Dimension of a minimal nilpotent orbit

We show that the dimension of the minimal nilpotent coadjoint orbit for a complex simple Lie algebra is equal to twice the dual Coxeter number minus two. Let g be a finite dimensional complex simple Lie algebra. We fix a Cartan subalgebra h, a root system ∆ ⊂ h and a set of positive roots ∆+ ⊂ ∆. Let ρ be half the sum of all positive roots. Denote by θ the highest root and normalize the Killing form ( , ) : g × g → C by the condition (θ, θ) = 2. The dual Coxeter number h can be defined as h = (ρ, θ) + 1 (cf. [K]). This intrinsic number of the Lie algebra g plays an important role in representation theory(cf. e.g. [K]). As is well known there exists a unique nonzero nilpotent (co)adjoint orbit of minimal dimension. A coadjoint orbit can be identified with an adjoint one by means of the Killing form. For more detail on the nilpotent orbits, we refer to the excellent exposition [CM] and the references therein. Our result of this short note is the following theorem. Theorem 1 The dimension of the minimal nonzero nilpotent orbit equals 2h − 2. We start with the following well-known lemma, cf. for example, Lemma 4.3.5, [CM]. Lemma 1 The dimension of the minimal nonzero nilpotent orbit is equal to one plus the number of positive roots not orthogonal to θ. ∗1991 Mathematics Subject Classification. Primary 22E10; Secondary 17B20. Partially supported by NSF grant DMS-9304580.