Journal of Lie Theory Version of May 4, 1995 Volume 4 (1994) 185–235 C ◯1994 Heldermann Verlag

The geometry of nilpotent coadjoint orbits of convex type in hermitian Lie algebras

Joachim Hilgert, Karl-Hermann Neeb ^{∗}, and Bent Ørsted

Communicated by K. H. Hofmann

Introduction

The unitarizable highest weight representations of simple Lie groups have been classified in an algebraic manner more than ten years ago (cf. [34], [5], [16], [17], [6]). Nevertheless, no uniform analytic way of constructing these representations is known. Pioneering work in this direction has been done by Rossi and Vergne [33]. Other approaches are given for example in [7], [4], [18] and [27].

It is our goal to construct the unitary highest weight representations in an analytic fashion which is based on the orbit method. The coadjoint orbits we want to use are the orbits of convex type. These are the coadjoint orbits which are contained in a proper generating cone invariant under the coadjoint action. The basic idea is that unitary highest weight modules can be extended to contraction representations of certain complex semigroups having the original group as a kind of Shilov boundary. This leads to a one sided boundedness for the spectrum of the operators given by the derived representation. Following the idea that the spectrum of the representation has a close connection to the spectrum of its classical counterpart, i.e., the range of the Hamiltonian functions on the coadjoint orbit, one is led to the consideration of coadjoint orbits of convex type. This connection has been studied in detail in [24]. So far we are only able to carry out the program for nilpotent orbits. In that case it results in analytic realizations of all highest weight modules whose lowest K -types are one dimensional.

For the construction of the representations one proceeds in several steps. The first step is to use the detailed knowledge of the geometry of nilpotent orbits of convex type to construct a polarization of a suitable open dense subset of the orbit. This subset is in its own right a coadjoint orbit for a subgroup Q_{red} for

ISSN 0940–2268 / $2.50 ^{_{C} ◯} Heldermann Verlag which the polarization is invariant. The next step is to construct a Hilbert space and a representation of Q_{red} following the scheme of geometric quantization. This Hilbert space turns out to be a space of holomorphic functions on a Siegel domain of the second kind. It is a key observation that this Hilbert space can be reinterpreted as a space of holomorphic functions on a Siegel domain of the third kind on which the big group acts. The last step is to identify the resulting representation of Q_{red} as the restriction of a highest weight representation by comparing the reproducing kernels coming from the construction with the ones predicted by the general realization theory given in [25]. The first step is described in the present paper. The two others are dealt with in [14] and [15]. The paper is organized as follows:

∗ The first two authors would like to thank Odense University for its kind hospitality

I. Cayley homomorphisms of reductive Lie algebras II. (H_{1})-homomorphisms and invariant cones III. The classification of the nilpotent orbits of convex type IV. The fine structure of the Jacobson-Morosow parabolics V. Moment maps VI. The symplectic geometry of O_{k} and M_{k} VII. Examples

We explain the contents in a little more detail. In Section I we establish some refinements of well known results on 𝔰𝔩_{2} -triples which will be used to describe the geometry of conjugacy classes of real nilpotent elements. The main point is to keep track of the Cartan decompositions when studying homomorphisms between real reductive Lie algebras. Homomorphisms preserving the Cartan decomposition are called Cayley homomorphisms. In the case of quasihermitean Lie algebras, which are characterized by the fact that there exists a complex structure on 𝔭 induced by a central element of 𝔨, one can also consider (H_{1})-homomorphisms. These are Cayley homomorphisms preserving also the complex structure on the 𝔭-parts. Quasihermitean Lie algebras are singled out as the class of algebras which admit generating Ad-invariant convex cones with compactly embedded edge. The key results of Section II describe how invariant cones behave under (H_{1})-homomorphisms. We show that this description can be used to see that nilpotent elements of convex type correspond to 𝔰𝔩_{2} -triples which are at the same time (H_{1})-homomorphisms. The Sekiguchi correspondence between nilpotent orbits and K_{ℂ} -orbits in 𝔭_{ℂ} ([32]) then yields a bijection between nilpotent orbits of convex type and K_{ℂ} -orbits in 𝔭^{+} ∪ 𝔭^{−}.

In the third section we use the results of Section II to give a complete classification of the nilpotent orbits of convex type in a hermitean simple Lie algebra 𝔤. It turns out that there are exactly 2r + 1 such orbits, where r is the real rank of 𝔤 and each invariant cone in 𝔤 contains r + 1 such orbits. The ordering of the orbits lying in an invariant cone via inclusion of closures is linear. Using the Sekiguchi correspondence, we see that the nilpotent orbits lying in a fixed cone correspond to K_{ℂ} -orbits in 𝔭^{+} which have been classified by Muller, Rubenthaler and Schiffmann in [22].

Section IV gives a detailed account of the structure of the parabolic subalgebras associated to 𝔰𝔩_{2} -triples which arise from nilpotent orbits of convex type via the Jacobson-Morosow Theorem. These structural results are mostly based on the work of Koranyi and Wolf ([19], [36]). Our presentation follows Satake’s book [29] from which we extract and reorganize all the material we will need in this and the papers [14,15].

The structure of the Jacobson-Morosow parabolics then leads to a series of natural moment maps between various subalgebras. A closer inspection of these maps restricted to coadjoint orbits shows the existence of an open dense submanifold M of the nilpotent orbit which is a symplectic homogeneous space in its own right and is later shown to admit an invariant polarization. We show in particular that an invariant polarization for the whole group exists only in very special cases (when the orbit of convex type is maximal and G/K is a tube domain).

In Section VI we provide explicit information on the symplectic geometry of the nilpotent orbit and the manifold M. In particular we show that M has a symplectic potential and determine an invariant complex polarization for M. The final section contains various examples illustrating the theory.

As a standard reference we use [29]; several of the results there we reinterpret and strengthen to suit our point of view, in particular with our application to nilpotent orbits of convex type in mind.

I. Cayley homomorphisms of reductive Lie algebras

Definition I.1. Let (𝔤, θ) and (𝔤̃, θ̃) be two reductive Lie algebras with a fixed Cartan involutions. A homomorphism κ: 𝔤 → 𝔤̃ is called a Cayley homomorphism if κ ∘ θ = θ̃∘ κ.

Let 𝔤 = 𝔨 + 𝔭 and 𝔤̃ = 𝔨̃ + 𝔭̃ be the Cartan decompositions associated to θ and θ̃. Then κ: 𝔤 → 𝔤̃ is a Cayley homomorphism if and only if one has κ(𝔨) ⊆ 𝔨̃ and κ(𝔭) ⊆ 𝔭̃.

The following lemma which is a generalization of a theorem of Mostow ([21, Th. 6]) shows that the inclusion of a semisimple Lie algebra into a reductive one can always be made a Cayley inclusion if one chooses the Cartan involution on the bigger algebra in an appropriate way.

Recall that a subalgebra 𝔥 of a Lie algebra 𝔤 is called compactly embedded if the group of inner automorphisms of 𝔤 generated by e^{ad 𝔥} is relatively compact in Aut(𝔤).

Lemma I.2. Let 𝔤̃ be a semisimple Lie algebra and 𝔤 ⊆ 𝔤̃ a reductive subalgebra with the Cartan decomposition 𝔤 = 𝔨 + 𝔭 such that 𝔷(𝔤) is compactly embedded. Then there exists a Cartan decomposition 𝔤̃ = 𝔨̃ + 𝔭̃ with 𝔨 ⊆ 𝔨̃ and 𝔭 ⊆ 𝔭̃.

Proof. First we use Mostow’s Theorem ([21, Th. 6]) to see that there exists a Cartan decomposition 𝔤̃ = 𝔨̃ + 𝔭̃ extending the Cartan decomposition of the commutator algebra 𝔤′ = [𝔤, 𝔤]. Let θ̃ denote the corresponding involution.

Set 𝔞 := 𝔷_{𝔤}([𝔤, 𝔤]). Then 𝔞 ⊆ 𝔤̃ is a θ̃-invariant, and hence reductive, subalgebra which intersects 𝔤 in 𝔷(𝔤). The orthogonal (w.r.t. the Killing form) projection p: 𝔤̃ → [𝔞, 𝔞] commutes with θ̃ and its restriction to 𝔞 is a surjective homomorphism with kernel 𝔷(𝔞). Therefore p(𝔷(𝔤)) is a compactly embedded abelian subalgebra of [𝔞, 𝔞]. Using [10, Ex. VI.A.8(ii)], we now find a Cartan decomposition 𝔞 = 𝔨_{𝔞} + 𝔭_{𝔞} such that 𝔷(𝔤) ⊆ p(𝔷(𝔤)) + 𝔷(𝔞) ⊆ 𝔨_{𝔞} and 𝔭_{𝔞} ⊆ [𝔞, 𝔞]. On the other hand we have the Cartan decomposition 𝔞 = (𝔨̃∩ 𝔞) + (𝔭̃∩ 𝔞). Since Cartan decompositions of 𝔞 are conjugate under inner automorphisms, there exists γ ∈ ⟨e^{ad 𝔞}⟩ with γ(𝔨_{a}) = 𝔨̃∩ 𝔞 and γ(𝔭_{𝔞}) = 𝔭̃∩ 𝔞. In particular this implies γ(𝔷(𝔤)) ⊆ 𝔨̃.

Now γ(𝔨) = γ(𝔷(𝔤)) + (𝔨 ∩ 𝔤′) ⊆ 𝔨̃ and γ(𝔭) = 𝔭 ⊆ 𝔭̃ since 𝔭 ⊆ [𝔤, 𝔤] shows that the Cartan decomposition of γ(𝔤) induced by that of 𝔤 is compatible with the Cartan decomposition of 𝔤̃. Hence 𝔤̃ = γ^{−1}(𝔨̃) + γ^{−1}(𝔭̃) is a Cartan decomposition of 𝔤̃ satisfying the requirements of the lemma.

In this paper we will primarily be concerned with Cayley inclusions of 𝔰𝔩(2, ℝ) into some reductive Lie algebra 𝔤. So let 𝔤_{0} := 𝔰𝔩(2, ℝ) and

H = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , X = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , Y = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ,

T = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , U = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) .

Then

[U, T] = 2H, [U, H] = −2T, [H, T] = 2U, [H, X] = 2X, [H, Y] = −2Y, [X, Y] = H.

Every triple (H, X, Y) of elements in a Lie algebra 𝔤 satisfying the above commutation relations is called an 𝔰𝔩_{2} -triple. We fix the Cartan decomposition 𝔤_{0} = 𝔨_{0} + 𝔭_{0} with 𝔨_{0} = ℝU and 𝔭_{0} = ℝT + ℝH on 𝔤_{0} . An 𝔰𝔩_{2} -triple on which the Cartan involution of 𝔤 agrees with this involution will be called a Cayley triple.

Given a reductive Lie algebra 𝔤 and an element X ∈ 𝔤 we call X nilpotent if it belongs to the commutator algebra [𝔤, 𝔤] and is nilpotent there (in the usual sense). Then a trivial extension of the Jacobson-Morosow Theorem (cf. [35, Prop. 1.3.5.3]), says that a non-zero element X in a reductive Lie algebra 𝔤 is nilpotent if and only if it is part of an 𝔰𝔩_{2} -triple (H, X, Y). Using Lemma I.2 and the fact that the Cartan involutions on 𝔤 are conjugate, we see that the adjoint orbit of a nilpotent element in 𝔤 always contains an element X which belongs to a Cayley triple (H, X, Y).

Fix a Cayley triple (H, X, Y) in 𝔤 = 𝔨+𝔭 and choose a maximal abelian subspace 𝔞 of 𝔭 containing H. Then we have a system Σ = Σ(𝔤, 𝔞) of restricted roots and there is a positive system Σ^{+} such that μ(H) ≥ 0 for all μ ∈ Σ^{+} . This means that we declare a Weyl chamber containing H in its closure to be the positive one. Let 𝔤^{μ} be the root spaces and set 𝔫 = ∑_{μ∈Σ^{+}} 𝔤^{μ}. Then [H, X] = 2X implies that

X ∈ ∑_{μ(H)=2} 𝔤^{μ} ⊆ 𝔫.

Recall that the restriction of the adjoint representation to 𝔰 = ℝH + ℝX + ℝY is completely reducible and each irreducible summand is determined by a highest weight vector which is annihilated by X and an eigenvector for H. The corresponding eigenvalue is a non-negative integer and gives the isomorphy class of the simple 𝔰-module. Accordingly we decompose 𝔤 = 𝔤^{[0]} + 𝔤^{[1]} + … into isotypic 𝔰-submodules. In particular we see that ad H is diagonalizable with integer eigenvalues. Thus the action of H on 𝔤 gives a grading of 𝔤 via

𝔤(ad H; k) := {Z ∈ 𝔤: [H, Z] = kZ}, k ∈ ℤ.

We set

𝔮 := ∑_{j≥0} 𝔤(ad H; j), 𝔩 := 𝔤(ad H; 0), and 𝔲 := ∑_{j>0} 𝔤(ad H; j).

Then we have

𝔮 = ∑_{μ(H)≥0} 𝔤^{μ}, 𝔩 = ∑_{μ(H)=0} 𝔤^{μ}, and 𝔲 = ∑_{μ(H)>0} 𝔤^{μ},

and therefore 𝔮 is a parabolic subalgebra of 𝔤 with Levi decomposition 𝔮 = 𝔩 + 𝔲. We call it the Jacobson-Morosow parabolic associated to the Cayley triple (H, X, Y). Note here that a different choice of Σ^{+} with μ(H) ≥ 0 for μ ∈ Σ^{+} leads to the same parabolic.

Proposition I.3. Let (H, X, Y) be a Cayley triple, 𝔮 = 𝔩 + 𝔲 its Jacobson-Morosow parabolic and 𝔰 = ℝH + ℝX + ℝY . Then

(i) 𝔷_{𝔤}(H) ∩ 𝔷_{𝔤}(X) = 𝔷_{𝔤}(𝔰).

(ii) The centralizer 𝔷_{𝔤}(X) of X in 𝔤 is contained in 𝔮 and given by

𝔷_{𝔤}(X) = 𝔷_{𝔲}(X) + 𝔷_{𝔤}(𝔰) = ∑_{j≥0} 𝔷_{𝔤(ad H;j)}(X) = 𝔷_{𝔩}(X) + ∑_{j>0} 𝔷_{𝔤(ad H;j)}(X).

(iii) 𝔷_{𝔲}(X) = ∑_{j>0} 𝔷_{𝔤(ad H;j)}(X) = 𝔷_{𝔤}(X) ∩ [𝔤, X]. Proof. (i) Assume that Z ∈ 𝔷_{𝔤}(X)∩𝔷_{𝔤}(H). The general representation theory of 𝔰𝔩(2, ℝ) applied to the representation on 𝔤 then shows that also [Z, Y] = 0 so that Z ∈ 𝔷_{𝔤}(𝔰). (ii) We note first that obviously

𝔷_{𝔲}(X) + 𝔷_{𝔤}(𝔰) ⊆ 𝔷_{𝔤}(X) ∩ 𝔮.

For the converse we remark that the subalgebra 𝔷_{𝔤}(X) is invariant under ad H. Therefore we assume that Z ∈ 𝔷_{𝔤}(X) ∩ 𝔤(ad H; j). If j = 0, part (i) shows Z ∈ 𝔷_{𝔤}(𝔰). If j > 0 we have Z ∈ 𝔲 by definition. So it only remains to exclude the case j < 0. To that effect we only need to remark that 𝔷_{𝔤}(X) consists of highest weight vectors of the representation of 𝔰𝔩(2, ℝ) on 𝔤 with respect to the Cartan subalgebra ℝH and the corresponding weights are all non-negative. (iii) The first equality follows immediately from the proof of (ii) and the second from (ii) and the theory of 𝔰𝔩(2, ℝ)-representations applied to the representation on 𝔤.

Remark I.4. Suppose that 𝔤 is semisimple. For Z ∈ 𝔷_{𝔤}(X) the linear map ad X ad Z is nilpotent since ad X is is nilpotent and commutes with ad Z. Let B be the Killing form of 𝔤. Then B(X, Z) = tr(ad X ad Z) = 0 so that X and 𝔷_{𝔤}(X) are orthogonal under the Killing form. The non-degeneracy and the invariance of the Killing form show that 𝔷_{𝔤}(X) = [X, 𝔤]^{⊥}, where ^{⊥} denotes the orthogonal complement with respect to the Killing form. Thus we have

X ∈ 𝔷_{𝔲}(X) = 𝔷_{𝔤}(X) ∩ 𝔷_{𝔤}(X)^{⊥}.

Lemma I.5. Let (H, X, Y) be a Cayley triple in 𝔤 and 𝔮 = 𝔩 + 𝔲 its Jacobson-Morosow parabolic. Suppose that (H′, X, Y′) is another 𝔰𝔩_{2} -triple in 𝔤. Then there exists a Z ∈ 𝔷_{𝔲}(X) such that

e^{ad Z}.H = H′ and e^{ad Z}.Y = Y′.

Proof. [3, Ch. VIII, §11, no. 1, Lemme 4]

Remark I.6. It follows from Lemma I.5 that the Jacobson-Morosow parabolic of a Cayley triple (H, X, Y) only depends on the X -part of the triple.

Theorem I.7. Let G be a connected real reductive Lie group and (H, X, Y) be a Cayley triple in the Lie algebra 𝔤 of G. We set 𝔰 = ℝH + ℝX + ℝY and denote the Levi decomposition of the corresponding Jacobson-Morosow parabolic subgroup by Q = UL. Then

(i) Z_{U}(X) = exp (𝔷_{𝔲}(X)). (ii) Z_{G}(X) = Z_{U}(X)⋉Z_{G}(𝔰). (iii) Z_{G}(X) ⊆ Q.

Proof. (i) The connectedness of Z_{U}(X) follows from the fact that U is a unipotent algebraic group and that all algebraic subgroups of unipotent groups are connected. Since 𝔲 is nilpotent and hence U is exponential, (i) follows. (ii) It is clear that Z_{U}(X) and Z_{G}(𝔰) are contained in Z_{G}(X). The action of e^{ad ℝH} on Z_{U}(X) and Z_{G}(𝔰) shows that these two groups have trivial intersection. Moreover it is clear that Z_{U}(X) is normalized by Z_{G}(𝔰). Thus it only remains to show that Z_{G}(X) ⊆ Z_{U}(X)Z_{G}(𝔰). To this end we let g ∈ Z_{G}(X). Then (H, X, Y) and (Ad(g)H, X, Ad(g)Y) satisfy the hypotheses of Lemma I.5 so that one can find a Z ∈ 𝔷_{𝔲}(X) with e^{ad Z}H = Ad(g)H and e^{ad Z}Y = Ad(g)Y which implies exp(−Z)g ∈ Z_{G}(𝔰). This proves the claim. (iii) Let g ∈ Z_{G}(𝔰). Then g normalizes 𝔩 = ker(ad H) and also

𝔲 = ∑_{j>0} 𝔤(ad H; j).

Thus g normalizes 𝔮 and hence is in Q (cf. [35, Th. I.2.1.1]). Since Z_{U}(X) ⊆ U ⊆ Q holds by definition, the assertion follows from (ii).

Remark I.8. Theorem I.9 shows in particular that the adjoint orbit O_{X} = Ad(G).X of X has a bundle structure with compact base space G/Q and fibre isomorphic to Q/Z_{G}(X). More precisely we have

O_{X} ≅ G/Z_{G}(X) ≅ G ×_{Q} (Q/Z_{G}(X)),

where G×_{Q}(Q/Z_{G}(X)) is the fiber product with respect to the right action of Q on G via translation and the left action of Q on Q/Z_{G}(X) also via translation.

II. (H_{1})-homomorphisms and invariant cones

We call a reductive Lie algebra 𝔤 quasihermitean if it contains only compact or hermitean simple ideals. In this section 𝔤 always denotes a quasihermitean reductive Lie algebra and G a simply connected group with L(G) = 𝔤.

Definition II.1. (cf. [29]) Let 𝔤 be a reductive quasihermitean Lie algebra. An element H_{0} ∈ 𝔤 is called an H -element if 𝔷_{𝔤}(H_{0}) = ker ad H_{0} is a maximal compactly embedded subalgebra of 𝔤 and Spec(ad H_{0}) = {0, i, −i}. The pair (𝔤, H_{0}) is called a reductive Lie algebra of hermitean type.

Let (𝔤, H_{0}) be a reductive Lie algebra of hermitean type and set 𝔨 := ker ad H_{0} . Then 𝔭 := [H_{0}, 𝔤] is a uniquely determined complement for 𝔨 and 𝔤 = 𝔨+𝔭 is a Cartan decomposition. Note that 𝔷(𝔤) ⊆ 𝔨. In the complexification 𝔤_{ℂ}, the endomorphism ad H_{0} is diagonalizable and we obtain

𝔤_{ℂ} = 𝔭^{+} + 𝔨_{ℂ} + 𝔭^{−},

where 𝔭^{±} is the ±i-eigenspace of ad H_{0} .

Note that \frac{1}{2}U is an H -element in 𝔰𝔩(2, ℝ). In view of this example we should rather speak of U -elements, but we stick to Satake’s notation since we will have to refer to [29] on various occasions.

Remark II.2. Let 𝔤 = 𝔷(𝔤) ⊕ 𝔤_{0} ⊕ 𝔤_{1} ⊕ … ⊕ 𝔤_{k}, where 𝔷(𝔤) is the center, 𝔤_{0} is the maximal compact semisimple ideal, and 𝔤_{1}, …, 𝔤_{k} are the non-compact simple ideals. Then an element H_{z}+H_{0}+∑_{j=1}^{k} H_{j} is an H -element if and only if H_{0} = 0 and H_{j} is an H -element in 𝔤_{j}. It follows in particular that the number of H -elements associated to a fixed Cartan decomposition in the commutator algebra [𝔤, 𝔤] is 2^{k} since every simple hermitean Lie algebra contains exactly two H -elements associated to a fixed Cartan decomposition.

