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 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.

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 product

References in corpus (5)

• Whittaker rational structures and special values of the Asai $L$-function
• Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))
• Existence of Klyachko Models for GL(n;R) and GL(n;C)
• Invariant Functionals on the Speh representation
• Uniqueness of twisted linear periods and twisted Shalika periods

Cited by in corpus (11)

• Whittaker rational structures and special values of the Asai $L$-function
• On Period Relations for Automorphic L-functions II
• $L$-functions of $\mathrm{GL}(2n):$ $p$-adic properties and non-vanishing of twists
• Uniqueness of twisted linear periods and twisted Shalika periods
• Period Relations for Standard $L$-functions of Symplectic Type
• On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the ${\mathrm{GL}}_{2n}$ eigenvariety
• Archimedean Non-vanishing, Cohomological Test Vectors, and Standard $L$-functions of ${\mathrm{GL}}_{2n}$: Complex Case
• Nonvanishing of self-dual $L$-values via spectral decomposition of shifted convolution sums
• p-adic L-functions for GL(n)
• On Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms
• A rationality result for the exterior and the symmetric square $L$-function