Abstract
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 \(\mathfrak{g}^{*}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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
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)
Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Math., vol. 126. Springer, Berlin (1991)
Bourbaki, N.: Groupes et Algèbres de Lie. Hermann, Paris (1975). Chapitres VII, VIII
Carter, R.: Finite Groups of Lie Type. Wiley Classics Library. Wiley, New York (1993)
Chevalley, C.: Certains schémas de groupes semi-simples. In: Séminaire Bourbaki, 6, Exp. 219, pp. 219–234. Soc. Math. France, Paris (1995)
Digne, F., Michel, J.: Representations of Finite Groups of Lie Type. LMS Student Texts, vol. 21. Cambridge University Press, Cambridge (1991)
Dynkin, E.B.: The structure of semisimple Lie algebras. Transl. Am. Math. Soc. 9, 328–469 (1955)
Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Math., vol. 150. Springer, Berlin (1995)
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)
Haboush, W.: Reductive groups are geometrically reductive. Ann. Math. 102, 67–83 (1975)
Hesselink, W.H.: Uniform stability in reductive groups. J. Reine Angew. Math. 303(304), 74–96 (1978)
Hesselink, W.H.: Desingularizations of varieties of nullforms. Invent. Math. 55, 141–163 (1979)
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)
Humphreys, J.: Linear Algebraic Groups. Graduate Texts in Math., vol. 21. Springer, Berlin (1975)
Jantzen, J.C.: Representations of Algebraic Groups. Academic Press, San Diego (1987)
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)
Jantzen, J.C.: Nilpotent orbits in representation theory. In: Lie Theory. Progr. Math., vol. 228, pp. 1–211. Birkhäuser, Basel (2004)
Katz, N.: E-polynomials, zeta-equivalence, and polynomial-count varieties. Invent. Math. 174, 614–624 (2008)
Kempf, G.: Instability in invariant theory. Ann. Math. 108, 299–316 (1978)
Kirwan, F.C.: Cohomology of Quotients in Symplectic and Algebraic Geometry. Math. Notes, vol. 31. Princeton Univ. Press, Princeton (1984)
Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex semisimple Lie group. Am. J. Math. 81, 973–1032 (1959)
Kostant, B.: Groups over ℤ. In: Proc. Symposia in Pure Math., vol. 9, pp. 90–98 (1966)
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)
Lusztig, G.: On the finiteness of the number of unipotent classes. Invent. Math. 34, 201–213 (1976)
Lusztig, G.: Unipotent elements in small characteristic. Transform. Groups 10, 449–487 (2005)
Lusztig, G.: Unipotent elements in small characteristic. II. Transform. Groups 13, 773–797 (2008)
Lusztig, G.: Unipotent elements in small characteristic. IV. Transform. Groups 17, 921–936 (2010)
Lusztig, G.: Unipotent elements in small characteristic. III. J. Algebra 329, 163–189 (2011)
Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse Math., vol. 34. Springer, Berlin (1994)
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)
Mumford, D.: Geometric Invariant Theory. Ergebnisse Math., vol. 34. Springer, Berlin (1965)
Ness, L.: A stratification of the null-cone via the moment map. Am. J. Math. 106, 1281–1329 (1984)
Pommerening, K.: Über die unipotenten Klassen reduktiver Gruppen. J. Algebra 49, 525–536 (1977)
Pommerening, K.: Über die unipotenten Klassen reduktiver Gruppen. II. J. Algebra 65, 373–398 (1980)
Popov, V.L., Vinberg, È.B.: Invariant Theory. Algebraic Geometry IV. Encyclopedia Math. Sci., vol. 55. Springer, Berlin (1994)
Premet, A.: Support varieties of non-restricted modules over Lie algebras of reductive groups. J. Lond. Math. Soc. 55, 236–250 (1997)
Premet, A.: Nilpotent orbits in good characteristic and the Kempf-Rousseau theory. J. Algebra 260, 338–338336 (2003)
Premet, A., Skryabin, S.: Representations of restricted Lie algebras and families of associative \(\mathcal{L}\)-algebras. J. Reine Angew. Math. 507, 189–218 (1999)
Rousseau, G.: Immeubles sphériques et théorie des invariants. C. R. Math. Acad. Sci. Paris 286, 247–250 (1978)
Seshadri, C.S.: Geometric reductivity over arbitrary base. Adv. Math. 26, 225–274 (1977)
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)
Spaltenstein, N.: Classes Unipotentes et Sous-Groupes de Borel. Lecture Notes in Math., vol. 946. Springer, Berlin (1982)
Steinberg, R.: Lectures on Chevalley Groups. Yale University, New Haven (1968)
Steinberg, R.: Conjugacy Classes in Algebraic Groups. Lecture Notes in Math., vol. 366. Springer, Berlin (1974)
Springer, T.A.: The Steinberg function of finite Lie algebra. Invent. Math. 58, 211–215 (1980)
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)
Tsujii, T.: A simple proof of Pommerening’s theorem. J. Algebra 320, 2196–2208 (2008)
Xue, T.: Nilpotent elements in the dual of odd orthogonal Lie algebras. Transform. Groups (2012). doi:10.1007/S00031-012-9172-y
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-012-0401-8