Seminar Sophus Lie 2 (1992) 13–20

Convexity properties of Graßmannians

J. Hilgert^{∗}

In this note we study the images of various moment maps Φ_{u}: G_{p}(C^{n}) → 𝔲^{∗}, where G_{p}(C^{n}) is the Graßmann-manifold of p-planes in C^{n} and 𝔲^{∗} the (real) dual of the Lie algebra 𝔲 of a compact group U acting on G_{p}(C^{n}) in a Hamiltonian way.

More precisely, if we let 𝔤_{C} = gl(n, C) act on C^{n} via the identity representation then the p-fold wedge product of this representation is a highest weight module V = ∧^{p} C^{n} of G = Gl(n, C) with highest weight vector

v_{0} = e_{1} ∧ … ∧ e_{p}

and highest weight

λ_{0}( diag(a_{1}, …, a_{n})) = ∑_{j=1}^{p} a_{j},

where {e_{1}, …, e_{n}} is the canonical basis for C^{n} and the set 𝔱_{C} of diagonal matrices in gl(n, C) is used as Cartan subalgebra. The ordering of the root system Δ(𝔤_{C} , 𝔱_{C} ) is chosen such that the corresponding Borel algebra consists of the upper triangular matrices in gl(n, C). The stabilizer of the line [v_{0}] ∈ IP(V) generated by v_{0} is the maximal parabolic subgroup

P_{C} = \{( \begin{matrix}A & B \\ 0 & D \end{matrix} ) ∈ Gl(n, C): A ∈ Gl(p, C), D ∈ Gl(q, C)\}

and the action of Gl(n, C) induces an embedding

G_{p}(C^{n}) ≅ G_{C} /P_{C} ≅ G_{C} · [v_{0}] → IP(V).

The choice of a Hermitean metric on V gives a Fubini-Study metric on IP(V) which in turn induces a Kähler metric on G_{p}(C^{n}). We choose the metric in such a way that U(n) ⊆ Gl(n, C) acts unitarily on V . Then U(n) also preserves the Kähler metric on G_{p}(C^{n}). This in particular means that the action of U(n) preserves the symplectic structure of G_{p}(C^{n}), which is given by the imaginary part of the Kähler metric. Let φ̇: 𝔲(n) → X (G_{p}(C^{n})) be the derived action which associates with a Lie algebra element X the vector field on G_{p}(C^{n}) whose flow is given by exp tX. Moreover we consider the map

ψ: C^{∞}(G_{p}(C^{n})) → X (G_{p}(C^{n})),

∗ supported by a DFG Heisenberg grant

which maps a function to the corresponding Hamiltonian vector field. Choose a linear map

κ: 𝔤 → C^{∞}(G_{p}(C^{n}))

such that ψ ∘ κ = φ̇ and define

Φ: G_{p}(C^{n}) → 𝔲(n)^{∗}

via

⟨Φ(x), X⟩ = (κ(X))(x) X ∈ 𝔲(n).

Then Φ is called a moment map if it is U(n)-equivariant. Note at this point that the construction of the moment map just given of course works for any group action G × M → M preserving the symplectic structure and not only for U(n) × G_{p}(C^{n}) → G_{p}(C^{n}). Using the fact that moment maps are characterized by ψ∘κ = φ̇ and the equivariance it is not hard to show that the moment map for the action of U(n) on G_{p}(C^{n}) is just the restriction to G_{p}(C^{n}) of the moment map for the action of U(n) on IP(V) which is given by

⟨Φ([v]), X⟩ = i\frac{(X · v | v)}{(v | v)}

or, when we introduce homogeneous coordinates with respect to an orthonormal basis, by

Φ((z_{0} : …: z_{N})) = \frac{i}{∑_{k=0}^{N} |z_{k}|^{2}} (z_{j}\overline{z}_{k})_{j,k=0,...,N},

where dim V = N + 1. Moreover, when we restrict the action to a subgroup U of U(n), then the corresponding moment map Φ_{𝔲} is Φ followed by the canonical projection 𝔲(n)^{∗} → 𝔲^{∗}, where 𝔲 = L(U) ⊆ 𝔲(n).

