Abstract
Let \({\mathfrak{g}=W_1}\) be the p-dimensional Witt algebra over an algebraically closed field \({k=\overline{\mathbb{F}}_q}\), where p > 3 is a prime and q is a power of p. Let G be the automorphism group of \({\mathfrak{g}}\). The Frobenius morphism F G (resp. \({F_\mathfrak{g}}\)) can be defined naturally on G (resp. \({\mathfrak{g}}\)). In this paper, we determine the \({F_\mathfrak{g}}\) -stable G-orbits in \({\mathfrak{g}}\). Furthermore, the number of \({\mathbb{F}_q}\) -rational points in each \({F_\mathfrak{g}}\) -stable orbit is precisely given. Consequently, we obtain the number of \({\mathbb{F}_q}\) -rational points in the nilpotent variety.
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References
Chang H.-J.: Uber Wittsche Lie-Ringe. Abh. Math. Sem. Univ. Hamburg 14, 151–184 (1941)
Deng B., Du J.: Folding derived categories with Frobenius functors. J. Pure Appl. Algebra 208, 1023–1050 (2007)
Digne F., Michel J.: Representations of Finite Groups of Lie Type London Math. Soc. Student Texts, vol. 21. Cambridge University Press, Cambridge (1991)
Geck M.: An Introduction to Algebraic Geometry and Algebraic Groups. Oxford University Press, Oxford (2003)
Kaplansky I.: Nilpotent elements in Lie algebras. J. Algebra 133, 467–471 (1990)
Kawanaka N.: Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field. Invent. Math. 69, 411–435 (1982)
Mygind, M.: Orbit closure in the Witt algebra and its dual space. arXiv:1212.0740v1 (2012)
Premet A.A.: Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras. Math. USSR Sbornik 66(2), 555–570 (1990)
Premet A.A.: The theorem on restriction of invariants and nilpotent elements in W n . Math. USSR. Sbornik 73(1), 135–159 (1992)
Ree R.: On generalized Witt algebras. Trans. Am. Math. Soc. 83, 510–546 (1956)
Shu, B.: Representations of finite Lie algebras and geometry of reductive Lie algebras. In: Proceedings of the International Conference on Complex Geometry and Related Fields, pp. 277–287, AMS/IP (2007) (in: Stud. Adv. Math., vol. 39, Am. Math. Soc.)
Skryabin, S.: Nilpotent elements in the Jacobson-Witt algebra over a finite field. Private communication (2013)
Springer T.A.: The Steinberg function of a finite Lie algebra. Invent. Math. 58, 211–215 (1980)
Strade, H., Farnsteiner, R.: Modular Lie Algebras and Their Representations. Pure and Applied Mathematics, vol. 116. Marcel Dekker, Inc. New York (1988)
Wilson R.: Automorphisms of graded Lie algebra of Cartan type. Comm. Algebra 3(7), 591–613 (1975)
Yao Y.F., Shu B.: Nilpotent orbits in the Witt algebra W 1. Comm. Algebra 39(9), 3232–3241 (2011)
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This work is supported by the National Natural Science Foundation of China (Grant Nos. 11126062, 11201293, 11271130 and 11226327), the Innovation Program of Shanghai Municipal Education Commission (Grant Nos. 13YZ077 and 12ZZ038), and the Fund of ECNU and SMU for Overseas Studies.
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Chang, H., Yao, YF. On \({\mathbb{F}_q}\) -Rational Structure of Nilpotent Orbits in the Witt Algebra. Results. Math. 65, 181–192 (2014). https://doi.org/10.1007/s00025-013-0339-1
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DOI: https://doi.org/10.1007/s00025-013-0339-1