Closed Orbits of Semisimple Group Actions and the Real Hilbert-mumford Function

The action of a noncompact semisimple Lie group G on a finite dimensional real vector space V is said to be stable if there exists a nonempty Zariski open subset O of V such that the orbit G(v) is closed in V for all v ∈ O. We study a Hilbert-Mumford numerical function M : V → R defined by A. Marian that extends the corresponding function in the complex setting defined by D. Mumford and studied further by G. Kempf and L. Ness. The G-action may be stable on V if M ≥ 0 on V, as in the adjoint action of G on its Lie algebra G. However, we show that the G-action on V is always stable if M(v) < 0 for some v ∈ V. We show that M(v) < 0 ⇔ the orbit G(v) is closed in V and the stability subgroup Gv is compact. The subset of V where M is negative is open in the vector space topology of V but not necessarily open in the Zariski topology of V. We give criteria for M to be negative on a nonempty Zariski open subset of V, and we consider several examples.