We recall some basic facts about invariant cones (cf. [11, Ch. III] and [23]): As was mentioned in the introduction, quasihermitean reductive Lie algebras are precisely the class of reductive Lie algebras that admit invariant cones having the maximal compact ideal as largest vector subspace. More precisely, closed convex cones W with non-empty interior, invariant under inner automorphisms and such that W ∩ (−W) is the largest compact ideal in the algebra. Quasihermitean Lie algebras admit compactly embedded Cartan subalgebras.

Fix a quasihermitean Lie algebra 𝔤 and a compactly embedded Cartan subalgebra 𝔱 in 𝔤. Then there exists a unique Cartan involution θ on 𝔤 such that 𝔱 is contained in the corresponding maximal compactly embedded subalgebra 𝔨 of 𝔤. Let Δ = Δ(𝔤_{ℂ}, 𝔱_{ℂ}) be the root system associated to the pair (𝔤_{ℂ}, 𝔱_{ℂ}). A root α is called compact if the corresponding rootspace 𝔤_{ℂ}^{α} is contained in 𝔨_{ℂ} and non-compact otherwise. We write Δ_{k} for the set of compact roots and Δ_{p} for the set of non-compact roots. A positive system Δ^{+} of the root system Δ is called 𝔨-adapted, if the set Δ_{p}^{+} of positive non-compact roots is invariant under the Weyl group W_{𝔨} of the pair (𝔨, 𝔱). Such positive systems always exist. Given a 𝔨-adapted positive system we define two convex cones in 𝔱 via

C_{min} = cone\{i[\overline{X}, X]: X ∈ 𝔤_{ℂ}^{α}, α ∈ Δ_{p}^{+}\},

where cone(E) denotes the smallest closed convex cone containing E, and

C_{max} = {X ∈ 𝔱: (∀α ∈ Δ_{p}^{+}) iα(X) ≥ 0}.

Then C_{min} is actually contained in C_{max} . Invariant cones in 𝔤 are uniquely determined by their intersections with 𝔱 and each W_{𝔨} -invariant cone between C_{min} and C_{max} occurs as such an intersection. In particular, there is a minimal invariant cone W_{min} with C_{min} = W_{min} ∩ 𝔱 and a maximal invariant cone W_{max} with C_{max} = W_{max} ∩ 𝔱.

Proposition II.3. Let 𝔥 be a quasihermitean reductive Lie algebra.

(i) For each H -element H_{0} ∈ 𝔤 there exists one and only one maximal invariant cone W_{max}(H_{0}) containing H_{0} .

(ii) Each maximal invariant cone contains an H -element. Proof. (i) Let 𝔱 be a compactly embedded Cartan algebra containing H_{0} . Then we have 𝔨 = 𝔷_{𝔤}(H_{0}) and α(H_{0}) ∈ {i, −i} for all α ∈ Δ_{p}. Therefore we find a 𝔨-adapted positive system with iα(H_{0}) > 0 for all α ∈ Δ_{p}^{+} . This means that H_{0} ∈ int C_{max} ⊆ int W_{max} . The uniqueness follows from the fact that the interiors of two different maximal invariant cones are disjoint. (ii) It suffices to show that C_{max} contains an H -element. But that is an immediate consequence of the structure of quasihermitean reductive Lie algebras (cf. [11, III]).

Remark II.4.

(i) The choice of a maximal cone and the choice of an H -element are equivalent once one has fixed a compactly embedded Cartan algebra 𝔱. More precisely, we find that there is a unique H -element in [𝔤, 𝔤] in each maximal invariant cone contained in 𝔱. In particular a reductive quasihermitean Lie algebra containing k simple hermitean ideals contains exactly 2^{k} maximal and minimal cones.

(ii) In the proof of Proposition II.3 we used that for non-compact quasihermitean Lie algebras the choice of a 𝔨-adapted positive system also implies the choice of an H -element H_{0} ∈ 𝔱 (modulo the center) by asking that iα(H_{0}) > 0 for all α ∈ Δ_{p}^{+} . Conversely Δ_{p}^{+} for a 𝔨-adapted positive system is fixed by the choice of an H -element in 𝔱.

(iii) The definitions show that C_{min} contains the corresponding H -element

whenever it is contained in the commutator algebra of 𝔤. (iv) The choice of a maximal cone determines the projection of H_{0} onto the

commutator algebra.

We will write W_{max}(H_{0}) for the maximal cone containing H_{0} if we want to stress the relation with H_{0} . Similarly we write W_{min}(H_{0}) for the corresponding minimal cone.

Note here that we have compiled various examples in Section VII.

Definition II.5. (cf. [29]) Let (𝔤, H_{0}) and (𝔤̃, H̃_{0}) be two reductive Lie algebras of hermitean type.

(a) A homomorphism κ: 𝔤 → 𝔤̃ is called an (H_{1})-homomorphism if

κ ∘ ad H_{0} = ad H̃_{0} ∘ κ.

(b) A homomorphism κ: 𝔤 → 𝔤̃ is called an (H_{2})-homomorphism if κ(H_{0}) = H̃_{0} . It is clear that this implies in particular that κ is an (H_{1})-homomorphism.

Remark II.6. (a) Note that the (H_{1})-condition is equivalent to the condition that the complex linear extension κ: 𝔤_{ℂ} → 𝔤̃_{ℂ} satisfies κ(𝔨_{ℂ}) ⊆ 𝔨̃_{ℂ} and κ(𝔭^{±}) ⊆ 𝔭̃^{±}. It is also equivalent to the condition that κ is a Cayley homomorphism with the additional property that κ |_{𝔭}: 𝔭 → 𝔭̃ is complex linear with respect to the complex structures J = ad H_{0} |_{𝔭} and J̃ = ad H̃_{0} |_{𝔭}.

(b) Compositions of (H_{1})-homomorphisms yields (H_{1})-homomorphisms.

Proposition II.7. Let (𝔤, H_{0}) and (𝔤̃, H̃_{0}) be two reductive Lie algebras of hermitean type and κ: 𝔤 → 𝔤̃ an (H_{1})-homomorphism. Then κ(W_{min}(H_{0})) ⊆ W_{min}(H̃_{0}).

Proof. Since H̃_{0} ∈ 𝔷(𝔨̃), we may choose compactly embedded Cartan algebras 𝔱 and 𝔱̃ of 𝔤 and 𝔤̃ with H_{0} ∈ 𝔱 and H̃_{0} ∈ 𝔱̃ such that κ(𝔱) ⊆ 𝔱̃. Let p_{𝔱}: 𝔤 → 𝔱 resp. p_{𝔱}: 𝔤̃ → 𝔱̃ denote the orthogonal projection onto 𝔱 resp. 𝔱̃. Then

C_{min}(H_{0}) = cone (\{p_{𝔱}(i[\overline{X}, X]): X ∈ 𝔭^{+}\})

and

C_{min}(H̃_{0}) = cone (\{p_{𝔱}(i[\overline{X}, X]): X ∈ 𝔭̃^{+}\}).

Therefore κ(𝔱) ⊆ 𝔱̃ and κ(𝔭^{+}) ⊆ 𝔭̃^{+} imply κ(C_{min}(H_{0})) ⊆ C_{min}(H̃_{0}) and hence the claim.

Lemma II.8. Let 𝔤 and 𝔤̃ be two reductive quasihermitean Lie algebras of hermitean type with H -elements H_{0} and H̃_{0} and κ: 𝔤 → 𝔤̃ a Lie algebra homomorphism. Write p_{𝔷}, p_{𝔤_{n}}, and p_{𝔤′} for the projection onto the center, the sum 𝔤̃_{n} of all non-compact simple ideals, and the commutator algebra of 𝔤̃. Then the following are equivalent:

(1) κ is an (H_{1})-homomorphism.

(2) p_{𝔤′} ∘ κ is an (H_{1})-homomorphism with respect to the H -element

p_{𝔤′}(H̃_{0}) ∈ 𝔤̃′.

(3) p_{𝔤_{n}}∘κ is an (H_{1})-homomorphism with respect to the H -element p_{𝔤_{n}}(H̃_{0})

of 𝔤̃_{n}. For (H_{2})-homomorphisms the conclusions (1) ⇒ (2) ⇒ (3) hold. Proof. (1) ⇒ (2) ⇒ (3): Suppose that κ is an (H_{1})-homomorphism. Then (2) follows from the fact that the projection p_{𝔤′} is an (H_{2})-homomorphism and (3) from the fact that the projection p_{𝔤_{n}} |_{𝔤′}: 𝔤̃′ → 𝔤̃_{n} is an (H_{2})-homomorphism. The same argument applies if κ is an (H_{2})-homomorphism.

(3) ⇒ (1): Let 𝔤̃_{c} be the sum of all compact simple ideals in 𝔤̃. For X ∈ 𝔤 we write κ(X) = Y_{1} + Y_{2} with Y_{1} ∈ 𝔤̃_{c} and Y_{2} ∈ 𝔤̃_{n}. Then (3) implies

[H̃_{0}, κ(X)] = [H̃_{0}, Y_{2}] = p_{𝔤_{n}}(κ([H_{0}, X])).

On the other hand [H_{0}, X] ⊆ [𝔤, 𝔤] is a split element, which shows that p_{𝔷}(κ([H_{0}, X])) = 0 and p_{𝔤_{c}}(κ([H_{0}, X])) = 0. Therefore [H̃_{0}, κ(X)] = κ([H_{0}, X]), i.e., κ is an (H_{1})-homomorphism.

If W is a closed convex cone in a finite dimensional vector space, we write H(W) := W ∩ (−W) for the edge of the cone W.

Proposition II.9. Let 𝔤 and 𝔤̃ be two reductive quasihermitean Lie algebras and κ: 𝔤 → 𝔤̃ a Lie algebra homomorphism. Consider the following conditions:

(1) There exists a maximal invariant cone W̃_{max} ⊆ 𝔤̃ such that κ^{−1}(W̃_{max}) is generating in 𝔤 and κ(𝔷(𝔤)) is compactly embedded.

(2) There exist H -elements H_{0} ∈ 𝔤 and H̃_{0} ∈ 𝔤̃ such that κ is an (H_{1})-

homomorphism and W̃_{max} = W_{max}(H̃_{0}). Then the implication (1) ⇒ (2) holds and (2) ⇒ (1) holds if 𝔤 is semisimple without compact factors or if κ is an (H_{2})-homomorphism. Proof. Before we prove both implications, we make some reductions. Let W̃_{max} ⊆ 𝔤̃ be the maximal cone associated to an H -element H̃_{0} ∈ 𝔤̃_{n} and set W := κ^{−1}(W̃_{max}). Let κ̃ := p_{𝔤_{n}} ∘ κ, where p_{𝔤_{n}} is the projection onto the

semisimple ideal 𝔤̃_{n}. Then κ^{−1}(W̃_{max}) = κ^{−1}(W̃_{max}) and κ(𝔷(𝔤)) is compactly embedded in 𝔤 if and only if κ̃(𝔷(𝔤)) is compactly embedded in 𝔤_{n}. According to Lemma II.8, we may therefore w.l.o.g. assume that 𝔤̃ = 𝔤̃_{n} is semisimple and a sum of hermitean simple ideals. Note that this implies in particular that the cones W̃_{max} are pointed.

It is clear that ker κ ⊆ H(W). Further the fact that W̃_{max} is pointed yields that ker κ = H(W). Write 𝔤 = ker κ ⊕ 𝔟, where 𝔟 is a complementary ideal. Then W = ker κ+(W ∩𝔟) is generating if and only if W ∩𝔟 is generating. Moreover κ(𝔷(𝔟)) = κ(𝔷(𝔤)) and κ is an (H_{1})-homomorphism if and only this holds for κ |_{𝔟}. Hence it suffices to assume that κ is injective, i.e., that 𝔤 is a subalgebra of 𝔤̃.

(1) ⇒ (2): Let W̃_{max} be as in (1). We assume that 𝔤 ⊆ 𝔤̃ = 𝔤̃_{n}. Then (1) means that the cone W = 𝔤 ∩ W̃_{max} is a pointed generating invariant cone in the reductive Lie algebra 𝔤. According to the theory of invariant cones (cf. [23, III]), there exists an H -element H_{0} in 𝔤 such that

(II.1)  W_{min}(H_{0}) ⊆ W ⊆ W_{max}(H_{0}).

Let 𝔨 := ker ad H_{0} and θ the associated Cartan involution of 𝔤. Since κ is injective and, according to our assumption, the image of 𝔷(𝔤) is compactly embedded, it follows from Lemma I.2 that there exists a Cartan involution θ̃ of 𝔤̃ extending θ.

We conclude in particular that 𝔨 ⊆ 𝔨̃. Therefore we find a compactly embedded Cartan algebra 𝔱 of 𝔨 which is contained in a compactly embedded Cartan algebra 𝔱̃ of 𝔨̃. We choose 𝔨-adapted positive systems Δ_{p}^{+} and Δ̃_{p}^{+} such that 𝔭^{+} and 𝔭̃^{+} are the sum of the non-compact positive root spaces. Then (II.1) implies that

C̃_{max} = W̃_{max} ∩ 𝔱 = W ∩ 𝔱 ⊆ W_{max} ∩ 𝔱 = C_{max}.

If α ∈ Δ̃_{p}^{+} is a positive non-compact root, then α|_{𝔱} is zero or a non-compact root and iα is non-negative on the generating cone C̃_{max} ∩ 𝔱. Since no non-compact root changes its sign on iC_{max} , it follows that iα is non-negative on C_{max} , i.e., that iα|_{𝔱} ∈ Δ_{p}^{+} ∪ {0}.

On the other hand 𝔭_{ℂ} ⊆ 𝔭̃_{ℂ}, so that every root α in Δ_{p}^{+} is the restriction of a root in Δ̃_{p}^{+} . According to the above argument, every root β ∈ Δ̃_{p} with α = β |_{𝔱} must be contained in Δ̃_{p}^{+} . Hence 𝔭^{+} ⊆ 𝔭̃^{+} and similarly 𝔭^{−} ⊆ 𝔭̃^{−}, whence κ is an (H_{1})-homomorphism.

(2) ⇒ (1): Suppose that (𝔤, H_{0}) → (𝔤̃, H̃_{0}) is an (H_{1})-homomorphism. We may assume that 𝔤 ⊆ 𝔤̃ = 𝔤̃_{n}.

(a): Let us first assume that 𝔤 is semisimple. Then 𝔷(𝔤) = {0} and we only have to show that W = 𝔤 ∩ W̃_{max} is generating. Let 𝔱 ⊆ 𝔨 be a compactly embedded Cartan algebra of 𝔤. Since 𝔱 ⊆ 𝔨̃, there exists a compactly embedded Cartan algebra 𝔱̃ of 𝔤̃ contained in 𝔨̃ and containing 𝔱. It is clear that each element of the root system Δ of 𝔤 may be obtained as the restriction of an element in Δ̃, the root system of 𝔤̃. The (H_{1})-condition implies that Δ_{p}^{+} is obtained by restricting elements of Δ̃_{p}^{+} . Therefore C_{min} ⊆ C̃_{min} ⊆ C̃_{max} yields

W_{min} = \overline{Ad(G).C_{min}} ⊆ 𝔤 ∩ W̃_{max} = W.

Since 𝔤 is semisimple without compact factors, the cone W_{min} is generating, so that W is also generating.

(b): Now we assume that κ in an (H_{2})-homomorphism and that 𝔤 is not necessarily semisimple. Then κ(H_{0}) = H̃_{0} ∈ int W̃_{max} because 𝔤̃ = 𝔤̃_{n}. Therefore H_{0} ∈ int κ^{−1}(W̃_{max}) and it follows in particular that the cone κ^{−1}(W̃_{max}) is generating. That κ(𝔷(𝔤)) ⊆ 𝔨̃ follows from the assumption that κ is an (H_{1})-homomorphism.

Theorem II.10. Let 𝔤 be a quasihermitean semisimple Lie algebra and W_{max} ⊆ 𝔤 a maximal invariant cone. Then the following are equivalent:

(1) X is a nilpotent element contained in W_{max} .

(2) There exists an H -element H_{0} in 𝔤 with W_{max} = W_{max}(H_{0}) and a Cayley triple (H, X, Y) defining an (H_{1})-homomorphism (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0}).

Proof. (1) ⇒ (2): First we use the Jacobson-Morosow Theorem (cf. [35, Prop. 1.3.5.3]) to find an 𝔰𝔩_{2} -triple (H, X, Y). Let κ: 𝔰𝔩(2, ℝ) → 𝔤 denote the corresponding inclusion homomorphism and 𝔰 its image. Then κ^{−1}(W_{max}) = W_{max}∩𝔰 contains the element X. Therefore this is a non-zero closed convex pointed invariant cone in 𝔰, hence W_{max} ∩ 𝔰 = W_{max}(\frac{1}{2}U). Now we use Proposition II.9 to see that there exists an H -element H̃_{0} ∈ 𝔤 and an H -element \frac{1}{2} Ũ ∈ 𝔰 such that the inclusion of 𝔰 is an (H_{1})-homomorphism, W_{max} ∩ 𝔰 = W_{max}(\frac{1}{2} Ũ), and W_{max} = W_{max}(H̃_{0}). Since all H -elements in W_{max} ∩ 𝔰 are conjugate under inner automorphisms, there exists γ ∈ ⟨e^{ad 𝔰}⟩ with γ(Ũ) = U. Then H_{0} := γ(H̃_{0}) is H -element in 𝔤 with the same maximal cone and (𝔰, \frac{1}{2}U) → (𝔤, H_{0}) is an (H_{1})-homomorphism.

(2) ⇒ (1): It is clear that X is nilpotent and X ∈ W_{min} ⊆ W_{max} follows from Proposition II.7.

In view of [32, Th. I.9], one has a bijection between the set of all nilpotent orbits in 𝔤 and the set of all nilpotent K_{ℂ} -orbits in 𝔭_{ℂ}, where K_{ℂ} = ⟨e^{ad 𝔨_{ℂ}}⟩. More explicitly, this correspondence is obtained by assigning to a nilpotent element X belonging to the Cayley triple (H, X, Y) the K_{ℂ} -orbit of the element

X̃ = \frac{1}{2}(X + Y − iH) ∈ 𝔭_{ℂ}.

We will refer to this bijection as the Sekiguchi correspondence. We identify the orbits in 𝔭_{ℂ} corresponding to the nilpotent orbits in W_{max} .

Theorem II.11. For a nilpotent orbit O_{X} ⊆ 𝔤 we have:

(i) O_{X} is of convex type if and only if O_{X} is contained in 𝔭^{+} ∪ 𝔭^{−}.

(ii) O_{X} is contained in W_{max} if and only if O_{X} is contained in 𝔭^{+} .

Proof. Since the orbit O_{−X} corresponds to the orbit of \frac{1}{2}(−X − Y − iH) =

−\overline{X̃}, it follows that X̃ ∈ 𝔭^{+} is equivalent to (−X) ∈ 𝔭^{−}. Therefore it suffices to prove (ii).

If X is contained in W_{max} , then we use Theorem II.10 to obtain a Cayley-triple defining an (H_{1})-homomorphism κ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0}). Put 𝔤_{1} := 𝔰𝔩(2, ℝ). Then 𝔭_{1}^{+} = ℂ(X +Y −iH) and since the complex linear extension of κ maps 𝔭_{1}^{+} into 𝔭^{+} (cf. Remark II.6), it follows that X̃ ∈ 𝔭^{+} .

If, conversely, X̃ ∈ 𝔭^{+} , then \overline{X̃} ∈ 𝔭^{−}, so that the complex linear extension of κ maps 𝔭_{1}^{±} into 𝔭^{±}. Using Remark II.6, we see that κ is an (H_{1})-homomorphism (𝔤_{1}, \frac{1}{2}U) → (𝔤, H_{0}).

For later applications we provide a proof of Exercice II.8.1 in [29].

Lemma II.12. Let ρ: (𝔤, H_{0}) → (𝔤̃, H̃_{0}) be an (H_{1})-homomorphism of semisimple Lie algebras of hermitean type. Then ρ is an (H_{1})-homomorphism with respect to the H -element Ad(g).H̃_{0} , g ∈ G̃ if and only if g ∈ Z_{G}(ρ(𝔤))K̃.

Proof. First let g ∈ Z_{G}(ρ(𝔤))K̃ and g = xk, where k ∈ K̃ and x ∈ Z_{G}(ρ(𝔤)). Let H′_{0} := Ad(g).H̃_{0} = Ad(x).H̃_{0} . Then the corresponding Cartan decomposition is given by

𝔨′ := Ad(x).𝔨̃ and 𝔭′ := Ad(x).𝔭̃.

Since Ad(x) centralizes ρ(𝔤), we have ρ(𝔨) ⊆ 𝔨′ and ρ(𝔭) ⊆ 𝔭′. Moreover

ad ( Ad(g).H̃_{0}) ∘ ρ = ad ( Ad(x).H̃_{0}) ∘ ρ
= Ad(x) ∘ ad H̃_{0} ∘ Ad(x)^{−1} ∘ ρ
= Ad(x) ∘ ad H̃_{0} ∘ ρ
= Ad(x) ∘ ρ ∘ ad H_{0}
= ρ ∘ ad H_{0},

i.e., ρ is an (H_{1})-homomorphism with respect to H_{0} and Ad(g).H̃_{0} . If, conversely,

ad ( Ad(g).H̃_{0}) ∘ ρ = ρ ∘ ad H_{0},

then we may w.l.o.g. assume that g = exp Y with Y ∈ 𝔭̃ since G̃ = exp(𝔭̃)K̃. Then

ad ((e^{ad Y}).H̃_{0}) ∘ ρ = ρ ∘ ad H_{0}.

