On the Complement of the Richardson Orbit

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 $\geq 6$. In this paper, we describe the irreducible components of the complement. In particular, we show that there are at most $t-1$ irreducible components. We are also able to determine their codimensions.