On spherical ideals of Borel subalgebras
Let \(G\) be a connected reductive complex algebraic group, and \(B\) be a Borel subgroup of \(G\) with unipotent radical \(B_u\). Let \({\mathfrak g}\), \({\mathfrak b}\), and \({\mathfrak b}_u\) denote the Lie algebras of \(G\), \(B\), and \(B_u\) respectively. Of interest here are ad-nilpotent ideals of \({\mathfrak b}\), that is, ideals which are contained in \({\mathfrak b}_u\). In a previous work [Adv. Math. 159, 229--246 (2001; Zbl 0993.22016)], the authors investigated abelian ideals of \({\mathfrak b}\) (which are necessarily ad-nilpotent) and showed that the \(G\)-saturation (in \({\mathfrak g}\)) of such an ideal is a spherical \(G\)-variety (a variety on which \(B\) acts with a dense orbit). The goal of this work is to describe the non-abelian spherical ad-nilpotent ideals of \({\mathfrak b}\). Suppose that \(G\) is simple. If the root system is simply-laced or of type \(G_2\), the authors show that all spherical ad-nilpotent ideals are necessarily abelian. They then identify the maximal non-abelian spherical ad-nilpotent ideals in the remaining doubly-laced cases. The proofs use the authors aforementioned work as well as a general characterization of spherical orbits by the first author [Manuscr. Math. 83, 223--237 (1994; Zbl 0822.14024), Ann. Inst. Fourier 49, 1453--1476 (1999; Zbl 0944.17013)].
scientific article; zbMATH DE number 2162316