This leads to ρ(𝔨) ⊆ e^{ad Y} .𝔨̃, hence to e^{− ad Y} .ρ(𝔨) ⊆ 𝔨̃, so that [35, Lemma 1.1.3.7] implies that [Y, ρ(𝔨)] = {0}. Similarly e^{− ad Y} .ρ(𝔭) ⊆ 𝔭̃ implies [Y, ρ(𝔭)] = {0}, whence Y ∈ 𝔷_{𝔤}(ρ(𝔤)). This completes the proof.

III. The classification of nilpotent orbits of convex type

In this part we introduce the subject proper of this paper, the nilpotent orbits of convex type. We restrict our attention to simple hermitean Lie algebras so that we can identify 𝔤 and its dual 𝔤^{∗} via the Killing form B. Note right away that the more general considerations from Section I come to bear even here when we have to study various subalgebras.

Definition III.1. Let 𝔤 be a finite dimensional Lie algebra. A linear functional ω ∈ 𝔤^{∗} is said to be of convex type if the coadjoint orbit O_{ω} = Ad^{∗}(G).ω is contained in a pointed closed convex cone.

From now on we fix a simple hermitean Lie algebra 𝔤 and identify 𝔤 and 𝔤^{∗} via the Cartan Killing form of 𝔤.

As in Section I we let 𝔤_{0} := 𝔰𝔩(2, ℝ) and fix the elements H, X, Y, T, U. Moreover, we shall freely use the notation from Section I. Recall also that (𝔰𝔩(2, ℝ), \frac{1}{2}U) is a simple Lie algebra of hermitean type. In this case we have 𝔭^{±} = ℂ(T ∓ iH) and the invariant cone W_{max} = W_{min} containing U contains X in its boundary.

We have seen in Theorem II.10 how one can sharpen the Jacobson-Morozow theorem for nilpotent elements of convex type. This result shows that every ray ℝ^{+}X contained in the nilpotent orbit O_{X} in the boundary of the invariant cone W_{max} can be obtained as

lim_{t→∞} e^{t ad H}(ℝ^{+}U) = ℝ^{+}X,

where (H, X, Y) is an 𝔰𝔩_{2} -triple and U = X −Y . Moreover, this can be arranged in such a way that U ∈ 𝔨, H, T ∈ 𝔭 holds for a Cartan decomposition of 𝔤 and T = −JH = −[H_{0}, H] with respect to the complex structure J = ad H_{0} |_{𝔭} on 𝔭.

Definition III.2. Let 𝔤 and 𝔤̃ be Lie algebras. Two Lie algebra homomorphisms ρ_{1}, ρ_{2}: 𝔤 → 𝔤̃ are said to be orthogonal if the images commute with each other. In this case we can define the homomorphism

ρ_{1} + ρ_{2}: 𝔤 → 𝔤̃, X ↦ ρ_{1}(X) + ρ_{2}(X)

which is called the commutative sum of ρ_{1} and ρ_{2} .

Note that the commutative sum of two (H_{1})-homomorphisms between reductive Lie algebras of hermitean type is again an (H_{1})-homomorphism with respect to the same H -elements.

Lemma III.3. Let 𝔞 ⊆ 𝔭 be a maximal abelian subspace. Then 𝔰 := 𝔞 + J𝔞 + [𝔞, J𝔞] ≅ 𝔰𝔩(2, ℝ)^{r}, where r = dim 𝔞 is the real rank of 𝔤. Moreover 𝔰 can be obtained as the range of an (H_{1})-homomorphism 𝔰𝔩(2, ℝ)^{r} → 𝔤.

Proof. Using a system of r strongly orthogonal roots (cf. [29, p.109]), one obtains r orthogonal (H_{1})-homomorphisms κ_{1}, …, κ_{r}: 𝔰𝔩(2, ℝ) → 𝔤 such that 𝔞 = ∑_{j=1}^{r} ℝκ_{j}(H). Then

J𝔞 = [H_{0}, 𝔞] = ∑_{j=1}^{r} ℝκ_{j}([U, H]) = ∑_{j=1}^{r} ℝκ_{j}(T)

and

[𝔞, J𝔞] = ∑_{j=1}^{r} ℝκ_{j}(U).

Therefore the assertion follows from the fact that 𝔰 is the range of the (H_{1})- homomorphism κ_{1} × … × κ_{r}: 𝔰𝔩(2, ℝ)^{r} → 𝔤.

The crucial conclusion we draw from the preceding lemma is the following.

Proposition III.4. If κ: 𝔰𝔩(2, ℝ) → 𝔤 is an (H_{1})-homomorphism, then there exists an (H_{1})-homomorphism

ρ: 𝔰𝔩(2, ℝ) → 𝔰𝔩(2, ℝ)^{r}

and an (H_{1})-homomorphism κ̃: 𝔰𝔩(2, ℝ)^{r} → 𝔤 such that κ = κ̃ ∘ ρ.

Proof. First we extend ℝκ(H) to a maximal abelian subspace 𝔞 of 𝔭_{0} . Then κ(𝔰𝔩(2, ℝ)) = ℝκ(H) + ℝJκ(H) + ℝκ(U) and 2U = [JH, H]. Let 𝔰 = 𝔞 + J𝔞 + [𝔞, J𝔞] be as in Lemma III.3. Then 𝔰 ≅ 𝔰𝔩(2, ℝ)^{r} and the inclusion 𝔰 → 𝔤 is an (H_{1})-homomorphism. It is also clear that the corestriction of κ, ρ: 𝔰𝔩(2, ℝ) → 𝔰 is an (H_{1})-homomorphism. This proves the assertion.

Corollary III.5. Every nilpotent orbit of convex type meets the range of the (H_{1})-homomorphism κ: 𝔰 → 𝔤.

Proof. In view of the fact that different subalgebras 𝔰 are conjugate under Ad(K), this follows by combining Proposition III.4 with Theorem II.10.

Remark III.6. Theorem II.10 and Corollary III.5 have an interesting interpretation in terms of hermitean symmetric spaces. Since (H_{1})-homomorphisms 𝔤 → 𝔤̃ correspond to strongly equivariant holomorphic maps of the hermitean symmetric spaces G/K → G̃/K̃ (cf. [29]), Theorem II.10 shows that nilpotent orbits of convex type correspond to embeddings of the complex unit disc into G/K and Corollary III.5 shows that every embedding of a unit disc can be extended to an embedding of an r-dimensional polydisc.

Proposition III.7. Let H_{j}, T_{j}, U_{j}, j = 1, …, r denote the basis elements of 𝔰𝔩(2, ℝ)^{r} corresponding to the basis elements in the simple 𝔰𝔩(2, ℝ)-factors and set U^{r} := ∑_{j=1}^{r} U_{j}.

(i) Let ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔰𝔩(2, ℝ)^{r}, \frac{1}{2}U^{r}) be an (H_{1})-homomorphism. Then

ρ(U) = ∑_{j=1}^{r} λ_{j}U_{j}

with λ_{j} ∈ {0, 1}.

(ii) Conversely, for each collection λ_{1}, …, λ_{r} with λ_{j} ∈ {0, 1} there is an (H_{1})-homomorphism ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔰𝔩(2, ℝ)^{r}, \frac{1}{2}U^{r}) with ρ(U) = ∑_{j=1}^{r} λ_{j}U_{j}.

(iii) Let 𝔱 := ∑_{j=1}^{r} ℝU_{j}. Then e^{ad 𝔱} acts on the set of (H_{1})-homomorphisms (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔰𝔩(2, ℝ)^{r}, \frac{1}{2}U^{r}) by conjugation and the conjugacy class of ρ is determined by the r-tuple (λ_{1}, … λ_{r}) ∈ {0, 1}^{r}. In particular there are 2^{r} e^{ad 𝔱} -conjugacy classes.

Proof. (i) According to Proposition II.7, ρ(U) = ∑_{j=1}^{r} λ_{j}U_{j} is contained in the cone W_{max}(\frac{1}{2}U^{r}), and therefore λ_{j} ≥ 0 for j = 1, …, r. This element leaves the subspace ℝH_{j} + ℝT_{j} invariant and acts on it with the spectrum {±λ_{j}2i}. Hence ρ(H) and ρ(T) are contained in ∑_{λ_{j}=1}(ℝH_{j} + ℝT_{j}) and therefore [H, T] ∈ ℝU yields that λ_{j} = 1 whenever λ_{j} ≠ 0.

(ii) Given ∑_{j=1}^{r} λ_{j}U_{j} we set

U′ = ∑_{j=1}^{r} λ_{j}U_{j}, H′ = ∑_{j=1}^{r} λ_{j}H_{j}, T′ = ∑_{j=1}^{r} λ_{j}T_{j}

and note that these three elements define an (H_{1})-homomorphism

ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔰𝔩(2, ℝ)^{r}, \frac{1}{2}U^{r}) , via ρ(U) = U′, ρ(H) = H′, ρ(T) = T′.

(iii) It follows from the above arguments that there exist exactly 2^{r} (H_{1})- homomorphisms

ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔰𝔩(2, ℝ)^{r}, \frac{1}{2}U^{r})

with ρ(H) ⊆ ∑_{j=1}^{r} ℝ^{+}H_{j}. Since the image of H under an (H_{1})-homomor-phism is necessarily is contained in ∑_{λ_{j}=1}(ℝH_{j} + ℝT_{j}) it follows that each (H_{1})-homomorphism is e^{ad 𝔱} conjugate to such an (H_{1})-homomorphism. On the other hand it is clear that (H_{1})-homomorphisms with different (λ_{1}, …, λ_{r}) cannot be e^{ad 𝔱} -conjugate. This implies the claim.

Proposition III.7 shows that we can define an invariant m(ρ) := |{j: λ_{j} = 1}|, called the multiplicity (cf. [29, p.111]) for any

ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔰𝔩(2, ℝ)^{r}, \frac{1}{2}U^{r}) .

Let κ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0}) be an (H_{1})-inclusion. Then, according to Proposition III.4, it factors over an (H_{1})-inclusion

ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔰𝔩(2, ℝ)^{r}, \frac{1}{2}U^{r}) .

Proposition III.8(i) below shows that we may define the multiplicity m(κ) to be the multiplicity of ρ.

Proposition III.8. Let (𝔤, H_{0}) be a simple Lie algebra of hermitean type with real rank r.

(i) The multiplicity m(ρ) of an (H_{1})-inclusion ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0}) does not depend on the choice of the (H_{1})-inclusion of 𝔰𝔩(2, ℝ)^{r} in 𝔤 over which it factors.

(ii) The Ad(G)-conjugacy classes of (H_{1})-homomorphisms

(𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0})

are classified by their multiplicity m(ρ). (iii) The Ad(G)-conjugacy classes of (H_{1})-inclusions

ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0})

coincide with the Ad(K)-conjugacy classes of (H_{1})-inclusions

ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0}).

Proof. (cf. [29, p.111]) (i): Let 𝔰 and 𝔰_{1} be two copies of 𝔰𝔩(2, ℝ)^{r} which are (H_{1})-inclusions over which ρ factors. Choose maximal abelian subspaces 𝔞 and 𝔞_{1} of 𝔰 ∩ 𝔭 and 𝔰_{1} ∩ 𝔭 containing ρ(H). The centralizer 𝔤_{0} := ker(ad ρ(H)) of ρ(H) in 𝔤 is θ-invariant and hence a reductive subalgebra. Since 𝔞, 𝔞_{1} ⊆ 𝔭 ∩ 𝔤_{0} and 𝔤 has real rank r, we see that 𝔞 and 𝔞_{1} are maximal abelian also in 𝔭 ∩ 𝔤_{0} . Thus we can find k ∈ exp(𝔨 ∩ 𝔤_{0}) such that Ad(k)𝔞 = 𝔞_{1} . According to Lemma III.3,

𝔰 = 𝔞 + J𝔞 + [𝔞, J𝔞], 𝔰_{1} = 𝔞_{1} + J𝔞_{1} + [𝔞_{1}, J𝔞_{1}]

and Ad(k) commutes with J. Therefore Ad(k).𝔰 = 𝔰_{1} . Note that k ∈ exp 𝔨 now implies that Ad(k): 𝔰 → 𝔰_{1} is an (H_{1})-isomorphism. Let 𝔦 be the ideal complement of the largest ideal of 𝔰 contained in 𝔤_{0} and similarly 𝔦_{1} for 𝔰_{1} . Then Ad(k).𝔦 = 𝔦_{1} . Since the dimension of 𝔦 is three times the multiplicity of ρ calculated using 𝔰, the claim follows. (ii), (iii): We show first that m(ρ) = m(Ad(g) ∘ ρ) holds whenever Ad(g) ∘ ρ is an (H_{1})-homomorphism. Let 𝔰 ≅ 𝔰𝔩(2, ℝ)^{r} be such that ρ factors over the embedding of 𝔰. Assume that Ad(g) ∘ ρ is an (H_{1})-homomorphism. Then ρ: 𝔰𝔩(2, ℝ) → 𝔤 is an (H_{1})-homomorphism with respect to \frac{1}{2}U and Ad(g)^{−1}.H_{0} . The ideal of 𝔰 generated by the image of ρ is moved by Ad(g) into the ideal generated by the image of Ad(g) ∘ ρ in Ad(g).𝔰. Hence the dimensions of these ideals are the same and we see that both homomorphisms have the same multiplicity.

In view of Lemma II.12, g ∈ KZ_{G}(ρ(𝔰𝔩(2, ℝ))). Let g = k exp(Y) with k ∈ K and [Y, im ρ] = {0}. Then Ad(g) ∘ ρ = Ad(k) ∘ ρ and Ad(k).𝔰 is a subalgebra such that Ad(k) ∘ ρ factors over Ad(k).𝔰. This proves (iii).

It now suffices to show that if m(ρ) = m(ρ′), there exists k ∈ K with Ad(k) ∘ ρ = ρ′. Since the maximal abelian subspaces 𝔞 ⊆ 𝔭 are conjugate under K, we may w.l.o.g. assume that both ρ and ρ′ factor over the inclusion of 𝔰 ≅ 𝔰𝔩(2, ℝ)^{r}. Let 𝔞^{+} ⊆ 𝔞 be a Weyl chamber. Then we may also assume that ρ(H), ρ′(H) ∈ 𝔞^{+} . Let γ_{1}, …, γ_{r} denote the strictly orthogonal roots and assume that γ_{1} ≥ γ_{2} ≥ … ≥ γ_{r} in the ordering corresponding to the Weyl chamber. Write ρ(H) = ∑_{j=1}^{r} λ_{j}H_{j}. Then λ_{1} ≥ λ_{2} ≥ … ≥ λ_{r} and λ_{j} ∈ {0, 1}. Therefore ρ(H) = ∑_{j=1}^{k} H_{j}, where k = m(ρ) is the multiplicity. Since m(ρ) = m(ρ′) by assumption, it follows that ρ = ρ′.

We turn to the classification of the nilpotent orbits lying in an invariant cone W_{max} . Note first that there exist exactly 2^{r} nilpotent orbits lying in the cone W_{max}(\frac{1}{2}U^{r}) in the Lie algebra 𝔰𝔩(2, ℝ)^{r}. If 𝔤 is simple hermitean, then it contains exactly 2 conjugacy classes of H -elements. If ρ: 𝔰𝔩(2, ℝ) → 𝔤 is an (H_{1})-homomorphism with respect to some H_{0} and H′_{0} is another H -element, then there exists g ∈ G with Ad(g).H_{0} = H′_{0} or Ad(g).H_{0} = −H′_{0} . Hence it suffices to consider those (H_{1})-homomorphisms which are associated to a fixed H -element H_{0} .

Theorem III.9. Let 𝔤 be a simple hermitean Lie algebra of real rank r and W_{max} a maximal invariant cone. Then W_{max} contains exactly r + 1 nilpotent orbits O_{0}, …, O_{r}. They satisfy

\{0\} = O_{0} ⊆ \overline{O}_{1} ⊆ … ⊆ \overline{O}_{r}

and they are contained in W_{min} . Proof. We fix an H -element H_{0} contained in W_{max} , a maximal abelian subspace 𝔞 ⊆ 𝔭 together with the associated subalgebra 𝔰 ≅ 𝔰𝔩(2, ℝ)^{r} (cf. Lemma III.3), and a Weyl chamber 𝔞^{+} ⊆ 𝔞. Suppose that γ_{1} ≥ … ≥ γ_{r} is the corresponding order on the strictly orthogonal roots corresponding to 𝔰.

Let O_{X} ⊆ W_{max} be a nilpotent orbit and ρ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0}) the corresponding (H_{1})-homomorphism (cf. Theorem II.10). We may w.l.o.g. assume that ρ(H) ∈ 𝔞^{+} . Then ρ(H) = ∑_{j=1}^{k} H_{j} and m(ρ) = k. In view of Proposition III.8, this proves that every nilpotent orbit O_{X} in W_{max} can be obtained by an 𝔰𝔩_{2} -triple (H, X, Y) as above. Hence there exist at most r + 1 different nilpotent orbits in W_{max} which are given by X^{k} := ∑_{j=1}^{k} X_{j}. Since X^{k−1} ∈ \overline{O_{X^{k}}} follows from the fact that the nilpotent orbits in 𝔰𝔩(2, ℝ) contain 0 in their closure, we see that O_{X^{k−1}} ⊆ \overline{O_{X^{k}}}.

It remains to show that O_{X^{k}} ≠ O_{X^{k−1}} . Suppose that both orbits coincide. Then it follows from Lemma I.5 that both are contained in conjugate 𝔰𝔩_{2} -triples. Then the associated (H_{1})-homomorphisms must have the same rank which is impossible since k ≠ k − 1 (Propositions III.7, III.8).

Remark III.10. A second way to prove Theorem III.9 is to show that the dimension of the centralizers of X^{0} = 0, X^{1}, …, X^{r} are mutually different. This can be done by considering the 𝔰-module structure of 𝔤 (cf. Section IV). We will come back to this context when we study parabolics associated to the nilpotent orbits.

Corollary III.11. A simple hermitean Lie algebra 𝔤 contains exactly 2r + 1 nilpotent orbits of convex type.

Remark III.12. We have observed in Theorem II.11 that the nilpotent orbits contained in W_{max} correspond to K_{ℂ} -orbits in 𝔭^{+} under the Sekiguchi correspondence. These orbits have been classified by Muller, Rubenthaler and Schiffmann in [22].

IV. The fine structure of the Jacobson-Morosow parabolics

In this section we study the special structure of Jacobson-Morosow parabolics associated to (H_{1})-embeddings of 𝔰𝔩(2, ℝ) into some simple Lie algebra of hermitean type. So fix an (H_{1})-embedding κ: (𝔰𝔩(2, ℝ), \frac{1}{2}U) → (𝔤, H_{0}) and write 𝔰^{κ} for its image in 𝔤. Further we write H_{κ}, X_{κ}, Y_{κ}, U_{κ}, T_{κ} for the respective images of H, X, Y, U, T under κ. For the 𝔰𝔩(2, ℝ)-module 𝔤 defined by κ we have the isotypic decomposition

(IV.1)  𝔤 = 𝔤^{[0]_{κ}} + 𝔤^{[1]_{κ}} + 𝔤^{[2]_{κ}}

(cf. [29, p.90], Section I). The decomposition (IV.1) shows that the Jacobson-Morosow parabolic of the Cayley-triple (H_{κ}, X_{κ}, Y_{κ}) is

(IV.2)  U_{κ} + V_{κ} + 𝔷_{𝔤}(H_{κ})

with U_{κ} = 𝔤(ad H_{κ}; 2) and V_{κ} = 𝔤(ad H_{κ}; 1). In particular it follows that U_{κ} and V_{κ} commute. The space V_{κ} may or may not be trivial.

Proposition IV.1. (cf. [29, p.112]) Suppose that V_{κ} ≠ {0}. Then (i) the representation of 𝔷_{𝔤}(H_{κ}) on V_{k} is irreducible and faithful, and (ii) [V_{κ}, V_{κ}] = U_{κ}.

The spaces 𝔤^{[0]_{κ}} and 𝔤^{[even]_{κ}} = 𝔤^{[0]_{κ}} + 𝔤^{[2]_{κ}} are subalgebras of hermitean type with H -elements H_{0}− \frac{1}{2}U_{κ} and H_{0} , respectively (cf. [29, p.92]). Decompose 𝔤^{[even]_{κ}} in ideals

𝔤^{[even]_{κ}} = ⨁_{j=0}^{s_{κ}} 𝔰_{j}^{(κ)}

with 𝔰_{0}^{(κ)} the largest compactly embedded ideal and 𝔰_{j}^{(κ)} simple non-compact for j > 0. Further we order the ideals in such a way that ⨁_{j=1}^{s_{o,κ}} 𝔰_{j}^{(κ)} ⊆ 𝔤^{[0]_{κ}} and 𝔰_{j}^{(κ)} ⊈ 𝔤^{[0]_{κ}} for j > s_{o,κ}. Now we set

(IV.3)  𝔤_{1,κ} = ∑_{j=0}^{s_{o,κ}} 𝔰_{j}^{(κ)}.

Proposition IV.2. (cf. [29, p.94])

(i) 𝔤_{1,κ} is of hermitean type with H -element H_{0} − \frac{1}{2}U_{κ}.

(ii) ∑_{j=s_{o,κ}+1}^{s_{κ}} 𝔰_{j}^{(κ)} is of hermitean type with H -element \frac{1}{2}U_{κ}.
Consider the partial Cayley transform c_{κ} = e^{i \frac{π}{4} ad T_{κ}}. It satisfies
(IV.4)  c_{k}(U_{κ}) = −iH_{κ}, c_{κ}(H_{κ}) = −iU_{κ}, c_{κ}(T_{κ}) = T_{κ}.

