An infinite dimensional version of the Kostant convexity theorem. (Q1597995)

Let \(L\) be the unitary real form of a classical (infinite dimensional) Lie algebra, \(\mathcal U\) its unitary group, \(H\) a Cartan subspace (that is, a maximal abelian Lie subalgebra which can be simultaneously diagonalized), \(p\) the projection onto \(H\) and \(M\) the Weyl group, which is defined as the quotient \(N/Z,\) where \(N\) is the normalizer and \(Z\) is the centralizer of \(H\) in a Lie group with Lie algebra \(H.\) The group \(M\) operates canonically on \(H.\) The main result of the paper under review is that \(\overline{p(\{ U^\star XU: U\in {\mathcal U}\} )}= \overline{\text{ conv} (M.X)},\) for every \(X\in H.\) This is a generalization of the Kostant Convexity Theorem [\textit{B. Kostant}, Ann. Sci. Éc. Norm. Supér. (4) 6, 413--455 (1973; Zbl 0293.22019)].