Abstract
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 [8] 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.
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© 1983 Springer Science+Business Media New York
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Atiyah, M.F., Pressley, A.N. (1983). Convexity and Loop Groups. In: Artin, M., Tate, J. (eds) Arithmetic and Geometry. Progress in Mathematics, vol 36. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-9286-7_3
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DOI: https://doi.org/10.1007/978-1-4757-9286-7_3
Publisher Name: Birkhäuser, Boston, MA
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