Closed orbits and uniform 𝑆-instability in geometric invariant theory

Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Tange, Rudolf

doi:10.1090/S0002-9947-2012-05739-4

Closed orbits and uniform $S$-instability in geometric invariant theory
HTML articles powered by AMS MathViewer

by Michael Bate, Benjamin Martin, Gerhard Röhrle and Rudolf Tange PDF

Trans. Amer. Math. Soc. 365 (2013), 3643-3673 Request permission

Abstract:

In this paper we consider various problems involving the action of a reductive group $G$ on an affine variety $V$. We prove some general rationality results about the $G$-orbits in $V$. In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of $G$ for such general $G$-actions.

We apply our general rationality results to answer a question of Serre concerning the behaviour of his notion of $G$-complete reducibility under separable field extensions. Applications of our new optimality results also include a construction which allows us to associate an optimal destabilizing parabolic subgroup of $G$ to any subgroup of $G$. Finally, we use these new optimality techniques to provide an answer to Tits’ Centre Conjecture in a special case.

References

Michael Bate, Benjamin Martin, and Gerhard Röhrle, A geometric approach to complete reducibility, Invent. Math. 161 (2005), no. 1, 177–218. MR 2178661, DOI 10.1007/s00222-004-0425-9
Michael Bate, Benjamin Martin, and Gerhard Röhrle, Complete reducibility and commuting subgroups, J. Reine Angew. Math. 621 (2008), 213–235. MR 2431255, DOI 10.1515/CRELLE.2008.063
Michael Bate, Benjamin Martin, and Gerhard Röhrle, On Tits’ centre conjecture for fixed point subcomplexes, C. R. Math. Acad. Sci. Paris 347 (2009), no. 7-8, 353–356 (English, with English and French summaries). MR 2537229, DOI 10.1016/j.crma.2009.02.018
Michael Bate, Benjamin Martin, Gerhard Röhrle, and Rudolf Tange, Complete reducibility and separability, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4283–4311. MR 2608407, DOI 10.1090/S0002-9947-10-04901-9
Michael Bate, Benjamin Martin, Gerhard Röhrle, and Rudolf Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z. 269 (2011), no. 3-4, 809–832. MR 2860266, DOI 10.1007/s00209-010-0763-9
Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1–55. MR 0258838
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
A. Borel and J. Tits, Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. Math. 12 (1971), 95–104 (French). MR 294349, DOI 10.1007/BF01404653
Armand Borel and Jacques Tits, Compléments à l’article: “Groupes réductifs”, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 253–276 (French). MR 315007
Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
T. L. Gómez, A. Langer, A. H. W. Schmitt, and I. Sols, Moduli spaces for principal bundles in arbitrary characteristic, Adv. Math. 219 (2008), no. 4, 1177–1245. MR 2450609, DOI 10.1016/j.aim.2008.05.015
Wim H. Hesselink, Uniform instability in reductive groups, J. Reine Angew. Math. 303(304) (1978), 74–96. MR 514673, DOI 10.1515/crll.1978.303-304.74
Wim H. Hesselink, Desingularizations of varieties of nullforms, Invent. Math. 55 (1979), no. 2, 141–163. MR 553706, DOI 10.1007/BF01390087
Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689
George R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316. MR 506989, DOI 10.2307/1971168
Jason Levy, Rationality and orbit closures, Canad. Math. Bull. 46 (2003), no. 2, 204–215. MR 1981675, DOI 10.4153/CMB-2003-021-6
Martin W. Liebeck and Gary M. Seitz, Reductive subgroups of exceptional algebraic groups, Mem. Amer. Math. Soc. 121 (1996), no. 580, vi+111. MR 1329942, DOI 10.1090/memo/0580
Martin W. Liebeck and Gary M. Seitz, Variations on a theme of Steinberg, J. Algebra 260 (2003), no. 1, 261–297. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1973585, DOI 10.1016/S0021-8693(02)00649-X
Martin W. Liebeck and Donna M. Testerman, Irreducible subgroups of algebraic groups, Q. J. Math. 55 (2004), no. 1, 47–55. MR 2043006, DOI 10.1093/qjmath/55.1.47
Benjamin M. S. Martin, Reductive subgroups of reductive groups in nonzero characteristic, J. Algebra 262 (2003), no. 2, 265–286. MR 1971039, DOI 10.1016/S0021-8693(03)00189-3
Benjamin M. S. Martin, A normal subgroup of a strongly reductive subgroup is strongly reductive, J. Algebra 265 (2003), no. 2, 669–674. MR 1987023, DOI 10.1016/S0021-8693(03)00293-X
George McNinch, Completely reducible Lie subalgebras, Transform. Groups 12 (2007), no. 1, 127–135. MR 2308032, DOI 10.1007/s00031-005-1130-5
Bernhard Mühlherr and Jacques Tits, The center conjecture for non-exceptional buildings, J. Algebra 300 (2006), no. 2, 687–706. MR 2228217, DOI 10.1016/j.jalgebra.2006.01.011
D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978. MR 546290
Alexander Premet, Nilpotent orbits in good characteristic and the Kempf-Rousseau theory, J. Algebra 260 (2003), no. 1, 338–366. Special issue celebrating the 80th birthday of Robert Steinberg. MR 1976699, DOI 10.1016/S0021-8693(02)00662-2
R. W. Richardson, Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups, Duke Math. J. 57 (1988), no. 1, 1–35. MR 952224, DOI 10.1215/S0012-7094-88-05701-8
Guy Rousseau, Immeubles sphériques et théorie des invariants, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 5, A247–A250 (French, with English summary). MR 506257
Alexander H. W. Schmitt, A closer look at semistability for singular principal bundles, Int. Math. Res. Not. 62 (2004), 3327–3366. MR 2097106, DOI 10.1155/S1073792804132984
Gary M. Seitz, Abstract homomorphisms of algebraic groups, J. London Math. Soc. (2) 56 (1997), no. 1, 104–124. MR 1462829, DOI 10.1112/S0024610797005176
Jean-Pierre Serre, Semisimplicity and tensor products of group representations: converse theorems, J. Algebra 194 (1997), no. 2, 496–520. With an appendix by Walter Feit. MR 1467165, DOI 10.1006/jabr.1996.6929
—, La notion de complète réductibilité dans les immeubles sphériques et les groupes réductifs, Séminaire au Collège de France, résumé dans [37, pp. 93–98.] (1997).
—, The notion of complete reducibility in group theory, Moursund Lectures, Part II, University of Oregon, 1998, arXiv:math/0305257v1 [math.GR].
Jean-Pierre Serre, Complète réductibilité, Astérisque 299 (2005), Exp. No. 932, viii, 195–217 (French, with French summary). Séminaire Bourbaki. Vol. 2003/2004. MR 2167207
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
Jacques Tits, Théorie des groupes, Ann. Collège France 87 (1986/87), 89–98 (French). MR 1320558