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Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Part of: Lie algebras and Lie superalgebras (Co)homology theory Representation theory of groups Algebraic groups Birational geometry Linear algebraic groups and related topics

Published online by Cambridge University Press:  01 February 2011

Jean-Louis Colliot-Thélène ,
Boris Kunyavskiĭ ,
Vladimir L. Popov  and
Zinovy Reichstein
Jean-Louis Colliot-Thélène
Affiliation:
C.N.R.S., UMR 8628, Mathématiques, Bâtiment 425, Université Paris-Sud, F-91405, Orsay, France (email: jlct@math.u-psud.fr)
Boris Kunyavskiĭ
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel (email: kunyav@macs.biu.ac.il)
Vladimir L. Popov
Affiliation:
Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia (email: popovvl@mi.ras.ru)
Zinovy Reichstein
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada BC V6T 1Z2 (email: reichst@math.ubc.ca)
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Abstract

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Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let 𝔤 be its Lie algebra. Let k(G), respectively, k(𝔤), be the field of k-rational functions on G, respectively, 𝔤. The conjugation action of G on itself induces the adjoint action of G on 𝔤. We investigate the question whether or not the field extensions k(G)/k(G)G and k(𝔤)/k(𝔤)G are purely transcendental. We show that the answer is the same for k(G)/k(G)G and k(𝔤)/k(𝔤)G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type An or Cn, and negative for groups of other types, except possibly G2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Keywords

algebraic groupsimple Lie algebrarationality problemintegral representationalgebraic torusunramified Brauer group

