On a theorem of Ranee Brylinski

In her thesis [RB], Ranee Brylinski (then Gupta) studied the orbit structure of the projective variety of abelian subalgebras of a …xed dimension, k, in a simple Lie algebra, g, over C under its adjoint group, G. Fix a Borel subalgebra, b, of g and let B be the closed subgroup of G corresponding to b. Then the Borel …xed point theorem implies that the closed G-orbits are precisely the orbits of the abelian ideals in b of dimension k. Let l be the rank of g and …x h a Cartan subalgebra. The main result in [RB] is that the closed orbits are all contained in the closures of the orbits of the subspaces of h of dimension k. In particular, if k = l the closure of the orbit of h contains all of the closed orbits. In this paper we will only study the case when k = l (although we expect that similar methods will work for k < l) and we prove the following strengthening of Brylinski’s theorem.