Journal of Lie Theory Version of May 4, 1995 Volume 4 (1994) 139–183 C ◯1994 Heldermann Verlag

The classification of Lie algebras with invariant cones

Karl-Hermann Neeb

Communicated by J. Hilgert

Introduction

We say that a finite dimensional real Lie algebra 𝔤 contains invariant cones if there exists a pointed generating invariant closed convex cone W ⊆ 𝔤 which is invariant under the group of inner automorphisms. In this paper we obtain a classification of all Lie algebras which contain invariant cones.

This classification is the final stage of a development started in [3] and ranging over [4], [5], [6], [22] and [12]. The work of Hilgert and Hofmann on invariant cones in general Lie algebras which is completely documented in Chapter III of [6] provided the basic structure theory for Lie algebras with invariant cones.

We say that a subalgebra 𝔞 of the Lie algebra 𝔤 is compactly embedded if the closure of the group generated by e^{ad 𝔞} is compact. One central fact in [6] is that a Lie algebra with invariant cones always contains a compactly embedded Cartan algebra 𝔱. Associated to such a Cartan algebra 𝔱 is a uniquely determined maximal compactly embedded subalgebra 𝔨 containing 𝔱 ([6, A.2.40]). It was shown in [6] that a necessary condition for a Lie algebra 𝔤 to contain invariant cones is that 𝔤 is quasihermitean, i.e., that the centralizer of the center 𝔷(𝔨) of 𝔨 is not bigger than 𝔨, and that 𝔤 has cone potential, a property which can be formulated in terms of the root decomposition of 𝔤 with respect to 𝔱. The work of Spindler [22] contains a universal construction of a class of Lie algebras containing all those with cone potential. On the basis of this construction and the results from [6] we have given in [12] a characterization of the Lie algebras with invariant cones as those with the following properties:

(1) 𝔤 is not compact semisimple,

(2) all simple ideals contained in 𝔤 are either compact or hermitean, and (3) 𝔤 has strong cone potential, another property which can be defined in terms of the root decomposition and which implies cone potential. Unfortunately this characterization does not give an explicit description

of all Lie algebras with invariant cones. Every finite dimensional Lie algebra 𝔤

ISSN 0940–2268 / $2.50 ^{_{C} ◯} Heldermann Verlag with Levi decomposition 𝔤 = 𝔯× 𝔰 decomposes into a direct sum 𝔤 = 𝔤_{1} ⊕ 𝔰_{1} , where 𝔰_{1} is a maximal semisimple ideal of 𝔤 and the Levi complements of 𝔤_{1} act effectively on the radical of 𝔤. If 𝔤 contains invariant cones, then for 𝔤_{1} the results in [12] provide a homomorphism q: 𝔤 → 𝔥_{n} × 𝔰𝔭(n, IR) with central kernel, where 𝔥_{n} is the (2n + 1)-dimensional Heisenberg algebra, such that q maps the nilradical 𝔫 onto 𝔥_{n} and 𝔤 contains a reductive subalgebra 𝔩 with 𝔤 = 𝔫× 𝔩 such that 𝔩 is mapped into 𝔰𝔭(n, IR). In this paper we obtain an explicit description all Lie algebras with invariant cones which rests on these ideas.

The key ingredient in our classification is an observation relating the results in [14] to the classification problem. To state it, let 𝔤 = 𝔫× 𝔩 be a semidirect decomposition, where 𝔫 is the nilradical and 𝔩 is reductive. Then V := [𝔩, 𝔫] is an 𝔩-module which carries several invariant symplectic structures. The main point is that there exists a symplectic structure Ω on V such that the cone W_{V} := {X ∈ 𝔤: (∀v ∈ V)Ω(X.v, v) ≥ 0} has interior points.

In general we say that a symplectic vector space (V, Ω) on which a reductive Lie algebra 𝔤 acts via a homomorphism 𝔤 → 𝔰𝔭(V, Ω) is of convex type if

(1) the cone W_{V} has interior points, and

(2) the center acts by semisimple mappings with purely imaginary eigenvalues. The first three sections of this paper consist of a classification of all

symplectic modules of convex type. After some reductions one can use the closely related results of contraction representations from [14] which provide an explicit classification in the case where 𝔤 is simple hermitean and V is complex simple. In Section IV we also explain how these representations are related to strongly equivariant embeddings of bounded hermitean domains into the Siegel space which in turn can be characterized by so called (H_{1})-homomorphisms on the Lie algebra level (cf. [21], [20]).

In Section V we use these results to obtain a classification of all Lie algebras which contain invariant cones by constructing them from symplectic modules of convex type of reductive quasihermitean Lie algebras. The insights obtained by the point of view of Section III provide also new information on the structure of these Lie algebras which play a crucial role in the theory of highest weight representations of the associated groups and semigroups (cf. [15], [16]) and also in the complex geometry of certain coadjoint orbits which carry Kähler structures (cf. [13]).

The subsection on the relations between Lie algebras with invariant cones and parabolic subalgebras of hermitean Lie algebras is based on an observation of S. Sahi. We thank him for many illuminating discussions on the subject.

We also express our deep gratitude to the referee who made a lot of very valuable suggestions to improve the exposition of the paper and to clarify the arguments. The whole first section of the present version of this paper is a reaction on his comments. It contains some rather general result on semisimple symplectic representations of reductive Lie algebras without the convex type condition.

I. Symplectic representation theory

In this section we consider modules of Lie algebras which carry an invariant symplectic form. We are mainly interested in the special class of symplectic modules of convex type which will be studied in Section II. Our strategy is first to obtain some general results on the structure of symplectic modules that can be specialized to obtain sharper results for modules of convex type. In the following 𝔤 always denotes a finite dimensional real Lie algebra.

Definition I.1. (a) A symplectic 𝔤-module is a symplectic vector space (V, Ω) on which 𝔤 acts such that the symplectic form Ω is invariant, i.e.,

Ω(X.v, w) + Ω(v, X.w) = 0

holds for all v, w ∈ V and X ∈ 𝔤. An isomorphism of symplectic 𝔤-modules (V, Ω) and (V′, Ω′) is a 𝔤-module isomorphism φ: V → V′ with

Ω′(φ(v), φ(w)) = Ω(v, w)

for all v, w ∈ V .

(b) If (V_{j}, Ω_{j}), j = 1, …, n are symplectic 𝔤-modules, then the direct sum V := ⨁_{j=1}^{n} V_{j} is a symplectic 𝔤-module with respect to

Ω( ∑_{j=1}^{n} v_{j}, ∑_{k=1}^{n} w_{k}) := ∑_{j=1}^{n} Ω(v_{j}, w_{j}).

(c) A submodule W ⊆ V is called isotropic if it is an isotropic subspace of (V, Ω) and non-degenerate if the restriction Ω_{W} of Ω to W is non-degenerate.

(d) We say that a symplectic 𝔤-module (V, Ω) is indecomposable if it cannot be written as an orthogonal direct sum of two submodules different from {0} and V .

Remark I.2. In the following we only consider semisimple modules. We recall that if a Lie algebra 𝔤 has a faithful representation ρ: 𝔤 → 𝔤𝔩(V) such that V is a semisimple 𝔤-module, then 𝔤 is reductive (cf. [1, Ch. I, §6, No.4]). Therefore it is no loss of generality to assume that the Lie algebra under consideration is reductive.

Our first objective is to show that a semisimple symplectic 𝔤-module always decomposes as an orthogonal direct sum of indecomposable summands which are uniquely determined up to isomorphy, and to obtain a description of the indecomposable semisimple modules.

Lemma I.3. Let (V, Ω) be a symplectic 𝔤-module, W ⊆ V a simple submodule and Ω_{W} the restriction of Ω to W. Then either W is isotropic or nondegenerate. Proof. Let W_{0} := W^{⊥} ∩W = {w ∈ W: Ω(w, W) = {0}}. Then the invariance of Ω implies that W_{0} is a submodule of W. Hence W_{0} = {0} or W_{0} = W because W is simple. In the first case W is non-degenerate and in the second case W is isotropic.

Definition I.4. A simple 𝔤-module W is said to be of symplectic type if it carries a 𝔤-invariant symplectic form.

The following propositions collects some of the basic properties of symplectic 𝔤-modules.

Proposition I.5. (i) If W is a 𝔤-module, then

(W ⊕ W^{∗}, Ω_{W}^{∗}) with Ω_{W}^{∗} ((w, α), (w′, α′)) = α(w′) − α′(w)

is a symplectic 𝔤-module. (ii) If (V, Ω) is a symplectic 𝔤-module, W ⊆ V is maximal isotropic and a 𝔤-submodule, then there exists an isomorphism of symplectic 𝔤-modules

φ: (V, Ω) → (W ⊕ W^{∗}, Ω_{W}^{∗})

with φ|_{W} = id_{W} if and only if W has a module complement. (iii) If (W, Ω) is a symplectic 𝔤-module, then

(W ⊕ W^{∗}, Ω_{W}^{∗}) ≅ (W, Ω) ⊕ (W, −Ω).

(iv) If W is a simple 𝔤-module, then either W is of symplectic type and (W ⊕ W^{∗}, Ω_{W}^{∗}) decomposes into an orthogonal direct sum, or (W ⊕ W^{∗}, Ω_{W}^{∗}) is indecomposable.

Proof. (i) That W ⊕ W^{∗} is in fact a symplectic 𝔤-module follows from

Ω_{W}^{∗} (X.(w, α), (w′, α′)) = (X.α)(w′) − α′(X.w)
= −α(X.w′) + (X.α′)(w)
= −Ω_{W}^{∗} ((w, α), X.(w′, α′)).

(ii) If φ exists, then φ(W^{∗}) ⊆ V is a module complement for W.

Suppose, conversely, that U ⊆ V is a module complement of W. Since W is maximal isotropic, we have W^{⊥} = W and therefore

0 = V^{⊥} = W^{⊥} ∩ U^{⊥} = W ∩ U^{⊥}.

On the other hand dim W = \frac{1}{2} dim V yields dim U = dim U^{⊥}, so that we also have the direct module decomposition V = W ⊕ U^{⊥}. Let

U → V = W ⊕ U^{⊥}, u ↦ (γ(u), δ(u))

denote the corresponding embedding of U. We put

U′ := \{u − \frac{1}{2}γ(u) = (\frac{1}{2}γ(u), δ(u)): u ∈ U\}.

Since γ is a homomorphism of 𝔤-modules, U′ is a submodule of V . Moreover, U′ ∩ W = {0} because δ(u) = 0 implies u ∈ W ∩ U = {0}. We conclude that U′ is a module complement for W.

We claim that U′ is isotropic. In fact, since W is isotropic, we have

\begin{aligned}Ω(u − \frac{1}{2}γ(u), u′ − \frac{1}{2}γ(u′)) & = Ω(u, u′) − \frac{1}{2}Ω(γ(u), u′) − \frac{1}{2}Ω(u, γ(u′)) \\ & = Ω(u, u′) − \frac{1}{2}Ω(u, u′) − \frac{1}{2}Ω(u, u′) = 0. \end{aligned}

Now we define φ_{1}: U′ → W^{∗} by φ_{1}(u)(w) = Ω(u, w) and observe that since W is maximal isotropic, φ_{1} is an isomorphism of 𝔤-modules. Hence

φ: V = W ⊕ U′ → W ⊕ W^{∗}, (w, u) ↦ (w, φ_{1}(u))

is a homomorphism of 𝔤-modules extending id|_{W} . It remains to show that φ is an isomorphism of symplectic 𝔤-modules. This follows from

\begin{aligned}Ω_{W}^{∗} ((w, φ_{1}(u)), (w′, φ_{1}(u′))) & = φ_{1}(u)(w′) − φ_{1}(u′)(w) \\ & = Ω(u, w′) − Ω(u′, w) = Ω(w + u, w′ + u′). \end{aligned}

(iii) Put (V, Ω_{V}) := (W, Ω) ⊕ (W, −Ω). Then W̃ := {(w, w): w ∈ W } and U := {(w, −w): w ∈ W } are maximal isotropic submodules of V . Therefore (ii) implies that

(V, Ω_{V}) ≅ (W̃ ⊕ W̃^{∗}, Ω∗ ; _{W}) ≅ (W ⊕ W^{∗}, Ω_{W}^{∗}).

(iv) Suppose that (V, Ω) := (W ⊕ W^{∗}, Ω_{W}^{∗}) is decomposable. We show that W permits a 𝔤-invariant symplectic structure. Let V = U ⊕U^{⊥} denote a non-trivial orthogonal decomposition, where U is a non-degenerate submodule of V . Since V is a sum of two simple 𝔤-modules, it follows that U is simple. Since U is non-degenerate, we have U ∩ W = U ∩ W^{∗} = {0}. Therefore the projection p: U → W is an isomorphism of 𝔤-modules and consequently W is of symplectic type.

Conversely, if W is of symplectic type, then (iii) shows that (W ⊕ W^{∗}, Ω_{W}^{∗}) decomposes as asserted. This completes the proof.

Lemma I.6. Let V be a 𝔤-module and V^{∗} its dual module. Then the following are equivalent:

(i) V carries a symplectic form Ω such that (V, Ω) is a symplectic 𝔤-module. (ii) There exists an isomorphism of 𝔤-modules φ: V → V^{∗} such that φ^{∗} =

−φ.

Proof. To φ ∈ Hom(V, V^{∗}) we associate the bilinear form Ω_{φ}(v, w) = φ(v)(w) on V × V . The so obtained map Hom(V, V^{∗}) → Bil(V, IR) is 𝔤-equivariant with respect to the actions given by (X.φ)(v) = X.φ(v) − φ(X.v) on Hom(V, V^{∗}) and (X.Ω)(v, w) = −Ω(X.v, w) − Ω(v, X.w) on bilinear forms. Moreover, since φ^{∗}(v)(w) = φ(w)(v), the condition φ^{∗} = −φ is equivalent to the skew-symmetry of Ω_{φ} and φ is an isomorphism if and only if Ω_{φ} is non-degenerate. We conclude that a skew-symmetric isomorphism of 𝔤-modules V → V^{∗} exists if and only if V carries a skew-symmetric 𝔤-invariant bilinear form.

