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Epsilon Dichotomy for Linear Models: The Archimedean Case.
- Source :
-
IMRN: International Mathematics Research Notices . Oct2023, Vol. 2023 Issue 20, p17853-17891. 39p. - Publication Year :
- 2023
-
Abstract
- Let |$G=\textrm {GL}_{2n}({\mathbb {R}})$| or |$G=\textrm {GL}_n({\mathbb {H}})$| and |$H=\textrm {GL}_n({\mathbb {C}})$| 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$| -distinguished if |$\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 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$|. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 10737928
- Volume :
- 2023
- Issue :
- 20
- Database :
- Academic Search Index
- Journal :
- IMRN: International Mathematics Research Notices
- Publication Type :
- Academic Journal
- Accession number :
- 173152062
- Full Text :
- https://doi.org/10.1093/imrn/rnad110