MSC classification

Secondary: 14E08: Rationality questions 17B45: Lie algebras of linear algebraic groups 14L30: Group actions on varieties or schemes (quotients) 20C10: Integral representations of finite groups 20G15: Linear algebraic groups over arbitrary fields 14F22: Brauer groups of schemes
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 428 - 466
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[1]Alev, J., Ooms, A. and Van den Bergh, M., A class of counterexamples to the Gel’fand–Kirillov conjecture, Trans. Amer. Math. Soc. 348 (1996), 1709–1716.Google Scholar
[2]Artin, E., Geometric algebra (Interscience Publishers Inc., New York, 1957).Google Scholar
[3]Berhuy, G. and Favi, G., Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330.Google Scholar
[4]Białynicki-Birula, A., On homogeneous affine spaces of linear algebraic groups, Amer. J. Math. 85 (1963), 577–582.CrossRefGoogle Scholar
[5]Borel, A., Linear algebraic groups, second enlarged edition, Graduate Texts in Mathematics, vol. 126 (Springer, Berlin, 1991).CrossRefGoogle Scholar
[6]Bourbaki, N., Groupes et algèbres de Lie (Hermann, Paris, 1968), Chapters IV, V, VI.Google Scholar
[7]Colliot-Thélène, J.-L. and È Kunyavskiĭ, B., Groupe de Brauer et groupe de Picard des compactifications lisses d’espaces homogènes, J. Algebraic Geom. 15 (2006), 733–752.CrossRefGoogle Scholar
[8]Colliot–Thélène, J.-L. and Sansuc, J.-J., La R-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), 175–229.Google Scholar
[9]Colliot–Thélène, J.-L. and Sansuc, J.-J., Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), 148–205.CrossRefGoogle Scholar
[10]Colliot–Thélène, J.-L. and Sansuc, J.-J., La descente sur les variétés rationnelles, II, Duke Math. J. 54 (1987), 375–492.Google Scholar
[11]Colliot–Thélène, J.-L. and Sansuc, J.-J., The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), in Proceedings of the international colloquium on algebraic groups and homogeneous spaces, Mumbai, 2004, ed. Mehta, V. (Narosa Publishing House, TIFR Mumbai, 2007), 113–186.Google Scholar
[12]Cortella, A. and Kunyavskiĭ, B., Rationality problem for generic tori in simple groups, J. Algebra 225 (2000), 771–793.Google Scholar
[13]Demazure, M. and Grothendieck, A., Schémas en Groupes I, II, III, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Dirigé par M. Demazure et A. Grothendieck, Lecture Notes in Mathematics, vols. 151, 152, 153 (Springer, Berlin, 1970).Google Scholar
[14]Florence, M., On the essential dimension of cyclic p-groups, Invent. math. 171 (2008), 175–189.Google Scholar
[15]Fogarty, J., Fixed points schemes, Amer. J. Math. 95 (1973), 35–51.CrossRefGoogle Scholar
[16]Garibaldi, S. R., Merkurjev, A. and Serre, J.-P., Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28 (American Mathematical Society, Providence, RI, 2003).Google Scholar
[17]Grothendieck, A., Torsion homologique et sections rationnelles, Anneaux de Chow et Applications, Séminaire C. Chevalley (1958), exposé 5.Google Scholar
[18]Grothendieck, A., avec la collaboration de J. Dieudonné, Éléments de géométrie algébrique, IV, Étude locale des schémas et des morphismes de schémas, Publ. Math. Inst. Hautes Études Sci. 20 (1964), 24 (1965), 28 (1966), 32 (1967).CrossRefGoogle Scholar
[19]Correspondance Grothendieck–Serre, éd. par P. Colmez et J.-P. Serre, Documents mathématiques, vol. 2 (Soc. Math. France, 2001). Reprinted and translated into English in Grothendieck–Serre Correspondence, Bilingual Edition (P. Colmez, J.-P. Serre, eds) (Amer. Math. Soc., Soc. Math. France, 2004).Google Scholar
[20]Humphreys, J. E., Linear algebraic groups, Graduate Texts in Mathematics, vol. 21 (Springer, New York, 1975).Google Scholar
[21]Humphreys, J. E., Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43 (American Mathematical Society, Providence, RI, 1995).Google Scholar
[22]Igusa, J.-i., Geometry of absolutely admissible representations, in Number theory, algebraic geometry and commutative algebra (Kinokuniya, Tokyo, 1973), 373–452, in honor of Yasuo Akizuki.Google Scholar
[23]Kostant, B., Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.CrossRefGoogle Scholar
[24]Kottwitz, R. E., Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), 785–806.Google Scholar
[25]Kottwitz, R. E., Transfer factors for Lie algebras, Represent. Theory 3 (1999), 127–138.Google Scholar
[26]Lemire, N. and Lorenz, M., On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), 385–405.Google Scholar
[27]Lemire, N., Popov, V. L. and Reichstein, Z., Cayley groups, J. Amer. Math. Soc. 19 (2006), 921–967.Google Scholar
