Orbits of subsystem stabilizers
Let Φ be a reduced irreducible root system. We consider pairs (S, X (S)), where S is a closed set of roots, X(S) is its stabilizer in the Weyl group W(Φ). We are interested in such pairs maximal with respect to the following order: (S1, X (S1)) ≤ (S2, X (S2)) if S1 ⊆ S2 and X(S1) ≤ X(S2). The main theorem asserts that if Δ is a root subsystem such that (Δ, X (Δ)) is maximal with respect to the above order, then X (Δ) acts transitively both on the long and short roots in Φ \ Δ. This result is a wide generalization of the transitivity of the Weyl group on roots of a given length. Bibliography: 23 titles.
KeywordsRoot System Conjugacy Class Weyl Group Wreath Product Fundamental System
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- 2.N. Bourbaki, Lie Groups and Algebras [Russian translation], Chaps. IV-VI, Mir, Moscow 1972).Google Scholar
- 3.N. A. Vavilov, “Maximal subgroups of Chevalley groups which contain a split maximal torus,” in: Rings and Modules, Leningrad (1986), pp. 67–75.Google Scholar
- 4.N. A. Vavilov, “Weight elements of Chevalley groups,” Dokl. AN SSSR, 298, No. 3, 524–527 (1988).Google Scholar
- 8.N. A. Vavilov, A. Yu. Luzgarev, and I. M. Pevzner, “Chevalley groups of type E 6 in the 27-dimensional representation,” this issue, 5–68.Google Scholar
- 9.N. A. Vavilov and A. A. Semenov, “Long root semisimple elements in Chevalley groups,” Dokl. RAN, 338, No. 6, 725–727 (1994).Google Scholar
- 12.R. Carter, “Conjugacy classes in the Weyl group [Russian translation],” in: A Seminar on Algebraic Groups, Mir, Moscow (1973), pp. 288–306.Google Scholar
- 13.Yu. I. Manin, Cubic Forms: Algebra, Geometry, and Arithmetic [in Russian], Nauka, Moscow (1972).Google Scholar
- 21.N. A. Vavilov, “Intermediate subgroups in Chevalley groups,” in: Proceedings of the Conference on Groups of Lie Type and Their Geometries (Como-1993), Cambridge Univ. Press (1995), pp. 233–280.Google Scholar
- 22.N. A. Vavilov, “Do it yourself: structure theorems for Lie algebras of type El,” Zap. Nauchn. Semin. POMI, 281, 60–104 (2001).Google Scholar