representations of $GL_n(k)\times GL_r(k)$, with certain invariance properties. We show that up to scalar multiplication, these linear functionals are determined by the invariance properties