[28]Lorenz, M., Multiplicative invariant theory, in Invariant theory and algebraic transformation groups, VI, Encyclopaedia of Mathematical Sciences, vol. 135 (Springer, Berlin, 2005).Google Scholar
[29]Luna, D., Slices étales, Mém. Soc. Math. Fr. 33 (1973), 81–105.Google Scholar
[30]Matsushima, Y., Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J. 16 (1960), 205–218.Google Scholar
[31]Mumford, D., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Bd. 34 (Springer, Berlin, 1965), third enlarged edition; D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Bd. 34 (Springer, Berlin, 1994).Google Scholar
[32]Onishchik, A. L., Complex hulls of compact homogeneous spaces, Dokl. Akad. Nauk SSSR 130 (1960), 726–729; Engl. transl. Sov. Math. Dokl. 1 (1960), 88–91.Google Scholar
[33]Popov, V. L., On the stability of the action of an algebraic group on an algebraic variety, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 371–385; Engl. transl. Math. USSR Izv. 6 (1972), 367–379.Google Scholar
[34]Popov, V. L., Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Izv. Acad. Nauk SSSR Ser. Mat. 38 (1974), 294–322; Engl. transl. Math. USSR Izv. 8 (1974), 301–327.Google Scholar
[35]Popov, V. L., Groups, generators, syzygies, and orbits in invariant theory, Translations of Mathematical Monographs, vol. 100 (American Mathematical Society, Providence, RI, 1992).Google Scholar
[36]Popov, V. L., Sections in invariant theory, in The Sophus Lie memorial conference, Oslo, 1992 (Scand. University Press, Oslo, 1994), 315–361.Google Scholar
[37]Popov, V. L., Cross-sections, quotients, and representation rings of semisimple algebraic groups, available at arXiv:0908.0826 and http://www.math.uni-bielefeld.de/LAG/man/351.pdf.Google Scholar
[38]Popov, V. L. and Vinberg, E. B., Invariant theory, in Algebraic geometry IV, Encyclopaedia of Mathematical Sciences, vol. 55 (Springer, Berlin, 1994), 123–284.CrossRefGoogle Scholar
[39]Premet, A., Modular Lie algebras and the Gelfand–Kirillov conjecture, Invent. math. 181 (2010), 395–420.Google Scholar
[40]Réédition de SGA3, available at http://people.math.jussieu.fr/∼polo/SGA3/.Google Scholar
[41]Reichstein, Z., On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), 265–304.Google Scholar
[42]Richardson, R. W. Jr, Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. math. 16 (1972), 6–14.Google Scholar
[43]Rosenlicht, M., Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443.Google Scholar
[44]Rosenlicht, M., A remark on quotient spaces, An. Acad. Brasil. Ciênc. 35 (1963), 487–489.Google Scholar
[45]Sansuc, J.-J., Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math. 327 (1981), 12–80.Google Scholar
[46]Serre, J.-P., Espaces fibrés algébriques, in Anneaux de Chow et applications, Séminaire C. Chevalley, 1958, exposé 1. Reprinted in J.-P. Serre, Exposés de séminaires 1950–1999, deuxième édition, augmentée, Documents mathématiques, vol. 1 (Soc. Math. France, 2008), 107–140.Google Scholar
[47]Serre, J.-P., Cohomologie Galoisienne, Lecture Notes in Mathematics, vol. 5, 5ième édition (Springer, Berlin, 1994).Google Scholar
[48]Spanier, E. H., Algebraic topology (Springer, New York, 1981).CrossRefGoogle Scholar
[49]Springer, T. A., Aktionen reduktiver Gruppen auf Varietäten, in Algebraische Transformationsgruppen und Invariantentheorie, DMV Seminar, Bd. 13, Kraft, Herausgeb. H., Slodowy, P. and Springer, T. A. (Birkhäuser, Basel, 1989), 3–39.Google Scholar
[50]Springer, T. A., Linear algebraic groups, Progress in Mathematics, vol. 9, second edition (Birkhäuser, Boston, 1998).Google Scholar
[51]Steinberg, R., Regular elements of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 25 (1965), 49–80.Google Scholar
[52]Sumihiro, H., Equivariant completion, I, J. Math. Kyoto Univ. 14 (1974), 1–28.Google Scholar
[53]Thomason, R. W., Comparison of equivariant algebraic and topological K-theory, Duke Math. J. 55 (1986), 795–825.Google Scholar
[54]Voskresenskiĭ, V. E., Algebraicheskie tory (in Russian) [Algebraic tori] (Izdat. ‘Nauka’, Moscow, 1977).Google Scholar
[55]Voskresenskiĭ, V. E., Maximal tori without affect in semisimple algebraic groups, Mat. Zametki 44 (1988), 309–318; English transl. Math. Notes 44 (1988), 651–655.Google Scholar
[56]Voskresenskiĭ, V. E., Algebraic groups and their birational invariants (American Mathematical Society, Providence, RI, 1998).Google Scholar
[57]Weil, A., Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113 (1965), 1–87.Google Scholar