Lemma I.7. Let (V, Ω) be a semisimple symplectic 𝔤-module and W ⊆ V an isotropic simple submodule. Then there exists a simple isotropic submodule U ⊆ V such that U + W is non-degenerate and U + W ≅ W ⊕ W^{∗} as sympletic 𝔤-modules with respect to the restriction of Ω to U + W.

Proof. Since W^{⊥} ≠ V , the semisimplicity of V provides us with a module complement U satisfying V = W^{⊥} ⊕ U. Then the pairing

W × U → IR, (w, u) ↦ Ω(w, u)

is non-degenerate and 𝔤-invariant so that, as a 𝔤-module, U ≅ W^{∗}. Since

(U + W)^{⊥} ∩ (W + U) = U^{⊥} ∩ W^{⊥} ∩ (W + U) = U^{⊥} ∩ W = {0},

the submodule U + W is non-degenerate. Moreover W ⊆ W^{⊥} implies W ∩ U = {0}, hence dim(W + U) = 2 dim W, so that W ⊆ U + W is maximal isotropic. Now Lemma I.5(ii) implies that U + W ≅ W ⊕ W^{∗} as symplectic 𝔤-modules. We conclude in particular that U can be chosen isotropic.

Proposition I.8. Let (V, Ω) be a semisimple indecomposable symplectic 𝔤module which is not simple. Then there exists a simple 𝔤-module W such that (V, Ω) ≅ (W ⊕ W^{∗}, Ω_{W}^{∗}).

Proof. Let W ⊆ V be a non-zero submodule of minimal dimension. Then W is a simple 𝔤-module. If W is non-degenerate, then V = W ⊕ W^{⊥} is an orthogonal decomposition. Therefore V = W which in turn contradicts the assumption that V is not simple. Hence W is isotropic and Lemma I.7 applies. So we find a simple isotropic submodule U such that W + U is non-degenerate and isomorphic to W ⊕W^{∗}. Since V is indecomposable, we see that V = W +U, and this proves the assertion.

Remark I.9. We note that if W and U are isomorphic simple 𝔤-modules not of symplectic type, then the modules W ⊕ W^{∗} and U ⊕ U^{∗} are isomorphic as symplectic 𝔤-modules. If W ≅ U^{∗} via φ: W → U^{∗}, then we also obtain a symplectic isomorphism

ψ: W ⊕ W^{∗} → U ⊕ U^{∗}, (w, α) ↦ ((φ^{∗})^{−1}(α), −φ(w))

because

⟨−φ(w), (φ^{∗})^{−1}(α′)⟩−⟨−φ(w′), (φ^{∗})^{−1}(α)⟩ = −⟨w, α′⟩+⟨w′, α⟩ = α(w′)−α′(w).

If, conversely, W ⊕ W^{∗} and U ⊕ U^{∗} are isomorphic as 𝔤-modules, then either W ≅ U or W ≅ U^{∗}. It follows in particular that the indecomposable symplectic 𝔤-modules W ⊕ W^{∗} and U ⊕ U^{∗} are isomorphic as 𝔤-modules if and only if (W ⊕ W^{∗}, Ω_{W}^{∗}) ≅ (U ⊕ U^{∗}, Ω_{U}^{∗}) as symplectic 𝔤-modules.

We will see in the following subsection (cf. Proposition I.16) that if (W, Ω_{W}) and (U, Ω_{U}) are simple symplectic 𝔤-modules, then the isomorphy of W and U as 𝔤-modules does not necessarily imply that the symplectic 𝔤modules (W, Ω_{W}) and (U, Ω_{U}) are isomorphic.

Lemma I.10. Let (V, Ω) be a semisimple symplectic 𝔤-module and W, W′ two indecomposable non-degenerate submodules of V that are non-isomorphic as 𝔤modules. Then W and W′ are mutually orthogonal. Proof. Let us first assume that W′ is a simple module. If W is not orthogonal to W′, then Ω induces a 𝔤-invariant pairing W′ × W → IR which yields an embedding W′ → W^{∗} ≅ W.

If W is simple, then we obtain a contradiction to our assumption. If W is not simple, then W ≅ U ⊕U^{∗} (Proposition I.8), where U is a simple 𝔤-module which is not of symplectic type (Lemma I.5(iii)). The embedding of W′ into W now shows that either W′ ≅ U or W′ ≅ U^{∗}, contradicting the fact that W is of symplectic type.

Finally we assume that W′ = X ⊕ X^{∗}, where X is not symplectic (Proposition I.8). If W is not orthogonal to W′, then we may w.l.o.g. assume that X is not orthogonal to U^{∗} (otherwise we exchange X and X^{∗}). Then the pairing between the simple modules X and U^{∗} yields an isomorphism X ≅ (U^{∗})^{∗} ≅ U. Therefore W′ = X ⊕ X^{∗} ≅ U ⊕ U^{∗} ≅ W as symplectic 𝔤-modules (Remark I.9). This completes the proof.

The first part of the following structure theorem has been suggested by the referee. We note that a complex version of the second part has also appeared in [11].

Theorem I.11. (Structure theorem for semisimple symplectic modules) Let (V, Ω) be a semisimple symplectic 𝔤-module. Then the following assertions hold:

(i) V is an orthogonal direct sum of indecomposable symplectic 𝔤-modules ⨁_{j=1}^{m} V_{j} and the indecomposable summands are unique up to permutations and isomorphy of 𝔤-modules.

(ii) A semisimple indecomposable symplectic 𝔤-module is either simple or

isomorphic to a module of the type (W ⊕ W^{∗}, Ω_{W}^{∗}). Proof. (i) If V is indecomposable, there is nothing to show. We proof the assertion by induction over the dimension of V . Suppose that V is not indecomposable. Then V contains a non-degenerate submodule W. Then V ≅ W ⊕ W^{⊥} and we obtain the existence of the decomposition into indecomposable modules by applying induction on W and W^{⊥}.

Now let V = ⨁_{j=1}^{n_{0}} V_{j}^{n_{j}} = ⨁_{k=1}^{m_{0}} W_{k}^{m_{k}} be two decompositions into indecomposable summands, where the modules V_{1}, …, V_{n_{0}} are pairwise nonisomorphic as 𝔤-modules and the same holds for the modules W_{1}, …, W_{m_{0}} . If W_{k} is not isomorphic to V_{1} as a 𝔤-module, then W_{k}⊥V_{1} (Lemma I.10). Therefore there exists a uniquely determined index k_{0} such that W_{k_{0}} ≅ V_{1} and we have a non-degenerate pairing between W_{k_{0}}^{m_{k_{0}}} and V_{1}^{n_{1}} . Thus m_{k_{0}} = n_{1}

holds for dimensional reasons and we conclude from W_{k_{0}}^{m_{k_{0}}} ⊆ (∑_{j=2}^{n_{0}} V_{j}^{n_{j}})^{⊥} that

W_{k_{0}}^{m_{k_{0}}} = V_{1}^{n_{1}} . Now the assertion follows by induction applied to the orthogonal complement of this submodule. (ii) This follows from Lemma I.5 and Proposition I.8.

Remark I.12. Let 𝔤 be a real Lie algebra and write 𝔤̂ for the set of 𝔤module isomorphy classes of indecomposable finite dimensional symplectic 𝔤modules. Then the 𝔤-module isomorphy classes of finite dimensional symplectic 𝔤-modules are uniquely determined by a multiplicity function m: 𝔤̂ → IN_{0} with finite support.

In general a simple symplectic 𝔤-module carries a lot of non-equivalent symplectic forms so that the question whether two general semisimple symplectic 𝔤-modules are isomorphic as 𝔤-modules seems to be rather involved. Actually, in view of Remark I.9, it boils down to the case where V = W^{n} and W is simple symplectic. In the next subsection we will see how to classify the non-equivalent symplectic structures on a simple symplectic 𝔤-module. It will turn out that the general classification problem is much simpler for the class of symplectic modules of convex type because the convexity condition yields a coupling between the signs of the symplectic structures on the different submodules. In this case we will see that a module of the type W^{n} permits exactly two non-equivalent symplectic structures leading to modules of convex type (cf. Corollary II.9).

Symplectic structures on simple modules

Let V be a simple real 𝔤-module and ID := End_{𝔤}(V) the algebra of 𝔤-endomorphisms of V . Then ID is a skew-field over the real numbers so that ID = IR, C or IH (the quaternions). We want to determine how many invariant symplectic structures exist on V whenever V is symplectic, i.e., there exists at least one. In the following we write 1, I, J, and K for the basis elements of the quaternions.

Lemma I.13. Let σ: d ↦ d^{♯} be an involutive antiautomorphism of ID fixing IR. Then σ is IR-linear and the following cases occur:

(IR) ID = IR and σ = id_{IR} .

(C_{I}) ID =C and σ(z) = \overline{z}. (C_{II}) ID =C and σ = id_{C}. (IH_{I}) ID = IH and σ(z) = \overline{z}. (IH_{II}) ID = IH and σ(z) = a\overline{z}a^{−1} with \overline{a} = a^{−1} = −a.

Proof. Since σ leaves IR ⊆ ID pointwise fixed, we find for a ∈ IR and d ∈ ID that

σ(ad) = σ(d)σ(a) = σ(d)a = aσ(d),

i.e., σ is IR-linear.

We distinguish several cases. ID = IR: Then σ = id_{IR} holds trivially. ID =C: Then complex conjugation σ(z) = \overline{z} is an involutive antiautomorphism and also σ = id_{C}. On the other hand C is abelian so that σ is a field automorphism, hence there are only these two possibilities. ID = IH: (cf. [19, pp.180,181]) It is clear that conjugation σ(z) = \overline{z} is an involutive antiautomorphism of IH. Assume that σ is different from conjugation. Since IH is not commutative, σ ≠ id_{IH} . Moreover −σ: IH → IH is an isomorphism of the real Lie algebra IH with the bracket [x, y] := xy − yx. Therefore it preserves the commutator algebra IH′ := span{I, J, K} and consequently IH′ is invariant under σ. We conclude that σ commutes with conjugation, hence that α(d) := σ(\overline{d}) defines an involutive automorphism of IH leaving IR pointwise fixed. Put IH_{1} := {d ∈ IH: α(d) = d}. Then IH_{1} is a subalgebra of IH containing IR. The requirement that σ(d) ≠ \overline{d} for at least one d ∈ IH yields IH_{1} ≠ IH. Moreover α^{2} = id_{IH} shows that we have a direct vector space decomposition

IH = IH_{1} ⊕ {d ∈ IH′: α(d) = −d} = IH_{1} ⊕ {d ∈ IH′: σ(d) = d}.

We conclude further from σ ≠ id_{IH} that IH_{1} ≠ IR. Therefore IH_{1} is at least two dimensional. Thus there exists a ∈ IH_{1} ∩ IH′ with a^{2} = −1. Then IH_{1} ⊇ span{1, a} ≅ C and in this sense IH_{1} is a complex algebra. This shows that IH_{1} ≠ IH implies that IH_{1} = span{1, a} for dimensional reasons. Finally we conclude that z ↦ aza^{−1} is involutive, its fixed point set is IH_{1} , and aza^{−1} = −z for z in the orthogonal complement of IH_{1} . Hence α(z) = aza^{−1} beause α has the same eigenspaces for 1 and −1, and therefore σ(z) = a\overline{z}a^{−1} for all z ∈ IH.

Conversely, it is trivial that z ↦ a\overline{z}a^{−1} always defines an involutive antiautomorphism of IH.

According to the preceding lemma there are five different types of involutive antiautomorphism of skew-fields over IR.

Definition I.14. Let (V, Ω) be a symplectic 𝔤-module and ID = End_{𝔤}(V). For A ∈ End_{IR}(V) define A^{♯} ∈ End_{IR}(V) by

Ω(A.v, w) = Ω(v, A^{♯}.w)

for all v, w ∈ V . Then an easy calculation shows that ID^{♯} = ID so that d ↦ d^{♯} is an involutive antiautomorphism of the real algebra ID fixing IR pointwise. If V is simple, then ID = IR,C or IH and we say that V if of type IR, C_{I}, C_{II}, IH_{I} or IH_{II} if the involution on ID has the corresponding type (cf. Lemma I.13).

In the following we write ID^{∗} := ID ∩ Gl(V) for the set of units in the algebra ID = End_{𝔤}(V). Recall that d ∈ ID ∩ Gl(V) is in fact a unit in ID since the inverse of an intertwining operator intertwines also.

Lemma I.15. Let (V, Ω) be a symplectic 𝔤-module and ID = End_{𝔤}(V). Then the space of invariant symplectic structures on V is parametrized by {d ∈ ID^{∗}: d^{♯} = d} via d ↦ Ω_{d} and

Ω_{d}(v, w) := Ω(d.v, w).

Two symplectic 𝔤-modules (V, Ω_{a}) and (V, Ω_{b}) are isomorphic as symplectic 𝔤-modules if and only if there exists d ∈ ID with b = d^{♯}ad. Proof. Let Ω′ be a 𝔤-invariant symplectic structure on V . Then there exists d ∈ End_{IR}(V) such that Ω′(v, w) = Ω(d.v, w) for all v, w ∈ IR. Then d ∈ ID since Ω′ is invariant, d^{♯} = d follows from the requirement that Ω′ is skew-symmetric, and d ∈ Gl(V) from the non-degeneracy of Ω′. Conversely, it is clear that each form Ω_{d}, d = d^{♯} ∈ ID ∩ Gl(V) is a 𝔤-invariant symplectic form on V .

If Ω and Ω′ are two different invariant symplectic forms on V , then the corresponding symplectic 𝔤-modules are equivalent if and only if there exists d ∈ ID such that Ω′(v, w) = Ω(d.v, d.w) for all v, w ∈ V . Applying this to the forms Ω_{a} and Ω_{b}, the assertion follows.

