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The Hesselink stratification of nullcones and base change

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Let G be a connected reductive algebraic group over an algebraically closed field of characteristic p≥0. We give a case-free proof of Lusztig’s conjectures (Lusztig in Transform. Groups 10:449–487, 2005) on so-called unipotent pieces. This presents a uniform picture of the unipotent elements of G which can be viewed as an extension of the Dynkin–Kostant theory, but is valid without restriction on p. We also obtain analogous results for the adjoint action of G on its Lie algebra \(\mathfrak{g}\) and the coadjoint action of G on \(\mathfrak{g}^{*}\).

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References

  1. Bogomolov, F.A.: Holomorphic tensors and vector bundles on projective varieties. Izv. Akad. Nauk SSSR, Ser. Mat. 42, 1227–1287 (1978). [Russian]; English transl. in Math. USSR Izvestija, 13, 499–555, 1979

    MathSciNet  Google Scholar 

  2. Borel, A.: Properties and linear representations of Chevalley groups. In: Seminar on Algebraic Groups and Related Finite Groups. Lecture Notes in Math., vol. 131, pp. 1–55. Springer, Berlin (1970)

    Chapter  Google Scholar 

  3. Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Math., vol. 126. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  4. Bourbaki, N.: Groupes et Algèbres de Lie. Hermann, Paris (1975). Chapitres VII, VIII

    MATH  Google Scholar 

  5. Carter, R.: Finite Groups of Lie Type. Wiley Classics Library. Wiley, New York (1993)

    Google Scholar 

  6. Chevalley, C.: Certains schémas de groupes semi-simples. In: Séminaire Bourbaki, 6, Exp. 219, pp. 219–234. Soc. Math. France, Paris (1995)

    Google Scholar 

  7. Digne, F., Michel, J.: Representations of Finite Groups of Lie Type. LMS Student Texts, vol. 21. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  8. Dynkin, E.B.: The structure of semisimple Lie algebras. Transl. Am. Math. Soc. 9, 328–469 (1955)

    Google Scholar 

  9. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Math., vol. 150. Springer, Berlin (1995)

    MATH  Google Scholar 

  10. Goodwin, S., Röhrle, G.: Rational points of generalized flag varieties and unipotent conjugacy in finite groups of Lie type. Trans. Am. Math. Soc. 361, 177–201 (2009)

    Article  MATH  Google Scholar 

  11. Haboush, W.: Reductive groups are geometrically reductive. Ann. Math. 102, 67–83 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hesselink, W.H.: Uniform stability in reductive groups. J. Reine Angew. Math. 303(304), 74–96 (1978)

    MathSciNet  Google Scholar 

  13. Hesselink, W.H.: Desingularizations of varieties of nullforms. Invent. Math. 55, 141–163 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Holt, D.F., Spaltenstein, N.: Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristic. J. Aust. Math. Soc. 38, 330–350 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Humphreys, J.: Linear Algebraic Groups. Graduate Texts in Math., vol. 21. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  16. Jantzen, J.C.: Representations of Algebraic Groups. Academic Press, San Diego (1987)

    MATH  Google Scholar 

  17. Jantzen, J.C.: First cohomology groups for classical Lie algebras. In: Representation Theory of Finite Groups and Finite-Dimensional Algebras, Bielefeld, 1991. Progr. in Math., vol. 95, pp. 289–315. Birkhäuser, Basel (1991)

    Chapter  Google Scholar 

  18. Jantzen, J.C.: Nilpotent orbits in representation theory. In: Lie Theory. Progr. Math., vol. 228, pp. 1–211. Birkhäuser, Basel (2004)

    Chapter  Google Scholar 

  19. Katz, N.: E-polynomials, zeta-equivalence, and polynomial-count varieties. Invent. Math. 174, 614–624 (2008)

    Google Scholar 

  20. Kempf, G.: Instability in invariant theory. Ann. Math. 108, 299–316 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kirwan, F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Math. Notes, vol. 31. Princeton Univ. Press, Princeton (1984)

    MATH  Google Scholar 

  22. Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex semisimple Lie group. Am. J. Math. 81, 973–1032 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kostant, B.: Groups over ℤ. In: Proc. Symposia in Pure Math., vol. 9, pp. 90–98 (1966)

