Kostant's convexity theorem

In mathematics, Kostant's convexity theorem, introduced by Bertram Kostant (1973), 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, ... ,λ_n) is the convex polytope with vertices all permutations of the coordinates of Λ.

Kostant used this to generalize the Golden–Thompson inequality to all compact groups.

Compact Lie groups

Let K be a connected compact Lie group with maximal torus T and Weyl group W = N_K(T)/T. Let their Lie algebras be ${\mathfrak {k}}$ and ${\mathfrak {t}}$. Let P be the orthogonal projection of ${\mathfrak {k}}$ onto ${\mathfrak {t}}$ for some Ad-invariant inner product on ${\mathfrak {k}}$. Then for X in ${\mathfrak {t}}$, P(Ad(K)⋅X) is the convex polytope with vertices w(X) where w runs over the Weyl group.

Symmetric spaces

Let G be a compact Lie group and σ an involution with K a compact subgroup fixed by σ and containing the identity component of the fixed point subgroup of σ. Thus G/K is a symmetric space of compact type. Let ${\mathfrak {g}}$ and ${\mathfrak {k}}$ be their Lie algebras. Let ${\mathfrak {p}}$ be the −1 eigenspace of σ and let ${\mathfrak {a}}$ be a maximal Abelian subspace. Let Q be the orthogonal projection of ${\mathfrak {p}}$ onto ${\mathfrak {a}}$ for some Ad(K)-invariant inner product on ${\mathfrak {p}}$. Then for X in ${\mathfrak {a}}$, Q(Ad(K)⋅X) is the convex polytope with vertices the w(X) where w runs over the restricted Weyl group (the normalizer of ${\mathfrak {a}}$ in K modulo its centralizer).

The case of a compact Lie group is the special case where G = K × K, K is embedded diagonally and σ is the automorphism of G interchanging the two factors.

Proof for a compact Lie group

Kostant's proof is given in Helgason (1984). There is an elementary proof using similar ideas for compact Lie groups due to Wildberger (1993): it is based on a generalization of the Jacobi eigenvalue algorithm to compact Lie groups.

Other proofs

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 ${\mathfrak {t}}$, then the image of the moment map

$\displaystyle {M\rightarrow {\mathfrak {t}}^{*}}$

is a convex polytope with vertices in the image of the fixed point set of T (the image is a finite set). Taking for M a coadjoint orbit of K in ${\mathfrak {k}}^{*}$, the moment map for T is the composition

$\displaystyle {M\rightarrow {\mathfrak {k}}^{*}\rightarrow {\mathfrak {t}}^{*}.}$

Using the Ad-invariant inner product to identify ${\mathfrak {k}}^{*}$ and ${\mathfrak {k}}$, the map becomes

$\displaystyle {\mathrm {Ad} (K)\cdot X\rightarrow {\mathfrak {t}},}$

the restriction of the orthogonal projection. Taking X in ${\mathfrak {t}}$, the fixed points of T in the orbit Ad(K)⋅X are just the orbit under the Weyl group, W(X). So the convexity properties of the moment map imply that the image is the convex polytope with these vertices. Ziegler (1992) gave a simplified direct version of the proof using moment maps.

Duistermaat (1983) showed that a generalization of the convexity properties of the moment map could be used to treat the more general case of symmetric spaces. Let τ be a smooth involution of M which takes the symplectic form ω to −ω and such that t ∘ τ = τ ∘ t⁻¹. Then M and the fixed point set of τ (assumed to be non-empty) have the same image under the moment map. To apply this, let T = exp ${\mathfrak {a}}$, a torus in G. If X is in ${\mathfrak {a}}$ as before the moment map yields the projection map

$\displaystyle {\mathrm {Ad} (G)\cdot X\rightarrow {\mathfrak {a}}.}$

Let τ be the map τ(Y) = − σ(Y). The map above has the same image as that of the fixed point set of τ, i.e. Ad(K)⋅X. Its image is the convex polytope with vertices the image of the fixed point set of T on Ad(G)⋅X, i.e. the points w(X) for w in W = N_K(T)/C_K(T).

References

• Atiyah, M. F. (1982), "Convexity and commuting Hamiltonians", Bull. London Math. Soc., 14: 1–15
• Duistermaat, J. J. (1983), "Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution", Trans. Amer. Math. Soc., 275: 417–429
• Guillemin, V.; Sternberg, S. (1982), "Convexity properties of the moment mapping", Invent. Math., 67: 491–513
• Helgason, Sigurdur (1984), Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press, pp. 473–476, ISBN 0-12-338301-3
• Hilgert, Joachim; Hofmann, Karl Heinrich; Lawson, Jimmie D. (1989), Lie groups, convex cones, and semigroups, Oxford Mathematical Monographs, Oxford University Press, ISBN 0-19-853569-4
• Heckman, G. J. (1982), "Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups", Invent. Math., 67: 333–356
• Horn, Alfred (1954), "Doubly stochastic matrices and the diagonal of a rotation matrix", Amer. J. Math., 76: 620–630
• Kostant, Bertram (1973), "On convexity, the Weyl group and the Iwasawa decomposition", Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 6: 413–455, ISSN 0012-9593, MR0364552
• Schur, I. (1923), "Uber eine Klasse von Mittelbildungen mit Anwendungen auf der Determinanten Theorie", Sitzungsberichte der Berliner Mathematischen Gesellschaft, 22: 9–20
• Thompson, Colin J. (1972), "Inequalities and partial orders on matrix spaces", Indiana Univ. Math. J., 21: 469–480
• Wildberger, N. J. (1993), "Diagonalization in compact Lie algebras and a new proof of a theorem of Kostant", Proc. Amer. Math. Soc., 119: 649–655
• Ziegler, François (1992), "On the Kostant convexity theorem", Proc. Amer. Math. Soc., 115: 1111–1113