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ON JORDAN CLASSES FOR VINBERG’S θ-GROUPS

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Abstract

V. L. Popov has recently introduced an analogue of Jordan classes (packets or decomposition classes) for the action of a θ-group (G0, V), showing that they are finitely-many, locally-closed, irreducible unions of G0-orbits of constant dimension partitioning V. We carry out a local study of their closures showing that Jordan classes are smooth and that their closure is a union of Jordan classes. We parametrize Jordan classes and G0-orbits in a given class in terms of the action of subgroups of Vinberg’s little Weyl group, and include several examples and counterexamples underlying the differences with the symmetric case and the critical issues arising in the θ-situation.

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References

  1. Revêtements Étales et Groupe Fondamental (SGA 1), Documents Mathématiques (Paris), Vol. 3, Société Mathématique de France, Paris, 2003. Séminaire de Géométrie Algébrique du Bois Marie 1960–61, directed by A. Grothendieck, with two papers by M. Raynaud, updated and annotated reprint of the 1971 original [Lecture Notes in Math., Vol. 224, Springer, Berlin, 1971].

  2. A. Altman, S. Kleiman, Introduction to Grothendieck Duality Theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin, 1970.

  3. P. Bardsley, R. W. Richardson, Étale slices for algebraic transformation groups in characteristic p, Proc. London Math. Soc. s3-51 (1985), 295–317.

  4. W. Borho, Über Schichten halbeinfacher Lie-Algebren, Invent. Math. 65 (1981/82), no. 2, 283–317.

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Borho, H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), no. 1, 61–104.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Broer, Decomposition varieties in semisimple Lie algebras, Canad. J. Math. 50 (1998) no. 5, 929–971.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Bulois, Sheets of symmetric Lie algebras and Slodowy slices, J. Lie Theory 21 (2011), no. 1, 1–54.

    MathSciNet  MATH  Google Scholar 

  8. M. Bulois, P. Hivert, Sheets in symmetric Lie algebras and slice induction, Transform. Groups 21 (2016), no. 2, 355–375.

    Article  MathSciNet  MATH  Google Scholar 

  9. W. A. de Graaf, Computing representatives of nilpotent orbits of θ-groups, J. Symbolic Comput. 46 (2011), no. 4, 438–458.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. Yu. Galitski, D. A. Timashev, On classification of metabelian Lie algebras, J. Lie Theory 9 (1999), no. 1, 125–156.

    MathSciNet  MATH  Google Scholar 

  11. V. G. Kac, Some remarks on nilpotent orbits. J. Algebra 64 (1980) no. 1, 190–213.

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Levy, Vinberg’s θ-groups in positive characteristic and Kostant–Weierstrass slices, Transform. Groups 14 (2009), no. 2, 417–461.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Levy, KW-sections for Vinberg’s θ-groups of exceptional type, J. Algebra 389 (201), 78–97.

  16. D. Luna, Slices étales, Mémoires de la S.M.F. 33 (1973), 81–105.

  17. G. Lusztig, N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), no. 1, 41–52.

  18. G. Lusztig, Z. Yun, Z/m-graded Lie algebras and perverse sheaves, I, Represent. Theory 21 (2017), 277–321.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Lusztig, Z. Yun, Z/m-graded Lie algebras and perverse sheaves, III. Graded double affine Hecke algebra, Represent. Theory 22 (2018), 87–118.

  20. А. Г. Нурмиев, Орбиты и инварианты кубических матриц третьего порядка, Матем сб. 191 (2000), номер 5, 101–108. Engl. transl.: A. G. Nurmiev, Orbits and invariants of cubic matrices of order three, Sbornik: Mathematics 191 (2000), no. 5, 717–724.

  21. D. I. Panyushev, On invariant theory of θ-groups, J. Algebra 283 (2005) no. 2, 655–670.

    Article  MathSciNet  MATH  Google Scholar 

  22. D. I. Panyushev, Periodic automorphisms of Takiff algebras, contractions, and θ-groups, Transform. Groups 14 (2009) no. 2, 463–482.

    Article  MathSciNet  MATH  Google Scholar 

  23. Д. Д. Первушин, Инварианты и орбиты стандартного (SL4(C) × SL4(C) × SL2(C))-модуль, Изв. РАН. Сер. матем. 64 (2000), вып. 5, 133–146. Engl. transl.: D. D. Pervushin, Invariants and orbits of the standard (SL4(C) × SL4(C) × SL2(C))-module, Izvestiya: Mathem. 64 (2000), no. 5, 1003–1015.

