Some amazing properties of spherical nilpotent orbits

Abstract

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic zero. Write g for its Lie algebra. Let x ∈ g be a nilpotent element and G·x ⊂ g the corresponding nilpotent orbit. The maximal number m such that (adx) 6= 0 is called the height of x or of G·x, denoted ht(x). Recall that an irreducible G-variety X is called G-spherical if a Borel subgroup of G has an open orbit in X. It was shown in [Pa1] that G·x is G-spherical if and only if (adx) = 0. This means that the spherical nilpotent orbits are precisely the orbits of height 2 and 3. Unfortunately, whereas my proof for the orbits of height 2 and height ≥ 4 was completely general, the argument for the orbits of height 3 explicitly used their classification. In this paper, we give a proof of sphericity that does not rely on the classification of nilpotent orbits, see Theorem 3.3. We begin with some properties of invariants of symplectic representations. For instance, we prove that (1) if H ⊂ Sp(V ) is an irreducible representation without nonconstant invariants, then H = Sp(V ), and (2) if H has non-constant invariants, then it has an invariant of degree 4. Applying these results to nilpotent orbits, we prove that the centraliser zg(x) has a rather specific structure whenever ht(x) is odd. From this description, we deduce a conceptual proof of sphericity in case ht(x) = 3. As another application we compute the index of zg(x). It will be shown that ind zg(x) = rk g, if ht(x) = 3. This confirms Elashvili’s conjecture for such x (see [Pa5, Sect. 3] about this conjecture). In Section 4, we prove that if θ, the highest root of g, is fundamental, then g always has a specific orbit of height 3, which is denoted by O. This orbit satisfies several arithmetical relations. Namely, if g = ⊕ −3≤i≤3 g〈i〉 is the Z-grading associated with an sl2-triple containing e ∈ O, then dim g〈3〉 = 2 and dim g〈1〉 = 2 dim g〈2〉. Furthermore, the weighted Dynkin diagram of O can explicitly be described. Let β be the unique simple root that is not orthogonal to θ and let {αi} be all simple roots adjacent to β on the Dynkin diagram of g. Then one has to put ‘1’ at all αi’s and ‘0’ at all other simple roots (Theorem 4.5). It is curious that these properties of O enables us to give an intrinsic construction of G2-grading in each simple g whose highest root is fundamental.