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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

MSC:

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)
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