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Rationality and the Jordan–Gatti–Viniberghi Decomposition

Published online by Cambridge University Press:  20 November 2018

Jason Levy
Jason Levy*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N 6N5 e-mail: jlevy@uottawa.ca
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Abstract

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We verify our earlier conjecture and use it to prove that the semisimple parts of the rational Jordan–Kac–Vinberg decompositions of a rational vector all lie in a single rational orbit.

Keywords

20G1514L24reductive groupG-moduleJordan decompositionorbit closurerationality
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 97 - 104
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Bardsley, Peter and Richardson, R.W., Ètale slices for algebraic transformation groups incharacteristic p. Proc. London Math. Soc. 51 (1985), 295–317, 1985. http://dx.doi.org/10.1112/plms/s3-51.2.295 Google Scholar
[2] Bate, M., Martin, B., Röhrle, G., and Tange, R., Closed Orbits and uniform S-instability in GeometricInvariant Theory. Trans. Amer. Math. Soc., to appear. arxiv:0904.4853v4. Google Scholar
[3] Borel, Armand, Linear algebraic groups. Second edition. Graduate Texts in Math. 126, Springer-Verlag, New York, 1991.Google Scholar
[4] Borel, Armand and Tits, Jacques, Groupes réductifs. Inst. Hautes E´ tudes Sci. Publ. Math. 27 (1965), 55–150.Google Scholar
[5] Bremigan, Ralph J., Quotients for algebraic group actions over non-algebraically closed fields. J. Reine Angew. Math. 453 (1994), 21–47.Google Scholar
[6] Gatti, V. and Viniberghi, E., Spinors of 13-dimensional space. Adv. in Math. 30 (1978), 137–155. http://dx.doi.org/10.1016/0001-8708(78)90034-8 Google Scholar
[7] Kac, V. G., Infinite root systems, representations of graphs and invariant theory. II. J. Algebra 78 (1982), 141–162. http://dx.doi.org/10.1016/0021-8693(8290105-3) Google Scholar
[8] Kempf, George R., Instability in invariant theory. Ann. of Math. (2) 108 (1978), 299–316. http://dx.doi.org/10.2307/1971168 Google Scholar
[9] Levy, Jason, A truncated Poisson formula for groups of rank at most two. Amer. J. Math. 117 (1995), 1371–1408. http://dx.doi.org/10.2307/2375023 Google Scholar
[10] Levy, Jason, Rationality and orbit closures. Canad. Math. Bull. 46 (2003), 204–215. http://dx.doi.org/10.4153/CMB-2003-021-6 Google Scholar
[11] Richardson, R.W., Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke Math. J. 57 (1988), 1–35. http://dx.doi.org/10.1215/S0012-7094-88-05701-8 Google Scholar