J-invariant of linear algebraic groups
[J-invariant des groupes algébriques linéaires]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 1023-1053.

Soit G un groupe algébrique linéaire semi-simple de type intérieur sur un corps F et soit X un G-espace homogène projectif tel que le groupe G soit déployé sur le point générique de X. Nous introduisons le J-invariant de G qui caractérise le comportement motivique de X et généralise le J-invariant défini par A. Vishik dans le cadre des formes quadratiques. Nous utilisons cet invariant pour obtenir les décompositions motiviques de tous les G-espaces homogènes projectifs qui sont génériquement déployés, par exemple les variétés de Severi-Brauer, les quadriques de Pfister, la grassmannienne des sous-espaces totalement isotropes maximaux d’une forme quadratique, la variété des sous-groupes de Borel de G. Nous discutons également les relations avec les indices de torsion, la dimension canonique et les invariants cohomologiques du groupe G.

Let G be a semisimple linear algebraic group of inner type over a field F, and let X be a projective homogeneous G-variety such that G splits over the function field of X. We introduce the J-invariant of G which characterizes the motivic behavior of X, and generalizes the J-invariant defined by A. Vishik in the context of quadratic forms. We use this J-invariant to provide motivic decompositions of all generically split projective homogeneous G-varieties, e.g. Severi-Brauer varieties, Pfister quadrics, maximal orthogonal Grassmannians, varieties of Borel subgroups of G. We also discuss relations with torsion indices, canonical dimensions and cohomological invariants of the group G.

DOI : 10.24033/asens.2088
Classification : 14C25, 20G15
Keywords: motive, algebraic group, homogeneous variety
Mot clés : motif, groupe algébrique, espace homogène
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Petrov, Viktor; Semenov, Nikita; Zainoulline, Kirill. $J$-invariant of linear algebraic groups. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 6, pp. 1023-1053. doi : 10.24033/asens.2088. http://archive.numdam.org/articles/10.24033/asens.2088/

[1] F. W. Anderson & K. R. Fuller, Rings and categories of modules, second éd., Graduate Texts in Math. 13, Springer, 1992. | MR | Zbl

[2] J.-P. Bonnet, Un isomorphisme motivique entre deux variétés homogènes projectives sous l’action d’un groupe de type G 2 , Doc. Math. 8 (2003), 247-277. | EuDML | MR | Zbl

[3] P. Brosnan, Steenrod operations in Chow theory, Trans. Amer. Math. Soc. 355 (2003), 1869-1903. | MR | Zbl

[4] P. Brosnan, On motivic decompositions arising from the method of Białynicki-Birula, Invent. Math. 161 (2005), 91-111. | MR | Zbl

[5] B. Calmès, V. Petrov, N. Semenov & K. Zainoulline, Chow motives of twisted flag varieties, Compos. Math. 142 (2006), 1063-1080. | MR | Zbl

[6] V. Chernousov, A remark on the ( mod 5)-invariant of Serre for groups of type E 8 , Mat. Zametki 56 (1994), 116-121, 157. | MR | Zbl

[7] V. Chernousov, S. Gille & A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Math. J. 126 (2005), 137-159. | MR | Zbl

[8] V. Chernousov & A. Merkurjev, Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem, Transform. Groups 11 (2006), 371-386. | MR | Zbl

[9] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301. | EuDML | MR | Zbl

[10] M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. 7 (1974), 53-88. | Numdam | MR | Zbl

[11] H. Duan & X. Zhao, A unified formula for Steenrod operations in flag manifolds, Compos. Math. 143 (2007), 257-270. | MR | Zbl

[12] D. Edidin & W. Graham, Characteristic classes in the Chow ring, J. Algebraic Geom. 6 (1997), 431-443. | MR | Zbl

[13] R. Elman, N. Karpenko & A. Merkurjev, The algebraic and geometric theory of quadratic forms, to appear in AMS Colloquium Publications. | MR | Zbl

