B-orbits in abelian nilradicals of types B, C and D: towards a conjecture of Panyushev

TL;DR

This paper proves Panyushev's conjecture regarding the structure and dimensions of B-orbits in abelian nilradicals for types B, C, and D, confirming predictions about orbit closures and involutions in these cases.

Contribution

The paper confirms Panyushev's conjecture for types B, C, and D in the adjoint case, advancing understanding of B-orbit classifications in these Lie algebra types.

Findings

Confirmed conjecture for types B, C, D

Classified B-orbits in abelian nilradicals

Established orbit closure and dimension relations

Abstract

Let $B$ be a Borel subgroup of a semisimple algebraic group $G$ and let $\mathfrak m$ be an abelian nilradical in $\mathfrak b={\rm Lie} (B)$. Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to $\mathfrak m$, D. Panyushev \cite{Pan} gives in particular classification of $B-$orbits in $\mathfrak m$ and ${\mathfrak m}^*$ and states general conjectures on the closure and dimensions of the $B-$orbits in both $\mathfrak m$ and ${\mathfrak m}^*$ in terms of involutions of the Weyl group. Using Pyasetskii correspondence between $B-$orbits in $\mathfrak m$ and ${\mathfrak m}^*$ he shows the equivalence of these two conjectures. In this Note we prove his conjecture in types $B_n, C_n$ and $D_n$ for adjoint case.