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J-INVARIANT OF LINEAR ALGEBRAIC GROUPS OF OUTER TYPE

Part of: Linear algebraic groups and related topics Cycles and subschemes

Published online by Cambridge University Press:  04 December 2025

Nikita Geldhauser Open the ORCID record for Nikita Geldhauser [Opens in a new window]  and
Maksim Zhykhovich
Nikita Geldhauser*
Affiliation:
LMU Munich, Germany
Maksim Zhykhovich
Affiliation:
LMU Munich, Germany (zhykhovich@math.lmu.de)
*
geldhauser@math.lmu.de
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Abstract

We extend the notion of the J-invariant to arbitrary semisimple linear algebraic groups and provide complete decompositions for the normed Chow motives of all generically quasi-split twisted flag varieties. Besides, we establish some combinatorial patterns for normed Chow groups and motives and provide some explicit formulae for values of the J-invariant.

Keywords

Linear algebraic groupstwisted flag varietiesmotives

MSC classification

Primary: 20G15: Linear algebraic groups over arbitrary fields
Secondary: 14C15: (Equivariant) Chow groups and rings; motives

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 25 , Issue 2 , March 2026 , pp. 635 - 662
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

The work of the authors is supported by the DFG research grants SE 1721/4-1 and ZH 918/2-1.

