Symmetric varieties for endoscopic groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913322907992064 |
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| author | Leslie, Spencer |
| author_facet | Leslie, Spencer |
| contents | Given a quasi-split reductive group $G$ and a symmetric variety $X$, we introduce a notion of endoscopic varieties for $(G,X)$, and establish the foundational properties of these varieties such as matching of stable semi-simple orbits. To do this, we introduce certain automorphism groups of homogeneous spherical varieties, which encode the fine rational structure needed to work over non-algebraically closed fields. In particular, we establish the existence and uniqueness of the corresponding symmetric varieties under a mild restriction of the characteristic of the field of definition. We conjecture that this construction plays a role analogous to endoscopic groups in the context of the relative trace formula. As evidence, we show how our construction gives a pre-stabilization of regular elliptic terms of relative trace formulae for many pairs $(G,X)$.
When the cotangent bundle of the symmetric variety is hyperspherical, we relate our theory to the Hamiltonian variety of the Langlands dual group introduced by Ben-Zvi, Sakellaridis, and Venkatesh, proving some structural conjectures for this variety in the symmetric setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09156 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Symmetric varieties for endoscopic groups Leslie, Spencer Number Theory Algebraic Geometry Representation Theory Given a quasi-split reductive group $G$ and a symmetric variety $X$, we introduce a notion of endoscopic varieties for $(G,X)$, and establish the foundational properties of these varieties such as matching of stable semi-simple orbits. To do this, we introduce certain automorphism groups of homogeneous spherical varieties, which encode the fine rational structure needed to work over non-algebraically closed fields. In particular, we establish the existence and uniqueness of the corresponding symmetric varieties under a mild restriction of the characteristic of the field of definition. We conjecture that this construction plays a role analogous to endoscopic groups in the context of the relative trace formula. As evidence, we show how our construction gives a pre-stabilization of regular elliptic terms of relative trace formulae for many pairs $(G,X)$. When the cotangent bundle of the symmetric variety is hyperspherical, we relate our theory to the Hamiltonian variety of the Langlands dual group introduced by Ben-Zvi, Sakellaridis, and Venkatesh, proving some structural conjectures for this variety in the symmetric setting. |
| title | Symmetric varieties for endoscopic groups |
| topic | Number Theory Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2401.09156 |