Convexity and Loop Groups
The purpose of this paper is to extend certain convexity results associated with compact Lie groups to an infinite-dimensional setting, in which the Lie group is replaced by the corresponding loop group. To recall the finite-dimensional results which we shall generalize let G be a simply connected, compact Lie group, T a maximal torus of G and W its Weyl group. Consider the adjoint action of G on its Lie algebra L(G) and fix a G-invariant metric on L(C) so that we can define orthogonal projection. A result of Kostant  describes the images of the G-orbits in L(G) under the orthogonal projection onto L(T). To state it, recall that such G-orbits correspond to W-orbits in L(T). Then Kostant's result is:
(1.1). The orthogonal projection of a G-orbit onto L(T) coincides with the convex hull of the corresponding W-orbit.
KeywordsConvex Hull Weyl Group Symplectic Form Maximal Torus Loop Space
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