Epsilon dichotomy for linear models: the Archimedean case
Abstract
Let $D$ be a quaternion algebra over $\mathbb{R}$, $G=\mathrm{GL}_n(D)$ and $H=\mathrm{GL}_n(\mathbb{C})$ regarded as a subgroup of $G$. 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 TaklooBighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$orbits in a flag manifold of $G$ on 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 finitedimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finitedimensional moderate growth representation $\chi$ of $H$.
 Publication:

arXiv eprints
 Pub Date:
 July 2022
 arXiv:
 arXiv:2207.00743
 Bibcode:
 2022arXiv220700743S
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Representation Theory
 EPrint:
 25 pages