[14] R. S. Garibaldi, The Rost invariant has trivial kernel for quasi-split groups of low rank, Comment. Math. Helv. 76 (2001), 684-711. | MR | Zbl

[15] R. S. Garibaldi & H. P. Petersson, Groups of outer type E 6 with trivial Tits algebras, Transform. Groups 12 (2007), 443-474. | MR | Zbl

[16] P. Gille, Invariants cohomologiques de Rost en caractéristique positive, K-Theory 21 (2000), 57-100. | MR | Zbl

[17] A. Grothendieck, La torsion homologique et les sections rationnelles, in Anneaux de Chow et applications, Séminaire C. Chevalley, 2e année, 1958.

[18] A. J. Hahn & O. T. O'Meara, The classical groups and K-theory, Grund. Math. Wiss. 291, Springer, 1989. | MR | Zbl

[19] H. Hiller, Geometry of Coxeter groups, Research Notes in Math. 54, Pitman (Advanced Publishing Program), 1982. | MR | Zbl

[20] V. G. Kac, Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups, Invent. Math. 80 (1985), 69-79. | MR | Zbl

[21] N. Karpenko, Grothendieck Chow motives of Severi-Brauer varieties, St. Petersburg Math. J. 7 (1996), 649-661. | MR | Zbl

[22] N. Karpenko & A. Merkurjev, Canonical p-dimension of algebraic groups, Adv. Math. 205 (2006), 410-433. | MR | Zbl

[23] I. Kersten & U. Rehmann, Generic splitting of reductive groups, Tohoku Math. J. 46 (1994), 35-70. | MR | Zbl

[24] M.-A. Knus, A. Merkurjev, M. Rost & J.-P. Tignol, The book of involutions, AMS Colloquium Publ. 44 (1998). | MR | Zbl

[25] B. Köck, Chow motif and higher Chow theory of G/P, Manuscripta Math. 70 (1991), 363-372. | MR | Zbl

[26] Y. Manin, Correspondences, motives and monoidal transformations, Math. USSR Sbornik 6 (1968), 439-470. | Zbl

[27] A. Merkurjev, Rost invariants of simply connected algebraic groups, in Cohomological invariants in Galois cohomology, Univ. Lecture Ser. 28, Amer. Math. Soc., 2003, 101-158. | MR

[28] A. Merkurjev, I. A. Panin & A. R. Wadsworth, Index reduction formulas for twisted flag varieties 10 (1996), 517-596. | MR | Zbl

[29] M. Mimura & H. Toda, Topology of Lie groups. I, II, Translations of Mathematical Monographs 91, Amer. Math. Soc., 1991. | MR | Zbl

[30] S. Nikolenko, N. Semenov & K. Zainoulline, Motivic decomposition of anisotropic varieties of type F 4 into generalized Rost motives, to appear in J. of K-Theory. | MR | Zbl

[31] I. A. Panin, On the algebraic K-theory of twisted flag varieties, K-Theory 8 (1994), 541-585. | MR | Zbl

[32] M. Rost, The motive of a Pfister form, preprint http://www.mathematik.uni-bielefeld.de/~rost/data/motive.pdf, 1998.

[33] M. Rost, On the basic correspondence of a splitting variety, preprint http://www.mathematik.uni-bielefeld.de/~rost/data/bkc-c.pdf, 2006.

[34] J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., 1966, 33-62. | MR | Zbl

[35] A. Vishik, On the Chow groups of quadratic Grassmannians, Doc. Math. 10 (2005), 111-130. | MR | Zbl

[36] A. Vishik, Fields of u-invariant 2 r +1, in Algebra, Arithmetic and Geometry, Manin Festschrift, Birkhäuser, 2007. | Zbl

[37] V. Voevodsky, On motivic cohomology with /l-coefficients, preprint http://www.math.uiuc.edu/K-theory/0639/post_mot.pdf, 2003.

[38] K. Zainoulline, Canonical p-dimensions of algebraic groups and degrees of basic polynomial invariants, Bull. Lond. Math. Soc. 39 (2007), 301-304. | MR | Zbl

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