NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Let G be a semisimple complex algebraic group with Lie algebra g. For a nilpotent G-orbit O subset of g, let d(O) denote the maximal dimension of a subspace V subset of g that is contained in the closure of O. In this note, we prove that d(O )<= 1/2 dim O and this upper bound is attained if and only if O is a Richardson orbit. Furthermore, if V is B-stable and dim V = 1/2dim O, then V is the nilradical of a polarisation of O. Every nilpotent orbit closure has a distinguished B-stable subspace constructed via an sI(2) -triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits O such that the Dynkin ideal (1) has the minimal dimension among all B-stable subspaces c such that c boolean AND O is dense in c, or (2) is the only B-stable subspace c such that c boolean AND O is dense in c.