On the orbits of a Borel subgroup in abelian ideals

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.