The geometry of generalized Steinberg varieties.
Let \(G\) be a reductive algebraic group over an algebraically closed field. Assume the characteristic is zero or good, so that the variety \(\mathcal N\) of nilpotent elements of the Lie algebra \(\mathfrak g\) may be identified with the unipotent variety. The authors introduce a variety \(X^{\mathcal{P,Q}}_{c,d}\) that is designed to cover many earlier constructions. Here \(\mathcal{P,Q}\) are conjugacy classes of parabolic subgroups, \(c(P)\), \(d(Q)\) are closed subvarieties of \(\mathcal N\), depending \(G\)-equivariantly on \(P\in\mathcal P\), \(Q\in\mathcal Q\). One requires \({\mathfrak u}_P\subseteq c(P)\subseteq{\mathcal N}\cap\mathfrak p\) and \({\mathfrak u}_Q\subseteq d(Q)\subseteq{\mathcal N}\cap\mathfrak q\). One defines \(X^{\mathcal{P,Q}}_{c,d}\) to be the subset of \(\mathcal N\times\mathcal P\times\mathcal Q\) consisting of the \((x,P,Q)\) with \(x\in c(P)\cap d(Q)\). The cases where \(c\), \(d\) are both minimal (maximal) are best understood. For those it is shown that \(X^{\mathcal{P,Q}}_{c,d}\) is equidimensional and that the irreducible components are indexed by \((W_{\mathcal P},W_{\mathcal Q})\) double cosets in the Weyl group \(W\). One also gets a map that associates to one of these double cosets a nilpotent orbit in \(G{\mathfrak u}_P\cap G{\mathfrak u}_Q\).
scientific article; zbMATH DE number 2105048