Kostant's convexity theorem

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 and let σ also denote the corresponding involution of \mathfrak{g}. 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 for symmetric spaces is given in . There is an elementary proof just for compact Lie groups using similar ideas, due to : 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 \mathfrak{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

:\displaystyle{|P^\perp(Y_{n+1})|^2\le \left({m-1\over m}\right)|P^\perp(Y_n)|^2.}

Thus P⊥(Y**n) tends to 0 and

:\displaystyle{|P(Y_{n+1}-Y_n)|\le |P^\perp(Y_n)|.}

Hence X**n = P(Y**n) is a Cauchy sequence, so tends to X in \mathfrak{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.See:

) 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

gave another proof of the convexity theorem for compact Lie groups, also presented in . For compact groups, and 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. gave a simplified direct version of the proof using moment maps.

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 = NK(T)/CK(T).

Further directions

In 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 i \mathfrak{t}. The more general version of Kostant's theorem has also been generalized to semisimple symmetric spaces by . and gave a generalization for infinite-dimensional groups.

Notes

References

• {{citation|first=James E.|last= Humphreys|title=Introduction to Lie Algebras and Representation Theory|publisher=Springer|edition=2nd| year=1997|series=Graduate texts in mathematics|volume=9|isbn=978-3540900535}}