论文标题
线性模型的Epsilon二分法：Archimedean Case
Epsilon dichotomy for linear models: the Archimedean case
论文作者
论文摘要
令$ g = \ mathrm {gl} _ {2n}（\ mathbb {r}）$或$ g = \ mathrm {gl} _n（\ mathbb {h}）$和$ h = \ mathrm {gl}在这里，$ \ mathbb {h} $是$ \ mathbb {r} $的Quaternion division代数。对于$ \ mathbb {c}^\ times $上的字符$χ$，我们说如果$ \ mathrm {hom} _h（π，χ\\ circ \ circ \ circ \ det_h），不可减少可允许的可允许的可允许的中度中等增长代表$π$ $ g $ of $ g $。我们计算了$χ_h$ distindine的表示$π$的根号，该$π$由$χ$引起的表示形式扭曲。这证明了Prasad和Takloo-Bighash的猜想的Archimedean类似物（J. ReineAngew。Math。，2011）。证明是基于对$ h $ orbits在$ g $中的贡献的分析，对$ g $对主体系列表示的Schwartz同源性。该参数的很大一部分是针对内部类型的一般实际还原组开发的。特别是，我们证明了Schwartz同源性$ H_ \ ast（H，π\ otimesC）$是有限维度的，因此，Hausdorff的还原对称对对$（G，H）$和有限的二定值表示$χ$ H $ $ H $。
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 $χ$ on $\mathbb{C}^\times$, we say that an irreducible smooth admissible moderate growth representation $π$ of $G$ is $χ_H$-distinguished if $\mathrm{Hom}_H(π, χ\circ\det_H)\neq0$. We compute the root number of a $χ_H$-distinguished representation $π$ twisted by the representation induced from $χ$. 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 $H_\ast(H, π\otimesχ)$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $χ$ of $H$.
