Epsilon dichotomy for linear models: the Archimedean case

Let $G={\mathrm{G}\mathrm{L}}_{2n}(\mathbb{R})$ or $G={\mathrm{G}\mathrm{L}}_n(\mathbb{H})$ and $H={\mathrm{G}\mathrm{L}}_n(\mathbb{C})$ 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 ${\mathrm{H}\mathrm{o}\mathrm{m}}_H(\pi ,\chi \circ {\det }_H)\mathrm{\ne }0$. 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 $H$.