We set

(IV.5)  𝔤_{2,κ} = 𝔤 ∩ c_{κ}^{−1}((𝔨 ∩ ∑_{j=s_{o,κ}+1}^{s_{κ}} 𝔰_{j}^{(κ)})_{ℂ}).

Proposition IV.3. (cf. [29, p.96]) 𝔷_{𝔤}(H_{κ}) = 𝔤_{1,κ} ⊕ 𝔤_{2,κ}.

Proposition IV.4. (cf. [29, Th. III.2.3, Prop. III.4.4])

(i) The representation of 𝔤_{2,κ} on U_{κ} is faithful and irreducible.

(ii) 𝔤_{1,κ} acts trivially on U_{κ}.

(iii) Let G be a connected Lie group with Lie algebra 𝔤 and G_{2,κ} the analytic subgroup of G corresponding to 𝔤_{2,κ}. Then the orbit

Ω_{κ} := Ad(G_{2,κ}).X_{κ}

is an open convex cone in U_{κ} which is selfdual with respect to the euclidean inner product (· | ·) on 𝔤 defined by

(IV.6)  (X | Y) = −B(X, θY).

The group Ad_{U_{κ}}(G_{2,κ}) coincides with the identity component of the linear automorphism group G(Ω_{κ}) of the cone Ω_{κ}.

Proposition IV.5. (cf. [29, pp. 92, 102, 103, 112])

(i) θc_{κ}^{2} = c_{κ}^{−2}θ.

(ii) I_{κ} := ic_{κ}^{2}θ|_{V_{κ}} leaves V_{κ} invariant and defines a complex structure on

this space. (iii) ad(H_{0} − \frac{1}{2}U_{κ})|_{V_{κ}} = \frac{1}{2}I_{κ}. (iv) (v, v′) ↦ −\frac{i}{4}B(v, c_{κ}^{2}v′) = −\frac{1}{4}(X_{κ} | [v, v′]) defines a symplectic form on

V_{κ}.

(v) (𝔰𝔭(V_{κ}), \frac{1}{2}I_{κ}) is a simple Lie algebra of hermitean type.

(vi) The adjoint action of 𝔤 induces an (H_{2})-homomorphism 𝔤_{1,κ} → 𝔰𝔭(V_{κ}). (vii) The adjoint action of 𝔤 induces a homomorphism 𝔤_{2,κ} → 𝔤𝔩(V_{κ}, I_{κ}),

where 𝔤𝔩(V_{κ}, I_{κ}) denotes the complex linear endomorphisms of V_{κ} with respect to I_{κ}.

We define a skew symmetric bilinear form A_{u} on V_{κ} for any u ∈ U_{κ} via

(IV.7)  A_{u}(v, v′) = −\frac{1}{4}(u | [v, v′]).

and a bilinear map A: V_{κ} ×V_{κ} → U_{κ} by A(v, v′) = −\frac{1}{4}[v, v′]. Consider the group

(IV.8)  Sp(V_{κ}, A) = {g ∈ Gl(V_{κ}): A(g.v, g.v′) = A(v, v′)}

and its Lie algebra 𝔰𝔭(V_{κ}, A). Then it is clear that Sp(V_{κ}, A) ⊆ Sp(V_{κ}), where V_{κ} carries the symplectic form provided by Proposition IV.5(iv). But more is true:

Proposition IV.6. (cf. [29, p.132])

(i) 𝔰𝔭(V_{κ}, A) is of hermitean type with H -element \frac{1}{2}I_{κ}. (ii) The adjoint action of 𝔤 induces an (H_{2})-homomorphism

𝔤_{1,κ} → 𝔰𝔭(V_{κ}, A).

(iii) The real bilinear forms A_{u}I on V_{κ} given by (v, v′) ↦ (u | A(v, Iv′)) are symmetric for all u ∈ U_{κ} and positive definite for all u ∈ Ω_{κ}.

For the remainder of this section we assume that 𝔤 is simple hermitean of real rank r and the 𝔰𝔩_{2} -triples are the r ones occurring in the proof of Proposition III.7. More precisely, we fix a Cartan decomposition 𝔤 = 𝔨 + 𝔭, an H -element H_{0} ∈ 𝔷(𝔨), and a maximal abelian subspace 𝔞 ⊆ 𝔭 and then view 𝔞 + J𝔞 + [𝔞, J𝔞] as the range of an (H_{1})-homomorphism 𝔰𝔩(2, ℝ)^{r} → 𝔤, where J = ad(H_{0})|_{𝔭} (cf. Lemma III.3).

We write Σ for the set of restricted roots in 𝔞^{∗}. Then, according to a theorem of Moore, Σ is a not necessarily reduced root system of type (C_{r}) (G/K of tube type) or of type (BC_{r}) (cf. [29, p.110]).

Let γ_{1}, …, γ_{r} denote the strictly orthogonal roots. Then γ_{j} = 2ε_{j} and Σ is given by

Σ = {±2ε_{j}, ±(ε_{i} ± ε_{j}): 1 ≤ i < j ≤ r} type (C_{r})

and Σ = {±ε_{j}, ±2ε_{j}, ±(ε_{i} ± ε_{j}): 1 ≤ i < j ≤ r} type (BC_{r}).

As fundamental systems we choose

Υ = {μ_{1}, …, μ_{r}} := {ε_{1} − ε_{2}, …, ε_{r−1} − ε_{r}, 2ε_{r}} type (C_{r})

and

Υ = {μ_{1}, …, μ_{r}} := {ε_{1} − ε_{2}, …, ε_{r−1} − ε_{r}, ε_{r}} type (BC_{r}),

respectively. With respect to the ordering fixed by this choice of a fundamental system we have γ_{1} ≥ … ≥ γ_{r}.

Let H_{1}, …, H_{r} ∈ 𝔞 denote the coroots for the roots γ_{j}, i.e., γ_{j}(H_{i}) = 2δ_{ij}. The elements H_{j} span 𝔞. We have seen in the last section that the elements H^{k} := H_{1} + … + H_{k} belong to Cayley triples (H^{k}, X^{k}, Y^{k}) associated to the nilpotent orbits O_{X^{k}} of convex type, where X^{k} = X_{1} + … + X_{k}. Now we set 𝔰^{k} := ℝH^{k}+ℝX^{k}+ℝY^{k} and U^{k} := X^{k}−Y^{k}, T^{k} := X^{k}+Y^{k}. The corresponding (H_{1})-homomorphisms will be denoted by κ_{k}. To simplify the notation we will 𝔤^{[j]_{k}} instead of 𝔤^{[j]_{κ_{k}}} and similarly for 𝔤^{[even]} and other notations involving κ. In this situation we have a lot of additional information on the Jacobson-Morosow parabolic of (H^{k}, X^{k}, Y^{k}). We begin with a closer look at 𝔷_{𝔤}(H^{k}).

Lemma IV.7. Let μ ∈ Σ^{+} . Then μ(H^{k}) ∈ {0, 1, 2}. More precisely we have (i) μ(H^{k}) = 0 if and only if μ belongs to the following list:

\begin{matrix}ε_{i} − ε_{j} & i < j ≤ k, \\ ε_{i} − ε_{j} & k < i < j, \\ ε_{i} + ε_{j} & k < i < j, \\ 2ε_{i} & k < i, \\ ε_{i} & k < i. \end{matrix}

Here of course the last line only occurs if Σ is of type (BC_{r}). (ii) μ(H^{k}) = 1 if and only if it belongs to the following list:

ε_{i} ± ε_{j}, i ≤ k < j, ε_{i}, i ≤ k.

(iii) μ(H^{k}) = 2 if and only if it belongs to the following list:

ε_{i} + ε_{j}, i < j ≤ k, 2ε_{i}, i ≤ k.

Proof. Let 1 ≤ i < j ≤ r. Then we calculate

(ε_{i} − ε_{j})(H^{k}) = ∑_{s=1}^{k} δ_{is} − ∑_{s=1}^{k} δ_{js} = \{ \begin{matrix}1 & for i ≤ k < j \\ 0 & otherwise \end{matrix}

and

(ε_{i} + ε_{j})(H^{k}) = ∑_{s=1}^{k} δ_{is} + ∑_{s=1}^{k} δ_{js} = \{ \begin{matrix}0 & for k < i < j \\ 1 & for i ≤ k < j \\ 2 & for i < j ≤ k. \end{matrix}

Moreover, for 1 ≤ i ≤ r we have

ε_{i}(H^{k}) = ∑_{s=1}^{k} δ_{is} = \{ \begin{matrix}1 & for i ≤ k \\ 0 & otherwise. \end{matrix}

Now the claim follows by inspection.

Consider the subsets

(IV.9) Σ_{(k,±)} := Σ ∩ span{μ_{j} ∈ Υ: ±(k − j) < 0} Σ_{(k,j)} := {μ ∈ Σ: μ(H^{k}) = j}

of Σ. Note that Σ_{(k,+)} and Σ_{(k,−)} are the two root systems that one obtains from Σ by first deleting μ_{k} from the Dynkin diagram of Σ and then forming two root systems from the two remaining connected components.

Proposition IV.8.

(i) Σ_{(k,+)} is a root system of type (C_{(r−k)}) if Σ is of type (C_{r}). (ii) Σ_{(k,+)} is a root system of type (BC_{(r−k)}) if Σ is of type (BC_{r}). (iii) A system of fundamental roots for Σ_{(k,+)} is {μ_{k+1}, … μ_{r}}. (iv) Σ_{(k,−)} is a root system of type (A_{k−1}).

(v) A system of fundamental roots for Σ_{(k,−)} is {μ_{1}, … μ_{k−1}}.

(vi) (Σ_{(k,+)} + Σ_{(k,−)}) ∩ Σ = Ø.

(vii) (Σ_{(k,+)} + Σ_{(k,2)}) ∩ Σ = Ø.

(viii) Let μ ∈ Σ. Then μ(H^{k}) = 0 if and only if μ ∈ Σ_{(k,+)} ∪ Σ_{(k,−)} . Proof. An inspection of the list in Lemma IV.7 yields that

Σ_{(k,−)} = {±(ε_{i} − ε_{j}): i < j ≤ k}

which is a root system of type (A_{k−1}), whereas

Σ_{(k,+)} = {±2ε_{j}, ±(ε_{i} ± ε_{j}): k < i < j ≤ r} is of type (C_{(r−k)})

or

Σ_{(k,+)} = {±ε_{j}, ±2ε_{j}, ±(ε_{i} ± ε_{j}): k < i < j ≤ r} is of type (BC_{(r−k)})

depending on the type of Σ. The remaining claims now follow easily by inspection.

Remark IV.9. The dual base of {μ_{1}, …, μ_{r}} is {H^{1}, …, H^{r}} if Σ is of type (BC_{r}). It is \{H^{1}, …, H^{r−1}, \frac{1}{2}H^{r}\} if Σ is of type (C_{r}).

Set

𝔞_{(k,−)} := span{H_{1}, …, H_{k}} ∩ (H^{k})^{⊥} and 𝔞_{(k,+)} := span{H_{k+1}, …, H_{r}}.

Proposition IV.10.

(i) 𝔞_{(k,−)} = span{H^{k}, …, H^{r}}^{⊥}.

(ii) 𝔞_{(k,+)} = span{H^{1}, …, H^{k}}^{⊥}.

Proof. This follows from

span{H^{j}: k ≤ j ≤ r}^{⊥} = span{H_{k+1}, …, H_{r}, H^{k}}^{⊥}
= span{H_{j}: 1 ≤ j ≤ k} ∩ (H^{k})^{⊥}

and

span{H^{j}: 1 ≤ j ≤ k}^{⊥} = span{H_{j}: 1 ≤ j ≤ k}^{⊥}
= span{H_{j}: k + 1 ≤ j ≤ r}.

Proposition IV.11. Let μ ∈ Σ_{(k,±)} and 𝔪 = 𝔷_{𝔨}(𝔞). Then (i) μ(𝔞_{(k,∓)}) = {0}.

(ii) [𝔤^{μ}, 𝔤^{−μ}] ⊆ 𝔞_{(k,±)} + 𝔪.

Proof. (i) This follows immediately from Proposition IV.8, Remark IV.9 and Proposition IV.10. (ii) Let μ ∈ Σ_{(k,+)} , Z ∈ 𝔤^{μ}, Z′ ∈ 𝔤^{−μ} and B the Killing form on 𝔤. Then

B(H^{j}, [Z, Z′]) = B([H^{j}, Z], Z′) = μ(H^{j})B(Z, Z′)

which is zero for 1 ≤ j ≤ k. Thus

[𝔤^{μ}, 𝔤^{−μ}] ⊆ 𝔪 + span{H^{j}: 1 ≤ j ≤ k}^{⊥} = 𝔪 + 𝔞_{(k,+)}.

For μ ∈ Σ_{(k,−)} we argue similarly and find

[𝔤^{μ}, 𝔤^{−μ}] ⊆ 𝔪 + span{H^{j}: k ≤ j ≤ r}^{⊥} = 𝔪 + 𝔞_{(k,−)}.

Lemma IV.12. Let 𝔤_{(k,±)} be the subalgebra of 𝔤 generated by the root spaces 𝔤^{μ} with μ ∈ Σ_{(k,±)} .

(i) 𝔤_{(k,±)} is simple and θ-invariant with restricted root system Σ_{(k,±)} . (ii) 𝔤_{(k,+)} and 𝔤_{(k,−)} commute. In particular, being simple, they intersect

in {0}. (iii) 𝔤_{(k,±)} ∩ 𝔞 = 𝔞_{(k,±)} is maximal abelian in 𝔤_{(k,±)} ∩ 𝔭. (iv) We set 𝔨_{(k,±)} := 𝔨∩𝔤_{(k,±)} and 𝔫_{(k,±)} := 𝔫∩𝔤_{(k,±)} = ∑_{μ∈Σ_{(k,±)}^{+}} 𝔤^{μ}. Then

𝔤_{(k,±)} = 𝔨_{(k,±)} + 𝔞_{(k,±)} + 𝔫_{(k,±)}

is an Iwasawa decomposition.

Proof. (i) It is clear that 𝔤_{(k,±)} is θ-invariant and hence reductive. Therefore we can write

𝔤_{(k,±)} = ⨁_{j=0}^{s} 𝔥_{j}^{±}

with a compact factor 𝔥_{0}^{±} and non-compact θ-invariant simple factors 𝔥_{j}^{±}, j = 1, …, s. Then the restricted root system for 𝔤_{(k,±)} is the disjoint union of the restricted root systems for the 𝔥_{j}, j ≥ 1. On the other hand we know from Proposition IV.10 and Proposition IV.8 that

𝔤_{(k,±)} ⊆ 𝔪 + 𝔞_{(k,±)} + ∑_{μ∈Σ_{(k,±)}} 𝔤^{μ}.

Therefore the restricted root system for 𝔤_{(k,±)} is Σ_{(k,±)} . Thus there can only be one non-compact simple factor which we denote by 𝔥^{±}. But 𝔥^{±} contains all the root spaces 𝔤^{μ} with μ ∈ Σ_{(k,±)} . Thus it coincides with 𝔤_{(k,±)} . This proves (ii) It follows from Proposition IV.8 that the root spaces of 𝔤_{(k,+)} commute with the root spaces of 𝔤_{(k,−)} . Thus all of 𝔤_{(k,+)} commutes with 𝔤_{(k,−)} . (iii) Consider the algebra 𝔥̃^{±} := 𝔞 + 𝔤_{(k,±)} . It is θ-invariant, hence reductive, and 𝔞 is maximal abelian in 𝔥̃^{±} ∩ 𝔭. Since any maximal abelian subspace of 𝔥̃^{±} ∩ 𝔭 respects the decomposition into simple factors (plus the center), we see that 𝔞_{±} := 𝔞 ∩ 𝔤_{(k,±)} is maximal abelian in 𝔤_{(k,±)} ∩ 𝔭. From Proposition IV.11 we know that 𝔞_{±} ⊆ 𝔞_{(k,±)} . Conversely, the real rank of 𝔤_{(k,±)} coincides with the dimension of 𝔞_{(k,±)} . Thus 𝔞_{±} = 𝔞_{(k,±)} . (iv) This follows immediately from (i) and (iii).

Theorem IV.13. 𝔷_{𝔤}(H^{k}) = 𝔤̃_{1,k} ⊕ 𝔤̃_{2,k}, where 𝔤̃_{1,k} := 𝔤_{(k,+)} ⊕ 𝔪_{k}, 𝔤̃_{2,k} := ℝH^{k} ⊕ 𝔤_{(k,−)}, and 𝔪_{k} ⊆ 𝔪 = 𝔷_{𝔨}(𝔞).

Proof. We note first that 𝔷_{𝔤}(H^{k}) is θ-invariant hence reductive in 𝔤. Thus we may write

𝔷_{𝔤}(H^{k}) = ⨁_{j=0}^{s} 𝔥_{j}

with a compact factor 𝔥_{0} and non-compact θ-invariant simple factors 𝔥_{j}, j = 1, …, s. Then the restricted root system for 𝔷_{𝔤}(H^{k}) is the disjoint union of the restricted root systems for the 𝔥_{j}, j ≥ 1. On the other hand we know from Proposition IV.8 and Lemma IV.12 that

𝔷_{𝔤}(H^{k}) = ℝH^{k} ⊕ (𝔪 + (𝔤_{(k,+)} ⊕ 𝔤_{(k,−)})).

But 𝔪 leaves each root space 𝔤^{μ} invariant, so 𝔤_{(k,+)} and 𝔤_{(k,−)} are ideals in 𝔷_{𝔤}(H^{k}). Therefore the restricted root system for 𝔷_{𝔤}(H^{k}) is a disjoint union of Σ_{(k,+)} and Σ_{(k,−)} . Thus there can only be two non-compact simple factors and these have to be 𝔤_{(k,+)} and 𝔤_{(k,−)} . Now consider 𝔥_{0} . It is contained in 𝔷_{𝔤}(𝔞) since 𝔞 = ℝH^{k} + 𝔞_{(k,+)} + 𝔞_{(k,−)} , 𝔞_{(k,±)} ⊆ 𝔤_{(k,±)} and H^{k} is central in 𝔷_{𝔤}(H^{k}). Therefore the compact part 𝔪_{k} := 𝔨 ∩ 𝔥_{0} is contained in 𝔪. The non-compact part 𝔭 ∩ 𝔥_{0} is contained in 𝔞 and hence the claim follows.

We will show below (Remark IV.19) that 𝔤̃_{i,k} = 𝔤_{i,k}.

Proposition IV.14.

(i) 𝔤_{(k,+)} ⊆ 𝔤^{[0]_{k}} = 𝔷_{𝔤}(𝔰^{k}).

(ii) 𝔷_{𝔤}(H^{k}) ⊆ 𝔤^{[even]_{k}} (iii) H_{j} ∈ 𝔤^{[2]_{k}} for 1 ≤ j ≤ k.

Proof. (i) Since X_{j} ∈ 𝔤^{2ε_{j}} we have [X_{j}, 𝔤^{μ}] ⊆ 𝔤^{μ+2ε_{j}}. If μ ∈ Σ_{(k,+)} and 1 ≤ j ≤ k, then μ + 2ε_{j} ∉ Σ so that [X^{k}, 𝔤^{μ}] = 0, i.e., 𝔤^{μ} ⊆ 𝔷_{𝔤}(X^{k}). Therefore the algebra 𝔤_{(k,+)} which is generated by these root spaces is also contained in 𝔷_{𝔤}(X^{k}). Now Proposition I.3 implies the claim. (ii) This follows immediately from the definitions. (iii) H_{j} belongs to the 𝔰^{k} -submodule ℝH_{j} + ℝX_{j} + ℝY_{j} ⊆ 𝔤 which is equivalent to the adjoint representation of 𝔰^{k} and hence in 𝔤^{[2]_{k}}.

Proposition IV.15. 𝔤_{(k,+)} is an ideal in 𝔤^{[even]_{k}}. Proof. Note first that

𝔤^{[even]_{k}} = 𝔷_{𝔤}(H^{k}) + 𝔤(ad H^{k}; 2) + 𝔤(ad H^{k}; −2)

and 𝔤(ad H^{k}; −2) = θ𝔤(ad H^{k}; 2). Since 𝔤_{(k,+)} is a θ-invariant ideal in 𝔷_{𝔤}(H^{k}), it suffices to show that [𝔤_{(k,+)}, 𝔤(ad H^{k}; 2)] = {0}. But that follows from Proposition IV.8(vii) since

𝔤(ad H^{k}; 2) = ∑_{μ∈Σ_{(k,2)}} 𝔤^{μ}.

Recall from the beginning of this section that 𝔰_{0}^{(k)} is the maximal compactly embedded ideal in 𝔤^{[even]_{k}} and the 𝔰_{j}^{(k)} for j > 0 are non-compact simple ideals such that ⨁_{j=1}^{s_{o,k}} 𝔰_{j}^{(k)} ⊆ 𝔤^{[0]_{k}} and 𝔰_{j}^{(k)} ⊈ 𝔤^{[0]_{k}} for j > s_{o,k}. In particular this implies that the 𝔰_{j}^{(k)} for 1 ≤ j ≤ s_{o,k} are non-compact simple ideals of 𝔷_{𝔤}(H^{k}) which are contained in 𝔷_{𝔤}(𝔰^{k}). Thus Theorem IV.13, Proposition IV.14 and Proposition IV.15 show that s_{o,k} = 1 and 𝔰_{1}^{(k)} = 𝔤_{(k,+)} . We write 𝔰̃^{(k)} for ⨁_{j>1} 𝔰_{j}^{(k)} .