  24. V. L. Popov, Modality of representations, and packets for θ-groups, in: Lie Groups, Geometry, and Representation Theory, Progr. Math., Vol. 326, Birkhäuser/Springer, Cham, 2018, pp. 459–479.

  25. M. Reeder, P. Levy, J-K. Yu, B. H. Gross, Gradings of positive rank on simple Lie algebras, Transform. Groups 17 (2012) no. 4, 1123–1190.

  26. M. Reeder, J.-K. Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc. 27 (2014) no. 2, 437–477.

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, Vol. 815, Springer, Berlin, 1980.

  28. T. A. Springer, Linear Algebraic Groups, Progress in Mathematics, Vol. 9, Birkhäuser, Boston, Mass., 1981.

  29. P. Tauvel, R. W. T. Yu, Lie Algebras and Algebraic Groups, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005.

  30. A. Čap, J. Slovák, Parabolic Geometries. I. Background and General Theory, Mathematical Surveys and Monographs, Vol. 154, American Mathematical Society, Providence, RI, 2009.

  31. Э. Б. Винберг, Группа Вейля градуированной алгебры Ли, Из. АН СССР. Сер. матем. 40 (1976), вып. 3, 488–526. Engl. transl.: È. B. Vinberg, The Weyl group of a graded Lie algebra, Math. USSR-Izv. 10 (1977), 463–495.

  32. Э. Б. Винберг, Классификация однородных нильпотентных элементов полупростой градуированной алгебры Ли, Тр. сем. по вект. и тенз. анализу 19 (1979), 155–177. Engl. transl.: È. B. Vinberg, Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra, Sel. Math. Sov. 6 (1987), 15–35.

  33. Э. Б. Винберг, В. В. Горбацевич, А. Л. Онищик, Строение групп и алгебр Ли, Итоги науки и техн. Совр. пробл. математики. Фунд. направл. т. 41 (Группы и алгебры Ли–3), ВИНИТИ М. 1990, 5–257. Engl. transl: V. V. Gorbatsevich, A. L. Onishchik, È. B. Vinberg, Structure of Lie groups and Lie algebras, Encycl. of Math. Sci., Vol. 41 (Lie Groups and Lie Algebras III), Springer-Verlag, Berlin, 1994, pp. 1–248.

  34. Э. Б. Винберг, А. Г. Элашвили, Классификация тривекторов девятимерного пространства, Тр. сем. по вект. и тенз. анализу 18 (1978), 197–233. Engl. transl.: È. B. Vinberg, A. G. Èlašvili, The classification of the three-vectors of nine-dimensional space, Sel. Math. Sov. 7 (1988), no. 1, 63–98.

  35. Э. Б. Винберг, В. Л. Попов, Теория инвариантов, Итоги науки и техн., Соврем. пробл. матем., Фундам. направл., т. 55 (Алгебраическая геометрия–4), ВИНИТИ, М., 1989, 137–314. Engl. transl.: V. L. Popov, E. B. Vinberg, Invariant theory, in: Encyclopaedia of Mathematical Sciences, Vol. 55 (Algebraic Geometry, IV), Springer-Verlag, Berlin, 1994, pp. 123–284.

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Authors and Affiliations

  1. Dipartimento di Matematica, “Tullio Levi-Civita” (DM), Università degli Studi di Padova, Via Trieste 63, 35121, Padova, Italy

    GIOVANNA CARNOVALE & FRANCESCO ESPOSITO

  2. Department of Mathematics and Statistics, UiT the Arctic University of Norway, 9037, Tromsø, Norway

    ANDREA SANTI

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  1. GIOVANNA CARNOVALE
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  2. FRANCESCO ESPOSITO
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  3. ANDREA SANTI
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Correspondence to GIOVANNA CARNOVALE.

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Dedicated to Prof. È. B. Vinberg

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This research was partially supported by DOR1898721/18 and BIRD179758/17 funded by the University of Padova and FFABR2017 funded by MIUR. This project started when the third named author was holding a Type B post-doc fellowship at the University of Padova. The third named author acknowledges that the research leading to these results has received funding from the Norwegian Financial Mechanism 2014-2021 (project registration number 2019/34/H/ST1/00636).

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CARNOVALE, G., ESPOSITO, F. & SANTI, A. ON JORDAN CLASSES FOR VINBERG’S θ-GROUPS. Transformation Groups 28, 151–183 (2023). https://doi.org/10.1007/s00031-021-09675-8

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