Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
134 tokens/sec
GPT-4o
10 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Symmetric varieties for endoscopic groups (2401.09156v2)

Published 17 Jan 2024 in math.NT, math.AG, and math.RT

Abstract: 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.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (82)
  1. Groupoids, Geometric Induction and Gelfand Models. arXiv preprint arXiv:2012.15384, 2020.
  2. Jeffrey Adams. The real Chevalley involution. Compos. Math., 150(12):2127–2142, 2014.
  3. Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis’s theorem. Duke Math. J., 149(3):509–567, 2009. With an appendix by the authors and Eitan Sayag.
  4. On the SL⁢(2)SL2{\rm SL}(2)roman_SL ( 2 ) period integral. Amer. J. Math., 128(6):1429–1453, 2006.
  5. A local-global question in automorphic forms. Compos. Math., 149(6):959–995, 2013.
  6. Existence of equivariant models of spherical varieties and other G𝐺Gitalic_G-varieties. Int. Math. Res. Not. IMRN, (20):15932–16034, 2022.
  7. A. Borel. Automorphic L𝐿Litalic_L-functions. In Automorphic forms, representations and L𝐿Litalic_L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 27–61. Amer. Math. Soc., Providence, R.I., 1979.
  8. Mikhail Borovoi. Equivariant models of spherical varieties. Transform. Groups, 25(2):391–439, 2020.
  9. Raphaël Beuzart-Plessis. On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad. Invent. Math., 214(1):437–521, 2018.
  10. The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case. Publ. Math. Inst. Hautes Études Sci., 135:183–336, 2022.
  11. Isolation of cuspidal spectrum, with application to the Gan-Gross-Prasad conjecture. Ann. of Math. (2), 194(2):519–584, 2021.
  12. A local twisted trace formula for whittaker induction of coregular symmetric pairs: the geometric side. arXiv preprint arXiv:2312.10845, 2023.
  13. Michel Brion. Vers une généralisation des espaces symétriques. J. Algebra, 134(1):115–143, 1990.
  14. Michel Brion. Construction of equivariant vector bundles. arXiv preprint math/0410039, 2004.
  15. Groupes réductifs. Inst. Hautes Études Sci. Publ. Math., (27):55–150, 1965.
  16. Relative langlands duality. preprint, 2023.
  17. Unitary Friedberg–Jacquet periods. arXiv preprint arXiv:2108.04064, 2021.
  18. Pseudo-reductive groups, volume 17 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2010.
  19. Brian Conrad. Reductive group schemes. Lecture notes, 2010.
  20. Slices in the loop spaces of symmetric varieties and the formality conjecture. arXiv preprint arXiv:2310.20006, 2023.
  21. Pierre Deligne. Variétés de Shimura: interprétation modulaire, et techniques de construction de modeles canoniques. 1979.
  22. On representations distinguished by unitary groups. Publ. Math. Inst. Hautes Études Sci., 115:185–323, 2012.
  23. Multiplicative Higgs bundles and involutions. arXiv preprint arXiv:2304.02553, 2023.
  24. An introduction to automorphic representations. Available at website: https://www. math. duke. edu/~ hahn/GTM. pdf, 2019.
  25. Wee Teck Gan and Bryan Wang Peng Jun. Generalised Whittaker models as instances of relative Langlands duality. arXiv preprint arXiv:2309.08874, 2023.
  26. Oscar García-Prada. Vinberg pairs and Higgs bundles. arXiv preprint arXiv:2305.08202, 2023.
  27. Higgs bundles for real groups and the Hitchin-Kostant-Rallis section. Trans. Amer. Math. Soc., 370(4):2907–2953, 2018.
  28. Nearby cycle sheaves for symmetric pairs. arXiv preprint arXiv:1805.02794, 2018.
  29. Twisted relative trace formulae with a view towards unitary groups. Amer. J. Math., 136(1):1–58, 2014.
  30. Aloysius G. Helminck. Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces. Adv. in Math., 71(1):21–91, 1988.
  31. A. G. Helminck. On the classification of k𝑘kitalic_k-involutions. Adv. Math., 153(1):1–117, 2000.
  32. A class of parabolic k𝑘kitalic_k-subgroups associated with symmetric k𝑘kitalic_k-varieties. Trans. Amer. Math. Soc., 350(11):4669–4691, 1998.
  33. T. Hameister and B. Morrissey. The Hitchin fibration for symmetric pair. preprint, 2023.
  34. Johannes Hofscheier. Containment relations among spherical subgroups. arXiv preprint arXiv:1804.00378, 2018.
  35. On rationality properties of involutions of reductive groups. Adv. Math., 99(1):26–96, 1993.
  36. Hervé Jacquet. Sur un résultat de Waldspurger. Ann. Sci. École Norm. Sup. (4), 19(2):185–229, 1986.
  37. Tasho Kaletha. Rigid inner forms of real and p𝑝pitalic_p-adic groups. Ann. of Math. (2), 184(2):559–632, 2016.
  38. Tasho Kaletha. Lectures on the stable trace formula, with emphasis on SL⁢(2)SL2\mathrm{SL}(2)roman_SL ( 2 ). 2019. Lecture notes.
  39. Friedrich Knop. Automorphisms, root systems, and compactifications of homogeneous varieties. J. Amer. Math. Soc., 9(1):153–174, 1996.
