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

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.