Abstract
We study the geometry of the stratification induced by an affine hyperplane arrangement H on the quotient of a complex affine space by the action of a discrete group preserving H. We give conditions ensuring normality or normality in codimension 1 of strata. As an application, we provide the list of those categorical quotients of closures of Jordan classes and of sheets in all complex simple algebraic groups that are normal. In the simply connected case, we show that normality of such a quotient is equivalent to its smoothness.
Highlights
In [5, 4] the stratification of a semisimple Lie algebra by Jordan classes was introduced and studied in order to describe the sheets for the adjoint action of a semisimple algebraic group G on its Lie algebra g
Closure relations for Jordan classes were explicitly given and, for S a sheet, the topology of the orbit space S/G and the normalisation of the categorical quotient S//G were explicitly described. These quotients are the closures of Luna strata for g//G, as defined in [17, III.2]. It was proved in [15] that the orbit space S/G can be given the structure of a geometric quotient which is isomorphic to the quotient of an affine space modulo the action of a finite group
The same approach allowed to provide in [21, 7, 11] the complete list of those regular Jordan classes whose closure is normal and Cohen-Macaulay
Summary
In [5, 4] the stratification of a semisimple Lie algebra by Jordan classes ( called decomposition classes or packets) was introduced and studied in order to describe the sheets for the adjoint action of a semisimple algebraic group G on its Lie algebra g. In the case of Jordan classes in semisimple groups, minimal strata correspond to the Jordan classes consisting of one single conjugacy class, which is necessarily isolated, in the terminology of [18] Around such points the stratum is smoothly equivalent to the quotient of the closure of a Jordan class in a Lie algebra with automorphisms. The results obtained here have been applied in [1] to produce the complete list of regular Jordan classes in G semisimple and -connected whose closure is normal and Cohen-Macaulay.
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