Orbits of the Weyl group and a theorem of DeConcini and Procesi. (English) Zbl 0613.20029

C. DeConcini and C. Procesi (1981) proved a remarkable connection between dual \(SL_ n\) conjugacy classes of nilpotent matrices. The goal of this paper is to show that a weaker form of this connection holds for certain pairs of nilpotent conjugacy classes for an arbitrary semi-simple algebraic group G over \({\mathbb{C}}\). Let G, B and H be a semi-simple algebraic group over \({\mathbb{C}}\), a Borel subgroup and a maximal torus contained in B, respectively. For s of Lie(H), consider the orbit \(W\cdot s\) as a finite reduced variety with rings A(W\(\cdot s)\) of regular functions, where W is the Weyl group of (Lie(G), Lie(H)). The graded ring Gr A(W\(\cdot s)\) is canonically defined. Given s of Lie(H), let \(\tau\) be a regular nilpotent in the Levi subalgebra, centralizer of s in Lie(G) and let \(X^{\tau}\) denote the variety of Borel subalgebras of X containing \(\tau\). Let \(i_{\tau}: X^{\tau}\to X\) be the inclusion into the flag variety of all Borel subalgebras of Lie(G). Then there exists a W-equivariant \({\mathbb{C}}\)-algebra homomorphism \(\psi_ s: Gr A(W\cdot s)\to H^.(X^{\tau})\) doubling degree. The image of \(\psi_ s\) is \(i^*_{\tau}H^.(X)\). Moreover, \(\psi_ s\) is an isomorphism if and only if \(i^*_{\tau}\) is surjective.
Reviewer: Y.Asoo


20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20G10 Cohomology theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
Full Text: Numdam EuDML


[1] E. Akyildiz , J.B. Carrell and D.I. Lieberman : Zeros of holomorphic vector fields on singular varieties and intersection rings of Schubert varieties . Compositio Math. 57 (1986) 237-248. · Zbl 0613.14035
[2] W. Borho and H. Kraft : Über Bahnen und deren Deformation bei linearen Aktionen reduktiver Gruppen . Comment. Math. Helvetici 54 (1979) 62-104. · Zbl 0395.14013
[3] W.M. Beynon and N. Spaltenstein : Green functions of finite Chevalley groups of type E ( n = 6, 7, 8) , preprint. · Zbl 0539.20025
[4] J.B. Carrell : Vector fields and the cohomology of G/B, Mainfolds and Lie groups . Papers in honor of Y. Matsushima. Progress in Mathematics , Vol. 14 Birkhauser, Boston (1981). · Zbl 0482.57017
[5] J.B. Carrell and D.I. Lieberman : Holomorphic vector fields and compact Kaehler manifolds . Invent. Math. 21 (1973) 303-309. · Zbl 0253.32017
[6] J.B. Carrell and D.I. Lieberman : Vector fields and Chern numbers . Math. A nn . 225 (1977) 263-273. · Zbl 0365.32020
[7] J.B. Carrell and A.J. Sommese : Some topological aspects of C *-actions on compact Kaehler manifolds . Comment. Math. Helv. 54 (1979) 567-587. · Zbl 0466.32015
[8] C. Deconcini and C. Procesi : Symmetric functions, conjugacy classes, and the flag variety . Invent. Math., 64 (1981) 203-219. · Zbl 0475.14041
[9] J. Humphreys : Linear algebraic groups . Springer Verlag, Berlin and New York (1975). · Zbl 0325.20039
[10] B. Kostant : Lie group representations on polynomial rings. A mer . J. Math. 85 (1963) 327-404. · Zbl 0124.26802
[11] H. Kraft : Conjugacy classes and Weyl group representations; Tableaux de Young et foncteurs de Schur en algebre et geometrie (Conference international, Torun Pologne 1980). Asterisque 87-88 (1981) 195-205. · Zbl 0489.17002
[12] H. Kraft and C. Procesi : Closures of conjugacy classes of matrices are normal . Invent. Math. 53 (1979) 227-247. · Zbl 0434.14026
[13] N. Spaltenstein : The fixed point set of a unipotent transformation on the flag manifold . Nederl. Akad. Wetensch. Proc. Ser. A. 79 (1976) 452-456. · Zbl 0343.20029
[14] T.A. Springer : A construction of representations of Weyl groups . Invent. Math. 44 (1978) 279-293. · Zbl 0376.17002
[15] T. Tanisaki : Defining ideals of the closures of the conjugacy classes and representations of the Weyl groups . Tohuku Math. J. 34 (1982) 575-585. · Zbl 0544.14030
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