Thes moment maps and their images are closely related to subsemigroups of G = Gl(n, IR) which have been studied in the context of control theory and harmonic analysis. The common feature of the examples to be described below is the following:

Let G_{p}(IR^{n}) be the real Graßmannian of p-planes in IR^{n} which may be constructed as the set of real points of G_{p}(C^{n}) and H be a closed subgroup of Gl(n, IR). Further let U be a compact form of H, i.e., the complexifications 𝔥_{C} and 𝔲_{C} with 𝔥 = L(H) agree. Note that Gl(n, IR) acts transitively on G_{p}(IR^{n}). We assume that H has at least two open orbits in G_{p}(IR^{n}). Then our examples will satisfy the following conditions

(i) Φ_{𝔲}(G/P) is convex, where P = P_{C} ∩ Gl(n, IR).

(ii) Φ_{𝔲}(HP/P) is open in the linear span of Φ_{𝔲}(G/P) and dense in Φ_{𝔲}(G/P). (iii) Φ_{𝔲}|_{HP/P} is a diffeomorphism onto its image.

The semigroups alluded to earlier are given by

S = {g ∈ G: gHP/P ⊆ HP/P }.

We note in passing that the complex conjugation interchanges the Kähler metric with its complex conjugate, so it is antisymplectic, i.e., it interchanges the symplectic form with its negative. Moreover the moment map Φ anti-commutes with complex conjugation, i.e., it satisfies

Φ(\overline{x}) = −\overline{Φ(x)}.

In particular, the image of G/P is contained in i𝔤^{∗}.

Example 1. We let H be the subgroup of diagonal matrices and p = 1, then G_{p}(C^{n}) = IP(C^{n}). In this case the open H -orbits on G_{p}(IR^{n}) = IP(IR^{n}) are given by the sets

{(z_{0} : …: z_{n−1}) ∈ IP(IR^{n}): ε_{j}z_{j} > 0 j = 0, …, n − 1},

where the ε_{j} are fixed and equal to 1 or −1. All together we have 2^{n} open H - orbits and the statements (i) and (ii) are special cases of Duistermaat’s extension of the Atiyah-Guillemin-Sternberg convexity theorem which deals with fixed point sets of antisymplectic involutions. It is of course not necessary to appeal to this theorem here, since here 𝔲 = i𝔥 and the moment map is given by

Φ((z_{0} : …: z_{n−1})) = \frac{i}{∑_{k=0}^{N} |z_{k}|^{2}} (|z_{0}|^{2}, …, |z_{n−1}|^{2})

so that it is elementary to check that Φ_{𝔲}(IP(IR^{n})) = Φ_{𝔲}(IP(C^{n})) is an ndimensional simplex satisfying (i) and (ii).

The semigroup S in this case consists of all elements in Gl(n, IR) with non-negative entries (cf. [9] for more information).

Example 2. Let H = O(p, q), where n = p + q. We write elements of G and 𝔤 = gl(n, IR) as block matrices of the form

( \begin{matrix}A & B \\ C & D \end{matrix} )

with A of size p×p and D of size q×q. The usual Cartan involutions g ↦ (g^{⊤})^{−1} and X ↦ −X^{⊤} are both denoted by θ. Let τ be given by

g ↦ Jθ(g)J, X ↦ Jθ(X)J

on G and 𝔤 respectively, where

J = ( \begin{matrix}1_{p} & 0 \\ 0 & −1_{q} \end{matrix} ) .

Then H = G^{τ} = O(p, q) and G/H is a pseudo-Riemannian symmetric space. Let 𝔤 = 𝔥 + 𝔮 = 𝔨 + 𝔭 be the eigenspace decompositions of 𝔤 with respect to τ and θ for the eigenvalues 1 and −1. Then

𝔮 ∩ 𝔭 = \{( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ): A, D symmetric \},
𝔥 ∩ 𝔨 = \{( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ): A, D skew − symmetric \},
𝔥 ∩ 𝔭 = \{( \begin{matrix}0 & B \\ B^{⊤} & 0 \end{matrix} )\}

and

𝔞 = \{( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ): A, D diagonal \}

is maximal abelian in 𝔭 ∩ 𝔮, 𝔭 and 𝔮.

Note that 𝔲 := (𝔥 ∩ 𝔨) + i(𝔥 ∩ 𝔭) is a compact form of 𝔥 and the image of G/P under the corresponding moment map Φ_{𝔲} has to be contained in the dual of 𝔲 ∩ i𝔤 = i(𝔥 ∩ 𝔭). The key idea to prove the convexity of this image is to find a suitable maximal abelian subspace of i(𝔥 ∩ 𝔭), study the image of the moment map for the corresponding torus and then use equivariance properties to describe Φ_{𝔲}(G/P).

