2008/09/01 by Vyjayanthi Chari, Chari, Vyjayanthi, R. J. Dolbin +3
Mathematics · #17B20 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B20
paper · pdf · doi:10.48550/arxiv.0809.0245
14 pages
arxiv created 2008/09/01 · arxiv updated 2009/12/01
We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie algebra. Any such ideal determines an antichain in a set of positive roots of the simple Lie algebra. We give a necessary and sufficient condition for an antichain to determine an ad-nilpotent ideal of the parabolic. We write down all such antichains for the classical simple Lie algebras and in particular recover the results of D. Peterson. In section 2 of the paper we study the unique ideal in a parabolic which is irreducible as a module for the reductive part and give several equivalent statements that are satisfied by the corresponding subset of roots.