On the orbits of a Borel subgroup in abelian ideals

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 ${\mathfrak{a}}^{\ast }$. 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 ${\mathfrak{a}}^{\ast }$ and the invariant algebras $\mathbb{k}[\mathfrak{a}{]}^U$ and $\mathbb{k}[{\mathfrak{a}}^{\ast }{]}^U$, 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.

24 pages

References in corpus (1)

• Combinatorics of $B$-orbits and Bruhat--Chevalley order on involutions

Cited by in corpus (2)

• Spherical nilpotent orbits and abelian subalgebras in isotropy representations
• The Frobenius semiradical, generic stabilisers, and Poisson centre for nilradicals