Kostant's convexity theorem

Template:Short description In mathematics, Kostant's convexity theorem, introduced by Template:Harvs, can be used to derive Lie-theoretical extensions of the Golden–Thompson inequality and the Schur–Horn theorem for Hermitian matrices.

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 Template:Harvtxt, Template:Harvtxt and Template:Harvtxt 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.

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 $𝔨$ and $𝔱$. Let P be the orthogonal projection of $𝔨$ onto $𝔱$ for some Ad-invariant inner product on $𝔨$. Then for X in $𝔱$, 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 $𝔤$ and $𝔨$ be their Lie algebras and let σ also denote the corresponding involution of $𝔤$. Let $𝔭$ be the −1 eigenspace of σ and let $𝔞$ be a maximal Abelian subspace. Let Q be the orthogonal projection of $𝔭$ onto $𝔞$ for some Ad(K)-invariant inner product on $𝔭$. Then for X in $𝔞$, Q(Ad(K)⋅X) is the convex polytope with vertices the w(X) where w runs over the restricted Weyl group (the normalizer of $𝔞$ 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 for symmetric spaces is given in Template:Harvtxt. There is an elementary proof just for compact Lie groups using similar ideas, due to Template:Harvtxt: 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 $𝔨$ 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 particular the component in the α root space—its "α off-diagonal coordinate"—can be sent to 0. 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. Let m be the number of positive roots, half the dimension of K/T. Starting from an arbitrary Y₁ take the largest off-diagonal coordinate and send it to zero to get Y₂. Continue in this way, to get a sequence (Y_n). Then

$\Vert P^{\perp }(Y_{n+1}){\Vert }^2\le \left(\frac{m-1}{m}\right)\Vert P^{\perp }(Y_n){\Vert }^2.$

Thus P^⊥(Y_n) tends to 0 and

$\Vert P(Y_{n+1}-Y_n)\Vert \le \Vert P^{\perp }(Y_n)\Vert .$

Hence X_n = P(Y_n) is a Cauchy sequence, so tends to X in $𝔱$. Since Y_n = P(Y_n) ⊕ P^⊥(Y_n), Y_n tends to X. On the other hand, X_n lies on the line segment joining X_n+1 and its reflection in the root α. Thus X_n lies in the Weyl group polytope defined by X_n+1. These convex polytopes are thus increasing as n increases and hence P(Y) lies in the polytope for X. This can be repeated for each Z in the K-orbit of X. 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. (This matches a familiar picture from representation theory: if by duality X corresponds to a dominant weight λ, the other weights in the Weyl group polytope defined by λ are those appearing in the irreducible representation of K with highest weight λ. An argument with lowering operators shows that each such weight is linked by a chain to λ obtained by successively subtracting simple roots from λ.) 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).

Other proofs

Template:Harvtxt gave another proof of the convexity theorem for compact Lie groups, also presented in Template:Harvtxt. For compact groups, Template:Harvtxt and Template:Harvtxt showed that if M is a symplectic manifold with a Hamiltonian action of a torus T with Lie algebra $𝔱$, then the image of the moment map

$M\to {𝔱}^{*}$

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 ${𝔨}^{*}$, the moment map for T is the composition

$M\to {𝔨}^{*}\to {𝔱}^{*}.$

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

$\mathrm{A}\mathrm{d}(K)\cdot X\to 𝔱,$

the restriction of the orthogonal projection. Taking X in $𝔱$, 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. Template:Harvtxt gave a simplified direct version of the proof using moment maps.

Template:Harvtxt 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 $𝔞$, a torus in G. If X is in $𝔞$ as before the moment map yields the projection map

$\mathrm{A}\mathrm{d}(G)\cdot X\to 𝔞.$

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).

Further directions

In Template:Harvtxt 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 $i𝔱$. The more general version of Kostant's theorem has also been generalized to semisimple symmetric spaces by Template:Harvtxt. Template:Harvtxt gave a generalization for infinite-dimensional groups.

References

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