Convexity properties of the moment mapping

Abstract

Let G be a compact Lie group, $ its Lie algebra and X a compact, connected symplectic manifold. Let G • X-~X be a Hamiltonian action of G on X and let q~: X~q* be its associated moment mapping. (See w for definitions.) The set, @(X), is the union of co-adjoint orbits. The main result of this paper is a description of the orbit structure of this set. To describe this result, we first note that the coadjoint orbits in ,q* can be parametrized as follows: Let t be a Caftan subalgebra of g and let B be a positive-definite G-invariant bilinear form on $. By means of B we get a G-equivariant identification, ,q~*. Let t* be the subspace of ~q* corresponding to t. Then every co-adjoint orbit intersects i* in a Weyl-group orbit; so there is a one-one correspondence between co-adjoint orbits in ,q* and Weyl-group orbits in t*. To get an even sharper parametrization, let t* be a Weyl chamber in {*. Then every Weyl-group orbit intersects t* in a single point: so t* is a "moduli space" for the co-adjoint orbits.Two orbits in X are said to be of the same type if they are isomorphic as homogeneous G-spaces. Since X is compact, there are at most a finite number of different types of orbits. (See [1113 Let H be a closed subgroup of T of codimension r and let Cit be the set of all orbits in X of type G/H. Theorem 1. q)(Ctt)ni* is the union of a .finite number of open r-dimensional convex polytopes.We conjecture that qqx)nt* is itself a convex polytope. We have been able to prove this in some important special cases. (See below.)We also prove a number of related results. In w we show that if G=T=an r-dimensional torus, then X has a "symplectic stratification" such that q~ maps each stratum submersively onto a polyhedral subset of t*. The proof makes use of the following elementary sympleetic fact. (See also [2], Theorem 1.) Offprint requests to: S. Sternberg 009(I_001 O/R9 i(~NA?/Nd.Q I /~AA AN

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