Konstant's convexity theorem states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set.
It is a special case of a more general result for symmetric spaces.
Kostant's theorem is a generalization of a result of Schur (1923), Horn (1954) and Thompson (1972) for Hermitian matrices.
They proved that the projection onto the diagonal matrices of the space of all n by n complex self-adjoint matrices with given eigenvalues Λ = (λ1, ..., λn) is the convex polytope with vertices all permutations of the coordinates of Λ.
, P(Ad(K)⋅X) is the convex polytope with vertices w(X) where w runs over the Weyl group.
be their Lie algebras and let σ also denote the corresponding involution of
, Q(Ad(K)⋅X) is the convex polytope with vertices the w(X) where w runs over the restricted Weyl group (the normalizer of
There is an elementary proof just for compact Lie groups using similar ideas, due to Wildberger (1993): it is based on a generalization of the Jacobi eigenvalue algorithm to compact Lie groups.
Let K be a connected compact Lie group with maximal torus T. For each positive root α there is a homomorphism of SU(2) into K. A simple calculation with 2 by 2 matrices shows that if Y is in
and k varies in this image of SU(2), then P(Ad(k)⋅Y) traces a straight line between P(Y) and its reflection in the root α.
In performing this latter operation, the distance from P(Y) to P(Ad(k)⋅Y) is bounded above by size of the α off-diagonal coordinate of Y.
Starting from an arbitrary Y1 take the largest off-diagonal coordinate and send it to zero to get Y2.
On the other hand, Xn lies on the line segment joining Xn+1 and its reflection in the root α.
Thus Xn lies in the Weyl group polytope defined by Xn+1.
The limit is necessarily in the Weyl group orbit of X and hence P(Ad(K)⋅X) is contained in the convex polytope defined by W(X).
To prove the opposite inclusion, take X to be a point in the positive Weyl chamber.
Then all the other points Y in the convex hull of W(X) can be obtained by a series of paths in that intersection moving along the negative of a simple root.
[1]) Each part of the path from X to Y can be obtained by the process described above for the copies of SU(2) corresponding to simple roots, so the whole convex polytope lies in P(Ad(K)⋅X).
Heckman (1982) gave another proof of the convexity theorem for compact Lie groups, also presented in Hilgert, Hofmann & Lawson (1989).
For compact groups, Atiyah (1982) and Guillemin & Sternberg (1982) showed that if M is a symplectic manifold with a Hamiltonian action of a torus T with Lie algebra
, the moment map for T is the composition Using the Ad-invariant inner product to identify
Ziegler (1992) gave a simplified direct version of the proof using moment maps.
Then M and the fixed point set of τ (assumed to be non-empty) have the same image under the moment map.
The map above has the same image as that of the fixed point set of τ, i.e.
In Kostant (1973) the convexity theorem is deduced from a more general convexity theorem concerning the projection onto the component A in the Iwasawa decomposition G = KAN of a real semisimple Lie group G. The result discussed above for compact Lie groups K corresponds to the special case when G is the complexification of K: in this case the Lie algebra of A can be identified with
Kac & Peterson (1984) gave a generalization for infinite-dimensional groups.