On the orbits of a Borel subgroup in abelian ideals (Q2410922)

The Ad-action of a Borel subgroup \(B\) of a semisimple algebraic group \(G\) (over an algebraically closed field of characteristic 0) on abelian ideals \(A\) of the corresponding Lie algebra \(b\) (and on \(b^*\) too) is investigated. It is known that there are only a finite number of \(B\)-orbits in this case. In this article, a direct approach to the study of these \(B\)-orbits is presented. The representatives for \(B\)-orbits on \(A\) are described. The action of \(U=[B,B]\) on abelian ideals is investigated. Also, the numbers of \(B\)-orbits in the abelian nilradicals are counted.