Proposition I.16. Let (V, Ω) be a simple symplectic 𝔤-module. Then, according to the type of V , the following assertions hold:

(IR) The space of invariant symplectic forms is parametrized by IR^{∗} and there

are two equivalence classes of invariant forms. (C_{I}) The space of invariant symplectic forms is parametrized by IR^{∗} and there

are two equivalence classes of invariant forms. (C_{II}) The space of invariant symplectic forms is parametrized by C^{∗} and all

such forms are equivalent. (IH_{I}) The space of invariant symplectic forms is parametrized by IR^{∗} and there

are two equivalence classes of invariant forms. (IH_{II}) The space of invariant symplectic forms is parametrized by a^{⊥} \ {0} and

all such forms are equivalent. Proof. We use Lemma I.15 to check the different cases. If d^{♯} = \overline{d} is conjugation, then the space of invariant forms is parametrized by IR^{∗} and a, b ∈ IR^{∗} lead to equivalent forms Ω_{a} and Ω_{b} if and only if there exists d ∈ ID such that b = d^{♯}ad = |d|^{2}a. Therefore we have exactly two equivalence classes of forms. They are represented by 1 and −1.

In the case C_{II} we have d^{♯} = d for all d ∈C so that the space of invariant forms is parametrized by C^{∗}. The numbers a and b lead to equivalent forms if and only if there exists d ∈ C with b = d^{♯}ad = d^{2}a. Thus we have only one equivalence class.

In the case IH_{II}, we have a^{♯} = −a, and all elements in a^{⊥} are fixed by d ↦ d^{♯}. Therefore the space of invariant symplectic forms is parametrized by a^{⊥} \ {0}.

Let 0 ≠ x ∈ IH_{0} := {d ∈ ID: d^{♯} = d}. We claim that Ω_{x} is equivalent to Ω = Ω_{1} . Let A ⊆ IH_{0} be a two dimensional subalgebra containing IR and x. Then A ≅ C so that we find d ∈ A such that d^{2} = x^{−1} . Now d.x = dxd^{♯} = xd^{2} = 1.

Example I.17. (a) If V = IR^{2n} with the standard symplectic form Ω and 𝔤 = 𝔰𝔭(n, IR), then the commutant is ID = IR. Therefore the 𝔤-invariant symplectic forms are the multiples of Ω and the two different equivalence classes of such forms are represented by Ω and −Ω.

(b) If V = C^{n}, Ω(z, w) = Im ∑_{j} z_{j}\overline{w}_{j}, and 𝔤 = 𝔲(n), then (V, Ω) is a simple symplectic 𝔤-module and since i1 ∈ 𝔤, it is also simple as a real 𝔤-module. The commutant of 𝔤 is ID =C and since (i1)^{♯} = −i1, this module is of type C_{I}.

(c) If V = C^{2n} and Ω_{C} is the standard complex symplectic form, we put Ω := Re Ω_{C} and 𝔤 := 𝔰𝔭(n,C). Then V is a real simple 𝔤-module (this follows from (a)) and the commutant of 𝔤 is ID = C, where the involution on C is the identity. Therefore the space of 𝔤-invariant symplectic forms is C^{∗} and all such forms are equivalent.

(d) If V = IH^{n}, Ω(z, w) = Re ∑_{j} z_{j}J\overline{w}_{j}, and 𝔤 = 𝔰𝔬^{∗}(2n), then Ω is a symplectic 𝔤-invariant form on the real vector space V . We will see later (cf. Remark III.14) that V is a simple 𝔤-module over IR and that the commutant of 𝔤 is IH.

For d ∈ IH′, z, w ∈ IH we have

Re(dzJ\overline{w}) = Re(zJ\overline{w}d) = Re(zJ\overline{\overline{d}w})

showing that d^{♯} = \overline{d} for all d ∈ ID. Therefore V is a symplectic 𝔤-module of type IH_{I}.

(e) If V = IH^{n}, Ω_{IH}(z, w) = Im ∑_{j} z_{j}\overline{w}_{j}, and 𝔤 = 𝔰𝔭(n), then Ω_{IH} is a skew-symmetric IR-bilinear 𝔤-invariant form on V . Since the group Sp(n) = exp (𝔰𝔭(n)) acts transitively on the unit-sphere in V , the 𝔤-module V is simple as a real module and the commutant of 𝔤 is IH. Applying non-zero linear functions on IH′ = span{I, J, K} to Ω_{IH} , we obtain a three dimensional space of invariant symplectic forms, hence V is a symplectic 𝔤-module of type IH_{II}.

Remark I.18. Suppose that (V, Ω) is an isotypic symplectic 𝔤-module, i.e., (V, Ω) ≅ (W, Ω_{W})^{m}, where W is simple, and that W is of type IR,C_{I} or IH_{I}. Then ID = End_{𝔤}(V) ≅ IM(m, IK), where IK = End_{𝔤}(W) is IR,C or IH. Since the involution d ↦ d^{♯} coincides with the natural conjugation on IK, it follows easily that the involution d ↦ d^{♯} on ID ≅ IM(n, IK) coincides with the natural involution d ↦ d^{∗} induced by the positive definite hermitean form on IK^{n} given by ⟨v, w⟩ := ∑_{j=1}^{n} v_{j}\overline{w_{j}}. So we can identify the set of 𝔤-invariant symplectic forms on V with Herm(m, IK) = {d ∈ IM(m, IK): d^{∗} = d}, the space of m × mhermitean matrices over IK. The action of D^{∗} ≅ Gl(m, IK) on this space whose orbits are the equivalence classes of 𝔤-invariant symplectic structures is given by d.a = dad^{∗}. We recall that Herm(m, IK) is a Jordan algebra of rank m (cf. [2]). In this sense the orbits of Gl(m, IK) turn out to be the orbits of the structure group G(Ω), where Ω = int Herm^{+}(n, IK) is the open cone of positive definite hermitean matrices which in this case coincides with the orbit of 1 under this action. Now the theory of Jordan algebras implies in particular that there are at most m + 1 different ID^{∗} -orbits in ID^{∗} ∩ Herm(n, IK) (cf. [2, Ch. IV]).

II. Symplectic modules of convex type

In this section we specialize the results of the preceding section to symplectic modules of convex type. Here 𝔤 always denotes a real reductive Lie algebra and G a simply connected Lie group with L(G) = 𝔤.

If C is a closed convex cone in a vector space V , we write H(C) := C ∩ (−C) for the edge of the cone C and C^{⋆} := {α ∈ V^{∗}: α(C) ⊆ IR^{+}} for the dual cone. The latter notation is also used for arbitrary subsets C ⊆ V .

Definition II.1. (a) If (V, Ω) is a symplectic 𝔤-module, then we assign to each X ∈ 𝔤 the Hamiltonian function defined by

φ(X)(v) := \frac{1}{2}Ω(X.v, v).

Note that φ(X) = 0 is equivalent to X.V = {0}. The mapping

Φ: V → 𝔤^{∗}, v ↦ (X ↦ φ(X)(v))

is called the moment map. One checks easily that it is equivariant with respect to the actions of G on V and the coadjoint action of G on 𝔤^{∗} which is defined by Ad^{∗}(g).ω := ω ∘ Ad(g)^{−1} .

(b) A symplectic 𝔤-module (V, Ω) is said to be of convex type if the following two conditions are satisfied:

(i) The closed convex cone C_{V} ⊆ 𝔤^{∗} generated by the image Φ(V) of the moment map is pointed, i.e., contains no vector subspace. Note that, according to the duality of convex cones, this condition is equivalent to the condition that the cone W_{V} = C_{V}^{⋆} ⊆ 𝔤 mentioned in the introduction is generating.

(ii) The center 𝔷(𝔤) acts semisimply on V with purely imaginary eigenvalues. This means that under the homomorphism ρ: 𝔤 → 𝔰𝔭(V), the image of 𝔷(𝔤) is a compactly embedded subalgebra.

Lemma II.2. Let (V, Ω) be a symplectic 𝔤-module of convex type. Then the following assertions hold:

(i) Every non-degenerate submodule of V is of convex type.

(ii) V is a semisimple 𝔤-module.

Proof. (i) This is trivial. (ii) We have to show that V is a sum of simple submodules. To show this, we first decompose V as a direct sum V = ⊕_{j=1}^{k}V_{j} of isotypic 𝔷(𝔤)-modules. Then each V_{j} is invariant under 𝔤 and two cases occur. If 𝔷(𝔤) acts trivially on V_{j}, then V_{j} is a sum of simple 𝔤-modules which are exactly the simple [𝔤, 𝔤]-submodules (Weyl’s theorem). If 𝔷(𝔤) acts non-trivially, then there exists a complex structure I on V_{j} and a linear functional α_{j} on 𝔷(𝔤) with X.v = α_{j}(X)Iv for all v ∈ V_{j}. Then I.v = X.v for every X ∈ 𝔷(𝔤) with α_{j}(X) = 1 and therefore I commutes with 𝔤. This shows that the simple submodules of V_{j} are the complex simple [𝔤, 𝔤]-submodules which generate V_{j} by Weyl’s theorem.

In view of Lemma II.2, all the results of Section I on semisimple symplectic 𝔤-modules become available for symplectic 𝔤-modules of convex type. First we show that the indecomposable 𝔤-modules of type W ⊕ W^{∗} cannot occur in modules of convex type.

In the following we call a simple 𝔤-module W non-trivial if 𝔤.W ≠ {0}.

Lemma II.3. Let W be a non-trivial 𝔤-module. Then the symplectic 𝔤-module (W ⊕ W^{∗}, Ω_{W}^{∗}) cannot be contained in any symplectic 𝔤-module of convex type.

Proof. Since non-degenerate submodules of modules of convex type are of convex type, it suffices to show that (W ⊕ W^{∗}, Ω_{W}^{∗}) is not of convex type. For (w, α) ∈ W ⊕ W^{∗} and X ∈ 𝔤 we have

\begin{aligned}Φ(w, α)(X) & = \frac{1}{2}Ω(X.(w, α), (w, α)) = \frac{1}{2}Ω((X.w, X.α), (w, α)) \\ & = \frac{1}{2} ((X.α)(w) − α(X.w)) = \frac{1}{2} (−α(X.w) − α(X.w)) = −α(X.w). \end{aligned}

Replacing w by −w, it follows in particular that Φ(W ⊕ W^{∗}) = −Φ(W ⊕ W^{∗}). If W ⊕ W^{∗} is of convex type we therefore conclude that Φ(W ⊕ W^{∗}) = {0}, hence that 𝔤.W = {0}, contradicting the non-triviality of W.

Lemma II.4. If V is a symplectic 𝔤-module of convex type, then every nontrivial simple submodule W is non-degenerate.

Proof. Let W ⊆ V be a non-trivial simple submodule. We assume that W is isotropic and use Lemma I.7 to find a simple isotropic submodule U ⊆ V such that U +W ≅ W ⊕W^{∗} as symplectic 𝔤-modules. This contradicts Lemma II.3.

Proposition II.5. Let V be a symplectic 𝔤-module of convex type,

V_{fix} := {v ∈ V: 𝔤.v = {0}}, and V_{eff} := span{X.v: X ∈ 𝔤, v ∈ V }.

Then V = V_{fix} ⊕ V_{eff} is an orthogonal direct sum of symplectic 𝔤-modules, V_{eff} is a symplectic 𝔤-module of convex type, and every submodule of V_{eff} is nondegenerate.

Proof. The fact that V = V_{fix} ⊕ V_{eff} is a direct sum of 𝔤-modules follows from the semisimplicity of the 𝔤-module V (Lemma II.2).

Let W ⊆ V_{eff} be a submodule. We claim that W is non-degenerate. Suppose that W is degenerate. Then W ∩ W^{⊥} is a non-zero submodule of W. Let U ⊆ W ∩ W^{⊥} be a minimal non-zero submodule. Then U is a nontrivial simple 𝔤-module so that Lemma II.4 implies that U is non-degenerate, contradicting U ⊆ W ∩ W^{⊥} ⊆ U^{⊥}. This proves that W is non-degenerate.

We conclude in particular that V_{eff} is non-degenerate. Hence V_{eff}^{⊥} is a non-degenerate submodule of V complementary to V_{eff} . Thus V_{eff}^{⊥} ⊆ V_{fix} and by comparing dimensions we see that V_{fix} = V_{eff}^{⊥}. This completes the proof.

Corollary II.6. Every symplectic 𝔤-module V of convex type decomposes into an orthogonal direct sum of V_{fix} and simple symplectic 𝔤-modules of convex type.

Proof. This follows from Theorem I.11, Lemma II.4, and Proposition II.5.

Lemma II.7. Let (V, Ω) be a simple symplectic 𝔤-module of convex type. Then V is of type IR, C_{I} or IH_{I}.

Proof. The condition that V is of type IR, C_{I} or IH_{I} is equivalent to the statement that there exists no complex structure I ∈ End_{𝔤}(V) satisfying I^{♯} = I (cf. Proposition I.16). Suppose that I ∈ End_{𝔤}(V) with I^{♯} = I. Then

Φ(Iv)(X) = \frac{1}{2}Ω(X.Iv, Iv) = \frac{1}{2}Ω(IX.v, Iv) = \frac{1}{2}Ω(X.v, I^{2}.v) = −Φ(v)(X)

so that Φ(Iv) = −Φ(v). Therefore a non-trivial simple symplectic 𝔤-module of type C_{II} or IH_{II} cannot be of convex type.

Theorem II.8. Let (V, Ω) be a symplectic 𝔤-module of convex type and {V_{1}, …, V_{m}} a maximal set of pairwise non-equivalent non-trivial simple submodules. Let Ω_{j} := Ω |_{V_{j}×V_{j}}, Ω_{0} := Ω |_{V_{fix}×V_{fix}} and m_{j} denote the multiplicity of V_{j} in V . Then the symplectic 𝔤-module (V, Ω) is, as a symplectic 𝔤-module, equivalent the orthogonal direct sum

(V_{fix}, Ω_{0}) ⊕ ⨁_{j=1}^{m}(V_{j}, Ω_{j})^{m_{j}}.