The following proposition extends Proposition IV.2.

Proposition IV.16. ([29, Prop. III.1.3, III.1.5]) The following pairs are reductive Lie algebras of hermitean type:

(i) (𝔷_{𝔤}(𝔰^{k}), H_{0} − \frac{1}{2}U^{k}).

(ii) (𝔤^{[even]_{k}}, H_{0}).

(iii) (𝔰̃^{(k)}, \frac{1}{2}U^{k}).

(iv) (𝔰_{0}^{(k)} ⊕ 𝔤_{(k,+)}, H_{0} − \frac{1}{2}U^{k}).

Remark IV.17.

(i) Let 𝔰̃_{0}^{(k)} := 𝔤^{[0]_{k}} ∩ 𝔰̃^{(k)} and note that 𝔰̃_{0}^{(k)} ⊆ 𝔷_{𝔰^{(k)}}(U^{k}). Then 𝔰_{0}^{(k)} is compactly embedded in 𝔤^{[even]_{k}}, hence by Proposition IV.16(iii) annihilated by H_{0} . Thus it is contained in 𝔨. (ii) We have the following decompositions

(a) 𝔤^{[0]_{k}} = 𝔰_{0}^{(k)} ⊕ 𝔤_{(k,+)} ⊕ 𝔰̃_{0}^{(k)} .

(b) 𝔤^{[even]_{k}} = 𝔰_{0}^{(k)} ⊕ 𝔤_{(k,+)} ⊕ 𝔰̃^{(k)} .

(c) 𝔰̃^{(k)} = 𝔰̃_{0}^{(k)} ⊕ 𝔤^{[2]_{k}}.

Proposition IV.18.

(i) 𝔷_{𝔤}(U^{k}) = 𝔰_{0}^{(k)} ⊕ 𝔤_{(k,+)} ⊕ (𝔨 ∩ 𝔰̃^{(k)}).

(ii) 𝔷_{𝔤_{ℂ}}(H^{k}) = (𝔥_{0}^{(k)} ⊕ 𝔤_{(k,+)})_{ℂ} ⊕ c_{k}^{−1}(𝔨 ∩ 𝔰̃^{(k)})_{ℂ}.

(iii) 𝔷_{𝔤}(H^{k}) = 𝔰_{0}^{(k)} ⊕ 𝔤_{(k,+)} ⊕ (𝔤 ∩ c_{k}^{−1}(𝔨 ∩ 𝔰̃^{(k)})_{ℂ}).

Proof. (i) In view of Remark IV.17, we only have to show that 𝔷_{𝔤}(U^{k}) is contained in 𝔤^{[even]_{k}}. But this is clear since U^{k} acts as an injective mapping on 𝔤^{[1]_{k}}. (ii) and (iii) are an immediate consequence of (i) and c_{k}(H^{k}) = −iU^{k}.

Remark IV.19. Now we can identify 𝔤̃_{1,k} and 𝔤̃_{2,k} in terms of our root data. We have 𝔤_{1,k} = 𝔰_{0}^{(k)} ⊕ 𝔤_{(k,+)} . It is shown in [29, p.113] that

𝔤_{2,k} = ℝH^{k} ⊕ 𝔤_{(k,−)} = 𝔤̃_{2,k}

(cf. Th. IV.13). Combining this with Theorem IV.13 and Proposition IV.18, we see that 𝔰_{0}^{(k)} = 𝔪_{k} and

𝔤_{1,k} = 𝔪_{k} ⊕ 𝔤_{(k,+)} = 𝔤̃_{1,k}.

Remark IV.20. 𝔷_{𝔤_{(k,−)}}(X^{k}) = 𝔨_{(k,−)} = 𝔨_{2,k} = 𝔰̃_{0}^{(k)} , where the first equality follows from [29, Th. III.2.3] and the second is [29, Lemma III.2.2].

In our special situation it is also easy to determine whether we can apply Proposition IV.1.

Proposition IV.21. (cf. [29, p.112]) V_{k} = {0} if and only if k = r and Σ is of type (C_{r}).

Proposition IV.22. Let 𝔮_{k} = 𝔩_{k} + 𝔲_{k} be the Jacobson-Morosow parabolic for the Cayley triple (H^{k}, X^{k}, Y^{k}) and denote the generalized Heisenberg algebra U_{k} + V_{k} by 𝔥_{k}. Then

(i) 𝔮_{k} is a maximal parabolic in 𝔤.

(ii) [U_{k}, V_{k} + U_{k} + 𝔤_{(k,+)}] ⊆ [U_{k}, 𝔥_{k} + 𝔤_{1,k}] = {0}.

Proof. (i) μ_{k} is the only root in Υ which does not vanish on H^{k}. (ii) Proposition IV.8, (IV.1), and Proposition IV.15.

Proposition IV.23.

(i) 𝔷_{𝔤}(X^{k}) = 𝔷_{𝔩_{k}}(X^{k}) + 𝔷_{𝔲_{k}}(X^{k}).

(ii) 𝔷_{𝔩_{k}}(X^{k}) = 𝔰_{0}^{(k)} ⊕ 𝔤_{(k,+)} ⊕ 𝔰̃_{0}^{(k)} = 𝔤_{1,k} ⊕ 𝔨_{2,k}.

(iii) 𝔷_{𝔲_{k}}(X^{k}) = 𝔲_{k}. Proof. (i) This claim follows from the fact that 𝔷_{𝔤}(X^{k}) is invariant under ad H^{k} and Proposition I.3. (ii) 𝔷_{𝔩_{k}}(X^{k}) = 𝔤_{1,k} ⊕𝔨_{2,k} by Proposition IV.4 and 𝔰_{0}^{(k)} ⊕𝔤_{(k,+)} ⊕𝔰̃_{0}^{(k)} = 𝔤_{1,k} ⊕𝔨_{2,k} follows from Remark IV.19 and Remark IV.20. (iii) This follows from Proposition IV.22.

Lemma IV.24. The map ad(X^{k}): θU_{k} → 𝔭_{2,k} is a bijection. Proof. The representation theory of 𝔰𝔩(2, ℝ) shows that ad(X^{k}) is injective on the eigenspaces of ad H^{k} for negative eigenvalues and in particular on θU_{k}.

Moreover, also by 𝔰𝔩(2, ℝ)-representation theory, we have [U_{k}, θU_{k}] ⊆ 𝔩_{k} and therefore

ad X^{k}(θU_{k}) ⊆ 𝔤^{[2]} ∩ 𝔩_{k} = 𝔭_{2,k}

(cf. Proposition IV.23).

Now the assertion follows from dim 𝔭_{2,k} = dim U_{k} which in turn follows from the fact that the map 𝔤_{2} → U_{k}, Y ↦ [Y, X^{k}] is surjective with kernel 𝔨_{2,k}.

Remark IV.25. The argument in the above proof even yields

𝔤^{[2]_{k}} = θU_{k} + 𝔭_{2,k} + U_{k}.

For later use we record some facts concerning the transformation properties of the various spaces we encountered so far under the Cayley transform.

Proposition IV.26. (cf. [29, p.95])

(i) 𝔨_{ℂ}(c_{k}^{2}; −1) ⊆ 𝔤_{ℂ}^{[2]_{k}}.

(ii) 𝔭_{ℂ}(c_{k}^{2}; −1) ⊆ 𝔤_{ℂ}^{[2]_{k}}.

(iii) c_{k} induces a bijection 𝔨_{ℂ}(c_{k}^{2}; 1) → 𝔨_{ℂ}(c_{k}^{2}; 1).

(iv) c_{k} induces a bijection 𝔭_{ℂ}(c_{k}^{2}; 1) → 𝔭_{ℂ}(c_{k}^{2}; 1).

(v) c_{k} induces a bijection 𝔨_{ℂ}(c_{k}^{2}; −1) → 𝔭_{ℂ}(c_{k}^{2}; −1).

Let I_{κ} denote the complex structure in V_{κ} (Proposition IV.5). Then we write V_{κ}^{±} for the ±i-eigenspaces of the complex linear extension of I_{κ} to (V_{κ})_{ℂ}.

Proposition IV.27.

(i) c_{k}((U_{k})_{ℂ}) = 𝔭^{+} ∩ 𝔤_{ℂ}^{[2]_{k}}.

(ii) c_{k}(V_{k}^{+}) = 𝔭^{+} ∩ 𝔤_{ℂ}^{[1]_{k}}.

(iii) c_{k}(𝔭_{1,k}^{+} + V_{k}^{+} + (U_{k})_{ℂ}) = c_{k}((𝔭_{ℂ}^{+} ∩ 𝔷_{𝔤_{ℂ}}(𝔰^{k})) + V_{k}^{+} + (U_{k})_{ℂ}) = 𝔭^{+} .

(iv) c_{k}(V_{k}^{−} + θV_{k}^{+}) = 𝔨_{ℂ} ∩ 𝔤_{ℂ}^{[1]_{k}}.

(v) c_{k}((𝔤_{2,k} ∩ 𝔭)_{ℂ}) = c_{k}(ℂH^{k} + (𝔭 ∩ 𝔤_{(k,−)})_{ℂ}) = 𝔨_{ℂ} ∩ 𝔤_{ℂ}^{[2]_{k}}.

(vi) c_{k}((𝔨_{1,k})_{ℂ} + (V_{k}^{−} + θV_{k}^{+}) + (𝔤_{2,k})_{ℂ}) = 𝔨_{ℂ}. Proof. (i) [29, p.97], (ii) [29, p.101], (iii) [29, p.104], (iv) [29, p.105], (v) [29, p.96], (vi) [29, p.104].

Remark IV.28. 𝔷_{𝔨}(𝔰^{k}) = 𝔨∩(𝔪_{k}⊕𝔤_{(k,+)}⊕𝔨_{(k,−)}) = 𝔨_{1,k}+𝔨_{2,k} = 𝔩∩𝔨, according to Remark IV.17 and Proposition IV.18. Thus we have

c_{k}^{−1}(𝔨_{ℂ}) = (𝔨_{(k,+)} ⊕ 𝔪_{k} ⊕ 𝔤_{(k,−)} ⊕ ℝH^{k})_{ℂ} + V_{k}^{−} + θV_{k}^{+}

(cf. [29, p.104]).

Proposition IV.29. 𝔷_{𝔤_{ℂ}}(X^{k}) ∩ c_{k}^{−1}(𝔨_{ℂ}) = 𝔷_{𝔨_{ℂ}}(𝔰^{k}) + V_{k}^{−}. Proof. According to Remark IV.28 and Proposition IV.23, we have

𝔷_{𝔤_{ℂ}}(X^{k}) ∩ c_{k}^{−1}(𝔨_{ℂ}) = (𝔨_{(k,+)} ⊕ 𝔪_{k} ⊕ 𝔨_{(k,−)})_{ℂ} + V_{k}^{−} = 𝔷_{𝔨_{ℂ}}(𝔰^{k}) + V_{k}^{−}.

V. Moment maps

In this section we fix k ∈ {1, …, r}, the corresponding nilpotent orbit O_{k} = Ad(G).X^{k} and all the other entities depending on k (cf. Section IV). We will be interested in the following chain of subalgebras:

(V.1)  U_{k} ⟶ 𝔥_{k} ⟶ 𝔮_{k,red} := 𝔥_{k}⋉𝔤_{2,k} ⟶ 𝔮_{k} = 𝔥_{k}⋉𝔩_{k} ⟶ 𝔤,

where 𝔥_{k} := U_{k} + V_{k} and the corresponding chain

(V.1′)  𝔤^{∗} \underrightarrow{Φ_{𝔮_{k}}} 𝔮_{k}^{∗} \underrightarrow{Φ_{𝔮_{k,red}}} 𝔮_{k,red}^{∗} \underrightarrow{Φ_{𝔥_{k}}} 𝔥_{k}^{∗} \underrightarrow{Φ_{U_{k}}} U_{k}^{∗}

of restriction maps. In order to make this system of moment maps and its encoded symplectic information available for our study of O_{k}, we transfer it to 𝔤 using the Killing form B and the Cartan involution θ. We start with a few general observations: For the moment let 𝔤 be a real semisimple Lie algebra with Cartan involution θ and Killing form B. Then 𝔤 has an inner product (· | ·) defined by

(X | Y) = −B(X, θY) ∀X, Y ∈ 𝔤.

Moreover we have two linear isomorphisms ψ: 𝔤 → 𝔤^{∗} and φ: 𝔤 → 𝔤^{∗} defined by

⟨ψ(X), Y ⟩ = B(X, Y) ∀X, Y ∈ 𝔤

and

⟨φ(X), Y ⟩ = (X | Y) ∀X, Y ∈ 𝔤.

Then φ = −ψ ∘ θ and ψ is G-equivariant, where G acts on 𝔤 via the adjoint and on 𝔤^{∗} via the coadjoint action. This shows that

(V.2)  (Ad(g)X | Y) = (X | Ad(θ(g)^{−1})Y) ∀X, Y ∈ 𝔤, g ∈ G.

This equation implies

(V.3)  Ad^{∗}(g) ∘ φ = φ ∘ Ad (θ(g)^{−1}).

Now let 𝔟 ⊆ 𝔤 be a subspace. Then we may view 𝔟^{∗} as a subspace of 𝔤^{∗} extending ξ ∈ 𝔟^{∗} to 𝔤 by 0 on

𝔟^{⊥} = {Y ∈ 𝔤: (∀X ∈ 𝔥) (X | Y) = 0}.

A simple calculation shows 𝔟^{∗} = φ(𝔟). Let p_{𝔟}: 𝔤 → 𝔟 be the orthogonal projection w.r.t. (· | ·). It is clear that p_{𝔟} is selfadjoint w.r.t. the inner product. Now suppose that N ⊆ G is a subgroup such that Ad(N) leaves 𝔟 invariant. Then the selfadjointness of p_{𝔟} and equation (V.2) show that the map

ρ_{𝔟}: N × 𝔟 → 𝔟, (n, X) ↦ n.X := p_{𝔟} ∘ Ad(θn)(X)

is a group action. Let Φ_{𝔟}: 𝔤^{∗} → 𝔟^{∗} be the restriction map. We define a group action

ρ_{𝔟^{∗}}: N × 𝔟^{∗} → 𝔟^{∗}, (n, f) ↦ n.f := Φ_{𝔟} ∘ Ad^{∗}(n)(f).

Then one easily checks

(V.4)  φ(n.X) = n.φ(X).

and

(V.5)  Φ_{𝔟} ∘ φ = φ ∘ p_{𝔟}.

Proposition V.1. Let N ⊆ G be a subgroup and 𝔞 ⊆ 𝔟 subspaces of 𝔤. Suppose that 𝔞 and 𝔟 are Ad(N)-invariant. Then

(i) 𝔞^{∗} and 𝔟^{∗} are Ad^{∗} (θ(N))-invariant.

(ii) Φ_{𝔞}: 𝔟^{∗} → 𝔞^{∗} is equivariant w.r.t. the actions ρ_{𝔟^{∗}} and ρ_{𝔞^{∗}}.

(iii) p_{𝔞}: 𝔟 → 𝔞 is equivariant w.r.t. the actions ρ_{𝔟} and ρ_{𝔞}. Proof. This follows from the above by a simple calculation.

Now we return to our special situation and note that using φ we can replace (V.1′) by

(V.1′′)  𝔤 \underrightarrow{p_{𝔮_{k}}} 𝔮_{k} \underrightarrow{p_{k}} 𝔮_{k,red} \underrightarrow{p_{𝔥_{k}}} 𝔥_{k} \underrightarrow{p_{U_{k}}} U_{k} .

The kernels of the respective projections are given by the following proposition.

Proposition V.2.

(i) 𝔮_{k}^{⊥} = θ𝔥_{k}.

(ii) 𝔮_{k,red}^{⊥} ∩ 𝔮_{k} = 𝔤_{1,k}.

(iii) 𝔥_{k}^{⊥} ∩ 𝔮_{k,red} = 𝔤_{2,k}.

(iv) (U_{k})^{⊥} ∩ 𝔥_{k} = V_{k}.

Proof. (i) 𝔥_{k} is nilpotent, hence B vanishes on 𝔥_{k} × 𝔥_{k}. On the other hand θ𝔥_{k} and 𝔩_{k} are sums of different root spaces 𝔞, respectively and therefore orthogonal w.r.t. B. This shows “ ⊇ ” and equality follows for dimensional reasons. (ii) 𝔤_{1,k} and 𝔤_{2,k} are orthogonal which shows “ ⊇ ”. Equality follows again by counting dimensions. (iii) This follows from parts (i) and (ii). (iv) Again one uses that different root spaces are (· | ·)-orthogonal.

For the following we recall the parabolic subgroup Q_{k} of G associated to the parabolic subalgebra 𝔮_{k}. We write Q_{k} = H_{k}⋉L_{k} for the Levi decomposition of Q_{k} and G_{1,k}, G_{2,k}, H_{k} and Q_{red} for the analytic subgroups corresponding to 𝔤_{1,k}, 𝔤_{2,k}, 𝔥_{k} and 𝔮_{red} respectively. We also recall from [35, Lemma 1.2.4.5] that Q_{k} = Z_{K}(𝔞)(Q_{k})_{0} .

Proposition V.3. p_{𝔮_{k}}, p_{k}, p_{𝔥_{k}} and p_{U_{k}} are Q_{k} -equivariant w.r.t. the ρactions.

Proof. The only thing that remains to be checked is that 𝔮_{red} is invariant under Q_{k}. Since 𝔮_{red} is an ideal of 𝔮, it is invariant under (Q_{k})_{0} . Moreover 𝔥_{k} is invariant under Q_{k}, so that is suffices to show that 𝔤_{2,k} is invariant under Z_{K}(A). We have 𝔤_{2,k} = ℝH^{k} ⊕ 𝔤_{(k,−)} . The element H^{k} is fixed by Z_{K}(A), and since all the real root spaces are Z_{K}(A)-invariant, the same holds for the subalgebra 𝔤_{(k,−)} which is generated by certain real root spaces (Lemma IV.12).

Next we consider the G-orbit O_{k} of X^{k} in 𝔤. Note first that

Ad(G).X^{k} = ρ_{𝔤}(G, X^{k})

since G is θ-invariant. In order to take advantage of the above equivariance properties, we use the Bruhat decomposition of G w.r.t. Q_{k}:

(V.6)  G = ⋃_{w∈W(Σ)} Q_{k}m_{w}θH_{k} = ⋃_{w∈W(Σ)} H_{k}L_{k}m_{w}θH_{k},

where W(Σ) ≅ N_{K}(𝔞)/Z_{K}(𝔞) ≅ {−1, 1}^{r}⋉S_{r} is the Weyl group of the restricted root system Σ and m_{w} a representative of w in N_{K}(𝔞). We do not claim that (V.6) is a disjoint union. Now θH_{k} stabilizes X^{k} in the ρ_{𝔤} -action and the element w = ((ε_{1}, …, ε_{r}), σ) ∈ {−1, 1}^{r} × S_{r} acts on X^{k} = ∑_{j=1}^{k} X_{j} via

w.X^{k} = ∑_{{j∈\{1,...,k\} \atop ε_{σ(j)}=1}} X_{σ(j)} + ∑_{{j∈\{1,...,k\} \atop ε_{σ(j)}=−1}} θX_{σ(j)}.

Thus we have

p_{𝔮_{k}}(w.X^{k}) = ∑_{{j∈\{1,...,k\} \atop ε_{σ(j)}=1}} X_{σ(j)}.

The group G_{1,k} acts trivially on U_{k} and since the root space 𝔤^{2ε_{k}} is onedimensional, it is pointwise fixed by Z_{K}(A). So, in view of Q = Q_{red}G_{1,k}Z_{K}(A) and Proposition V.3, we obtain

(V.7)  p_{𝔮_{k}}(O_{k}) = ⋃_{j=0}^{k} ρ_{𝔮_{k}}(Q_{k,red}, X^{j}),

where X^{0} = 0.

Remark V.4.

(i) Recall the open cone Ω_{k} from Proposition IV.4 and its automorphism group G(Ω_{k}). It follows from [29, Th. III.2.3] that U_{k} is a euclidean Jordan algebra with unit X^{k} and Jordan frame (X_{1}…. , X_{k}). Here by a Jordan frame we mean a maximal set of pairwise orthogonal idempotents whose sum is the unit element.

(ii) Part (i) allows us to apply [7, Prop. IV.3.1] to Ω_{k}. It says in particular that

\overline{Ω}_{k} = ⋃_{j=0}^{k} G(Ω_{k})_{0}.X^{j}

and characterizes G(Ω_{k})_{0}.X^{j} as the set of elements Y ∈ \overline{Ω}_{k} whose rank (in the Jordan algebra sense) is j.

Lemma V.5.

(i) ρ_{𝔮_{k}}(L_{k}, X^{j}) = G(Ω_{k})_{0}.X^{j}.

(ii) ρ_{𝔮_{k}}(exp(X′), X) = X + [θX′, X] ∈ X + 𝔩_{k} for all X, X′ ∈ U_{k}.

(iii) ρ_{𝔮_{k}}(exp(Y), X + Z) = X + [θY, X] + (Z + \frac{1}{2}[θY, [θY, X]]) ∈ X + V_{k} + 𝔩_{k} for all Y ∈ V_{k}, X ∈ U_{k} and Z ∈ 𝔩_{k}.

Proof. (i) Since L_{k} is θ-invariant we see that Ad(θL_{k}).X^{j} ⊆ U_{k} so that ρ_{𝔮_{k}}(L_{k}, X^{j}) = Ad(L_{k}).X^{j} which in turn is equal to G(Ω_{k})_{0}.X^{j} since the groups Z_{K}(A) and G_{1,k} fix X^{k}. The assertions (ii) and (iii) follow from the eigenspaces decomposition of ad(H^{k}) and the fact that different root spaces are (· | ·)-orthogonal.

