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

TL;DR

This paper characterizes the structure of B-orbits of square-zero elements in the nilradical of the symplectic Lie algebra using symmetric link patterns, and analyzes their closures and intersections.

Contribution

It introduces a combinatorial description of B-orbits via symmetric link patterns and applies it to study orbital variety closures and their intersections.

Findings

Topology of orbits described by symmetric link patterns

Closures of orbital varieties of nilpotency order 2 computed

Intersections of codimension 1 are shown to be irreducible

Abstract

Let $SP_n(\mathbb{C})$ be the symplectic group and $\mathfrak{sp}_n(\mathbb{C})$ its Lie algebra. Let $B$ be a Borel subgroup of $SP_n(\mathbb{C} )$, $\mathfrak{b}={\rm Lie}(B)$ 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.