Proof. According to Corollary II.6, the symplectic 𝔤-module V is the orthogonal direct sum of simple symplectic 𝔤-modules of convex type. Let (W, Ω_{W}) and (U, Ω_{U}) be non-trivial simple summands occuring in such a decomposition of (V, Ω). Suppose that W ≅ U as 𝔤-modules. If (U, Ω_{U}) is not equivalent to (W, Ω_{W}) as a symplectic 𝔤-module, then (U, Ω_{U}) ≅ (W, −Ω_{W}) (Lemma II.7, Proposition I.16). Hence (W, Ω_{W}) ⊕ (W, −Ω_{W}) is isomorphic to a submodule of (V, Ω) and this is impossible if V is of convex type. Therefore any module decomposition V = V_{fix} ⊕ ⨁_{j=1}^{m} V_{j}^{m_{j}} which is in addition orthogonal with respect to Ω yields an isomorphism

(V, Ω) ≅ (V_{fix}, Ω_{0}) ⊕ ⨁_{j=1}^{m}(V_{j}, Ω_{j})^{m_{j}}

of symplectic 𝔤-modules.

Corollary II.9. Two symplectic 𝔤-modules (V, Ω) and (V′, Ω′) of convex type are isomorphic as symplectic 𝔤-modules if and only if V ≅ V′ as 𝔤-modules and C_{V} = C_{V′}.

Proof. If (V, Ω) and (V′, Ω′) are isomorphic as symplectic 𝔤-modules, then they are trivially isomorphic as 𝔤-modules and the cones C_{V} and C_{V′} coincide.

Suppose conversely that V ≅ V′ as 𝔤-modules and that C_{V} = C_{V′}. Then V and V′ contain the same types of irreducible submodules with the same multiplicities. First V_{fix} and V′_{fix} have the same dimension, so that they are isomorphic as symplectic vector spaces, hence as symplectic 𝔤-modules.

Let V_{j} be an irreducible submodule of V and V′_{j} ⊆ V′ an isomorphic submodule. We write Ω_{j} resp. Ω′_{j} for the induced symplectic stucture on V_{j} resp. V′_{j}. Then either (V′_{j}, Ω′_{j}) ≅ (V_{j}, Ω_{j}) or (V′_{j}, Ω′_{j}) ≅ (V_{j}, −Ω_{j}). Since Φ_{V′_{j}}(V′_{j}) ∪ Φ_{V_{j}}(V_{j}) ⊆ C_{V′} = C_{V} and C_{V} is a pointed cone, we see that the first possibility holds. Now we apply Theorem II.8 to complete the proof.

The preceding result shows that for modules of convex type it is rather easy to pass from module isomorphy classes to isomorphy classes of symplectic modules. The additional information one needs is the cone C_{V} which determines the choices of the class of the symplectic structure on each irreducible submodule. The following remark makes this a bit more precise.

Remark II.10. Let (V, Ω) be a symplectic 𝔤-module of convex type. We are interested in a description of the set of all 𝔤-invariant symplectic structures Ω′ on V such that (V, Ω′) is also of convex type.

Suppose first that V ≅ V_{1}^{m_{1}} is isotypic as a 𝔤-module. According to Lemma II.7, V_{1} permits exactly two classes of 𝔤-invariant symplectic structures represented by (V_{1}, Ω_{1}) and (V_{1}, −Ω_{1}), where Ω_{1} is the restriction of Ω to V_{1} . Therefore Theorem II.8 implies that V permits also two classes of invariant symplectic structures Ω′ such that (V, Ω′) is of convex type. Note that there are much more classes of symplectic structures if we do not impose this condition (cf. Remark I.18).

We conclude with Theorem II.8 that for any 𝔤-invariant symplectic structure Ω′ such that (V, Ω′) is of convex type, we have

(V, Ω′) ≅ (V_{fix}, Ω_{0}) ⊕ ⨁_{j=1}^{m}(V_{j}, ε_{j}Ω_{j})^{m_{j}},

where ε_{j} ∈ {1, −1}. The corresponding cone is given by

C′_{V} = ∑_{j=1}^{m} ε_{j}C_{V_{j}}.

The convex type condition for (V, Ω′) means that C′_{V} is pointed. This yields an additional coupling of the signs ε_{1}, …, ε_{m}.

This coupling condition is rather implicit. Anticipating Theorem III.5 and its consequences, we know that at most one simple non-compact ideal 𝔤_{j} acts on the simple module V_{j}. If V_{j} and V_{k} are modules for which 𝔤_{k} = 𝔤_{j}, then their symplectic structures are coupled in the sense that ε_{j} = ε_{k}. The following example illustrates what happens if no such coupling occurs.

Let V = IR^{2} be the canonical 𝔰𝔩(2, IR)-module and consider V ⊕ V as module of 𝔤 := 𝔰𝔩(2, IR)^{2} . Then (V, Ω) ⊕ (V, Ω) and (V, Ω) ⊕ (V, −Ω) are two different structure of symplectic 𝔤-modules of convex type on V ⊕ V . Here all choices (1, 1), (1, −1), (−1, 1), and (−1, −1) for ε_{1}, ε_{2} are possible and we obtain four classes of pairwise non-equivalent symplectic 𝔤-module structures of convex type on V ⊕ V .

We believe that the aforementioned coupling condition mentioned in Remark II.10 is not sufficient and we leave it as an open problem to make the coupling condition on the signs more explicit. We will see in Section V how this problem is related to the classification of Lie algebras with invariant cones.

Complexifications of modules of convex type

Lemma II.11. Let (V, Ω) be a symplectic 𝔤-module, V_{C} its complexification, v ↦ \overline{v} complex conjugation on V_{C}, and Ω_{C} the complex bilinear extension of Ω to V_{C}. Then the following assertions hold:

(i) The prescription B(v, w) := iΩ_{C}(v, \overline{w}) defines a pseudohermitean form on V_{C}.

(ii) Ω̃ := Im B is an invariant symplectic form on the real 𝔤-module V_{C} with

Ω̃(v_{1} + iv_{2}, w_{1} + iw_{2}) = Ω(v_{1}, w_{1}) + Ω(v_{2}, w_{2}),

i.e., (V_{C}, Ω̃) ≅ (V, Ω) ⊕ (V, Ω) is an orthogonal direct sum of symplectic 𝔤-modules.

Proof. (i) The sesquilinearity is clear. That it is pseudohermitean follows from

B(w, v) := iΩ_{C}(w, \overline{v}) = i\overline{Ω_{C}(\overline{w}, v)} = \overline{iΩ_{C}(v, \overline{w})} = \overline{B(v, w)}.

(ii) For v = v_{1} + iv_{2} and w = w_{1} + iw_{2} we have

Ω̃(v, w) = Im B(v_{1} + iv_{2}, w_{1} + iw_{2}) = Im iΩ_{C}(v_{1} + iv_{2}, w_{1} − iw_{2})
= Re Ω_{C}(v_{1} + iv_{2}, w_{1} − iw_{2}) = Ω(v_{1}, w_{1}) + Ω(v_{2}, w_{2}).

This proves (ii).

Proposition II.12. Let (V, Ω) be a symplectic 𝔤-module and (V_{C}, Ω̃) its complexification considered as a real 𝔤-module. Then (V, Ω) is of convex type if and only if (V_{C}, Ω̃) is of convex type and in this case C_{V} = C_{VC}.

Proof. From Lemma II.11(ii) we infer that Φ_{VC}(v_{1} +iv_{2}) = Φ_{V} (v_{1})+Φ_{V} (v_{2}). Therefore the sets Φ_{V} (V) and Φ_{VC}(V_{C}) generate the same cone in 𝔤^{∗}.

Example II.13. Let V = IR^{2} be the standard 𝔰𝔩(2, IR)-module of dimension 2. Then the invariant symplectic form is given by Ω(v, w) = det(v, w). Since G = Sl(2, IR) acts transitively on IR^{2} \ {0}, the image of the moment mapping is the closure of one coadjoint orbit. This shows that Φ(IR^{2}) is the closure of a nilpotent orbit in 𝔰𝔩(2, IR)^{∗}. It follows in particular that IR^{2} is a symplectic module of convex type.

Let V_{C} =C^{2} be the complexification. Then Φ_{VC}(V_{C}) = Φ_{V} (V) + Φ_{V} (V) is the closed convex invariant cone generated by Φ_{V} (V).

Proposition II.14. Recall the cone

W_{V} = C_{V}^{⋆} = {X ∈ 𝔤: (∀α ∈ C_{V})α(X) ≥ 0}.

Then the following are equivalent:

(i) X ∈ W_{V} .

(ii) The Hamiltonian function defined by φ(X)(v) = \frac{1}{2}Ω(X.v, v) is non-ne-

gative on V . (iii) B(iX.v, v) ≤ 0 holds for all v ∈ V_{C}.

Proof. The equivalence of (i) and (ii) follows immediately from the definition of C_{V} .

Let v = v_{1} + iv_{2} ∈ V_{C} and X ∈ 𝔤. Then iX is a B -hermitean operator on V_{C}. Therefore B(iX.v, v) is real and consequently

−B(iX.v, v) = Im B(X.v, v) = Ω̃(X.v, v) = 2Φ_{VC}(v)(X).

which immediately implies the equivalence of (i) and (iii).

Proposition II.14 links the symplectic modules of convex type to those complex modules which carry invariant pseudohermitean forms and for which the cone W_{V} is generating. Such modules have been studied in detail in [14].

Remark II.15. If W is a complex vector space endowed with a pseudohermitean form B and

S = {s ∈ Gl(W): (∀w ∈ W)B(s.w, s.w) ≤ B(w, w)}

is the semigroup of B -contractions, then the group of units of S is the group U(B) of B -unitary mapping and the tangent cone

L(S) := {X ∈ 𝔤𝔩(W): e^{IR^{+}X} ⊆ S}

can be written as L(S) = 𝔲(B) + iC_{W} , where C_{W} = {X ∈ 𝔤𝔩(W): (∀w ∈ W)B(iX.w, w) ≤ 0} (cf. [14]).

In the light of these observations, the cone W_{V} consists exactly of those elements X in the Lie algebra 𝔤 for which iX generates a one-parameter semigroup of B -contractions on V_{C}.

Example II.16. We consider V = IR^{2n} endowed with the usual symplectic structure given by

Ω(v, w) = ⟨v, J.w⟩,

where J = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) and 𝔤 = 𝔰𝔭(n, IR). Then V_{C} =C^{2n} and

B(v, w) = iΩ_{C}(v, \overline{w}) = i⟨v, J.w⟩,

where ⟨·, ·⟩ denotes the natural scalar product on C^{2n}. For X ∈ 𝔤 we see that X ∈ W_{V} means that

Ω(X.v, v) = ⟨X.v, J.v⟩ = −⟨JX.v, v⟩ ≥ 0

for all v ∈ V which in turn means that the symmetric operator JX is negative semidefinite on IR^{2n}. Recall that X ∈ 𝔰𝔭(n, IR) means that JX is a symmetric matrix.

The space 𝔱 of all those elements in 𝔰𝔭(n, IR) for which JX is a diagonal matrix is a compactly embedded Cartan subalgebra of 𝔰𝔭(n, IR).

Weight spaces in symplectic modules of convex type

In this subsection 𝔤 denotes a reductive real Lie algebra which has a compactly embedded Cartan algebra 𝔱.

Definition II.17. Let V be a 𝔤-module and V_{C} its complexification. We fix a compactly embedded Cartan algebra 𝔱 ⊆ 𝔤. For α ∈ 𝔱C^{∗} we write

VC^{α} = {v ∈ V_{C}: (∀X ∈ 𝔱_{C})X.v = α(X)v}

for the weight space of weight α. We write P_{V} = {α ∈ 𝔱C^{∗}: VC^{α} ≠ 0} for the set of all weights of V .

If, in addition, V is a semisimple 𝔤-module, then 𝔷(𝔤) acts by diagonalizable operators on V_{C} and hence V_{C} is the direct sum of the weight spaces. If, moreover, V is of convex type, then 𝔷(𝔤), and therefore 𝔱 acts on V with purely imaginary eigenvalues, hence α(𝔱) ⊆ iIR holds for every weight α ∈ P_{V} .

Definition II.18. For this definition we drop the assumption that 𝔤 is reductive. We merely assume that 𝔱 ⊆ 𝔤 is a compactly embedded Cartan subalgebra.

We recall the root space decomposition 𝔤_{C} = 𝔱_{C} ⊕ ⨁_{α∈Δ} 𝔤C^{α}, of 𝔤_{C} with respect to 𝔱_{C}, where

𝔤C^{α} = {Y ∈ 𝔤_{C}: (∀X ∈ 𝔱_{C})[X, Y] = α(X)Y }

and Δ = {α ∈ 𝔱C^{∗} \ {0}: 𝔤C^{α} ≠ {0}}. Let 𝔨 ⊇ 𝔱 denote the unique maximal compactly embedded subalgebra containing 𝔱 ([6, A.2.40]). We say that a root α ∈ Δ is compact if 𝔤C^{α} ⊆ 𝔨_{C}, otherwise we say that α is non-compact. We write Δ_{k} resp. Δ_{p} for the sets of compact resp. non-compact roots. If 𝔯 ⊆ 𝔤 is the solvable radical, then we also set Δ_{r} := {α ∈ Δ: 𝔤C^{α} ∩ 𝔯_{C} ≠ {0}}.

We define the Weyl group W_{𝔨} := N_{G}(𝔱)/Z_{G}(𝔱) = N_{G}(𝔱)/ exp 𝔱. A positive system of roots is a subset Δ^{+} of Δ for which there exists a regular element X_{0} ∈ i𝔱 such that Δ^{+} = {α ∈ Δ: α(X_{0}) > 0}.

We say that a positive system Δ^{+} is 𝔨-adapted if Δ_{p}^{+} is invariant under the Weyl group. Note that this condition is equivalent to the existence of an element H ∈ 𝔷(𝔨) such that Δ_{p}^{+} = {α ∈ Δ: iα(H) > 0}. We refer to [15, Prop. II.7] for the result that a 𝔨-adapted positive system exists if and only if 𝔷_{𝔤}(𝔷(𝔨)) = 𝔨. If this condition is satisfied, then 𝔤 is called quasihermitean.