Similarly we find Lemma V.6.

(i) ρ_{𝔮_{k,red}}(L_{k}, X^{j}) = G(Ω_{k})_{0}.X^{j}.

(ii) ρ_{𝔮_{k,red}}(exp(X′), X) = X + [θX′, X] ∈ X + 𝔤_{2,k} for all X, X′ ∈ U_{k}.

(iii) ρ_{𝔮_{k,red}}(exp(Y), X + Z) = X + [θY, X] + (Z + \frac{1}{2}p_{𝔮_{k,red}}([θY, [θY, X]])) ∈ X + V_{k} + 𝔤_{2,k} for all Y ∈ V_{k}, X ∈ U_{k} and Z ∈ 𝔤_{2,k}.

Proof. The only thing we still have to note is that [θX′, X] ∈ 𝔤_{2,k} for all X, X′ ∈ U_{k}. But that follows from [29, (3.41)].

The action ρ_{𝔮_{k,red}} will be the one we use primarily. Therefore we abbreviate it by ρ_{k}: Q_{k,red} × 𝔮_{k,red} → 𝔮_{k,red}. Similarly, we write p_{k} for p_{𝔮_{k,red}} .

Lemma V.7.

(i) ρ_{𝔥_{k}}(L_{k}, X^{j}) = G(Ω_{k})_{0}.X^{j}.

(ii) ρ_{𝔥_{k}}(exp(X′), X) = X for all X, X′ ∈ U_{k}.

(iii) ρ_{𝔥_{k}}(exp(Y), X) = X + [θY, X] ∈ X + V_{k} for all Y ∈ V_{k} and X ∈ U_{k}.

Lemma V.8.

(i) ρ_{U_{k}}(L_{k}, X^{j}) = ρ_{U_{k}}(G_{2,k}, X^{j}).

(ii) ρ_{U_{k}}(exp(X′), X) = X for all X, X′ ∈ U_{k}.

(iii) ρ_{U_{k}}(exp(Y), X) = X for all Y ∈ V_{k} and X ∈ U_{k}.

Proposition V.9.

(i) p_{U_{k}}(O_{k}) = \overline{Ω}_{k}.

(ii) The set O′_{k} := p_{U_{k}}^{−1}(Ω_{k}) ∩ O_{k} = Ad(θQ_{k,red}).X^{k} is open and dense in

O_{k}.

Proof. (i) According to Lemma V.8 and equation (V.6) we have in view of Remark V.4

p_{U_{k}}(O_{k}) = p_{U_{k}}(ρ_{𝔤}(G, X^{k})) = ⋃_{j=0}^{k} G_{2,k}.X^{j} = \overline{Ω}_{k}.

(ii) The formula follows from (i), (V.7), and Lemma V.8. Moreover it is clear that the set is open in O_{k} since Ω_{k} is open in U_{k}. To show that it is also dense it suffices to show that the open Bruhat cell B = Q_{k}θH_{k} satisfies p_{U_{k}}(ρ_{𝔤}(B, X^{k})) = Ω_{k}. But that is clear since θH_{k} fixes X^{k} and

p_{U_{k}}(ρ_{𝔤}(Q_{k}, X^{k})) = ρ_{U_{k}}(Q_{k}, X^{k}) = ρ_{U_{k}}(L_{k}, X^{k}) = Ω_{k}.

We want to gain more insight into the geometry of O_{k} via the study of the fibers and images of the various moment maps. We will essentially restrict ourselves to the dense open subset of O_{k} given in Proposition V.9.

Proposition V.10.

(i) p_{U_{k}}^{−1}(X) ∩ (U_{k} + V_{k}) = X + V_{k} ⊆ p_{𝔥_{k}}(O_{k}) for all X ∈ Ω_{k}.

(ii) p_{𝔥_{k}}(O_{k}) ⊇ Ω_{k} + V_{k}.

(iii) p_{𝔥_{k}}(O′_{k}) = Ω_{k} + V_{k}.

Proof. (i) Since p_{U_{k}} is Q_{k} -equivariant, the fibers of p_{U_{k}} are permuted by Q_{k}. In particular we have

p_{U_{k}}^{−1}(X) = l.p_{U_{k}}^{−1}(X^{k})

for X = l.X^{k}. Thus it suffices to prove the claim for X = X^{k}.

According to Lemma V.7, we have to show that ad(X^{k}): θV_{k} → V_{k} is surjective. But that follows immediately from 𝔰𝔩(2)-theory applied to the 𝔰𝔩_{2} - triple (H^{k}, X^{k}, Y^{k}). (ii) follows directly from (i). (iii) Again the inclusion “ ⊇ ” is clear and for the converse we calculate

Ω_{k} + V_{k} ⊆ p_{𝔥_{k}}(O_{k} ∩ p_{U_{k}}^{−1}(Ω_{k})) = p_{𝔥_{k}}(O′_{k}).

Lemma V.11. Let Y ∈ V_{k} and Ỹ ∈ V_{k} be the uniquely determined element with ad(X^{k})θỸ = Y . Then

p_{k}(O_{k}) ∩ (X^{k} + Y + 𝔤_{2,k}) = X^{k} + Y + p_{k}(ad(θỸ)^{2}X^{k}) + ad(X^{k})θU_{k}.

Proof. According to Lemma V.6 and the Bruhat decomposition (V.6) any element of p_{k}(O_{k}) ∩ p_{U_{k}}^{−1}(X^{k}) can be written as

X^{k} + [θX′, X^{k}] + [θY′, X^{k}] + p_{k}([θY′, [θY′, X^{k}]])

with X′ ∈ U_{k} and Y′ ∈ V_{k} since p_{k} is equivariant w.r.t. the ρ-actions. But this element gets mapped to X^{k} +[θY′, X^{k}] under p_{𝔥_{k}}, i.e., its 𝔥_{k} -component is X^{k} + [θY′, X^{k}]. If [θY′, X^{k}] = Y , then Y′ = −Ỹ. On the other hand [θX′, X^{k}] does not effect the p_{𝔥_{k}} projection and can be varied freely.

Proposition V.12. The fibers of the map

p_{𝔥_{k}}: p_{k}(O′_{k}) → p_{𝔥_{k}}(O′_{k}) = Ω_{k} + V_{k}

are affine spaces of dimension dim U_{k}. More precisely, for Z ∈ p_{k}(O′_{k}) with p_{U_{k}}(Z) = X we have

p_{k}(O′_{k}) ∩ p_{𝔥_{k}}^{−1}(p_{𝔥_{k}}(Z)) = Z + ad(X)θU_{k}.

Proof. As in the proof of Proposition V.10, we may restrict ourselves to the case X = X^{k}. Then Z ∈ X^{k} + Y + 𝔤_{2,k} and according to Lemma V.11, the fiber in question is given by

X^{k} + Y + p_{k}(ad(θỸ)^{2}X^{k}) + ad(X^{k})θU_{k} = Z + ad(X^{k})θU_{k}.

The statement about the dimension now follows from Lemma IV.24.

Remark V.14. dim O_{k} = 2 dim U_{k} + dim V_{k}. In fact, we calculate

dim O_{k} = dim 𝔤 − dim 𝔷_{𝔤}(X^{k})
= (dim 𝔩_{k} + 2 dim 𝔲_{k}) − (dim 𝔷_{𝔩_{k}}(X^{k}) + dim 𝔲_{k})
= (dim 𝔩_{k} − dim 𝔷_{𝔩_{k}}(X^{k})) + dim 𝔲_{k}
= dim U_{k} + (dim V_{k} + dim U_{k})
= 2 dim U_{k} + dim V_{k}.

Now we consider the diagram

O′_{k} \underrightarrow{p_{U_{k}}} p_{U_{k}}(O′_{k}) = Ω_{k}
↓p_{k}  ↑p_{U_{k}}
p_{k}(O′_{k}) \underrightarrow{p_{𝔥_{k}}} p_{𝔥_{k}}(O′_{k}) = Ω_{k} + V_{k}

It clearly commutes and the dimension of the manifolds involved are

dim O′_{k} = 2 dim U_{k} + dim V_{k}
dim Ω_{k} = dim U_{k}
dim(Ω_{k} + V_{k}) = dim U_{k} + dim V_{k}
dim p_{k}(O′_{k}) = 2 dim U_{k} + dim V_{k}

Thus p_{k}: O′_{k} → p_{k}(O′_{k}) is a surjective submersion of equidimensional manifolds. In fact, more is true:

It follows from (V.4) that we can use the map φ: 𝔤 → 𝔤^{∗} to transport the natural G-invariant Poisson structure on 𝔤^{∗} over to a Poisson structure on 𝔤 which is invariant under the ρ_{𝔤} action. Therefore the symplectic leaves of 𝔤 are precisely the G-orbits w.r.t. ρ_{𝔤}. Similarly we introduce a Q_{k,red} -invariant Poisson structure on 𝔮_{k,red} for which the symplectic leaves are the Q_{k,red} -orbits w.r.t. ρ_{𝔮_{k,red}} . Then φ is a Poisson isomorphism and the map p_{k} is a Poisson morphism since Φ_{𝔮_{k,red}} is one. Therefore the G-orbit O_{k} and the Q_{k,red} -orbit p_{k}(O′_{k}) are symplectic submanifolds of 𝔤 and 𝔮_{k,red} , respectively. Moreover O′_{k} is a symplectic submanifold of O_{k}.

Proposition V.15. The map p_{k}: O′_{k} → p_{k}(O′_{k}) is a symplectic diffeomorphism. Proof. We first recall that p_{k} is Q_{k,red} -equivariant w.r.t. the ρ-actions. Moreover Q_{k,red} acts transitively on both O′_{k} and p_{k}(O′_{k}) and the symplectic structures are both Q_{k,red} -invariant. Finally the map p_{k} can be viewed (via φ) as the moment map of the Q_{k,red} -action on O′_{k}. Thus [9, p.185] implies that p_{k}: O′_{k} → p_{k}(O′_{k}) is a symplectic covering. To conclude the argument we note that the stabilizers of X^{k} in Q_{k,red} w.r.t. the ρ_{𝔤} and theρ_{𝔮_{k,red}} actions agree, whence the equivariance shows that the fibers are trivial.

Since the symplectic manifold p_{k}(O′_{k}) will play an important role in this paper we abbreviate it by M_{k} and denote its symplectic form by ω^{(k)} .

Note that Proposition V.15 obviously implies that also

p_{𝔮_{k}}: O′_{k} → p_{𝔮_{k}}(O′_{k}) and p_{k}: p_{𝔮_{k}}(O′_{k}) → p_{k}(O′_{k})

are symplectic diffeomorphisms.

The above results on the structure of O_{k} should be compared to Proposition V.16. O_{k} ∩ 𝔥_{k} = Ω_{k}. Proof. Since O_{k} is of convex type, we know that it is contained in a proper generating closed convex invariant cone W ⊆ 𝔤. In particular it contains X^{k} ∈ U_{k} and therefore its G(Ω_{k})_{0} -orbit Ω_{k}. We apply [14, Lemma I.14] to W ∩ 𝔥_{k} and find that W ∩ 𝔥_{k} = \overline{Ω}_{k}.

Let c_{j} := X_{j} = X^{j} − X^{j−1} for j = 1, …, r. Then (c_{1}, …, c_{k}) is a Jordan frame in U_{k}. Suppose that Y ∈ O_{k} ∩ Ω_{k}. Then there exists an element g ∈ G(Ω_{k})_{0} and j ∈ {0, …, k} such that g.Y = X^{j} (Remark V.4). Since ρ_{U}: G_{2,k} → G(Ω_{k})_{0} is surjective, we conclude that Y ∈ O_{j}, hence that j = k and therefore that Y ∈ G(Ω_{k})_{0}.X^{k} = Ω. Thus O_{k} ∩ U_{k} = Ω_{k}.

Remark V.17. The proof of Proposition V.16 even shows that the nilpotent orbits of convex type O_{j} with j > k do not intersect 𝔥_{k} and that for j ≤ k the intersection consists of the set of all elements of rank j in the closed cone \overline{Ω}_{k}.

VI. The symplectic geometry of O_{k} and M_{k}

In this section we give details concerning the symplectic structure of O_{k} and its various projections. We start with a few general facts on the symplectic structure of coadjoint orbits.

Lemma VI.1. Let H be a Lie group and f ∈ 𝔥^{∗}. Then the following statements are equivalent:

(1) f vanishes on the Lie algebra 𝔥_{f} of the stabilizer H_{f} of f under the coadjoint action.

(2) The left invariant 1-form α^{(f)} on H defined by f is the pullback of an H -invariant 1-form α^{(f)} on the coadjoint orbit O_{f} = Ad^{∗}(H).f under the orbit map s: H → O_{f}, h ↦ h.f := Ad^{∗}(h).f.

Proof. If such an α^{(f)} exists, then it has to satisfy

(∗)  α_{h.f}^{(f)}(h.(ad^{∗}(X).f)) = ⟨f, X⟩ ∀h ∈ H, X ∈ 𝔥

since the orbit map s is H -equivariant. So, if (1) holds, the formula

α_{f}^{(f)}(ad^{∗}(X).f) = ⟨f, X⟩

actually defines a linear functional α_{f} on T_{f}(O_{f}) = ad^{∗}(𝔥).f. Since

(VI.1)  Ad^{∗}(h) ∘ ad^{∗}(X) = ad^{∗}(Ad(h).X) ∘ Ad^{∗}(h)

and

⟨f, X⟩ = ⟨Ad^{∗}(h).f, Ad(h)X⟩,

we can move this form around with the H -action to define a 1-form α^{(f)} on O_{f} via (∗). It follows from its invariance that this form is smooth and has α^{(f)} as its pullback under s. This proves (2).

Conversely, suppose that (2) holds. Then X ∈ 𝔥_{f} shows ad^{∗}(X).f = 0 so that (∗) implies ⟨f, X⟩ = 0, i.e., (1).

Consider the canonical Poisson structure on the dual 𝔥^{∗} of a Lie algebra 𝔥. Given f ∈ 𝔥^{∗} the cotangent space T_{f}^{∗}(𝔥^{∗}) is identified with 𝔥 and the Poisson tensor Λ_{f} at f is given by

Λ_{f}(X, Y) = ⟨f, [X, Y]⟩ ∀X, Y ∈ 𝔥.

Therefore the associated bundle map

Λ^{♯}: T^{∗}(𝔥^{∗}) ≅ 𝔥^{∗} × 𝔥 → T (𝔥^{∗}) ≅ 𝔥^{∗} × 𝔥^{∗}

is given by

Λ^{♯}(f, X) = (f, − ad^{∗}(X).f)

and we also write Λ_{f}^{♯}(X) = − ad^{∗}(X).f for this fact. The characteristic distribution {C_{f} ⊆ T_{f}(𝔥^{∗}): f ∈ 𝔥^{∗}} is given by

C_{f} = Λ_{f}^{♯}(T_{f}(𝔥^{∗})) = ad^{∗}(𝔥).f.

We denote the annihilator of C_{f} in T_{f}^{∗}(𝔥^{∗}) ≅ 𝔥 by AnnC_{f} and note that

AnnC_{f} = ker Λ_{f}^{♯} = {X ∈ 𝔥: ad^{∗}(X).f = 0} = 𝔥_{f}.

Thus we have a canonical isomorphism C_{f}^{∗} ≅ 𝔥/𝔥_{f}. The map Λ_{f}^{♯} induces an isomorphism ℓ_{f}: C_{f}^{∗} → C_{f} and one has a symplectic form ω_{f} on C_{f} defined by

ω_{f}(a, b) = Λ_{f}(ℓ_{f}^{−1}(a), ℓ_{f}^{−1}(b)), ∀a, b ∈ C_{f}.

This means that the corresponding symplectic form on C_{f}^{∗} is given by

ω̃_{f}(X + 𝔥_{f}, Y + 𝔥_{f}) = ⟨f, [X, Y]⟩.

The symplectic forms ω_{f} for f in a coadjoint orbit of 𝔥 form the H -invariant symplectic form ω^{(f)} on that orbit which turns it into a symplectic leaf of the Poisson manifold 𝔥^{∗}.

Lemma VI.2. Let H be a Lie group and f ∈ 𝔥^{∗}. Suppose that 𝔥_{f} ⊆ ker f. Then the form α^{(f)} constructed in Lemma VI.1 satisfies

(i) α_{f′}^{(f)}(ad^{∗}(X).f′) = ⟨f′, X⟩ for all f′ ∈ O_{f}.
(ii) −2dα^{(f)} = ω^{(f)} .

Proof. (i) Let f′ = h.f. Then, using (VI.1), we calculate

α_{f′}^{(f)}(ad^{∗}(X).f′) = α_{h.f}^{(f)}(ad^{∗}(X).(h.f))
= α_{f}^{(f)}(h^{−1}.(ad^{∗}(X)(h.f)))
= α_{f}^{(f)}(ad^{∗}(h^{−1}.X).f)
= ⟨f, h^{−1}.X⟩
= ⟨f′, X⟩.

(ii) We recall the orbit map s: H → O_{f} from Lemma VI.1. The definitions show that

(s^{∗}ω^{(f)})_{1}(X, Y) = ⟨f, [X, Y]⟩.

Since s is a submersion, it suffices to show that s^{∗}(dα^{(f)}) = d(s^{∗}α^{(f)}) = s^{∗}ω^{(f)} . On the other hand the H -invariance shows that we have to test d(s^{∗}α^{(f)}) and s^{∗}ω^{(f)} only against left invariant vector fields, i.e., elements of 𝔥:

\begin{aligned}2d(s^{∗}α^{(f)})(X_{1}, X_{2}) & = X_{1}.α(X_{2}) − X_{2}.α(X_{1}) − α([X_{1}, X_{2}]) \\ & = −α([X_{1}, X_{2}]) = −⟨f, [X_{1}, X_{2}]⟩. \end{aligned}

This proves the lemma.

Now we consider a semisimple Lie algebra 𝔤 with Cartan involution θ and Killing form B. We retain the notation from Section V and transport the Poisson structure from 𝔥^{∗} to 𝔥 via φ: 𝔥 → 𝔥^{∗}. Here T_{X}^{∗}(𝔥) gets identified with 𝔥^{∗} and hence the Poisson tensor on 𝔥 is given by

Λ_{X}(a, b) = ⟨φ(X), [φ^{−1}(a), φ^{−1}(b)]⟩, ∀X ∈ 𝔥, a, b ∈ 𝔥^{∗}.

Lemma VI.3. The bundle map

Λ^{♯}: T^{∗}(𝔥) ≅ 𝔥 × 𝔥^{∗} → T (𝔥) ≅ 𝔥 × 𝔥

associated to Λ is given by

Λ^{♯}(X, f) = (X, − ad(X)(θ(φ^{−1}(f)))) , ∀X ∈ 𝔥, f ∈ 𝔥^{∗}.

Proof. Let a, b ∈ 𝔥^{∗} and X ∈ 𝔥. Then we calculate

⟨Λ_{X}^{♯}(a), b⟩ = Λ_{X}(a, b)
= ⟨φ(X), [φ^{−1}(a), φ^{−1}(b)]⟩
= (X | ad (φ^{−1}(a))φ^{−1}(b))
= (ad (θ(φ^{−1}(a)))X | φ^{−1}(b))
= ⟨ad (θ(φ^{−1}(a)))X, b⟩
= −⟨ad(X)(θ(φ^{−1}(a))), b⟩.

From Lemma VI.3 it is clear that

(VI.2)  C_{X} = ad(θ𝔥)X = ad(X)θ𝔥.

We will write C_{X,𝔥} for C_{X} when we want to emphasize the subalgebra 𝔥 for which we consider the Poisson structure. The annihilator AnnC_{X,𝔥} of C_{X,𝔥} in T_{X}^{∗}(𝔥) ≅ 𝔥^{∗} is

AnnC_{X,𝔥} = ker Λ_{X}^{♯} = {f ∈ 𝔥^{∗}: θ𝔥 ⊆ ker ( ad^{∗}(X).f)}.

Proposition VI.4. Let (φ^{−1})^{∗}: 𝔥^{∗} → (𝔥^{∗})^{∗} ≅ 𝔥 denote the dual map of φ^{−1}: 𝔥^{∗} → 𝔥. Then we have (φ^{−1})^{∗} ∘ Λ_{φ(X)}^{♯} = Λ_{X}^{♯} ∘ φ.

Proof. For f ∈ 𝔥^{∗} and Y ∈ 𝔥 we calculate

⟨f, Λ_{X}^{♯} ∘ φ(Y)⟩ = Λ_{X}(φ(Y), f)
= ⟨φ(X), [Y, φ^{−1}(f)]⟩
= ⟨− ad^{∗}(Y)(φ(X)), φ^{−1}(f)⟩
= ⟨Λ_{φ(X)}^{♯}(Y), φ^{−1}(f)⟩
= ⟨f, (φ^{−1})^{∗} ∘ Λ_{φ(X)}^{♯}(Y)⟩.

Proposition VI.4 implies that φ^{∗} induces an isomorphism φ^{∗}: C_{φ(X)} → C_{X} and φ an isomorphism φ: AnnC_{φ(X)} → AnnC_{X}. In fact, we have a commutative diagram with exact lines

0 ⟶ AnnC_{X} ⟶ 𝔥^{∗}  \underrightarrow{Λ_{X}^{♯}}  C_{X} ⟶ 0
↑φ  ↑φ  ↑φ^{∗}
0 ⟶ AnnC_{φ(X)} ⟶ 𝔥 \underrightarrow{Λ_{φ(X)}^{♯}} C_{φ(X)} ⟶ 0

which then gives a commutative diagram

C_{X}^{∗}  \underrightarrow{ℓ_{X}}  C_{X}
↑\overline{φ}  ↑φ^{∗}
C_{φ(X)}^{∗} \underrightarrow{ℓ_{φ(X)}} C_{φ(X)}

where \overline{φ} is the map induced by φ on C_{φ(X)}^{∗} = 𝔥/AnnC_{φ(X)} .