    Google Scholar 

  24. Kac, V., Weisfeiler, B.: Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p. Indag. Math. 38, 135–151 (1976)

    MathSciNet  Google Scholar 

  25. Lusztig, G.: On the finiteness of the number of unipotent classes. Invent. Math. 34, 201–213 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lusztig, G.: Unipotent elements in small characteristic. Transform. Groups 10, 449–487 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lusztig, G.: Unipotent elements in small characteristic. II. Transform. Groups 13, 773–797 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lusztig, G.: Unipotent elements in small characteristic. IV. Transform. Groups 17, 921–936 (2010)

    Article  MathSciNet  Google Scholar 

  29. Lusztig, G.: Unipotent elements in small characteristic. III. J. Algebra 329, 163–189 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse Math., vol. 34. Springer, Berlin (1994)

    Book  Google Scholar 

  31. Mizuno, K.: On the conjugate classes of unipotent classes of the Chevalley groups E 7 and E 8. Tokyo J. Math. 3, 391–461 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mumford, D.: Geometric Invariant Theory. Ergebnisse Math., vol. 34. Springer, Berlin (1965)

    Book  MATH  Google Scholar 

  33. Ness, L.: A stratification of the null-cone via the moment map. Am. J. Math. 106, 1281–1329 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pommerening, K.: Über die unipotenten Klassen reduktiver Gruppen. J. Algebra 49, 525–536 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pommerening, K.: Über die unipotenten Klassen reduktiver Gruppen. II. J. Algebra 65, 373–398 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  36. Popov, V.L., Vinberg, È.B.: Invariant Theory. Algebraic Geometry IV. Encyclopedia Math. Sci., vol. 55. Springer, Berlin (1994)

    Google Scholar 

  37. Premet, A.: Support varieties of non-restricted modules over Lie algebras of reductive groups. J. Lond. Math. Soc. 55, 236–250 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Premet, A.: Nilpotent orbits in good characteristic and the Kempf-Rousseau theory. J. Algebra 260, 338–338336 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Premet, A., Skryabin, S.: Representations of restricted Lie algebras and families of associative \(\mathcal{L}\)-algebras. J. Reine Angew. Math. 507, 189–218 (1999)

    MathSciNet  MATH  Google Scholar 

  40. Rousseau, G.: Immeubles sphériques et théorie des invariants. C. R. Math. Acad. Sci. Paris 286, 247–250 (1978)

    MathSciNet  MATH  Google Scholar 

  41. Seshadri, C.S.: Geometric reductivity over arbitrary base. Adv. Math. 26, 225–274 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  42. Slodowy, P.: Die Theorie der optimalen Einparametergruppen für instabile Vektoren. In: Algebraic Transformation Groups and Invariant Theory. DMV Sem., vol. 13, pp. 115–131. Birkhäuser, Basel (1989)

    Chapter  Google Scholar 

  43. Spaltenstein, N.: Classes Unipotentes et Sous-Groupes de Borel. Lecture Notes in Math., vol. 946. Springer, Berlin (1982)

    MATH  Google Scholar 

  44. Steinberg, R.: Lectures on Chevalley Groups. Yale University, New Haven (1968)

    MATH  Google Scholar 

  45. Steinberg, R.: Conjugacy Classes in Algebraic Groups. Lecture Notes in Math., vol. 366. Springer, Berlin (1974)

    MATH  Google Scholar 

  46. Springer, T.A.: The Steinberg function of finite Lie algebra. Invent. Math. 58, 211–215 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  47. Springer, T.A., Steinberg, R.: Conjugacy classes. In: Seminar on Algebraic Groups and Related Finite Groups. Lecture Notes in Math., vol. 131, pp. 167–266. Springer, Berlin (1970)

    Chapter  Google Scholar 

  48. Tsujii, T.: A simple proof of Pommerening’s theorem. J. Algebra 320, 2196–2208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Xue, T.: Nilpotent elements in the dual of odd orthogonal Lie algebras. Transform. Groups (2012). doi:10.1007/S00031-012-9172-y

    Google Scholar 

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Authors and Affiliations

  1. Trinity College, Cambridge, CB2 1TQ, UK

    Matthew C. Clarke

  2. School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK

    Alexander Premet

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  1. Matthew C. Clarke
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Clarke, M.C., Premet, A. The Hesselink stratification of nullcones and base change. Invent. math. 191, 631–669 (2013). https://doi.org/10.1007/s00222-012-0401-8

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