For a 𝔨-adapted positive system Δ^{+} we define the cones

C_{min} = cone(\{i[\overline{X}_{α}, X_{α}]: α ∈ Δ_{p}^{+}, X_{α} ∈ 𝔤C^{α}\}) and C_{max} = (iΔ_{p}^{+})^{⋆},

where cone(E) denotes the smallest closed convex cone containing the set E. We say that 𝔤 has cone potential, if [\overline{X}_{α}, X_{α}] ≠ 0 holds for all non-zero elements X_{α} ∈ 𝔤C^{α}, α ∈ Δ_{p} and that 𝔤 has strong cone potential if, in addition, there exists a 𝔨-adapted positive system such that the cone C_{min} is pointed.

If (V, Ω) is an effective module of convex type for the reductive Lie algebra 𝔤, then the cone C_{V} ⊆ 𝔤^{∗} is pointed and generating. Therefore the Lie algebra 𝔤 is quasihermitean (cf. [6, Th. III.5.16]) and it follows in particular that it contains a compactly embedded Cartan subalgebra. Thus the assumption that 𝔤 contains a compactly embedded Cartan subalgebra is simply for technical convenience whenever we study only modules of convex type.

Lemma II.19. Let (V, Ω) be a symplectic 𝔤-module of convex type. Then the following assertions hold:

(i) For α ≠ β in P_{V} the weight spaces VC^{α} and VC^{β} are orthogonal with respect to B.

(ii) For α ∈ P_{V} , v ∈ VC^{α} we have that

Φ_{VC}(v) = \frac{1}{2}B(v, v)(−iα) = \frac{1}{2}Ω_{C}(v, \overline{v})α,

where we identify 𝔱^{∗} with the subspace [𝔱, 𝔤]^{⊥} in 𝔤^{∗}. (iii) For λ ∈C and v ∈ V_{C} we have Φ_{VC}(λv) = |λ|^{2}Φ_{VC}(v).

Proof. (i) This follows from the invariance of B under 𝔱. (ii) Let β ∈ Δ and X_{β} ∈ 𝔤C^{β}. Then X_{β}.v ∈ VC^{α+β} and therefore B(X_{β}.v, v) = 0 by (i). Similarly B(\overline{X}_{β}.v, v) = 0 holds for \overline{X}_{β} ∈ 𝔤C^{−β}. Therefore

\begin{aligned}Φ_{V} (v)(X_{β} + \overline{X}_{β}) & = \frac{1}{2} Im B((X_{β} + \overline{X}_{β}).v, v) \\ & = \frac{1}{2} Im B(X_{β}.v, v) + \frac{1}{2} Im B(\overline{X}_{β}.v, v) = 0. \end{aligned}

Let 𝔤^{[β]} := (𝔤C^{β} ⊕ 𝔤C^{−β}) ∩ 𝔤. Then the above argument shows that

Φ_{V} (v) ∈ 𝔤^{[β]}^{⊥}. Since [𝔱, 𝔤] = ⨁_{α∈Δ} 𝔤^{[α]} , we conclude that Φ_{V} (v) ∈ 𝔱^{∗}. To calculate this functional explicitly, we only have to evaluate it on 𝔱. For X ∈ 𝔱 we obtain

Φ_{VC}(v)(X) = \frac{1}{2} Im B(X.v, v) = \frac{1}{2} Im (α(X)B(v, v))

= \frac{1}{2}B(v, v) Im α(X) = \frac{1}{2}B(v, v)( − iα(X)).

(iii) This is a simple consequence of the sesquilinearity of B.

Proposition II.20. If (V, Ω) is a symplectic 𝔤-module of convex type and 0 ≠ α ∈ P_{V} , then two cases arise:

(i) B is positive definite on VC^{α} and −iα ∈ C_{V} .

(ii) B is negative definite on VC^{α} and iα ∈ C_{V} .

Proof. First we note that Lemma II.19(i) together with V_{C} = ⨁_{α∈P_{V}} VC^{α} implies that the restriction of B to VC^{α} is non-degenerate. Since C_{V} is pointed, Lemma II.19(ii) now shows that B is either positive or negative definite on VC^{α} depending on whether −iα of iα is contained in C_{V} .

Corollary II.21. If (V, Ω) is a symplectic 𝔤-module of convex type and 0 ≠ v ∈ VC^{α} with Φ_{VC}(v) = 0, then α = 0.

Proof. In view of Proposition II.20 and Lemma II.19(ii),

0 = Φ_{C}(v) = \frac{1}{2}B(v, v)(−iα)

implies that α = 0.

We draw some important conlusions in the real setting.

Proposition II.22. Let 𝔤 be a reductive Lie algebra with compactly embedded Cartan algebra 𝔱 and (V, Ω) a symplectic 𝔤-module of convex type. Then the following assertions hold:

(i) V_{fix} = Φ_{V}^{−1}(0).

(ii) If V_{fix} = {0}, then 0 ∉ P_{V} .

Proof. (i) The inclusion V_{fix} ⊆ Φ_{V}^{−1}(0) holds trivially. Suppose conversely that Φ_{V} (v) = 0. We write v = ∑_{α∈P_{V}} v_{α} with v_{α} ∈ VC^{α}. Then Lemma I.18(i),(ii) imply that for X ∈ int C_{V}^{⋆} ∩ 𝔱 we have

0 = Φ_{V} (v)(X) = ∑_{α∈P_{V}} Φ_{VC}(v_{α})(X).

Since X ∈ int C_{V}^{⋆} and all functionals Φ_{VC}(v_{α}) lie in the pointed cone C_{V} , we conclude that all summands are 0, hence that v_{α} ∈ Φ_{VC}^{−1}(0). Now Corollary II.21 shows that v_{α} = 0 for α ≠ 0 and that v ∈ VC^{0} ∩ V .

This proves that Φ_{V}^{−1}(0) = VC^{0} ∩ V . On the other hand, the equivariance of Φ_{V} shows that Φ_{V}^{−1}(0) is invariant under G, hence under 𝔤. Since 𝔱 acts trivially on VC^{0} ∩ V , the same holds for the whole Lie algebra 𝔤 because 𝔤 is the smallest ideal containing 𝔱. This means that VC^{0} ∩ V = V_{fix} . (ii) This follows from (i) and Lemma II.19(ii).

Proposition II.23. Let (V, Ω) be a symplectic 𝔤-module with V_{fix} = {0} and where 𝔷(𝔤) acts semisimply with purely imaginary eigenvalues. Then the following are equivalent:

(i) V is of convex type.

(ii) The cone W_{V} = C_{V}^{⋆} has interior points.

(iii) There exists X ∈ 𝔷(𝔨) such that Ω(X.v, v) > 0 for all v ∈ V \ {0}.

(iv) There exists X ∈ 𝔷(𝔨) such that α(X) ≠ 0 holds for all α ∈ P_{V} and B

is positive definite on VC^{α} if and only if iα(X) < 0. Proof. The equivalence of (i) and (ii) follows from the duality theory of cones which implies that C_{V} is pointed if and only if its dual cone W_{V} is generating. (ii) ⇒ (iii): If (ii) holds, then the cone W_{V} is a generating invariant cone in 𝔤. Therefore there exists an element X ∈ 𝔷(𝔨) ∩ int W_{V} (cf. [6, Prop. III.5.5]). Then Φ_{V} (v)(X) > 0 holds for all v ∈ V with Φ_{V} (v) ≠ 0. Finally Φ_{V}^{−1}(0) = V_{fix} = {0} follows from Proposition II.22(i) and this implies (iii). (iii) ⇒ (iv): Using (iii), we choose X ∈ 𝔷(𝔨) such that Ω(X.v, v) > 0 holds for all v ∈ V \ {0}. Then also

Im B(X.v, v) = Ω̃(X.v, v) > 0

for all v ∈ V_{C} \ {0}. For v ∈ VC^{α} we find in particular that

Im B(X.v, v) = B(v, v)(−iα)(X) > 0.

Hence B is positive definite on VC^{α} for iα(X) < 0 and negative definite if iα(X) > 0. (iv) ⇒ (iii): Let v ∈ V and write it as v = ∑_{α∈P_{V}} v_{α} with v_{α} ∈ VC^{α}. Then

Ω(X.v, v) = ∑_{α∈P_{V}} B(v_{α}, v_{α})(−iα)(X) > 0

for v ≠ 0 follows from (iv). (iii) ⇒ (i): Let S ⊆ V be a sphere for some norm on V . Then K := conv Φ_{V} (S) is a compact convex set and ⟨X, α⟩ > 0 holds for all α ∈ K. Therefore C_{V} = IR^{+}K is a pointed closed convex cone in 𝔤^{∗}.

Complex structures on simple modules

We have already seen that a symplectic 𝔤-module of convex type breaks up into a symplectic direct sum of simple modules of convex type. Our next objective is to show that the classification problem for these modules can be translated into a classification problem for certain simple complex 𝔤-modules. This problem is much easier to deal with since the complex simple 𝔤-modules can easily be classified by their highest weights (cf. Section III).

Lemma II.24. Let (V, Ω) be a symplectic 𝔤-module and I ∈ End_{𝔤}(V) an invariant complex structure with I^{♯} = −I, i.e., I ∈ Sp(V, Ω). Then

B(v, w) := Ω(I.v, w) + iΩ(v, w)

defines a 𝔤-invariant pseudohermitean form on the complex vector space (V, I) such that Ω = Im B.

Proof. For v, w ∈ V we have

B(I.v, w) = Ω(I^{2}.v, w) + iΩ(I.v, w) = −Ω(v, w) + iΩ(I.v, w) = iB(v, w)

and

\begin{aligned}B(w, v) & = Ω(I.w, v) + iΩ(w, v) = Ω(w, −I.v) − iΩ(v, w) \\ & = Ω(I.v, w) − iΩ(v, w) = \overline{B(v, w)}. \end{aligned}

Thus B is a pseudohermitean form on V which is easily seen to be invariant under 𝔤.

We recall that if (V, Ω) is a non-trivial simple 𝔤-module of convex type, then every invariant complex structure I ∈ End_{𝔤}(V) is skew-symmetric (cf. Lemma II.7), hence that the construction of the preceding lemma works for End_{𝔤}(V) =C, IH.

Definition II.25. Let 𝔤 be a finite dimensional real Lie algebra. A complex symplectic 𝔤-module is a triple (V, Ω, I), where (V, Ω) is a symplectic 𝔤-module and I ∈ Sp(V, Ω) ∩ End_{𝔤}(V) is a 𝔤-invariant complex structure. We say that (V, Ω, I) is simple if it is simple as a complex 𝔤-module.

An isomorphism of two complex symplectic 𝔤-modules is required to be an isomorphism of symplectic 𝔤-modules which is in addition complex linear or antilinear.

Theorem II.26. Let (V, Ω) be a simple 𝔤-module of convex type and ID = End_{𝔤}(V). If ID = IR we put (Ṽ, Ω̃, I) := (V_{C}, Im Ω_{C}, i id) and for ID = IH,C we put (Ṽ, Ω̃) = (V, Ω, I), where I is any invariant complex structure on V . Then this prescription associates to each isomorphy class [(V, Ω)] of simple symplectic 𝔤-modules of convex type an isomorphy class [(Ṽ, Ω̃, I)] of simple complex symplectic 𝔤-modules. It yields a bijection between the classes of simple symplectic 𝔤-modules of convex type and the classes of simple complex symplectic 𝔤-modules of convex type.

Proof. First we show that the assignment is well defined on the level of classes. For ID = IR this follows from Lemma II.10. For ID = C we have to choose between the two complex structures I and −I. Since the identitiy map (V, I) → (V, −I) is complex antilinear, the two complex symplectic 𝔤-modules (V, Ω, I) and (V, Ω, −I) are isomorphic.

Now let ID = IH. Then V is of type IH_{I} (Lemma II.7). If I and Ĩ ∈ ID are two different complex structures, then there exists an element d ∈ ID with |d| = 1 such that Ĩ = dId^{−1} . Since d^{♯} = \overline{d} = d^{−1} , it follows that d ∈ Sp(V, Ω). Hence the map

d: (V, Ω, I) → (V, Ω, Ĩ)

is a complex linear isomorphism between complex symplectic 𝔤-modules. This shows that our prescription yields a well defined map [(V, Ω)] ↦ [(Ṽ, Ω̃, I)] on the level of isomorphy classes.

We claim that this map is injective. So assume that [(Ṽ, Ω̃_{V} , I_{V})] = [(W̃, Ω̃_{W} , I_{W})]. Let ID := End_{𝔤}(V) and ID′ := End_{𝔤}(W). If ID = IR, then Ṽ is reducible as a real 𝔤-module and otherwise ID = End_{𝔤}(Ṽ). Therefore ID = ID′. We distinguish three cases: ID = IR: In the following we write Ω_{C} for the complex bilinear extension of a symplectic form Ω on V to V_{C}. Suppose that [(V_{C}, Im Ω_{V,C})] = [(W_{C}, Im Ω_{W,C})]. Since V_{C} ≅ V ⊕ V as real 𝔤-modules, we observe that V ≅ W. If [(V, Ω_{V})] ≠ [(W, Ω_{W})], then [(W, Ω_{W})] = [(V, −Ω_{V})]. In that case we obtain a contradiction to C_{V} = C_{VC} = C_{WC} = −C_{V} . ID =C, IH: If [(V, Ω_{V} , I_{V})] = [(W, Ω_{W} , I_{W})], then V ≅ W as real 𝔤-modules. If [(V, Ω_{V})] ≠ [(W, Ω_{W})], we have [(W, Ω_{W})] = [(V, −Ω_{V})] and we obtain the same contradiction as above.

This proves that our assignment is injective. Next we show that it is also surjective. Let (W, Ω_{W} , I) be a simple complex symplectic 𝔤-module of convex type.

If W is simple as a real module, then (W, Ω_{W}) is a simple 𝔤-module of convex type and since it carries an invariant complex structure, End_{𝔤}(W) ≅ C or IH. Therefore [(W, Ω, I)] is the image of the class [(W, Ω)].

