Cohomologically induced distinguished representations and cohomological test vectors

TL;DR

This paper develops methods to construct non-vanishing linear functionals on cohomologically induced representations of real reductive groups, aiding the study of special L-function values through modular symbols.

Contribution

It introduces a new construction of $oldsymbol{ ext{chi}}$-invariant linear functionals on cohomologically induced representations and proves their non-vanishing on bottom layers.

Findings

Constructed $oldsymbol{ ext{chi}}$-invariant linear functionals

Proved non-vanishing on bottom layers of representations

Established archimedean non-vanishing assumptions

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 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.