Epsilon dichotomy for linear models: the Archimedean case

TL;DR

This paper proves an Archimedean analogue of a conjecture relating to distinguished representations of certain real reductive groups, analyzing their root numbers and Schwartz homology contributions.

Contribution

It computes root numbers for distinguished representations in the Archimedean case and develops a general framework for analyzing Schwartz homology in real reductive groups.

Findings

Computed root numbers for distinguished representations.

Proved finiteness and Hausdorff property of Schwartz homology.

Extended analysis to general real reductive groups of inner type.

Abstract

Let $G=\mathrm{GL}_{2n}(\mathbb{R})$ or $G=\mathrm{GL}_n(\mathbb{H})$ and $H=\mathrm{GL}_n(\mathbb{C})$ regarded as a subgroup of $G$. Here, $\mathbb{H}$ is the quaternion division algebra over $\mathbb{R}$. For a character $\chi$ on $\mathbb{C}^\times$, we say that an irreducible smooth admissible moderate growth representation $\pi$ of $G$ is $\chi_H$-distinguished if $\mathrm{Hom}_H(\pi, \chi\circ\det_H)\neq0$. We compute the root number of a $\chi_H$-distinguished representation $\pi$ twisted by the representation induced from $\chi$. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast(H, \pi\otimes\chi)$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $\chi$ of $H$.