References

Borel, A (1991) Linear Algebraic Groups. Second Edition. Springer New York, NY.CrossRefGoogle Scholar
Brosnan, P (1999) Steenrod operations in Chow theory. Trans. AMS 355, 1869–1903.CrossRefGoogle Scholar
Carter, R (1972) Simple Groups of Lie Type. London, New York, Sydney, Toronto: John Wiley & Sons.Google Scholar
Chernousov, V (2003) The kernel of the Rost invariant, Serre’s Conjecture II and the Hasse principle for quasi-split groups ${}^{3,6}{\mathrm{D}}_4$ , ${\mathrm{E}}_6$ , ${\mathrm{E}}_7$ . Math. Ann. 326, 297–330.CrossRefGoogle Scholar
Demazure, M (1974) Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. 7, 53–88.CrossRefGoogle Scholar
Edidin, D and Graham, W (1998) Equivariant intersection theory (with an appendix by A. Vistoli: The Chow ring of ${M}_2$ ), Invent. Math. 131(3), 595–634.CrossRefGoogle Scholar
Elman, R, Karpenko, N and Merkurjev, A (2008) The Algebraic and Geometric Theory of Quadratic Forms. Colloquium Publications, vol. 56. Providence, RI: American Mathematical Society.Google Scholar
Fino, R (2019) $\mathrm{J}$ -invariant of hermitian forms over quadratic extensions. Pacific J. of Math. 300(2), 375–404.CrossRefGoogle Scholar
Fulton, W (1998) Intersection Theory . Second Edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics 2, Berlin: Springer-Verlag.Google Scholar
Garibaldi, S (2001) The Rost invariant has trivial kernel for quasi-split groups of low rank. Comm. Math. Helv. 76, 684–711.CrossRefGoogle Scholar
Garibaldi, S, Merkurjev, A and Serre, J-P (2003) Cohomological invariants in Galois cohomology. Providence, RI: AMS.CrossRefGoogle Scholar
Garibaldi, S, Petrov, V and Semenov, N (2016) Shells of twisted flag varieties and the Rost invariant. Duke Math. J. 165(2), 285–339.CrossRefGoogle Scholar
Garibaldi, S and Semenov, N (2010) Degree $5$ invariant of ${\mathrm{E}}_8$ . Int. Math. Res. Not. 19, 3746–3762.Google Scholar
Gille, S and Zainoulline, K (2012) Equivariant pretheories and invariants of torsors. Transf. Groups 17, 471–498.CrossRefGoogle Scholar
Grothendieck, A 1958 La torsion homologique et les sections rationnelles, Exposé 5 in Anneaux de Chow et applications, Séminaire Claude Chevalley.Google Scholar
Kac, V (1985) Torsion in cohomology of compact Lie groups and Chow rings of reductive algebraic groups. Invent. Math. 80, 69–79.CrossRefGoogle Scholar
Karpenko, N (2000) Weil transfer of algebraic cycles. Indag. Math. 11(1), 73–86.CrossRefGoogle Scholar
Karpenko, N (2010) Upper motives of outer algebraic groups. In: Quadratic Forms, Linear Algebraic Groups, and Cohomology, Dev. Math. 18, 249–258. New York: Springer.Google Scholar
Karpenko, N (2010), Canonical dimension, Proceedings of the International Congress of Mathematicians, vol. II, 146–161. New Delhi: Hindustan Book Agency.Google Scholar
Karpenko, N (2012) Sufficiently generic orthogonal grassmannians. J. Algebra 372, 365–375.CrossRefGoogle Scholar
Karpenko, N (2012) Incompressibility of quadratic Weil transfer of generalized Severi-Brauer varieties. J. Inst. Math. Jussieu 11(1), 119–131.CrossRefGoogle Scholar
Karpenko, N (2012), Unitary grassmannians. J. Pure Appl. Algebra 216(12), 2586–2600.CrossRefGoogle Scholar
Karpenko, N (2013) Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties. J. Reine Angew. Math. 677, 179–198.Google Scholar
Karpenko, N (2015) Incompressibility of products of Weil transfers of generalized Severi–Brauer varieties. Math. Z. 279(3–4), 767–777.CrossRefGoogle Scholar
Karpenko, N and Merkurjev, A (2006) Canonical $p$ -dimension of algebraic groups. Adv. Math. 205, 410–433.CrossRefGoogle Scholar
Karpenko, N and Zhykhovich, M (2013) Isotropy of unitary involutions. Acta Math. 211(2), 227–253.CrossRefGoogle Scholar
Knus, M-A, Merkurjev, A, Rost, M and Tignol, J-P (1998) The Book of Involutions, Colloquium Publications, vol. 44, AMS.Google Scholar
Levine, M and Morel, F (2007) Algebraic Cobordism , Springer Monographs in Mathematics, Berlin Heidelberg: Springer-Verlag.Google Scholar
Manin, Y (1968) Correspondences, motives and monoidal transformations. Math. USSR Sbornik 6, 439–470.CrossRefGoogle Scholar
Merkurjev, A, Panin, I and Wadsworth, A (1996) Index reduction formulas for twisted flag varieties. I. K-Theory 6(3), 517–596.CrossRefGoogle Scholar
Merkurjev, A and Tignol, J-P (1995) The multipliers of similitudes and the Brauer group of homogeneous varieties. J. Reine Angew. Math. 461, 13–47.Google Scholar
Nikolenko, S, Semenov, N and Zainoulline, K (2009) Motivic decomposition of anisotropic varieties of type ${F}_4$ into generalized Rost motives. J. of K-Theory 3(1), 85–102.CrossRefGoogle Scholar
Panin, I (1994) On the algebraic $K$ -theory of twisted flag varieties. K-Theory 8(6), 541–585.CrossRefGoogle Scholar
Petrov, V and Semenov, N (2010) Generically split projective homogeneous varieties. Duke Math. J. 152, 155–173.CrossRefGoogle Scholar
Petrov, V and Semenov, N (2012) Generically split projective homogeneous varieties. II. J. of K-theory 10(1), 1–8.CrossRefGoogle Scholar
Petrov, V and Semenov, N (2017) Rost motives, affine varieties, and classifying spaces. J. London Math. Soc. 95(3), 895–918.CrossRefGoogle Scholar
Petrov, V and Semenov, N (2021) Hopf-theoretic approach to motives of twisted flag varieties. Compos. Math. 157(5), 963–996.CrossRefGoogle Scholar
Petrov, V, Semenov, N and Zainoulline, K (2008) $\mathrm{J}$ -invariant of linear algebraic groups. Ann. Sci. École Norm. Sup. 41, 1023–1053.CrossRefGoogle Scholar
Primozic, E (2020) Motivic Steenrod operations in characteristic $p$ . Forum Math. Sigma 8, e52, 1–25.CrossRefGoogle Scholar
Rost, M, The motive of a Pfister form, Preprint 1998, available from http://www.math.uni-bielefeld.de/~rost Google Scholar
Schofield, A and Van den Bergh, M (1992) The Index of a Brauer class on a Brauer–Severi variety. Trans. AMS 333(2), 729–739.CrossRefGoogle Scholar
Semenov, N (2016) Motivic construction of cohomological invariants. Comment. Math. Helv. 91(1), 163–202.CrossRefGoogle Scholar
Springer, TA (2009) Linear Algebraic Groups. Second Edition, Boston: Birkhäuser.Google Scholar
Tamagawa, T (1976) Representation Theory and the Notion of the Discriminant , Algebraic Number Theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Kyoto: Univ. Kyoto), 219–227. Japan Soc. Promotion Sci., Tokyo, 1977.Google Scholar
Tits, J (1966) Classification of algebraic semisimple groups. In: Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. 33–62. Providence, RI: Amer. Math. Soc.CrossRefGoogle Scholar
Totaro, B (1999) The Chow ring of a classifying space. In: Algebraic K-Theory, ed. Raskind, W. and Weibel, C., Proceedings of Symposia in Pure Mathematics, vol. 67, 249–281. Providence, Rhode Island: American Mathematical Society.Google Scholar
Vishik, A (2005) On the Chow groups of quadratic Grassmannians. Doc. Math. 10, 111–130.CrossRefGoogle Scholar
Vishik, A (2007) Fields of $u$ -invariant ${2}^r+1$ . In: Algebra, Arithmetic and Geometry, Manin Festschrift: Birkhäuser.Google Scholar
Vishik, A (2024) Isotropic and numerical equivalence for Chow groups and Morava $K$ -theories. Inventiones Math. 237, 779–808.CrossRefGoogle Scholar
Voevodsky, V (2003) Reduced power operations in motivic cohomology. Publ. Math ., Inst. Hautes Étud. Sci. 98, 1–57.CrossRefGoogle Scholar
Voevodsky, V (2003) Motivic cohomology with $\mathbb{Z}/2$ -coefficients. Publ. Math., Inst. Hautes Étud. Sci. 98, 59–104.CrossRefGoogle Scholar
Zhykhovich, M (2024) The $J$ -invariant over splitting fields of Tits algebras. Comp. Math. 160(9), 2100–2114.CrossRefGoogle Scholar