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.