On the Nonlinear Convexity Theorem of Kostant

A classical result of Schur and Horn [Sc, Ho] states that the set of diagonal elements of all n x n Hermitian matrices with fixed eigenvalues is a convex set in IRn. Kostant [Kt] has generalized this result to the case of any semisimple Lie group. This is often referred to as the linear convexity theorem of Kostant: picking up the diagonal of a Hermitian matrix is a linear operation. This result was later put into the framework of symplectic geometry by Atiyah [Ay] , Guillemin-Sternberg [Gu-St] and Duistermaat [Du1], the key argument being that the map that picks up the diagonal of a Hermitian matrix is a moment map for a Hamiltonian torus action and that the image of such a moment map is always a convex polytope. There is also a nonlinear version of the convexity theorem of Kostant. It has not been connected so far with symplectic geometry. The purpose of this paper is to make this connection. We first describe the nonlinear convexity theorem for the case of SL(n, q (see §2 for more details). The Gram-Schmidt orthonormalization process in linear algebra asserts that every nonsingular n x n complex matrix g can be uniquely written as the product g = kan for some unitary matrix k, positive diagonal matrix a, and strictly upper-triangular matrix n. We use A to denote the group of all positive diagonal matrices, and we call the matrix a in the above decomposition the A-component of g. The nonlinear convexity theorem of Kostant now asserts that the set of A-components of g as g runs through all positive definite Hermitian matrices with fixed eigenvalues a = (aJ ' ••• , an) is the convex hull of the set of all points obtained by permuting the coordinates of a. Here we identify A with its Lie algebra a via the exponential map; thus convexity in A makes sense. For a general semisimple Lie group G, the Gram-Schmidt decomposition is replaced by an Iwasawa decomposition G = KAN of G, and the permutation group that appeared above is replaced by the Weyl group W of (K, A). If an element g EGis written as the product g = kan for k E K, a E A,