Suppose that W is not simple as a real module. Let V ⊆ W be a simple non-zero submodule. Then V + IV is a complex submodule of W so that W = V + IV . Since W is simple and different from V , we also have V ∩ IV = {0}. Therefore W ≅ V_{C} as complex 𝔤-modules. The fact that W is of convex type implies that V is a non-degenerate submodule (Lemma II.4). We claim that IV = V^{⊥}. Let B(v, w) := Ω(Iv, w). Then B is a real symmetric bilinear form on V which is invariant under 𝔤 (cf. Lemma II.24). Hence there exists d ∈ End_{𝔤}(V) with d^{♯} = −d such that B(v, w) = Ω_{V} (d.v, w). On the other hand End_{𝔤}(V) = IR consists of symmetric elements, so that B = 0, consequently V ⊥IV and hence V^{⊥} = IV . We conclude that

\begin{aligned}Ω_{W} (v + Iw, v′ + Iw′) & = Ω_{W} (v, v′) + Ω_{W} (Iw, Iw′) \\ & = Ω_{V} (v, v′) + Ω_{V} (w, w′) = Ω_{VC}(v + Iw, v′ + Iw′). \end{aligned}

Thus we also have surjectivity.

For the following lemma we note that a compact Lie algebra which has a non-trivial symplectic module of convex type cannot be semisimple because the dual of a semisimple compact Lie algebra contains no non-zero pointed invariant cone ([6, Prop. III.2.2]).

Proposition II.27. Let 𝔤 be a compact Lie algebra with non-trivial center and V an irreducible symplectic 𝔤-module. Then the following are equivalent:

(1) V is of convex type.

(2) The center 𝔷(𝔤) acts semisimply non-trivially with purely imaginary

eigenvalues. In this case End_{𝔤}(V) =C or IH.

Proof. (1) ⇒ (2): This follows directly from the definitions.

(2) ⇒ (1): Let π: 𝔤 → 𝔰𝔭(V) ⊆ 𝔤𝔩(V) be the Lie algebra homomorphism defining the symplectic module structure on V . If (2) is satisfied, then the group generated by e^{π(𝔤)} has compact closure in Sp(V). Now the fact that U(n) := Sp(n, IR)∩Gl(n,C) is maximal compact in Sp(n, IR) ⊆ Gl(2n, IR) shows that there exists a 𝔤-invariant complex structure I on V such that Ω(v, I.v) > 0 holds for all v ∈ V \{0}. Pick X ∈ 𝔷(𝔤) which acts non-trivially on V . Then this mapping is complex linear and by Schur’s lemma we may assume that X.v = I.v holds for all v ∈ V . Now Proposition II.23 implies that (V, Ω) is a symplectic module of convex type and that V admits an invariant complex structure so that either End_{𝔤}(V) =C or IH.

We will see later that this result does not hold in general for a noncompact Lie algebra. This will be a byproduct of the classification. A typical example is the Sp(n, IR)-module IR^{2n} which does not admit any invariant complex structure.

In many applications it is possible to obtain information on modules of reductive Lie algebras by looking at their 𝔰𝔩(2, IR)-subalgebras. The following result paves the way to such a reduction in the context of modules of convex type.

Proposition II.28. Let V be a symplectic module of convex type of 𝔤, 𝔱 ⊆ 𝔤 a compactly embedded Cartan algebra and 𝔰 ⊆ 𝔤 a 𝔱-invariant subalgebra isomorphic to 𝔰𝔩(2, IR). Then V is a symplectic 𝔰-module of convex type, and every non-trivial irreducible 𝔰-submodule of V is non-degenerate, of convex type and 2-dimensional.

Proof. We may w.l.o.g. assume that 𝔤 acts effectively on V . Then the cone W_{V} is pointed and generating. Fix a compactly embedded Cartan algebra 𝔱 ⊆ 𝔤. Then there exists a positive 𝔨-adapted positive system of roots Δ^{+} such that

C_{min} ⊆ W_{V} ∩ 𝔱 ⊆ C_{max}

(cf. Definition II.18, [12, Prop. III.15]).

We set 𝔱_{s} := 𝔱 ∩ 𝔰. Since 𝔰 is invariant under 𝔱 and all derivations of 𝔰 are inner, it follows that there exists an element U ∈ 𝔰 such that [𝔱, U] = {0}.

Hence 𝔱_{s} = IRU ≠ {0} is a compactly embedded Cartan subalgebra of 𝔰. Moreover, since 𝔰 is invariant under 𝔱 and three dimensional, we then have 𝔰_{C} as 𝔱_{s}_{C} + 𝔤C^{β} + 𝔤C^{−β} for a root β ∈ Δ_{p}^{+} and choose a non-zero element X_{β} ∈ 𝔤C^{β}. Then X_{0} := i[\overline{X}_{β}, X_{β}] ∈ C_{min} ⊆ W_{V} and this means in particular that X_{0} ∈ C_{V}^{⋆} . Hence the restriction mapping 𝔤^{∗} → 𝔰^{∗} maps C_{V} into a pointed invariant cone and thus V is an 𝔰-module of convex type.

Therefore the results from above apply to the 𝔰-module V . We conclude that every non-trivial irreducible submodule E is non-degenerate (Proposition II.5) and therefore a symplectic 𝔰-module of convex type. Thus W_{E} ≠ {0} and consequently dim E = 2 by Theorem V.8 in [14] and Remark II.15.

III. The classification of simple modules of convex type

In the preceding section we have analyzed the structure of symplectic 𝔤modules of convex type. Theorem II.8 and Corollary II.9 provide effective tools to reduce the classification of general modules of convex type to the simple case. On the other hand we have Theorem II.26 which translates the classification problem of the real simple modules of convex type to the problem of classifying simple complex symplectic modules of convex type modulo complex linear or antilinear isomorphisms of symplectic 𝔤-modules. In this section we adopt this point of view that will permit us by using results obtained in [14] to obtain a complete classification of the simple modules of convex type.

Before we turn directly towards the classification we first have to use tensor product decompositions to reduce the problem to the case where the Lie algebra under consideration is simple hermitean.

Tensor products of complex modules as real modules

Lemma III.1. Let V resp. W be semisimple modules of the reductive IK-Lie algebras 𝔞 resp. 𝔟. Then End_{𝔞⊕𝔟}(V ⊗_{IK} W) ≅ End_{𝔞}(V) ⊗_{IK} End_{𝔟}(W).

Proof. Since the map A ⊗ B ↦ A ⊗ B induces an 𝔞 ⊕ 𝔟-equivariant algebra isomorphism

End_{IK}(V) ⊗_{IK} End_{IK}(W) → End_{IK}(V ⊗_{IK} W),

we only have to show that for a semisimple 𝔞-module X and a semisimple 𝔟module Y we have

(X ⊗ Y)_{fix} = X_{fix} ⊗ Y_{fix},

where X ⊗ Y is considered as (𝔞 ⊕ 𝔟)-module. Then we apply this to X = End_{IK}(V) and Y = End_{IK}(W).

Since X and Y decompose as X = X_{fix} ⊕ X_{eff} and Y = Y_{fix} ⊕ Y_{eff} , it follows that

X ⊗ Y ≅ (X_{fix} ⊗ Y_{fix}) ⊕ (X_{eff} ⊗ Y_{fix}) ⊕ (X ⊗ Y_{eff}).

On the other hand, we see that for x ∈ X, y ∈ Y and b ∈ 𝔟 that

x ⊗ (y − b.y) = x ⊗ y − b.(x ⊗ y) ∈ (X ⊗ Y)_{eff}.

We conclude that

(X ⊗ Y_{eff}) + (X_{eff} ⊗ Y) ⊆ (X ⊗ Y)_{eff}

so that the assertion follows from X ⊗ Y = (X ⊗ Y)_{fix} ⊕ (X ⊗ Y)_{eff} .

Let V be a complex simple 𝔤-module. We consider V as a real 𝔤-module. Then there are three possibilities: V is of real type: V is reducible over IR. Then V = (V^{0})_{C} for a real simple 𝔤-module V^{0} . In this case End_{𝔤}(V) ≅ IM(2, IR). V is of complex type: V is simple over IR and End_{𝔤}(V) =C. V is of quaternionic type: V is simple over IR and End_{𝔤}(V) = IH.

Lemma III.2. Let V resp. W be two complex simple modules for the real Lie algebra 𝔞 resp. 𝔟. We consider X := V ⊗_{C} W as a real module for the Lie algbera 𝔤 := 𝔞 ⊕ 𝔟. Then X is simple as a complex 𝔤-module and the type of X can be determined as follows:

(1) If V is of real type, then X has the same type as W.

(2) If V is of complex type and W is not of real type, then X is of complex type.

(3) If V and W are of quaternionic type, then X is of real type.

Proof. We distinguish several cases: If V is of real type with V = (V^{0})_{C}, then the map V^{0} ⊗_{IR} W → V ⊗_{C} W is a module isomorphism. Hence Lemma III.1 implies that

End_{𝔤}(V ⊗_{C} W) ≅ End_{𝔤}(V^{0} ⊗_{IR} W) ≅ End_{𝔞}(V^{0}) ⊗_{IR} End_{𝔟}(W) ≅ End_{𝔟}(W).

Since this is a division algebra, we see that V ⊗_{C} W is simple as a real 𝔤-module and that its type is the type of W. Suppose that V and W are not of real type. Then the map V ⊗_{IR} W → V ⊗_{C} W has a non-trivial kernel. Moreover

End_{𝔤}(V ⊗_{IR} W) ≅ End_{𝔞}(V) ⊗_{IR} End_{𝔟}(W).

If V and W are of complex type, then End_{𝔤}(V ⊗_{IR} W) ≅C⊗_{IR}C ≅C⊕C possesses one idempotent different from 0 and 1. Therefore V ⊗_{IR}W decomposes into a direct sum of two simple modules so that V ⊗_{C} W must be simple over IR and of complex type.

If V is of complex and W of quaternionic type, then

End_{𝔤}(V ⊗_{IR} W) ≅C ⊗_{IR} IH ≅ IM(2,C)

as one easily verifies by viewing IH as a subalgebra of IM(2,C) (cf. [19, p.190]). Therefore V ⊗_{IR}W is a direct sum of two equivalent submodules with commutant C. Thus V ⊗_{C} W must be simple over IR and of complex type.

We claim that if V and W are of quaternionic type, then

End_{𝔤}(V ⊗_{IR} W) ≅ IH ⊗_{IR} IH ≅ IM(4, IR).

In fact, using the left and right regular representation of the IR-algebra IH, we obtain an algebra homomorphism IH ⊗_{IR} IH → IM(4, IR) which turns out be surjective since the IH ⊗_{IR} IH-module IH is simple with commutant isomorphic to IR (Wedderburn’s Theorem). Hence, counting dimensions yields IH ⊗_{IR} IH ≅ IM(4, IR) (cf. [19, p.190]).

We conclude that V ⊗_{IR} W is a direct sum of four equivalent simple submodules and counting dimensions shows that V ⊗_{C} W is reducible over IR.

Towards the classification of modules of convex type

Lemma III.3. Let (V, Ω_{V} , I_{V}) resp. (W, Ω_{W} , I_{W}) be a complex symplectic 𝔤module. Then (V ⊗_{C} W, Ω, I) is a complex symplectic 𝔤-module, where

Ω(v ⊗ w, v′ ⊗ w′) = Ω_{V} (v, v′)Ω_{W} (I_{W} .w, w′) + Ω_{W} (w, w′)Ω_{V} (I_{V} .v, v′).

Proof. According to Lemma II.24, the forms (v, v′) ↦ Ω_{V} (I_{V} .v, v′) and (w, w′) ↦ Ω_{W} (I_{W} .w, w′) are symmetric so that

(v ⊗ w, v′ ⊗ w′) ↦ Ω_{V} (v, v′)Ω_{W} (I_{W} .w, w′) + Ω_{W} (w, w′)Ω_{V} (I_{V} .v, v′)

is a well defined skew symmetric map on V ⊗_{IR} W. To see that this map factors to the quotient V ⊗_{C}W, we have to show that it vanishes if one of the arguments has the form v ⊗ w + I_{V} .v ⊗ I_{W} .w. For the first argument this follows from

Ω_{V} (v, v′)Ω_{W} (I_{W} .w, w′) + Ω_{W} (w, w′)Ω_{V} (I_{V} .v, v′)
− Ω_{V} (I_{V} .v, v′)Ω_{W} (w, w′) − Ω_{W} (I_{W} .w, w′)Ω_{V} (v, v′) = 0.

This shows that Ω defines a skew-symmetric real bilinear form on V ⊗_{C}W which is invariant under the complex structure I. To see that this form is nondegenerate, we note that it is the imaginary part of the pseudo-hermitean form B on V ⊗_{C} W that arises from the pseudo-hermitean forms B_{V} resp. B_{W} on V resp. W (cf. Lemma II.24) by

(3.1)  B((v ⊗ w), (v′ ⊗ w′)) := B_{V} (v, v′)B_{W} (w, w′).

Since B_{V} and B_{W} are non-degenerate, the same follows for B and hence for Ω by choosing orthogonal bases in V resp. W. The 𝔤-invariance of Ω is verified with (3.1) by an easy calculation.

Definition III.4. If (V, Ω_{V} , I_{V}) resp. (W, Ω_{W} , I_{W}) are complex symplectic 𝔤-modules, then the complex symplectic 𝔤-module (V ⊗_{C} W, Ω, I) defined in Lemma III.3 is called the tensor product of the complex symplectic 𝔤-modules (V, Ω_{V} , I_{V}) and (W, Ω_{W} , I_{W}). It is denoted by (V, Ω_{V} , I_{V}) ⊗ (W, Ω_{W} , I_{W}).

We recall the following result which is Theorem IV.13 in [14]. Here we state it in the terms relevant for our present purposes.

