On spherical ideals of Borel subalgebras
type: Item
instance of: scholarly article
title: On spherical ideals of Borel subalgebras
zbMATH Open document ID: 1076.14061
author: Gerhard Röhrle
author: Dmitri I. Panyushev
published in: Archiv der Mathematik
publication date: 2005-04-28T00:00:00Z
review text: 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)].
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 17B45
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 2162316
zbMATH Keywords: linear algebraic groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: ad-nilpotent ideals
zbMATH Keywords: abelian ideals
MaRDI profile type: Publication
full work available at URL: s00013-004-1092-1
OpenAlex ID: W2029161169
DOI: 10.1007/S00013-004-1092-1
description: scientific article; zbMATH DE number 2162316