Consider the elements

X_{i} = ( \begin{matrix}0 & B_{i} \\ B_{i}^{⊤} & 0 \end{matrix} ) ,

where all entries of B_{i} are zero except for the entry at the i-th row and the i-th column which is 1. Then

𝔟 := ∑_{i=1}^{min(p,q)} IRX_{i}

is maximal abelian in 𝔥∩𝔭. But 𝔥 = (𝔨∩𝔥)+(𝔭∩𝔥) is the Cartan decomposition associated with θ|_{𝔥} so there belongs a Weyl group to the data (𝔥, 𝔟). This Weyl group which we denote by W_{𝔟} is isomorphic to the symmetric group on min(p, q) elements, extended by the multiplications with −1 in each coordinate with the obvious action by permutation of the entries.

If p ≤ q we write the elements of 𝔤 as block matrices with nine entries, where the diagonal entries are of size p × p, p × p and (q − p) × (q − p) (the case p > q can be treated analogously and we will not mention that case explicitly). Then conjugation by

\frac{1}{\sqrt{2}} ( \begin{matrix}−1_{p} & 1_{p} & 0 \\ 1_{p} & 1_{p} & 0 \\ 0 & 0 & 1_{(q_{p})} \end{matrix} )

is an automorphism of 𝔤_{C} which maps 𝔞_{C} onto a Cartan algebra 𝔱_{C}^{♯} containing 𝔟 via

( \begin{matrix}A & 0 & 0 \\ 0 & B & 0 \\ 0 & 0 & C \end{matrix} ) ↦ \frac{1}{2} ( \begin{matrix}B + A & B − A & 0 \\ B − A & B + A & 0 \\ 0 & 0 & 2C \end{matrix} ) .

There is another Weyl group, denoted by W^{♯}, associated to the pair (𝔤_{C} , 𝔱_{C}^{♯}). This Weyl group is of course isomorphic to the Weyl group for the pair (𝔤_{C} , 𝔱_{C} ) which is simply the permutation group on n letters. Each element of W_{𝔟} can be obtained from an element of W^{♯} by restriction to 𝔟. Let 𝔱^{♯} = 𝔲(n)∩𝔱_{C}^{♯} and

p: 𝔲(n)^{∗} → i𝔟^{∗}

then the crucial property of our Weyl groups is

(∗)  p(conv(W^{♯} · μ)) = conv(W_{𝔟} · p(μ))

for all μ ∈ i(𝔱^{♯})^{∗}.

Let T^{♯} and T_{𝔟} be the tori in U(n) with Lie algebras 𝔱^{♯} and i𝔟. Note the following inclusions:

𝔱^{♯} ⟵ i𝔟
↓  ↓
𝔲(n) ⟵ 𝔲.

We collect the corresponding moment maps in a commutative diagramm

\begin{aligned}& & (𝔱^{♯})^{∗} \\ & ↗ & ↑ Φ^{♯} & ↘ \\ 𝔲(n)^{∗} & ⟵^{Φ} & G_{p}(C^{n}) & ⟶^{Φ_{𝔟}} & i𝔟^{∗} \\ & ↘ & ↓ Φ_{𝔲} & ↗ \\ & & 𝔲^{∗} \end{aligned} .

As in the case of Example 1

Φ^{♯}(G_{C} /P_{C} ) = Φ^{♯}(G/P)

is a simplex and hence Φ_{𝔟}(G/P), being a projection of Φ^{♯}(G/P) is also a convex polytop. In fact, we may again apply Duistermaat’s results and this time [2], Proposition 4.2, implies

Φ_{𝔟}(G/P) = Φ_{𝔟}(\overline{B · [v_{0}]}),

where B is the analytic subgroup of G with Lie algebra 𝔟. Moreover the general theory says that Φ^{♯}(G/P) is the convex hull of the Weyl group orbit W^{♯} · λ^{♯} for any extremal weight λ^{♯} of the module V with respect to 𝔱_{C}^{♯} . But such a weight is given by

λ^{♯}: ( \begin{matrix}A & B & 0 \\ B & A & 0 \\ 0 & 0 & C \end{matrix} ) ↦ ∑_{i=1}^{p}(b_{i} − a_{i}) + ∑_{k=1}^{q} c_{k},

where A = diag(a_{1}, …, a_{p}), B = diag(b_{1}, …, b_{p}) and C = diag(c_{1}, ..., c_{q}). Applying the projection p and (∗) we find

(∗∗)  Φ_{𝔟}(G/P) = conv(W_{𝔟} · p(λ^{♯})).

