Cartan decomposition of the moment map

Abstract

Abstract. We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna's slice theorem as well as a version of the Hilbert-Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.

Spherical gradient manifolds

“…Since G is real-reductive, we may apply the techniques developed in [9] to solve the second problem.…”

“…Since the minimal parabolic subgroup Q 0 = M AN is not compatible, we cannot apply the theory developed in [9] Proof. -Recall that the twisted product G × Q X is by definition the quotient space of G×X by the Q-action q ·(g, The claim follows now from the fact that the map , gQ), is a G-equivariant diffeomorphism with respect to the diagonal G-action on X × (G/Q).…”

“…First we observe that X contains an open Q 0 -orbit if and only if (G/Q 0 ) × X contains an open G-orbit with respect to the diagonal action of G. The gradient map µ p on X induces a gradient map µ p on (G/Q 0 ) × X. Now we are in a situation where we can apply the methods introduced in [9]. These allow us to show that open G-orbits correspond to isolated minimal K-orbits of the norm squared of µ p .…”

A quotient restriction theorem for actions of real reductive groups

“…Let X ⊂ V be a G-semistable space. From [HS07b] we recall the construction of a slice at a point x ∈ M p in the case X = V . First, the action of G on X induces an isotropy representation of G x on the tangent space T x X.…”

Semistable points with respect to real forms

“…In [HS05] it is shown that S G (M ip ) is open when U is abelian. Here we give a proof of a more general result.…”