Geometric invariant theory and the moment map

First we need some generalities on the relationship between the geometric invariant theory and the moment map. General reference of this section is [60, Chapter 8]. (See also [71] for reference to the geometric invariant theory.) 

Let H be a vector space over C with a hermitian metric, G a connected Lie subgroup of U(H), and its complexification. The Lie algebra of G is denoted by g, and its dual 

Theorem 3.3 ([60, 1.2]). Let Wi and W2 be two elosed invariant subsets ofV. Then W and W2 are disjoint if and only if there exists an invariant funetion f G such 

The if direction is clear. Let W and W2 be two disjoint closed invariant subsets of V. There exists / G AfV) (not necessarily invariant) such that f wi = 1 and f w2 = 0. Averaging by the action of the eompaet group G, we may assume that / is invariant under the action of G. Since / is holomorphic, it is automatically invariant under the action of  

The second statement of Theorem 3.1 follows from the following fact the closure of a G -orbit is a union of orbits of smaller dimensions. Hence any orbit contains a closed G -orbit in its closure. (Moreover, it is unique by Theorem 3.3.) 

In order to illustrate how the closedness of the orbit is determined, we consider the case Gc is the complex torus Tc = (C )r. We choose a basis aq. xn of V so that Tc is contained in the group of nonsingular diagonal matrices. Then we have distinguished 

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This function plays a fundamental role in the relationship between the moment map and the geometric invariant theory as seen in the following proposition. 

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