But the projection p(λ^{♯}) = μ_{0} corresponds to the element

( \begin{matrix}0 & B_{0} \\ B_{0}^{⊤} & 0 \end{matrix} ) ∈ 𝔟

with B_{0} = ∑_{i=1}^{min(p,q)} B_{i}. Thus the image of Φ_{𝔟} is the cube which one obtains

from ( \begin{matrix}0 & B_{0} \\ B_{0}^{⊤} & 0 \end{matrix} ) upon the action of the Weyl group W_{𝔟}.

Now we are ready to use equivariance: Let K_{H} = O(p) × O(q) be the maximal compact subgroup of H contained in U(n) and consider the Cartan decomposition

H = K_{H}BK_{H}.

Note that K_{H} ⊆ P so that the U -equivariance of Φ implies

Φ(H · [v_{0}]) = Φ(K_{H}B · [v_{0}]) = Ad^{∗}(K_{H})Φ(B · [v_{0}]).

Analogously, we have

Φ_{𝔲}(H · [v_{0}]) = Ad^{∗}(K_{H})Φ_{𝔟}(B · [v_{0}]) = Ad^{∗}(K_{H})Φ_{𝔟}(B · ·[v_{0}]).

We set

D := Ad^{∗}(K_{H})Φ_{𝔟}(G/P) = Φ_{𝔲}(\overline{H · [v_{o}]}) ⊆ i(𝔥 ∩ 𝔭)^{∗}.

Let μ ∈ D. Then

(†)  conv( Ad^{∗}(K_{H})μ) ⊆ D.

In fact, let μ′ ∈ conv( Ad^{∗}(K_{H})μ). Then μ′ corresponds to an element of i(𝔥 ∩ 𝔭)^{∗} so that there exists an element k ∈ K_{H} such that Ad^{∗}(k)μ′ ∈ i𝔟^{∗}. Let [v] ∈ G/P be such that μ = Φ([v]) then

p(conv( Ad^{∗}(K_{H})μ) = conv(p(Φ(K_{H} · [v])))
⊆ conv(Φ_{𝔟}(K_{H} · [v]))
⊆ conv(Φ_{𝔟}(G/P))
= Φ_{𝔟}(G/P) ⊆ D

and hence

Ad^{∗}(k)μ′ = p(Ad^{∗}(k)μ′) ∈ D.

Thus we have

μ′ ∈ Ad^{∗}(K_{H})D = D

which proves (†). Now let μ ∈ Φ_{𝔟}(G/P) ⊆ i(𝔥 ∩ 𝔭)^{∗}, and k ∈ K_{H} be such that Ad^{∗}(k)μ ∈ i𝔟^{∗}. But then

Ad^{∗}(k)μ ∈ i𝔟^{∗} ∩ Φ_{𝔲}(G/P) ⊆ Φ_{𝔟}(G/P) ⊆ D

and hence μ ∈ Ad^{∗}(k)D ⊆ D. Thus we have

Φ_{𝔟}(G/P) = Φ_{𝔟}(\overline{H · [v_{0}]}) = D.

Using (†), (∗) and (∗∗) we calculate

D = Ad^{∗}(K_{H})Φ_{𝔟}(G/P)
= Ad^{∗}(K_{H})conv(W_{𝔟} · μ_{0})
⊆ conv(Ad^{∗}(K_{H}) · μ_{0})
⊆ D,

since the action of W_{𝔟} is induced by N_{K_{H}} (𝔟). Thus, in particular D is convex and we have shown (i) and the density part of (ii) for this example. We can even say more:

Φ_{𝔟}(G/P) = \{( \begin{matrix}0 & gBh \\ h^{−1}B^{⊤}g^{−1} & 0 \end{matrix} ): g ∈ O(p), h ∈ O(q), B ∈ Φ_{𝔟}(G/P)\}

which is linearly isomorphic to the set of linear contractive mappings from C^{q} to C^{p} with the usual norm. This can be seen by mapping the image of the linear operator gBh isometrically to C^{q} and then considering the polar decomposition.

In order to show (iii) we study Φ_{𝔲}|_{B·[v_{0}]} in more detail. Consider the elements

v_{i} := e_{1} ∧ … ∧ e_{i−1} ∧ e_{p+i} ∧ e_{i+1} ∧ … ∧ e_{p}

then

(exp tX_{i}) · v_{i} = (cosh t)v_{i} + (sinh t)v_{0}.

