Cohomologically induced distinguished representations and cohomological test vectors

Abstract

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 we show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two Archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain $L$-functions via modular symbols.