  40. Friedrich Knop. Functoriality properties of the dual group. Doc. Math., 24:47–64, 2019.
  41. Robert E. Kottwitz. Rational conjugacy classes in reductive groups. Duke Mathematical Journal, 49(4):785–806, 1982.
  42. Robert E. Kottwitz. Stable trace formula: cuspidal tempered terms. Duke Math. J., 51(3):611–650, 1984.
  43. Robert E. Kottwitz. Stable trace formula: elliptic singular terms. Math. Ann., 275(3):365–399, 1986.
  44. Robert E. Kottwitz. Tamagawa numbers. Ann. of Math. (2), 127(3):629–646, 1988.
  45. The dual group of a spherical variety. Transactions of the Moscow Mathematical Society, 78:187–216, 2017.
  46. Jean-Pierre Labesse. Cohomologie, stabilisation et changement de base. Astérisque, (257):vi+161, 1999. Appendix A by Laurent Clozel and Labesse, and Appendix B by Lawrence Breen.
  47. R. P. Langlands. Stable conjugacy: definitions and lemmas. Canadian J. Math., 31(4):700–725, 1979.
  48. Spencer Leslie. The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods. preprint, 2019.
  49. Spencer Leslie. Endoscopy for unitary symmetric spaces. preprint arXiv:1910.09685, 2019.
  50. Spencer Leslie. An analogue of the Grothendieck-Springer resolution for symmetric spaces. Algebra Number Theory, 15(1):69–107, 2021.
  51. Spencer Leslie. On the stabilization of relative trace formulae: descent and the fundamental lemma. Adv. Math., 394:Paper No. 108026, 68, 2022.
  52. Spencer Leslie. On the stabilization of relative trace formulae: elliptic terms. forthcoming, 2024.
  53. Paul Levy. Involutions of reductive lie algebras in positive characteristic. Advances in Mathematics, 210(2):505–559, 2007.
  54. L𝐿{L}italic_L-indistinguishability for SL2subscriptSL2\mathrm{SL}_{2}roman_SL start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-periods. in progress.
  55. Ivan V. Losev. Uniqueness property for spherical homogeneous spaces. Duke Math. J., 147(2):315–343, 2009.
  56. Periods of Eisenstein series: the Galois case. Duke Math. J., 120(1):153–226, 2003.
  57. R. P. Langlands and D. Shelstad. On the definition of transfer factors. Math. Ann., 278(1-4):219–271, 1987.
  58. Conjugates of Shimura varieties. In Hodge cycles, motives, and Shimura varieties, pages 280–356. Springer, 1982.
  59. David Nadler. Perverse sheaves on real loop Grassmannians. Invent. Math., 159(1):1–73, 2005.
  60. Bao Chau Ngo. Fibration de Hitchin et endoscopie. Invent. Math., 164(2):399–453, 2006.
  61. Bao Châu Ngô. Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci., (111):1–169, 2010.
  62. Bao Châu Ngô and B. Morrissey. forthcoming, 2024.
  63. Dipendra Prasad. Generalizing the MVW involution, and the contragredient. Trans. Amer. Math. Soc., 372(1):615–633, 2019.
  64. Dipendra Prasad. A “relative” local Langlands correspondence. arXiv preprint arXiv:1512.04347 (updated on personal website), 2020.
  65. On the residue method for period integrals. Duke Math. J., 170(7):1457–1515, 2021.
  66. N. Ressayre. Spherical homogeneous spaces of minimal rank. Adv. Math., 224(5):1784–1800, 2010.
  67. R. W. Richardson. Orbits, invariants, and representations associated to involutions of reductive groups. Invent. Math., 66(2):287–312, 1982.
  68. Roberto Rubio. On the Gelfand property for complex symmetric pairs. to appear in Trans. Amer. Math. Soc., 2022.
  69. Tonny Albert Springer et al. Some results on algebraic groups with involutions. In Algebraic groups and related topics, pages 525–543. Mathematical Society of Japan, 1985.
  70. Yiannis Sakellaridis. The Schwartz space of a smooth semi-algebraic stack. Selecta Math. (N.S.), 22(4):2401–2490, 2016.
  71. Yiannis Sakellaridis. Spherical varieties, functoriality, and quantization. arXiv preprint arXiv:2111.03004, 2021.
  72. Jiro Sekiguchi. The nilpotent subvariety of the vector space associated to a symmetric pair. Publ. Res. Inst. Math. Sci., 20(1):155–212, 1984.
  73. T. A. Springer. The classification of involutions of simple algebraic groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34(3):655–670, 1987.
  74. Robert Steinberg. Endomorphisms of linear algebraic groups. Memoirs of the American Mathematical Society, No. 80. American Mathematical Society, Providence, R.I., 1968.
  75. Periods and harmonic analysis on spherical varieties. Astérisque, (396):viii+360, 2017.
  76. Dmitry A. Timashev. Homogeneous spaces and equivariant embeddings, volume 138 of Encyclopaedia of Mathematical Sciences. Springer, Heidelberg, 2011. Invariant Theory and Algebraic Transformation Groups, 8.
  77. J. Tits. Classification of algebraic semisimple groups. In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pages 33–62. Amer. Math. Soc., Providence, RI, 1966.
  78. Thierry Vust. Opération de groupes réductifs dans un type de cônes presque homogenes. Bulletin de la Societe Mathematique de France, 102:317–333, 1974.
  79. Thierry Vust. Plongements d’espaces symétriques algébriques: une classification. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17(2):165–195, 1990.
  80. Chen Wan. On a multiplicity formula for spherical varieties. J. Eur. Math. Soc. (JEMS), 24(10):3629–3678, 2022.
  81. Zhiwei Yun. The fundamental lemma of Jacquet and Rallis. Duke Math. J., 156(2):167–227, 2011. With an appendix by Julia Gordon.
  82. Wei Zhang. Automorphic period and the central value of Rankin-Selberg L-function. J. Amer. Math. Soc., 27(2):541–612, 2014.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now