The map

(t_{1}, …, t_{min(p,q)}) ↦ ∏_{i=1}^{min(p,q)} exp t_{i}X_{i} · [v_{0}]

is a diffeomorphism IR^{min(p,q)} → B·[v_{0}]. If we use it as coordinate chart for B·[v_{0}] and further identify i𝔟^{∗} with IR^{min(p,q)} via the dual basis of {X_{1}, …, X_{min(p,q)}}, then Φ_{𝔲}|_{B·[v_{0}]} is given by

(t_{1}, …, t_{s}) ↦ (tanh \frac{t_{1}}{2} , …, tanh \frac{t_{s}}{2} ),

where s = min(p, q). In order to prove (iii) it now suffices to consider the mappings

φ_{1} = ((k, b) ↦ khP ): K_{H} × B → HP/P

and

φ_{2} = ((k, f) ↦ Ad^{∗}(k)f): K_{H} × i𝔟^{∗} → i(𝔥 ∩ 𝔭)^{∗}.

In fact,

φ_{2}(k, p_{𝔥} ∘ Φ(b · [v_{0}])) = Φ_{𝔥} ∘ φ_{1}(k, b · [v_{0}])

and the claim follows since the fibers of φ_{1} and of φ_{2} are both given by the Weyl group W_{𝔟}.

The semigroup S in this case is given by

\overline{H exp c_{max}H}

with

c_{max} = {diag(a_{1}, …, a_{n}): (∀j ≤ p < k) a_{j} ≥ a_{k}}

(cf. [6, §5.7).

Example 3. We conclude this report with an example which indicates that the simultaneous occurence of the properties (i)-(iii) and semigroups as shown in Examples 1 and 2 is not purely accidental. Let H = Gl(n, IR), and G = Gl(n, C) embedded as the diagonal in G_{C} = Gl(n, C) × Gl(n, \overline{C}), where \overline{C} indicates that on this copy of C we have the opposite complex structure. As maximal parabolic in 𝔤 we choose

P = \{( \begin{matrix}a & v^{⊤} \\ 0 & D \end{matrix} ) ∈ sl(n, C): a ∈ C, v ∈ C^{n}\}.

Then G/P = IP(C^{n}), G_{C} /P_{C} = IP(C^{n}) × IP(\overline{C}^{n}), and a compact form of H is U(n). The moment map Φ_{𝔲}: G_{C} /P_{C} → 𝔲(n)^{∗} associated with this action is

Φ_{𝔥}(z_{0} : …: z_{n−1}), (w_{0} : …: w_{n−1})) = \frac{i}{2}(\frac{z_{j}\overline{z}_{k}}{∑_{k=0}^{n−1} |z_{k}|^{2}} + \frac{\overline{w_{j}}w_{k}}{∑_{k=0}^{n−1} |w_{k}|^{2}} )_{j,k=0,...,n−1}

and, when restricted to G/P,

Φ_{𝔥}((z_{0} : …: z_{n−1})) = \frac{i}{∑_{k=0}^{n−1} |z_{k}|^{2}} (Re z_{j}\overline{z}_{k}).

Whereas the image is a disk for n = 2 it is non-convex for n = 3. This can be seen from a somewhat tedious comutation using affine coordinates on the projective plane. Note that for n ≥ 3 no proper subsemigroup S of Gl(n, C) is known which has non-empty interior and satisfies Gl(n, IR) ⊆ S.

References

[1] Atiyah, M.F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1–15.

[2] Duistermaat, J.J., Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution, Trans. Amer. Math. Soc. 275 (1983), 417–429.

[3] Gelfand, I.M., and V.V. Serganova, Combinatorial geometries and torus strata on homogeneous compact manifolds, Russ. Math. Surveys 42 (1987), 133–168.

[4] Hilgert, J., A convexity theorem for boundaries of ordered symmetric spaces, submitted.

[5] Neeb, K.H., A convexity theorem for semisimple symmetric spaces, submitted.

[6] Olafsson, G., Causal symmetric spaces, Mathematica Gottingensis 15 (1990).

[7] — , Symmetric spaces of Hermitean type, Differential Geometry and Appl. 1 (1991), 195–233.

[8] Ol’shanskiĭ, G. I., Convex cones in symmetric Lie algebras, Lie semigroups, and invariant causal (order) structures on pseudo-Riemannian symmetric spaces, Sov. Math. Dokl. 26 (1982), 97–101.

[9] San Martin, L., Two results on maximal subsemigroups of Lie groups, preprint, (1991).

J. Hilgert Mathematisches Institut Universität Erlangen Bismarckstr. 1 \frac{1}{2} D-8520 Erlangen, Germany

Received April 9, 1992 and in final form May 27, 1992