B-orbits of square zero in nilradical of the symplectic algebra

Abstract

Let $S{P}_n(\mathbb{C})$ be the symplectic group and ${\mathfrak{sp}}_n(\mathbb{C})$ its Lie algebra. Let $B$ be a Borel subgroup of $S{P}_n(\mathbb{C})$, $\mathfrak{b}=\mathrm{Lie}(B)$ and $\mathfrak{n}$ its nilradical. Let $\mathcal{X}$ be a subvariety of elements of square 0 in $\mathfrak{n}.$ $B$ acts adjointly on $\mathcal{X}$. In this paper we describe topology of orbits $\mathcal{X}/B$ in terms of symmetric link patterns. Further we apply this description to the computations of the closures of orbital varieties of nilpotency order 2 and to their intersections. In particular we show that all the intersections of codimension 1 are irreducible.

arXiv comment36 pages