Theorem III.5. Let 𝔤 be a reductive Lie algebra and V a simple complex symplectic 𝔤-module such that W_{V} ≠ {0} and 𝔤 acts effectively. Then 𝔤 ≅ 𝔤_{1} ⊕ 𝔤_{2} , where 𝔤_{2} is a compact semisimple Lie algebra, the commutator algebra of 𝔤_{1} is either trivial or a hermitean simple ideal, and

(V, Ω, I) ≅ (V_{1}, Ω_{1}, I_{1}) ⊗ (V_{2}, Ω_{2}, I_{2})

is a tensor product of simple complex symplectic 𝔤-modules, where V_{1} resp. V_{2} is a 𝔤_{1} resp. 𝔤_{2} -module, W_{V_{1}} ≠ {0}, and the canonical pseudo-hermitean form B_{2} on V_{2} is positive definite.

If, conversely, the simple complex symplectic 𝔤_{1} -module (V_{1}, Ω_{1}, I_{1}) and the simple complex symplectic 𝔤_{2} -module (V_{2}, Ω_{2}, I_{2}) have these properties, then their tensor product V := V_{1} ⊗_{C} V_{2} is a simple complex symplectic 𝔤-module and

W_{V} ∩ 𝔤_{1} = W_{V_{1}} ≠ {0}.

Proof. To see that this is the statement of Theorem IV.13 in [14], we only have to note that, in view of Lemma II.24, the concept of a complex symplectic 𝔤-module is the same as that of a complex 𝔤-module with an invariant nondegenerate pseudo-hermitean form.

To link Theorem III.5 to modules of convex type, we need the following lemma.

Lemma III.6. Let 𝔤 be a non-zero quasihermitean reductive Lie algebra and (V, Ω) an effective simple symplectic 𝔤-module such that 𝔷(𝔤) acts semisimply with purely imaginary spectrum. Then the following are equivalent

(1) W_{V} ≠ {0}

(2) W_{V} is generating. Proof. (1) ⇒ (2): Suppose that W_{V} ≠ {0}. Since V is an effective 𝔤-module C_{V}^{⊥} = H(W_{V}) = {0}, i.e., W_{V} is pointed. Put 𝔞 := W_{V} − W_{V} . Then 𝔞 is a non-zero ideal of 𝔤 containing the pointed generating invariant cone W_{V} . Let us consider V as a symplectic 𝔞-module and let p_{𝔞}: 𝔤^{∗} → 𝔞^{∗} denote the restriction map. Then

Φ_{𝔞}(V) = p_{𝔞}(Φ_{𝔤}(V)) ⊆ p_{𝔞}(C_{V}) ≅ C_{V} /H(C_{V})

lies in a pointed cone, whence (V, Ω) is a symplectic 𝔞-module of convex type.

Moreover the Lie algebra 𝔞 contains a pointed generating invariant cone so that it is quasihermitean (cf. [6, Th. III.5.16]). Let 𝔨 ⊆ 𝔞 be a maximal compactly embedded subalgebra and Z ∈ 𝔷(𝔨) ∩ int C_{V}^{⋆} . We claim that the function φ(Z) on V is positive on V \{0}. Suppose that φ(Z)(v) = Φ(v)(Z) = 0. Since Φ(v) ∈ C_{V} , we conclude that Φ(v) ∈ H(C_{V}) = 𝔞^{⊥}. Therefore Φ_{𝔞}(v) = 0. Now Proposition II.22 applies since 𝔞 is quasihermitean and we see that 𝔞.v = {0}. For X ∈ 𝔤 we then obtain 𝔞.(X.v) ⊆ [X, 𝔞].v + X.(𝔞.v) = {0}, so that 𝔤.v = {0}. Thus v = 0 by the simplicity of V . We conclude that φ(Z) is positive on V \ {0}, so that Proposition II.23 entails that V is of convex type, i.e., W_{V} is generating.

(2) ⇒ (1): Since 𝔤 ≠ {0}, the assumption that W_{V} is generating implies that W_{V} ≠ {0}.

Remark III.7. The preceding result applies equally well to the complexification (V_{C}, Ω̃) of a simple symplectic module (V, Ω) because the cones corresponding to both modules are the same. Hence the assertion remains true for simple complex symplectic 𝔤-modules.

Remark III.8. In view of Remark III.7, Theorem III.5 shows that the classification of simple complex symplectic modules V of convex type of the reductive Lie algebra 𝔤 boils down to the classification of those simple complex symplectic 𝔤_{1} -modules V_{1} of convex type of a reductive Lie algebra 𝔤_{1} whose commutator algebra is simple hermitean and whose center is at most one-dimensional (Schur’s Lemma). To find the corresponding real simple module of convex type one has to apply Lemma III.2 to see whether V is of real type or not.

To do this, we need some information on the type of a complex irreducible representation of a compact semisimple Lie algebra.

Definition III.9. A simple complex 𝔤-module V is called orthogonal resp. symplectic if it admits a non-degenerate 𝔤-invariant complex bilinear symmetric resp. skew-symmetric form.

Since the 𝔤-invariance of a complex bilinear form is equivalent to the 𝔤_{C}- invariance, we will be able to draw the information we need out of the following result which is essentially due to Malcev ([11]):

Proposition III.10. Let 𝔤 be a complex Lie algebra with Cartan algebra 𝔥 and root decomposition 𝔤 = 𝔥 ⊕ ⨁_{α∈Δ} 𝔤^{α}. Let V_{λ} denote the irreducible 𝔤-module with highest weight λ, write w_{0} ∈ W for the unique element of the Weyl group with w_{0}(Δ^{+}) = −Δ^{+} and put m := ∑_{α∈Δ^{+}} λ(α̌). Then one of the following three mutually exclusive cases occurs:

(1) w_{0}(λ) ≠ −λ and V_{λ} ≅ V̸_{λ}^{∗}.

(2) w_{0}(λ) = −λ and m is even. Then V_{λ} is orthogonal.
(3) w_{0}(λ) = −λ and m is odd. Then V_{λ} is symplectic.

Proof. This is contained in [1, Ch. VIII, §7, no. 5, Prop. 12]. We only note that since λ(α̌) ∈ IN_{0} for all α ∈ Δ^{+} the number m is a non-negative integer.

To apply this result to determine the types of representations of compact semisimple Lie algebras, one needs the following:

Proposition III.11. Let 𝔤 be a compact semisimple Lie algebra and V a simple complex 𝔤-module. Then the following assertions hold:

(i) V is of real type if and only if V is orthogonal. (ii) V is of complex type if and only if V ≅ V̸^{∗}. (iii) V is of quaternionic type if and only if V is symplectic.

Proof. Since representations of semisimple compact Lie algebras are in oneto-one correspondence with representations of the associated simply connected group which is compact, the assertions follow from [1, Ch. 9, App. 2, no. 2, Prop.

Combining Propositions II.10 and II.11 one can determine the type of each representation of a compact semisimple Lie algebra, so that we have access to the type of the V_{2} -factor in Theorem III.5. Later we will also derive a similar result for the V_{1} -factor (Proposition III.16).

Remark III.12. Let us return to the situation of Theorem III.5. Here dim 𝔷(𝔤_{1}) ≤ 1 is a consequence of Schur’s Lemma and the injectivity of the representation on V . Then the module V_{1} is a simple module for the commutator algebra because the center acts by scalar multiples of the identity. If 𝔤_{1} = 𝔷(𝔤_{1}) ≅ IR, then every one-dimensional complex symplectic module is of convex type and if 𝔤_{1} is not abelian, then the condition W_{V_{1}} ≠ {0} implies that W_{V_{1}} ∩ [𝔤_{1}, 𝔤_{1}] ≠ {0} (cf. Lemma III.6, [6, III]). On the other hand this condition trivially implies W_{V_{1}} ≠ {0} so that it only remains to consider the case where 𝔤_{1} is simple hermitean.

For the case where 𝔤 is a simple hermitean Lie algebra, the classification has been carried out in [14] where it was based on the following result ([14, Th. V.12]).

Theorem III.13. Let 𝔤 be a simple hermitean Lie algebra, 𝔱 ⊆ 𝔨 a compactly embedded Cartan algebra, Z ∈ i𝔷(𝔨) with Spec(ad Z) = {0, 1, −1}, and V a simple complex 𝔤_{C}-module. Let λ denote the highest weight of V with respect to a 𝔨-adapted positive system of roots. Then W_{V} ≠ {0} if and only if the following two conditions are satisfied:

(1) λ(Z) ∈]0, 1[.

(2) If W denotes the Weyl group of (𝔤_{C}, 𝔱_{C}), then (W.λ)(Z) consists only of two elements.

To obtain further general information on the simple modules, we need the following proposition. Note that in view of [14, Th. V.3], every simple complex module of a reductive quasihermitean Lie algebra on which 𝔷(𝔤) acts by purely imaginary multiples of the identity admits an invariant pseudohermitean form.

Proposition III.14. Let V be a simple complex 𝔤-module, where 𝔤 is a quasihermitean non-compact reductive Lie algebra acting effectively on V , and B an invariant non-degenerate pseudohermitean form on V . Then the following are equivalent:

(1) (V, Im B) is of convex type.

(2) V contains exactly two K -types, i.e., V decomposes under 𝔨 into two simple 𝔨-modules.

Proof. We first use Theorem III.5 to write V as V_{1} ⊗_{C} V_{2} and 𝔤 = 𝔤_{1} ⊕ 𝔤_{2} , where 𝔤_{2} is compact semisimple and 𝔤_{1} is reductive with exactly one simple hermitean ideal because 𝔤 is non-compact. Similarly 𝔨 = 𝔨_{1} ⊕ 𝔤_{2} and the simple 𝔨-modules can be obtained as V_{1}^{j} ⊗ V_{2} , where V_{1}^{j} is a simple 𝔨_{1} -module in V_{1} . Therefore the numbers of K -types in V and V_{1} coincide. In view of Remark III.12, this reduces the problem to the case where 𝔤 is simple hermitean since the center acts by purely imaginary multiples of the identity. From now on we assume that 𝔤 is simple hermitean.

(1) ⇒ (2): This follows from Proposition VII.1 in [14].

(2) ⇒ (1): We choose Z ∈ i𝔷(𝔨) such that Spec(ad Z) = {1, 0, −1}. Let V = V^{+} ⊕ V^{−} be the decomposition of V into simple complex 𝔨-modules and write λ resp. μ for the highest weights of V^{+} resp. V^{−} with respect to a positive system Δ_{k}^{+} . We assume that Δ_{p}^{+} = {α ∈ Δ: α(Z) > 0}, that λ ist the highest weight of the 𝔤_{C}-module V , and decompose the set P_{V} as P_{V^{+}} ∪P_{V^{−}}. Since the 𝔨-modules V^{+} and V^{−} are simple, the center of 𝔨 acts on each space by a scalar multiple of the identity. Hence λ(Z) = α(Z) for all α ∈ P_{V^{+}} and μ(Z) = α(Z) for all α ∈ P_{V^{−}}.

Let W denote the big Weyl group of 𝔤 acting on 𝔱. Since 𝔤 is simple, 𝔱 is a simple W -module and conv P_{V} = conv(W.λ) ⊆ i𝔱^{∗}. The center of mass of W.λ is a W -fixed point, hence equal to 0 and the affine subspace generated by W.λ is i𝔱^{∗}. Hence 0 lies in the relative interior of conv P_{V} in i𝔱^{∗} and we see that this set contains a weight which is positive on Z and one which is negative on Z. So we find in particular that λ(Z) > 0 > μ(Z). Moreover μ(Z) = λ(Z) − 1 because α(Z) = 1 for all positive non-compact roots since 𝔤C^{β}.V^{+} = {0} for β ∈ Δ_{p}^{+} and 𝔤C^{β}.V^{+} ⊆ V^{−} for β ∈ −Δ_{p}^{+} . Therefore λ(Z) ∈]0, 1[ and ⟨W.λ, Z⟩ = {λ(Z), μ(Z)} contains only two elements. Now Theorem III.13 applies.

We recall the explicit classification from [14]. We do this by writing down the highest weights which correspond to those simple complex modules which are of convex type. These are always fundamental weights which are denoted by ω_{1}, …, ω_{n}, where n is the rank of 𝔤_{C} (cf. [1, Ch. VI, VIII] for more details on roots and weights).

Theorem III.15. For the simple hermitean Lie algebra 𝔤 the following highest weights correspond to modules of convex type:

(A_{n}) 𝔤 = 𝔰𝔲(p, q), 𝔤_{C} = 𝔰𝔩(p + q,C), n = p + q − 1: ω_{1} and ω_{n}.

(A_{n}) 𝔤 = 𝔰𝔲(n, 1), 𝔤_{C} = 𝔰𝔩(n + 1,C): ω_{1}, …, ω_{n}.

(B_{n}) 𝔤 = 𝔰𝔬(2n − 1, 2), 𝔤_{C} = 𝔰𝔬(2n + 1,C): ω_{n}.

(C_{n}) 𝔤 = 𝔰𝔭(n, IR), 𝔤_{C} = 𝔰𝔭(n,C): ω_{1} .

(D_{n}) 𝔤 = 𝔰𝔬(2n − 2, 2), 𝔤_{C} = 𝔰𝔬(2n,C): ω_{n−1} and ω_{n}.

(D_{n}) 𝔤 = 𝔰𝔬^{∗}(2n), 𝔤_{C} = 𝔰𝔬(2n,C): ω_{1} .

(E_{6}) 𝔤 = 𝔢_{(6,−14)} : no weight.

(E_{7}) 𝔤 = 𝔢_{(7,−25)} : no weight.

One remarkable consequence of this classification is that the exceptional hermitean simple algebras have no symplectic modules of convex type. Next we study the types of these representations and the possible antilinear isomorphisms because antiisomorphic symplectic 𝔤-modules correspond to the same real module (cf. Theorem II.26).

Proposition III.16. Let W be a non-zero simple complex symplectic 𝔤module of convex type. Then

(i) W is of real type if and only if W is symplectic, i.e., W carries an invariant complex symplectic form.

(ii) W is of quaternionic type if and only if W is orthogonal, i.e., carries an invariant non-degenerate symmetric complex bilinear form.