Lemma VI.5. Λ_{X}^{♯} ∘ φ = ad(X) ∘ θ = −θ ∘ ad(θX).

Proof. Λ_{X}^{♯} ∘ φ(Y) = − ad(X) (θ(φ^{−1}(φ(Y)))) = ad(X)(θ(Y)).

Proposition VI.6. The symplectic form ω_{X} on C_{X,𝔥} is given by

ω_{X}(ad(X)θY, ad(X)θZ) = −B(X, θ[Y, Z]), ∀X, Y, Z ∈ 𝔥.

Proof.

ω_{X}(ad(X)θY, ad(X)θZ) = Λ_{X} (ℓ_{X}^{−1}(ad(X)θY), ℓ_{X}^{−1}(ad(X)θZ))
= Λ_{X} (ℓ_{X}^{−1}(Λ_{X}^{♯} ∘ φ(Y)), ℓ_{X}^{−1}(Λ_{X}^{♯} ∘ φ(Z)))
= Λ_{X} (φ(Y) + AnnC_{X}, φ(Z) + AnnC_{X})
= Λ_{φ(X)} (Y + AnnC_{φ(X)}, Z + AnnC_{φ(X)})
= ⟨φ(X), [Y, Z]⟩
= (X | [Y, Z])
= −B(X, θ[Y, Z]).

Lemma VI.7. Suppose that X ∈ 𝔤 is nilpotent. Then the functional φ(X) ∈ 𝔤^{∗} vanishes on 𝔤_{φ(X)} = {Y ∈ 𝔤: ad^{∗}(Y).φ(X) = 0}. Proof. Let Y ∈ 𝔤. Then ⟨φ(X), Y ⟩ = −B(X, θY). If Y ∈ 𝔤_{φ(X)} , then for all Z ∈ 𝔤 we have

\begin{aligned}0 & = ⟨ad^{∗}(Y).φ(X), Z⟩ = ⟨φ(X), [Z, Y]⟩ \\ & = −B(X, [θZ, θY]) = −B([X, θZ], θY). \end{aligned}

But the Jacobson-Morosow Theorem says in particular that there exists a θZ ∈ 𝔤 with X = [X, θZ]. Therefore ⟨φ(X), Y ⟩ = −B(X, θY) = 0 for all Y ∈ 𝔤_{φ(X)} .

Proposition VI.8. Let X ∈ 𝔤 be nilpotent and ω^{(X)} the symplectic form on O_{X} = Ad(G).X induced by the canonical symplectic form ω^{(φ(X))} on the coadjoint orbit O_{φ(X)} = φ(O_{X}) = Ad^{∗}(G).φ(X). Then there exists a Ginvariant 1-form α^{(X)} on O_{X} defined by

α_{X}^{(X)}(ad(X)θY) = B(X, θY)

such that −2dα^{(X)} = ω^{(X)} . Proof. According to Lemma VI.7 and Lemma VI.2 there exists a form α^{(φ(X))} on O_{φ(X)} with −2dα^{(φ(X))} = ω^{(φ(X))} . Then α^{(X)} is the pullback of α^{(φ(X))} under φ. A simple calculation shows that φ(ad(X)θY) = − ad^{∗}(Y)φ(X) so the claim follows from

α_{φ(X)}^{(φ(X))}( − ad^{∗}(Y)φ(X)) = −⟨φ(X), Y ⟩ = B(X, θY)

We note here that we cannot expect to be able to prove an analogue of Proposition VI.8 for the H -orbits since the centralizer 𝔥_{φ(X)} of φ(X) for X ∈ 𝔥 in 𝔥 may not be contained in 𝔤_{φ(X)} and the conclusion of Lemma VI.7 depended on the Jacobson-Morosow Theorem which is not available for non-semisimple algebras.

Now we return to our special situation and apply the preceding results to O_{k} and M_{k}.

Lemma VI.9. C_{X^{k},𝔤} = C_{X^{k},𝔮_{k,red}} = 𝔥_{k} + 𝔭_{2,k}. Proof. This can be derived from Proposition V.15 by calculating the Q_{k,red} - orbit of X^{k}. We give a more direct argument: 𝔰𝔩_{2} -theory shows that the image of ad(X^{k}): 𝔤 → 𝔤 is

C_{X^{k},𝔤} = 𝔥_{k} + [X^{k}, θU_{k}] = 𝔥_{k} + 𝔭_{2,k} ⊆ 𝔮_{k,red}

(Lemma IV.24).

It is obvious now that p_{𝔥_{k}}(C_{X^{k},𝔤}) = 𝔥_{k} and p_{U_{k}}(C_{X^{k},𝔤}) = U_{k}.

Proposition VI.10. The symplectic form ω^{(k)} on M_{k} is exact. More precisely, there exists a Q_{k,red} -invariant 1-form α^{(k)} on M_{k} with −2dα^{(k)} = ω^{(k)} and

α_{X^{k}}^{(k)}(ad(X^{k})θY) = B(X^{k}, θY)

for all Y ∈ 𝔮_{k,red} . Proof. This follows immediately from Proposition V.15, Proposition VI.8 and Lemma VI.9 by taking pullbacks.

Lemma VI.11.

(i) L_{+,k}(X^{k}) := (U_{k})_{ℂ} + V_{k}^{+} is a Lagrangian subspace of (C_{X^{k},𝔮_{k,red}})_{ℂ}.

(ii) L_{−,k}(X^{k}) := (𝔭_{2,k})_{ℂ} + V_{k}^{−} is a Lagrangian subspace of (C_{X^{k},𝔮_{k,red}})_{ℂ}.

Proof. For dimensional reasons it suffices to show that ω_{X^{k}} vanishes on the two subspaces. Let X ∈ (U_{k})_{ℂ}, Y^{±} ∈ V_{k}^{±} and Z ∈ (𝔭_{2,k})_{ℂ}. Then we find X̃, Ỹ^{±}, Z̃ ∈ 𝔮_{k,red} with

X = ad(X^{k})θX̃, Y^{±} = ad(X^{k})θỸ^{±}, Z = ad(X^{k})θZ̃.

Then X̃ ∈ (𝔩_{k})_{ℂ}, Ỹ^{±} ∈ V_{k}^{±} and Z̃ ∈ (U_{k})_{ℂ}. Now the claim follows since the below commutator spaces all belong to the orthogonal complement of X^{k} w.r.t. the Killing form B.

[𝔩_{k}, 𝔩_{k}] ⊆ 𝔩_{k}, [𝔩_{k}, V_{k}^{+}] ⊆ V_{k}^{+}, [V_{k}^{+}, V_{k}^{+}] = {0},
[U_{k}, U_{k}] = {0}, [U_{k}, V_{k}^{−}] = {0}, [V_{k}^{−}, V_{k}^{−}] = {0}.

Recall that Q_{k,red} acts transitively on O′_{k} and M_{k}. The stabilizer of X^{k} in Q_{k,red} is K_{2,k}. Note that K_{2,k} leaves both Lagrangian subspaces from Lemma VI.11 invariant (the ρ-action of K_{2,k} coincides with the adjoint action). Since the Poisson structure is Q_{k,red} -invariant we can define two Q_{k,red} -invariant Lagrangian distributions L_{±,k} on M_{k} using the action ρ_{k}.

Using the semidirect product structure of Q_{k} = H_{k}⋉L_{k} we obtain additional information on the form α^{(k)} :

Consider the diffeomorphism

γ: H_{k} × Ω_{k} → M_{k}, (h, X) ↦ ρ_{k}(h, X).

The Q_{k} -action on H_{k} × Ω_{k} induced from ρ_{k} via γ is given by the formula

(h, l).(h′, X) = (hlh′l^{−1}, l.X), ∀h, h′ ∈ H_{k}, l ∈ L_{k}, X ∈ Ω_{k}.

Pulling back α^{(k)} to H_{k} × Ω_{k} via γ yields an invariant form on H_{k} × Ω_{k} and since the action of H_{k} on H_{k} × Ω_{k} simply consists of left translation in the fist argument, it is given by elements of 𝔥_{k}^{∗} on the H_{k} -orbits. We identify T^{∗}H_{k} with H_{k} × 𝔥_{k}^{∗} and T^{∗}(H_{k} × Ω_{k}) with (H_{k} × Ω_{k}) × (𝔥_{k}^{∗} × U_{k}^{∗}).

Proposition VI.12.

(i) The derivative of γ at (1, X) ∈ H_{k} × Ω_{k} is given by

dγ_{(1,X)}(X′ + Y′, X′′) = X′′ − ad(X)θ(X′ + Y′) ∀X′, X′′ ∈ U_{k}, Y′ ∈ V_{k}.

(ii) The pullback form γ^{∗}α^{(k)} is given by

γ^{∗}α_{(h,X)}^{(k)}(X′ + Y′, X′′) = ⟨φ(X), X′⟩ + α_{X}^{(k)}(X′′),

for all X′, X′′ ∈ U_{k}, Y′ ∈ V_{k}.

Proof. (i) follows immediately from

γ(exp Y′ exp X′, X) = X − ad(X).θ(X′ + Y′) + \frac{1}{2}p_{k}(ad(θY′)^{2}.X)

for Y′ ∈ V_{k} and X′, X ∈ U_{k} which in turn is an immediate consequence of Lemma V.5. (ii) We calculate

γ^{∗}α_{(h,X)}^{(k)}(X′ + Y′, X′′) = γ^{∗}α_{h.(1,X)}^{(k)}(X′ + Y′, X′′)
= γ^{∗}α_{(1,X)}^{(k)}(X′ + Y′, X′′)
= α_{X}^{(k)}(dγ_{(1,X)}(X′ + Y′, X′′))
= α_{X}^{(k)}(X′′ − ad(X)θ(X′ + Y′))
= α_{X}^{(k)}(X′′) − B(X, θ(X′ + Y′))
= α_{X}^{(k)}(X′′) − B(X, θX′)
= α_{X}^{(k)}(X′′) + ⟨φ(X), X′⟩.

We now turn to the study of polarizations. Let 𝔥 be a Lie algebra and f ∈ 𝔥^{∗}. Recall that a subalgebra 𝔟(f) of 𝔥_{ℂ} is called an algebraic polarization if (cf. [2, p.54])

(a) 𝔟(f) + \overline{𝔟(f)} is a subalgebra of 𝔥_{ℂ}.

(b) 𝔟(f) is maximal isotropic in 𝔥_{ℂ} w.r.t. the form (X, Y) ↦ ⟨f, [X, Y]⟩.

Note (cf. [37, p.103]) that the maximality implies that (𝔥_{f})_{ℂ} ⊆ 𝔟(f). Moreover an isotropic subalgebra containing 𝔥_{f} is maximal iff

dim_{ℂ} (𝔟(f)/(𝔥_{f})_{ℂ}) = dim_{ℂ} (𝔥_{ℂ}/𝔟(f)).

Let f ∈ 𝔮_{k,red}^{∗} be the linear functional given by

(VI.3)  f(u + v + X) = −(X^{k} | u), ∀u ∈ U_{k}, v ∈ V_{k}, X ∈ 𝔤_{2,k}

Lemma VI.13. The subalgebra 𝔟(f) := (U_{k})_{ℂ} + V_{k}^{−} + (𝔨_{2,k})_{ℂ} is an algebraic polarization of f ∈ 𝔮_{k,red}^{∗}.

Proof. It is clear that 𝔟(f) + \overline{𝔟(f)} = (U_{k})_{ℂ} + (V_{k})_{ℂ} + (𝔨_{2,k})_{ℂ} is a subalgebra. Since [𝔨_{2,k}, V_{k}^{−}] ⊆ V_{k}^{−}, it is also clear that 𝔟(f) is a subalgebra of (𝔮_{k,red})_{ℂ}. Moreover

[𝔟(f), 𝔟(f)] ⊆ (𝔨_{2,k})_{ℂ} + V_{k}^{−} + [𝔨_{2,k}, U_{k}]_{ℂ} ⊆ f^{⊥}

because 𝔨_{2,k} ⊆ (𝔮_{k})_{f} implies f([𝔨_{2,k}, U_{k}]) = {0}. This means that 𝔟(f) is isotropic. Obviously we have (𝔮_{k,red})_{f} = 𝔨_{2,k} ⊆ 𝔟(f). So to prove maximality, we simply have to check that it has the right dimension. Since (𝔮_{k,red})_{ℂ}/𝔟(f) ≅ (V_{k}^{+} + (𝔤_{2,k})_{ℂ}) /(𝔨_{2,k})_{ℂ} and 𝔟(f)/(𝔮_{k,red} ∩ (𝔮_{k})_{f})_{ℂ} ≅ (U_{k})_{ℂ} + V_{k}^{−}, the assertion follows from dim U_{k} = dim 𝔤_{2,k} − dim 𝔨_{2,k} which in turn follows from the fact that 𝔤_{2,k} acts effectively on U_{k} with 𝔤_{2,k}.X^{k} = U_{k} and 𝔨_{2,k} = (𝔤_{2,k})_{X^{k}} (cf. Proposition IV.4).

Remark VI.14. Suppose that V_{k} = {0} (Proposition IV.21). Put 𝔟 := (𝔤_{2,r} + θU_{r})_{ℂ}. We claim that 𝔟 is an algebraic polarization in f, considered as an element of 𝔤^{∗}.

Since 𝔟 = \overline{𝔟}, it is clear that 𝔟 + \overline{𝔟} is a subalgebra of 𝔤_{ℂ}. Moreover [𝔟, 𝔟] ⊆ 𝔤′_{2,r} + θU_{r} ⊆ ker f shows that 𝔟 is isotropic. Since

dim(𝔟 ∩ 𝔤) − dim 𝔤_{f} = dim 𝔤_{2,r} − dim 𝔨_{2,r} = dim U_{r} = dim 𝔤 − dim(𝔟 ∩ 𝔤),

it follows that 𝔟 is maximal isotropic, hence an algebraic polarization in f.

Theorem VI.15. If V_{k} ≠ {0}, then f ∈ 𝔤^{∗} has no algebraic polarization for 𝔤.

Proof. For the sake of simplicity we omit the indices k in the following proof, f.i. we write 𝔤_{2} instead of 𝔤_{2,k} etc.

Suppose that 𝔟 ⊆ 𝔤_{ℂ} is a polarization in f. We recall that

𝔤_{f} = 𝔨_{2} + 𝔤_{1} + θV + θU.

In view of the fact that 𝔟 is maximal isotropic, it follows that (𝔤_{f})_{ℂ} ⊆ 𝔟. It suffices to show that 𝔟 = 𝔤_{ℂ}. First we claim that 𝔤_{2} ⊆ 𝔟. As a 𝔤_{1} -module the quotient 𝔤_{ℂ}/(𝔤_{f})_{ℂ} is isomorphic to (𝔭_{2})_{ℂ} ⊕V_{ℂ} ⊕U_{ℂ}.

The subspace (𝔭_{2})_{ℂ} ⊕ U_{ℂ} is annihilated by 𝔤_{1} and the action of 𝔤_{1} on V_{ℂ} is effective because V is an irreducible module for 𝔤_{1} + 𝔤_{2} (cf. Proposition IV.1). Let π: 𝔤_{ℂ} → 𝔤_{ℂ}/(𝔤_{f})_{ℂ} denote the quotient map. Then π(𝔟) is a 𝔤_{1} -submodule and therefore

π(𝔟) = (π(𝔟) ∩ V_{ℂ}) + (π(𝔟) ∩ (U_{ℂ} + (𝔭_{2})_{ℂ})).

The natural symplectic form on 𝔤_{ℂ}/(𝔤_{f})_{ℂ} is given by

(Y + (𝔤_{f})_{ℂ}, Z + (𝔤_{f})_{ℂ}) ↦ f([Y, Z]).

Hence V is orthogonal to 𝔭_{2} + U. Therefore the fact that 𝔟 is maximal isotropic shows that π(𝔟) ∩ V_{ℂ} and π(𝔟) ∩ (U_{ℂ} + (𝔭_{2})_{ℂ}) are maximal isotropic.

If π(𝔟)∩(U_{ℂ} +(𝔭_{2})_{ℂ}) ⊆ (𝔭_{2})_{ℂ}, then counting dimensions therefore shows that π(𝔟)∩(U_{ℂ}+(𝔭_{2})_{ℂ}) = (𝔭_{2})_{ℂ}, hence that 𝔭_{2} ⊆ 𝔟 which in turn implies 𝔤_{2} ⊆ 𝔟.

Suppose that π(𝔟) ∩ (U_{ℂ} + (𝔭_{2})_{ℂ}) ⊈ (𝔭_{2})_{ℂ}. The kernel for the mapping ãdθX induced by ad θX on 𝔤_{ℂ}/(𝔤_{f})_{ℂ} is given by (𝔭_{2})_{ℂ} + V_{ℂ} and since ad θX: U → 𝔭_{2} is bijective (Lemma IV.24), the same holds for ãdθX. Thus our assumption and the invariance of π(𝔟) under ãdθX yield π(𝔟) ∩ (𝔭_{2})_{ℂ} ≠ {0}, i.e., 𝔟 ∩ (𝔭_{2})_{ℂ} ≠ {0}.

Since 𝔤_{2} = ℝH ⊕ 𝔤′_{2} , where 𝔤′_{2} is a simple real Lie algebra (Lemma IV.12, Theorem IV.13]), the 𝔨_{2} -fixed elements in (𝔭_{2})_{ℂ} are ℂH and the effective submodule is (𝔭′_{2})_{ℂ}, where 𝔭′_{2} := 𝔤′_{2} ∩ 𝔭_{2} . Now we use the 𝔨_{2} -invariance of 𝔟 to see that

𝔟 ∩ (𝔭_{2})_{ℂ} = (𝔟 ∩ ℂH) ⊕ (𝔟 ∩ (𝔭′_{2})_{ℂ}).

Hence one of the two summands must be non-zero. If the first one is non-zero, then H ∈ 𝔟 and we can decompose 𝔟 into ad H eigenspaces

𝔟 = θU_{ℂ} + θV_{ℂ} + (𝔤_{1})_{ℂ} + (𝔟 ∩ (𝔤_{2})_{ℂ}) + (𝔟 ∩ V_{ℂ}) + (𝔟 ∩ U_{ℂ}).

If 𝔤_{2} ⊈ 𝔟, then 𝔟∩(𝔭_{2})_{ℂ} = ℂH and consequently 𝔟∩U_{ℂ} has codimension 1 since 𝔟 is maximal isotropic. Then we apply the injectivity of ad θX: U → 𝔭_{2} a second time to conclude that dim U = dim 𝔭_{2} ≤ 2. Now 𝔭′_{2} is at most one-dimensional, so that p′_{2} = {0} and therefore 𝔤_{2} = ℝH ⊆ 𝔟. This proves that 𝔤_{2} ⊆ 𝔟 if H ∈ 𝔟.

Next we assume that 𝔟 ∩ (𝔭_{2})_{ℂ} = 𝔟 ∩ (𝔭′_{2})_{ℂ} ≠ {0}. The element X is orthogonal to 𝔟 with respect to the form (Y, Z) ↦ f([Y, Z]) since f([X, Z]) = B(θX, [X, Z]) = B([θX, X], Z) = −B(H, Z) and therefore X ∈ 𝔟 because 𝔟 is maximal isotropic. Thus [X, θX] = H ∈ 𝔟, a contradiction.

Taking all these cases together, we have proved that 𝔤_{2} ⊆ 𝔟. Hence

𝔟 = θU_{ℂ} + θV_{ℂ} + (𝔤_{1})_{ℂ} + (𝔤_{2})_{ℂ} + (𝔟 ∩ V_{ℂ}).

Next we use the fact that 𝔟 + \overline{𝔟} is a subalgebra invariant under complex conjugation, so that it can be written as 𝔥_{ℂ} for 𝔥 := (𝔟 + \overline{𝔟}) ∩ 𝔤. Then the subalgebra 𝔥 ⊆ 𝔤 satisfies

θU + θV + 𝔤_{1} + 𝔤_{2} ⊆ 𝔥 ⊆ θU + θV + 𝔤_{1} + 𝔤_{2} + V

and therefore

(VI.4)  𝔥 = θU + θV + 𝔤_{1} + 𝔤_{2} + (V ∩ 𝔥).

In view of Proposition IV.1, the (𝔤_{1} + 𝔤_{2})-module V is simple, hence 𝔥 ∩ V = V holds because it is non-zero. Finally [V, V] = U (Proposition IV.1) contradicts (VI.4).

So far we have seen that f has no algebraic polarization for 𝔤 if V_{k} ≠ {0} and that 𝔟 = (𝔤_{2} + θU)_{ℂ} is a polarization if this is not the case. In [14] we will construct representations by applying the orbit method to the polarization in Lemma VI.13. The representations associated to the polarization in Remark VI.14 have been studied in [31] and also in [30].

The notion of algebraic polarization is closely related to the notion of strongly integrable complex polarizations (in the terminology of [37, p.92]) which is defined in terms of Lagrangian distributions.

Lemma VI.16. The Lagrangian distributions L_{±,k} on M_{k} are involutive.

Proof. Consider the derivative δ: 𝔮_{k,red} → C_{X^{k},𝔮_{k,red}} of the orbit map

Q_{k,red} → M_{k}, q ↦ ρ_{𝔮_{k,red}}(q, X^{k})

at the identity. According to [8, p.235], it suffices to show that the preimages of L_{±,k}(X^{k}) in 𝔮_{k,red} under the complex linear extension δ_{ℂ}: (𝔮_{k,red})_{ℂ} → (C_{X^{k},𝔮_{k,red}})_{ℂ} is a subalgebra of (𝔮_{k,red})_{ℂ}. But from Lemma V.6 we see that

δ(X) = − ad(X^{k}) ∘ θ ∀X ∈ 𝔮_{k,red}.

