Knop, Friedrich

Functoriality Properties of the Dual Group

Doc. Math. 24, 47-64 (2019)
DOI: 10.25537/dm.2019v24.47-64

Summary

Let \(G\) be a connected reductive group. Previously, it was shown that for any \(G\)-variety \(X\) one can define the dual group \(G^\vee_X\) which admits a natural homomorphism with finite kernel to the Langlands dual group \(G^\vee\) of \(G\). Here, we prove that the dual group is functorial in the following sense: if there is a dominant \(G\)-morphism \(X\to Y\) or an injective \(G\)-morphism \(Y\to X\) then there is a unique homomorphism with finite kernel \(G^\vee_Y\to G^\vee_X\) which is compatible with the homomorphisms to \(G^\vee\).

Mathematics Subject Classification

17B22, 14L30, 11F70

Keywords/Phrases

spherical variety, Langlands dual group, root system, algebraic group, reductive group

References

  • 1. Ahiezer, Dmitry, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom., 1, 49-78, (1983); DOI 10.1007/BF02329739; zbl 0537.14033; MR0739893.
  • 2. Bravi, Paolo; Luna, Domingo, An introduction to wonderful varieties with many examples of type $\rm F_4$, J. Algebra, 329, 4-51, (2011); DOI 10.1016/j.jalgebra.2010.01.025; zbl 1231.14040; MR2769314; arxiv 0812.2340.
  • 3. Bravi, Paolo; Pezzini, Guido, Primitive wonderful varieties, Math. Z., 282, 1067-1096, (2016); DOI 10.1007/s00209-015-1578-5; zbl 1356.14041; MR3473657; arxiv 1106.3187.
  • 4. Brion, Michel, On spherical varieties of rank one (after D. Ahiezer, A. Huckleberry, D. Snow). In: CMS Conf. Proc., 10, Amer. Math. Soc., Providence, RI, Group actions and invariant theory, (1988); zbl 0702.20029; MR1021273.
  • 5. Brion, Michel, Vers une généralisation des espaces symétriques, J. Algebra, 134, 115-143, (1990); DOI 10.1016/0021-8693(90)90214-9; zbl 0729.14038; MR1068418.
  • 6. Gaitsgory, Dennis; Nadler, David, Spherical varieties and Langlands duality, Mosc. Math. J., 10, 65-137, 271, (2010); zbl 1207.22013; MR2668830; arxiv math/0611323.
  • 7. Knop, Friedrich, Weylgruppe und Momentabbildung, Invent. Math., 99, 1-23, (1990); DOI 10.1007/BF01234409; zbl 0726.20031; MR1029388.
  • 8. Knop, Friedrich, The Luna-Vust theory of spherical embeddings. In: Manoj Prakashan, Madras, Proceedings of the Hyderabad Conference on Algebraic Groups, (1989); zbl 0812.20023; MR1131314.
  • 9. Knop, Friedrich, Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind, Math. Ann., 295, 333-363, (1993); DOI 10.1007/BF01444891; zbl 0789.14040; MR1202396.
  • 10. Knop, Friedrich, The asymptotic behavior of invariant collective motion, Invent. Math., 116, 309-328, (1994); DOI 10.1007/BF01231563; zbl 0802.58024; MR1253195.
  • 11. Knop, Friedrich, Automorphisms, root systems, and compactifications of homogeneous varieties, J. Amer. Math. Soc., 9, 153-174, (1996); DOI 10.1090/S0894-0347-96-00179-8; zbl 0862.14034; MR1311823.
  • 12. Knop, Friedrich; Krötz, Bernhard, Reductive group actions, Preprint, 62 pp., (2016); arxiv 1604.01005.
  • 13. Knop, Friedrich; Schalke, Barbara, The dual group of a spherical variety, Preprint, 30 pp. pp., (2017); DOI 10.1090/mosc/270; zbl 06929240; MR3738085; arxiv 1702.08264.
  • 14. Losev, Ivan, Uniqueness property for spherical homogeneous spaces, Duke Math. J., 147, 315-343, (2009); DOI 10.1215/00127094-2009-013; zbl 1175.14035; MR2495078; arxiv 0904.2937.
  • 15. Luna, Domingo, Variétés sphériques de type $A$, Publ. Math. Inst. Hautes Études Sci., 94, 161-226, (2001); DOI 10.1007/s10240-001-8194-0; zbl 1085.14039; MR1896179.
  • 16. Luna, Domingo; Vust, Thierry, Plongements d'espaces homogènes, Comment. Math. Helv., 58, 186-245, (1983); DOI 10.1007/BF02564633; zbl 0545.14010; MR0705534.
  • 17. Panyushev, Dmitri, On homogeneous spaces of rank one, Indag. Math. (N.S.), 6, 315-323, (1995); DOI 10.1016/0019-3577(95)93199-K; zbl 0838.53041; MR1351150.
  • 18. Rosenlicht, Maxwell, Some basic theorems on algebraic groups, Amer. J. Math., 78, 401-443, (1956); DOI 10.2307/2372523; MR0082183.
  • 19. Sakellaridis, Yiannis; Venkatesh, Akshay, Periods and harmonic analysis on spherical varieties, 296 p. pp., (2017); zbl 06847674; arxiv 1203.0039v4.

Affiliation

Knop, Friedrich
Department Mathematik, FAU Erlangen-Nürnberg, Cauerstraße 11, D-91058 Erlangen, Germany

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