On the orbits of a Borel subgroup in abelian ideals

TL;DR

This paper classifies the orbits of a Borel subgroup acting on abelian ideals of its Lie algebra, providing explicit descriptions, counting orbits, and exploring invariant algebras and orbit closures, with applications to classical groups.

Contribution

It offers an explicit classification of B-orbits in abelian ideals using root subsets and establishes a connection with Weyl group involutions, extending prior work.

Findings

Finitely many B-orbits in abelian ideals and their duals.

Explicit classification via root subsets.

Conjecture relating orbit closure and Weyl group involutions.

Abstract

Let $B$ be a Borel subgroup of a semisimple algebraic group $G$, and let $\mathfrak a$ be an abelian ideal of $\mathfrak b=Lie(B)$. The ideal $\mathfrak a$ is determined by certain subset $\Delta_{\mathfrak a}$ of positive roots, and using $\Delta_{\mathfrak a}$ we give an explicit classification of the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$. Our description visibly demonstrates that there are finitely many $B$-orbits in both cases. We also describe the Pyasetskii correspondence between the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$ and the invariant algebras $\Bbbk[\mathfrak a]^U$ and $\Bbbk[\mathfrak a^*]^U$, where $U=(B,B)$. As an application, the number of $B$-orbits in the abelian nilradicals is computed. We also discuss related results of A.Melnikov and others for classical groups and state a general conjecture on the closure and dimension of the $B$-orbits in the abelian nilradicals, which exploits a relationship between between $B$-orbits and involutions in the Weyl group.