Moreover Lemma IV.24 together with [29, III(3.27)] shows that

δ(𝔤_{2,k}) = U_{k}, δ(U_{k}) = 𝔭_{2,k}, δ_{ℂ}(V_{k}^{±}) = V_{k}^{±}.

Since δ_{ℂ}^{−1}(0) = (𝔨_{2,k})_{ℂ} we find

δ_{ℂ}^{−1}(L_{+,k}(X^{k})) = V_{k}^{+} + (𝔤_{2,k})_{ℂ}

and

δ_{ℂ}^{−1}(L_{−,k}(X^{k})) = (U_{k})_{ℂ} + V_{k}^{−} + (𝔨_{2,k})_{ℂ}

which proves the claim in view of Proposition IV.5(vii).

Consider

D_{±,k} = L_{±,k} ∩ \overline{L_{±,k}} ∩ T (M_{k})

and

E_{±,k} = (L_{±,k} + \overline{L_{±,k}}) ∩ T (M_{k}).

Since D_{±,k}(q.X^{k}) = q.D_{±,k}(X^{k}), we see that D_{±,k} are distributions and hence the distributions L_{±,k} are complex polarizations in the sense of Woodhouse (cf. [37, p.92] and note that the definition of a polarization in [8] is weaker). Similarly we see that E_{±,k} are distributions. We recall that the polarization L_{±,k} is called strongly integrable if E_{±,k} is involutive.

Proposition VI.17.

(i) L_{+,k} is strongly integrable iff V_{k} = {0}.

(ii) L_{−,k} is strongly integrable. Proof. We use the same reasoning as in the proof of Lemma VI.16 but for E_{±,k} instead of L_{±,k}. In fact, we see that

δ^{−1}(E_{+,k}(X^{k})) = V_{k} + 𝔤_{2,k} and δ^{−1}(E_{−,k}(X^{k})) = 𝔥_{k} + 𝔨_{2,k}. Since V_{k} + 𝔤_{2,k} is a Lie algebra iff V_{k} = {0} whereas 𝔥_{k} + 𝔨_{2,k} always is a Lie algebra, the claim follows.

Recall that integrable real distributions of a real manifold are the same as real foliations.

Proposition VI.18.

(i) The leaves of the foliation E_{−,k} are precisely the fibers of p_{U_{k}}: M_{k} → Ω_{k}. (ii) The leaves of the foliation D_{−,k} are precisely the fibers of p_{𝔥_{k}}: M_{k} →

Ω_{k} + V_{k}. Proof. (i) Let E = p_{U_{k}}^{−1}(X^{k})∩M_{k}. Then H_{k} acts transitively on E (Lemma V.6) and for any q ∈ exp(U_{k}) exp(V_{k}) we have

T_{q.X^{k}}(E) = q.T_{X^{k}}(E) = q.(𝔭_{2,k} + V_{k}) = q.E_{−,k}(X^{k}) = E_{−,k}(q.X^{k}). This shows that E is an integral manifold for E_{−,k}. Since G_{2,k} permutes the fibers and E_{−,k} is G_{2,k} invariant by definition, this proves the claim. (ii) According to Proposition V.12, the fiber through Z ∈ M_{k} is Z + ad(X)θU_{k} with X = p_{U_{k}}(Z). In particular the fiber D over X^{k} is X^{k} + 𝔭_{2,k} and the tangent space T_{X^{k}}(D) is 𝔭_{2,k} = D_{−,k}(X^{k}). As in the proof of (i), now the invariance of D_{−,k} and the equivariance of p_{𝔥_{k}} prove the claim.

We denote the space of leaves of D_{−,k} by M_{k}/D_{−,k} and note that it is just Ω_{k} + V_{k}. Similarly the space of leaves of E_{−,k} gets identified with Ω_{k}.

Remark VI.19. The polarization L_{−,k} is non-positive in the sense of [37, p.91]. In fact, the symplectic vector space E_{−,k}(X^{k})/D_{−,k}(X^{k}) is isomorphic to V_{k} with the symplectic form ω_{X^{k}}^{(k)}|_{V_{k}×V_{k}}. But Proposition VI.12 shows that

\begin{aligned}γ^{∗}ω_{X^{k}}^{(k)}(Y′, Y′′) & = γ^{∗}α_{X^{k}}^{(k)}([Y′, Y′′], 0) = ⟨φ(X^{k}), [Y′, Y′′]⟩ \\ & = −B(X^{k}, θ[Y′, Y′′]) = −4A_{X^{k}}(Y′, Y′′), \end{aligned}

where A_{X}(Y′, Y′′) = −\frac{1}{4}(X | [Y′, Y′′]) (cf. [29, p.101] ). Since A_{X^{k}}I_{k} + iA_{X^{k}} is positive definite (cf. [loc.cit.]), the claim follows.

VII. Examples

Let 𝕂 = ℝ, ℂ, ℍ and E a 𝕂-left vector space endowed with a positive definite hermitean form (· | ·), i.e.,

\overline{(v | w)} = (w | v), λ(v | w) = (λv | w) and (v | λw) = (v | w)\overline{λ}

and (v | v) > 0 for all v ≠ 0. The standard form on 𝕂^{n} is

(x | y) = ∑_{j=1}^{n} x_{j}\overline{y}_{j}.

We write U(E) for the group of 𝕂-linear isometries of E.

In addition to (· | ·) we consider a non-degenerate skewhermitean form B(·, ·), i.e.,

\overline{B(v, w)} = −B(w, v), λB(v, w) = B(λv, w) and B(v, λw) = B(v, w)\overline{λ}.

Here the standard forms are (1) 𝕂 = ℝ, E = ℝ^{2n}, B(x, y) = ∑_{j=1}^{n}(x_{j}y_{j+n} − x_{j+n}y_{j})

(2) 𝕂 = ℂ, E = ℂ^{p+q}, B(x, y) = i (∑_{j=1}^{p} x_{j}\overline{y}_{j} − ∑_{j=p+1}^{p+q} x_{j}\overline{y}_{j})

(3) 𝕂 = ℍ, E = ℍ^{n}, B(x, y) = ∑_{j=1}^{n} x_{j}J \overline{y}_{j}, where 1, I, J , K are the standard basis for ℍ. We write U(B) for the group of 𝔨-linear B -isometries of E. The Lie

algebras corresponding to the groups introduced above will be denoted by 𝔲(E) and 𝔲(B).

Consider the endomorphism φ_{B} ∈ End_{𝕂}(E) defined by

(VII.1)  B(v, w) = (φ_{B}v | w)

as well as the transpose operations ∗ and ♯ for (· | ·) and B(·, ·). Then we have

(VII.2)  𝔲(B) = {X ∈ 𝔤𝔩(E): X^{♯} = −X}

and

(VII.3)  φ_{B}^{∗} = −φ_{B}, X^{♯} = φ_{B}^{−1}X^{∗}φ_{B}, φ_{B}^{♯} = −φ_{B}.

In the standard cases, where elements of End_{𝕂}(E) are given by right multiplication of matrices on row vectors we have

(1) φ_{B} = ( \begin{matrix}0 & 1_{n} \\ −1_{n} & 0 \end{matrix} ) and 𝔲(B) = 𝔰𝔭(n, ℝ).

(2) φ_{B} = i ( \begin{matrix}1_{p} & 0 \\ 0 & −1_{q} \end{matrix} ) and 𝔲(B) = 𝔲(p, q).

(3) φ_{B} = J 1_{n} and 𝔲(B) = 𝔰𝔬^{∗}(2n).

For each X ∈ 𝔲(B) we have a symmetric real bilinear form B_{X} on E which is given by

(VII.4)  B_{X}(v, w) = −B(X.v, w).

We set

(VII.5)  W_{B} := {X ∈ 𝔲(B): B_{X} positive semidefinite}

and

(VII.6)  W_{max,B} := {X ∈ 𝔲(B): (∀v ∈ E)B(v, v) = 0 ⇒ B(X.v, v) ≤ 0}.

Then W_{B} ⊆ W_{max,B} are closed convex cones which are invariant under conjugation by elements from U(B). We assume that

(VII.7)  φ_{B}^{2} = − id .

Then φ_{B} ∈ W_{B}, so that the cones W_{B} and W_{max,B} are non-trivial.

Note that (VII.7) is true for the standard cases. It follows from (VII.7) that ♯ and ∗ commute. In fact, up to a renormalization of B the two conditions are equivalent. From this it follows that \frac{1}{2}φ_{B} is an H -element of the quasihermitean reductive algebra 𝔲(B). To see this, note that 𝔲(B) is invariant under ∗ and θ(X) = −X^{∗} defines a Cartan involution on 𝔲(B). Therefore the centralizer

(VII.8)  𝔨(B) := 𝔲(B) ∩ 𝔲(E)

of φ_{B} in 𝔲(B) is a maximal compactly embedded subalgebra of 𝔲(B). Then the equality

(VII.9)  (ad φ_{B})^{2}X = −2(X^{∗} + X) ∀X ∈ 𝔲(B)

proves the claim.

The complex structure on

(VII.10)  𝔭(B) = {X ∈ 𝔲(B): X^{∗} = X}

is simply given as multiplication by φ_{B}.

For v ∈ E let P_{v} be the associated rank one operator, which is defined by P_{v}w = (w | v)v. Choose an orthonormal basis {v_{1}, …, v_{m}} for E and consider the map

τ: ∑_{j=1}^{m} ℝv_{j} → 𝔲(B), ∑_{j=1}^{m} x_{j}v_{j} ↦ \frac{1}{2} ∑_{j=1}^{m} x_{j}(φ_{B}P_{v_{j}} + P_{v_{j}}φ_{B}).

Then 𝔱 := τ(E) is a Cartan algebra of 𝔨(B) and 𝔲(B). Note that τ is injective for 𝕂 = ℂ and 𝕂 = ℍ but not for 𝕂 = ℝ.

Using the embedding Mat(n × n, ℍ) → Mat(2n × 2n, ℂ) given by

A + J B ↦ ( \begin{matrix}A & −B \\ \overline{B} & \overline{A} \end{matrix} )

for A, B ∈ Mat(n × n, ℂ) ⊆ Mat(n × n, ℍ) one has the following matrix representations for our standard cases:

(1)

𝔰𝔭(n, ℝ) = \{( \begin{matrix}A & B \\ C & −A^{t} \end{matrix} ): A, B, C ∈ Mat(n × n, ℝ), B^{t} = B, C^{t} = C\}
𝔨 = \{( \begin{matrix}A & B \\ −B & A \end{matrix} ) ∈ 𝔰𝔭(n, ℝ): A^{t} = −A, B^{t} = B\}
𝔱 = \{( \begin{matrix}0 & B \\ −B & 0 \end{matrix} ) ∈ 𝔰𝔭(n, ℝ): B diagonal\}

(2) Here we have complex block matrices according to the partion (p, q) of p + q.

𝔲(p, q) = \{( \begin{matrix}A & B \\ B^{∗} & D \end{matrix} ): A^{∗} = −A, D^{∗} = −D, \}
𝔨 = \{( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ) ∈ 𝔲(p, q)\}
𝔱 = \{( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ) ∈ 𝔲(p, q): A, D diagonal\}

(3)

𝔰𝔬^{∗}(2n) = \{( \begin{matrix}A & −B \\ \overline{B} & \overline{A} \end{matrix} ): A, B ∈ Mat(n × n, ℂ), A^{t} = −A, B^{∗} = B\}
𝔨 = \{( \begin{matrix}A & −B \\ B & A \end{matrix} ) ∈ 𝔰𝔬^{∗}(2n): A = \overline{A} = −A^{⊤}, B = \overline{B} = B^{⊤}\}
𝔱 = \{( \begin{matrix}0 & −B \\ B & 0 \end{matrix} ) ∈ 𝔨: B diagonal\}

Choose a maximal B -isotropic subspace F ⊆ E and an orthonormal basis {v_{1}, …, v_{r}} of F. The number r is the real rank of 𝔲(B). We have

(VII.11)  F ⊕ φ_{B}F = (𝕂v_{1} ⊕ 𝕂φ_{B}v_{1}) ⊕ … ⊕ (𝕂v_{r} ⊕ 𝕂φ_{B}v_{r}).

The (𝕂v_{j} ⊕ 𝕂φ_{B}v_{j}) are pairwise orthogonal w.r.t. (· | ·) and B. The matrix of

B on (𝕂v_{j} ⊕ 𝕂φ_{B}v_{j}) w.r.t. the basis \{v_{j}, −φ_{B}v_{j}\} is ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). All B -isotropic

vectors are contained in F ⊕ φ_{B}F since

(VII.12)  E_{red} := (F ⊕ φ_{B}F)^{⊥} = (F ⊕ φ_{B}F)^{⊥_{B}}.

We have E_{red} = {0} iff 𝔲(E) is of tube type.

In the standard cases we have

(1) 𝔰𝔭(n, ℝ), r = n, E_{red} = {0}.

(2) 𝔲(p, q), p ≥ q, r = q, E_{red} = ℂ^{p−q}.
(3) 𝔰𝔬^{∗}(2n), r = [\frac{n}{2}], E_{red} = \{ \begin{matrix}\{0\} & for n even \\ ℍ & for n odd \end{matrix} .

For the skewhermitean planes 𝕂^{2} := (𝕂v_{j} ⊕ 𝕂φ_{B}v_{j}), the real rank of 𝔲(B|_{𝕂^{2}}) is 1 and the corresponding (H_{1})-homomorphism κ_{j}: 𝔰𝔩(2, ℝ) → 𝔲(B|_{𝕂^{2}}) is the natural inclusion. We have in particular

H_{κ_{j}} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) and X_{κ_{j}} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) .

Similarly (VII.11) gives an inclusion 𝔲(B|_{𝕂^{2}})^{r} → 𝔲(B) which provides in particular an inclusion of 𝔰𝔩(2, ℝ)^{r}. This makes it rather transparent how the different 𝔰𝔩(2, ℝ)-subalgebras 𝔰^{k} look like, namely exactly like they are embedded into 𝔰𝔩(2, ℝ)^{r}. Using Proposition II.7, we see that all the X_{κ_{j}} are contained in −W_{min}(\frac{1}{2}φ_{B}) and in fact generate it as an invariant closed convex cone. We claim that

(VII.13)  W_{max,B} = W_{max} (\frac{1}{2}φ_{B}) .

The inclusion ⊆ follows from φ_{B} ∈ W_{max,B}. For the converse let Y ∈ W_{max}(\frac{1}{2}φ_{B}) and v ∈ E be non-zero B -isotropic. Then there exists a g ∈ U(B) with g.v = v_{1} and we can extend {g.v, −φ_{B}g.v} to a orthogonal basis {w_{1}, w_{2}, …, w_{m}} for E. Then X_{κ_{1}} ∈ End_{𝕂}(E) satisfies X_{κ_{1}}.w_{2} = w_{1} , X_{κ_{1}}.w_{l} = 0 for l ≠ 2. Thus Ad(g^{−1}).X_{κ_{1}} = g^{−1}X_{κ_{1}}g ∈ −W_{min}(\frac{1}{2}φ_{B}). But now we have

B(Y.v, v) = B(gY g^{−1}.(g.v), g.v)
= (φ_{B}gY g^{−1}.(g.v) | g.v)
= −(gY g^{−1}.(g.v) | φ_{B}g.v)
= (gY g^{−1}.w_{1} | w_{2})
= (gY g^{−1}X_{κ_{1}}.w_{2} | w_{2})
= tr(gY g^{−1}X_{κ_{1}}) ≤ 0

which implies the claim.

Now we turn the structure of the Jacobson-Morosow parabolics. The (H_{1})-embedding described above shows that

(VII.14)  𝔷_{𝔲(B)}(H^{k}) = 𝔷_{𝔲(B|_{𝕂^{2k}})}(H^{k}) ⊕ 𝔲(B|_{𝕂^{2(r−k)}⊕E_{red}}).

Note that H^{k} belongs to the 𝔰𝔩_{2} -triple of maximal rank in 𝔲(B|_{𝕂^{2k}}). In view of the simple structure of H^{k}, as an endomorphism of 𝕂^{2k}, it is

clear that its commutant consists of those linear mappings leaving the eigenspaces of H^{k} invariant. Writing accordingly 𝕂^{2k} = 𝕂^{k} ⊕ 𝕂^{k}, we see that

𝔷_{𝔲(B|_{𝕂^{2k}})}(H^{k}) = \{( \begin{matrix}A & 0 \\ 0 & −A^{∗} \end{matrix} )\}

so that we find

𝔷_{𝔲(B|_{𝕂^{2k}})}(H^{k}) ≅ 𝔤𝔩(k, 𝕂).

Putting all this together we get

(VII.15)  𝔷_{𝔲(B)}(H^{k}) ≅ 𝔲(B|_{E_{k}}) ⊕ 𝔤𝔩(k, 𝕂) ≅ 𝔤_{1,k} ⊕ 𝔤_{2,k},

where E_{k} = 𝕂^{2(r−k)} ⊕ E_{red}.

The corresponding parabolic 𝔮_{k} is given by

(VII.16)  𝔮_{k} = {X ∈ 𝔲(B): X.E(H^{k}; 1) ⊆ E(H^{k}; 1)},

where one has to read X.v as the matrix product v · X since we are dealing with a skew-field. To prove (VII.16) one checks that 𝔮_{k} as above contains the minimal parabolic, hence is parabolic, and then that it has the correct Levi algebra.

The corresponding flag manifold M_{k} := SU(E)/Q_{k} is the space of all k -dimensional isotropic subspaces of E. For k = 1 we find in particular the isotropic part of the projective space and for k = r the space of all Lagrangian subspaces. The minimal non-zero nilpotent orbit of convex type O_{1} is the image of the map

{v ∈ V \ {0}: B(v, v) = 0} → 𝔲(B), v ↦ X_{v},

where X_{v} is defined by

(VII.17)  B(Y v, v) = tr(Y X_{v}), ∀Y ∈ 𝔲(B)

as can be seen from the considerations leading to (VII.13).

Note that the above class of examples covers (ignoring the trivial central factor in 𝔲(p, q)) all simple hermitean Lie algebras up to 𝔰𝔬(n, 2) and the two exceptional ones. We conclude with some explicit information on those cases (cf. [29, pp.115–119]).

Example VII.1. For 𝔤 = 𝔰𝔬(2, n) we have r = 2.

For k = 1 we have 𝔷_{𝔤}(H^{k}) ≅ 𝔰𝔩(2, ℝ) ⊕ ℝH^{k} ⊕ 𝔰𝔬(n − 2), V_{1} is a real module of dimension 2(n−2). More precisely, it is the tensor product ℝ^{2}⊗ℝ^{n−2} , where the factors are endowed with the natural representations.

For k = 2 we have 𝔷_{𝔤}(H^{k}) ≅ ℝH^{k} ⊕ 𝔰𝔬(1, n − 1), V_{2} = {0}, and U_{2} ≅ ℝ^{n} is n-dimensional Minkowski space. Here the parabolic corresponds to the identification of 𝔤 with the Lie algebra 𝔤(U_{2}), where we consider U_{2} as a Jordan algebra.

Example VII.2. For the exceptional hermitean algebra 𝔤 = 𝔢_{(6,−14)} one has r = 2.

For k = 1 we have 𝔷_{𝔤}(H^{k}) ≅ ℝH^{k} ⊕ 𝔰𝔲(5, 1), and V_{1} is a real module of dimension 20, hence a real form of ⋀^{3}(ℂ^{6}).

For k = 2 we have 𝔷_{𝔤}(H^{k}) ≅ ℝ ⊕ ℝH^{k} ⊕ 𝔰𝔬(7, 1), dim U_{2} = 8 and dim V_{2} = 16. Here V_{2} = V_{1} ⊗_{ℂ} V_{2} , where ℝ acts on V_{1} = ℂ by multiplication by i and 𝔰𝔬(7, 1) acts on V_{2} ≅ ℂ^{8} according to the spin representation.

Example VII.3. For the exceptional hermitean algebra 𝔤 = 𝔢_{(7,−25)} one has r = 3.

For k = 1 we have 𝔷_{𝔤}(H^{k}) ≅ ℝH^{k} ⊕ 𝔰𝔬(10, 2), and that V_{1} is a real module of dimension 32, hence the spin representation.

For k = 2 we have 𝔷_{𝔤}(H^{k}) ≅ 𝔰𝔩(2, ℝ) ⊕ ℝH^{k} ⊕ 𝔰𝔬(9, 1), dim U_{2} = 10 and dim V_{2} = 32. Here V_{2} = ℝ^{2} ⊗_{ℝ} ℝ^{16} , where 𝔰𝔬(9, 1) acts on ℝ^{16} according to the spin representation.

For k = 3 we have 𝔷_{𝔤}(H^{k}) ≅ ℝH^{k} ⊕𝔢_{(6,−26)}, dim U_{3} = 27 and V_{3} = {0}. This is the Jordan algebra situation, where U_{3} = Herm(3, IO).

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Joachim Hilgert Mathematisches Institut Technische Universität Clausthal Erzstr. 1 D-38678 Clausthal-Zellerfeld Germany Karl-Hermann Neeb Mathematisches Institut Universität Erlangen-Nürnberg Bismarckstr. 1 \frac{1}{2} D-91054 Erlangen Germany

Bent Ørsted

Matematisk Institut Odense Universitet Campusvej 55 DK-5230 Odense Danmark

Received October 14, 1994 and in final form February 22, 1995