On the complement of the Richardson orbit

TL;DR

This paper studies the structure of the complement of the Richardson orbit in the nilradical of parabolic subgroups, revealing the number of irreducible components and their properties for different Levi factors.

Contribution

It characterizes the irreducible components of the complement of the Richardson orbit, showing there are at most t-1 components when the Levi part has t factors.

Findings

For t ≥ 6, the complement has infinitely many orbits.

The complement has at most t-1 irreducible components.

The structure of the complement varies with the number of Levi factors.

Abstract

We consider parabolic subgroups of a general algebraic group over an algebraically closed field $k$ whose Levi part has exactly $t$ factors. By a classical theorem of Richardson, the nilradical of a parabolic subgroup $P$ has an open dense $P$-orbit. In the complement to this dense orbit, there are infinitely many orbits as soon as the number $t$ of factors in the Levi part is $\ge 6$. In this paper, we describe the irreducible components of the complement. In particular, we show that there are at most $t-1$ irreducible components.