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 $\mathfrak b=\mathrm{Lie} (B)$. Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to $\mathfrak m$, D. Panyushev [1] 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.