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Quotients for sheets of conjugacy classes. (English) Zbl 1436.20089

Gorelik, Maria (ed.) et al., Representations and nilpotent orbits of Lie algebraic systems. In honour of the 75th birthday of Tony Joseph. Based on the 2017 conferences “Algebraic Modes of Representations”, Weizmann Institute of Science, Rehovot, Israel, July 16–18, 2017 and University of Haifa, Haifa, Israel, July 19–13, 2017. Cham: Birkhäuser. Prog. Math. 330, 73-90 (2019).
Summary: We provide a description of the orbit space of a sheet \(S\) for the conjugation action of a complex simple simply connected algebraic group \(G\). This is obtained by means of a bijection between \(S/G\) and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to [W. Borho and H. Kraft, Comment. Math. Helv. 54, 61–104 (1979; Zbl 0395.14013)]. We also describe the normalisation of the categorical quotient \(\overline{S}//G\) for arbitrary simple \(G\) and give a necessary and sufficient condition for \(\overline{S}//G\) to be normal in analogy to results of Borho and Kraft [loc. cit.] and R. W. Richardson [Lect. Notes Math. 1271, 243–264 (1987; Zbl 0632.14011)]. The example of \(G_2\) is worked out in detail.
For the entire collection see [Zbl 1432.17001].

MSC:

20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20G07 Structure theory for linear algebraic groups
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