On Lusztig's map for spherical unipotent conjugacy classes. (Q2865911)

Let \(G\) be a connected reductive linear algebraic group over an algebraically closed field \(k\), with Weyl group \(W\). Let \(\underline G\) (resp. \(\underline W\)) denote the set of unipotent conjugacy classes in \(G\) (resp. conjugacy classes in \(W\)). \textit{G. Lusztig} [Represent. Theory 15, 494-530 (2011; Zbl 1263.20045); Represent. Theory 16, 189-211 (2012; Zbl 1264.20043)], has described a surjective map \(\varphi\colon\underline W\to\underline G\), and a right inverse \(\Psi\) to this map which conjecturally coincides (in characteristic zero) with an earlier map from \(\underline G\) to \(\underline W\) defined by \textit{D. Kazhdan} and \textit{G. Lusztig} [Isr. J. Math. 62, No. 2-3, 129-168 (1988; Zbl 0658.22005)].NEWLINENEWLINE The map \(\Psi\) seems to be rather hard to get hold of directly, and the purpose of the paper under review is to give a direct description for spherical unipotent conjugacy classes (recall that a conjugacy class is called \textit{spherical} if it contains a dense \(B\)-orbit for a Borel subgroup \(B\) of \(G\)). It follows from earlier work of the two authors and others (see references) that given a spherical unipotent class \(\gamma\) of \(G\), there exists a unique element \(w_\gamma\in W\) such that \(\gamma\cap Bw_\gamma B\) is dense in \(\gamma\). The main result of the paper can now be stated: Let \(\gamma\) be a spherical unipotent conjugacy class in \(G\). Then \(\Psi(\gamma)=W\cdot w_\gamma\).