A conjectural generalization of the n! result to arbitrary groups

The aim of this paper is to formulate a conjecture for an arbitrary simple Lie algebra g in terms of the geometry of principal nilpotent pairs. When g is specialized to sln, this conjecture readily implies the n! result and it is very likely that, in fact, it is equivalent to the n! result in this case. In addition, this conjecture can be thought of as generalizing an old result of Kostant. In another direction, we show that to prove the validity of the n! result for an arbitrary n and an arbitrary partition of n, it suffices to show its validity only for the staircase partitions.