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 > 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.