Proof. (i) If W is of real type, then W = V_{C} holds for a real symplectic 𝔤-module V (Lemma II.4) and Proposition II.5 and Lemma II.11 provide an invariant non-degenerate symplectic complex bilinear form (Ω_{V})_{C} on W. Hence W is symplectic.

If, conversely, W is symplectic and Ω_{C} is an invariant complex symplectic form, then we find an antilinear operator A on W such that

Ω_{C}(v, w) = B(v, Aw)

holds for all v, w ∈ W, where B is the pseudo-hermitean form on W with Ω = Im B. Then A commutes with 𝔤 and the fact that Im Ω_{C} is skew-symmetric yields that A^{♯} = A with respect to the real symplectic form Ω.

Suppose that W is not of real type, i.e., W is simple over IR. Then W is of type C_{I} or IH_{I} as a simple real 𝔤-module of convex type (cf. Lemma II.7). Thus A^{♯} = A implies that A ∈ IR1, contradicting the fact that A is antilinear. (ii) A real bilinear symmetric form B′: V × V →C can be written as

B′(v, w) = Ω(a.v, w) + iΩ(b.v, w),

where a, b ∈ Hom_{𝔤}(V) satisfy a^{♯} = −a and b^{♯} = −b. Moreover, the complex bilinearity of such a form is equivalen to the condition

Ω(aI.v, w) + iΩ(b.Iv, w) = iΩ(a.v, w) − Ω(b.v, w),

which means that b = −aI. Now b^{♯} = −b entails −aI = (aI)^{♯} = −Ia^{♯} = Ia.

This shows that V is orthogonal if and only if there exists a ∈ End_{𝔤}(V) with a^{♯} = −a and Ia = −aI. Since, according to Lemma II.7, V is of real type or of type C_{I} or IH_{I}, we see that V is orthogonal if and only if it is of type IH_{I}.

If V is of type IH_{I}, then the form B′(v, w) = Ω(J.v, w) + iΩ(K.v, w) satisfies all the requirements.

Lemma III.17. Let V be a simple complex 𝔤-module. Then \overline{V} ≅ V if and only if V is of real or quaternionic type. If V is of complex type and a complex symplectic 𝔤-module, then \overline{V} ≅ V^{∗}.

Proof. (cf. [1, Ch. IX, App. 2, no. 1]). If V is of real type, then V = W_{C} for a simple real 𝔤-module W. Hence V ≅ W ⊕ W as a real 𝔤-module and therefore \overline{V} ≅ W ⊕ W ≅ V .

If V is of complex or quaternionic type, then V ≅ \overline{V} if and only if End_{𝔤}(V) contains two anticommuting complex structures which is the case if and only if V is of quaternionic type.

Suppose that V is a simple complex symplectic 𝔤-module. Then we use Lemma II.23 to obtain a 𝔤-invariant pseudohermitean form on V . Hence V^{∗} ≅ \overline{V} .

The preceding results provide the information we need to use the tables on simple complex modules to determine their type.

Remark III.18. Here is some additional information on the representations which occur in Theorem III.15.

(A_{n}) V_{ω_{1}} = C^{n+1} and V_{ω_{n}} = ⋀^{n}(C^{n+1}) are representations which are also simple over IR. For 𝔤 = 𝔰𝔲(p, q), n = p + q − 1 with p ≥ q ≥ 2, these representations are of complex type. Here \overline{V_{ω_{1}}} ≅ V_{ω_{1}}^{∗} ≅ V_{ω_{n}}. For 𝔤 = 𝔰𝔲(p, 1) we have V_{ω_{k}} = ⋀^{k}(C^{p+1}). This module is of complex type for k ≠ \frac{p+1}{2} , and for k = \frac{p+1}{2} it is of real type if k is odd and of quaternionic type otherwise. Therefore \overline{V_{ω_{\frac{p+1}{2}}}} ≅ V_{\frac{p+1}{2}} and for k ≠ \frac{p+1}{2}

we obtain \overline{V_{ω_{k}}} ≅ V_{ω_{k}}^{∗} ≅ V_{ω_{n+1−k}} ([1, Ch. VIII, §13, no. 1]).

(B_{n}) V_{ω_{n}} = ⋀^{∗}(C^{n}) ≅ \overline{V_{ω_{n}}} is the spin representation. According to [1, Ch. VIII, §13, no. 2], the module V_{ω_{n}} is orthogonal for n = 0, 3 mod 4 and symplectic for n = 1, 2 mod 4. Hence it is of real type for n = 1, 2 mod 4 and for n = 0, 3 mod 4 it is quaternionic.

(C_{n}) V_{ω_{1}} = C^{2n} is of real type because IR^{2n} carries no Sp(n, IR)-invariant

complex structure ([1, Ch. VIII, §13, no. 3]). (D_{n}) V_{ω_{n−1}} and V_{ω_{n}} are the simple constituents of the spin representation

on ⋀^{∗}(C^{n}) which are given by the odd and the even exterior powers of C^{n}. Here the situation is more complicated than for B_{n} ([1, Ch. VIII, §13, no. 4]). For n = 0, 3 mod 4 there exists an invariant symmetric complex bilinear form on the spin representation and a symplectic form for n = 1, 2 mod 4. If n is even, then the restriction of these forms to the simple modules V_{ω_{n−1}} and V_{ω_{n}} is non-degenerate and if n is odd, then both are isotropic and in duality. In the latter case both are neither orthogonal nor symplectic. It follows from Proposition III.16 that these modules are of real type for n = 2 mod 4, of quaternionic type for n = 0 (mod 4), and of complex type for n odd. For n even we therefore have \overline{V_{ω_{n}}} ≅ V_{ω_{n}}, \overline{V_{ω_{n−1}}} ≅ V_{ω_{n−1}} , and for n odd we obtain \overline{V_{ω_{n−1}}} ≅ V_{ω_{n−1}}^{∗} ≅ V_{ω_{n}}.

(D_{n}) V_{ω_{1}} is the standard representation of 𝔰𝔬^{∗}(2n) on C^{2n} ≅ IH^{n}. It is of quaternionic type ([1, Ch. VIII, §13, no. 4]).

IV. (H_{1})-homomorphisms of quasihermitean Lie algebras

In this section we explain how modules of convex type are related to equivariant embedding of bounded symmetric domains into the Siegel space, i.e., the hermitean symmetric space of the symplectic group Sp(n, IR). We will always write 𝔤 for a quasihermitean reductive Lie algebra and G for a simply connected group with L(G) = 𝔤.

Definition IV.1. 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 module complement for 𝔨 in 𝔤 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} .

Remark IV.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}. Since every simple hermitean Lie algebra contains exactly 2 H -elements associated to a fixed Cartan decomposition, the number of H -elements associated to a fixed Cartan decomposition in the commutator algebra 𝔤′ = [𝔤, 𝔤] is 2^{k}.

Let (𝔤, H_{0}) be a reductive Lie algebra of hermitean type, 𝔤_{c} the maximal compact ideal, and 𝔤_{n} its complementary ideal. There exists a maximal closed convex invariant cone W_{max} = W_{max}(H_{0}) ⊆ 𝔤 with H(W_{max}) = 𝔤_{c}, H_{0} ∈ W_{max} , and W_{max} ∩ 𝔱 = C_{max} for every compactly embedded Cartan algebra 𝔱 containing H_{0} , where the positive system of roots is chosen in such a way that Δ_{p}^{+} = {α ∈ Δ: iα(H_{0}) > 0} (cf. [12, III]). Note that a reductive quasihermitean Lie algebra containing k simple hermitean ideals contains exactly 2^{k} maximal cones corresponding to the conjugacy classes of H -elements in 𝔤′ and that the cone W_{max}(H_{0}) determines the projection of H_{0} onto the commutator algebra.

There also exist minimal cones W_{min} = W_{min}(H_{0}) which are contained in the sum of the hermitean simple ideals of 𝔤 and which are uniquely determined by the condition that they contain the 𝔤′ -component of the H -element H_{0} . Therefore we have exactly 2^{k} such cones and W_{min} ∩ 𝔱 = C_{min} ([12, III]).

Definition IV.3. 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.

Note that this 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, i.e., it respects the Cartan involutions, 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) 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.

Note that composition of (H_{1})-homomorphisms yields (H_{1})-homomor-phisms. We recall from [21] that (H_{1})-homomorphisms of Lie algebras of hermitean type (without compact factors) are in one-to-one correspondence with strongly equivariant holomorphic maps of the corresponding bounded symmetric domains. This correspondence is set up by assigning to the (H_{1})-homomorphism κ: 𝔤 → 𝔤̃ the corresponding induced map G/K → G̃/K̃.

Lemma IV.4. Let (𝔤, H_{0}) and (𝔤̃, H̃_{0}) be two reductive Lie algebras of hermitean type and κ: 𝔤 → 𝔤̃ a Lie algebra homomorphism. Write p_{𝔷}, p_{𝔤_{n}} resp. p_{𝔤′} for the projection onto the center, 𝔤̃_{n} resp. the commutator algebra 𝔤̃′ 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. [8, Lemma II.8]

Proposition IV.5. 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. [8, Proposition II.9]

The preceding proposition can be used as a bridge between (H_{1})-homo-morphisms and symplectic modules of convex type.

Theorem IV.6. Let 𝔤 be a reductive quasihermitean Lie algebra and (V, Ω) be a symplectic 𝔤-module of convex type. Then we can choose H -elements in such a way that κ: 𝔤 → 𝔰𝔭(V, Ω) is an (H_{1})-homomorphism. The converse is true if 𝔤 is semisimple without compact factors or if κ is an (H_{2})-homomorphism.

Proof. Let (V, Ω) be a symplectic vector space and Sp(V, Ω) the corresponding symplectic group. Let further I ∈ Sp(V) denote a complex structure on V such that IΩ: (v, w) ↦ Ω(v, I.w) is positive definite. Then H̃_{0} := \frac{1}{2}I is an H - element in the hermitean Lie algebra 𝔰𝔭(V, Ω). The associated invariant cones are

W̃_{min}(H̃_{0}) = W̃_{max}(H̃_{0}) = {X ∈ 𝔰𝔭(V, Ω): φ(X) ≥ 0},

where φ(X) = \frac{1}{2}Ω(X.v, v) is the Hamiltonian function associated to X (cf. Section II). Now let 𝔤 be a reductive quasihermitean Lie algebra and κ: 𝔤 → 𝔰𝔭(V, Ω) a homomorphism. Then W_{V} = κ^{−1}(W̃_{max}), so that V is a symplectic 𝔤-module of convex type if and only if κ^{−1}(W̃_{max}) is generating.

Now the assertion is immediate from Proposition IV.5.

Note that Theorem IV.6 explains why the classification in Theorem III.15 and the classification in [20] of the (H_{1})-homomorphisms of simple hermitean Lie algebras into 𝔰𝔭(V, Ω) yield the same representations.

Remark IV.7. The converse of the conclusion in Theorem IV.6 is false in general. One simply can take a compactly embedded one-dimensional subalgebra 𝔤 of 𝔰𝔭(V, Ω) such that 𝔤 ∩ W_{max} = {0}. Then the inclusion of 𝔤 is trivially an (H_{1})-homomorphism but V is not a symplectic 𝔤-module of convex type.

V. The classification of Lie algebras with invariant cones

First we build a Lie algebra with invariant cones from basic building blocks. We need:

• a reductive quasihermitean Lie algebra 𝔩,

• a real vector space 𝔷_{1} ,

• a symplectic 𝔩-module of convex type (V, Ω) with V_{fix} = {0} such that 𝔷(𝔩) acts effectively on V , and

• an 𝔩-invariant skew-symmetric form q: V × V → 𝔷_{1} .

We define the vector space

𝔤 := 𝔤(𝔩, V, 𝔷_{1}, q) := V × IR × 𝔷_{1} × 𝔩

and endow it with the Lie bracket

(5.1) [(v, t, z, X), (v′, t′, z′, X′)] = (X.v′ − X′.v, Ω(v, v′), q(v, v′), [X, X′]).

Theorem V.1. The space (𝔤, [·, ·]) is a Lie algebra containing invariant cones. It is reductive if and only if V = {0}, and every non-reductive Lie algebra containing invariant cones arises in that way.

Proof. Let 𝔱_{𝔩} ⊆ 𝔩 be a compactly embedded Cartan algebra. According to Proposition II.22, the only element of V fixed by 𝔱_{𝔩} is 0. Therefore Proposition II.21 in [12] shows that 𝔤 is a Lie algebra and that

𝔷(𝔤) = {0} × IR × 𝔷_{1} × {0}
𝔫 = V × IR × 𝔷_{1} × {0} is the nilradical
𝔯 = V × IR × 𝔷_{1} × 𝔷(𝔩) is the radical, and
𝔱 = {0} × IR × 𝔷_{1} × 𝔱_{𝔩}

is a compactly embedded Cartan algebra.

We choose a regular element X ∈ 𝔱 such that α(X) ≠ 0 holds for all α ∈ P_{V} and B is positive definite on VC^{α} if and only if iα(X) < 0 (Proposition II.23). Let V^{+} denote the sum of all those weight spaces for which iα(X) < 0. Then the mapping v ↦ v + \overline{v}, V^{+} → V is a bijection and we can use it to define a complex structure I on V by I.(v + \overline{v}) := i(v − \overline{v}). Let v ∈ VC^{α} ⊆ V^{+} . Then

Ω(I.(v + \overline{v}), v + \overline{v}) = Ω(i(v − \overline{v}), v + \overline{v}) = 2iΩ_{C}(v, \overline{v}) = 2B(v, v) > 0

for v ≠ 0. In view of Proposition III.13 in [12], this proves that 𝔤 has strong cone potential, so that Theorem III.39 in [12] shows that 𝔤 contains invariant cones.

If, conversely, 𝔤 is a non-reductive Lie algebra containing invariant cones, let 𝔱 ⊆ 𝔤 be a compactly embedded Cartan algebra, 𝔫 the nilradical of 𝔤, 𝔰 a 𝔱-invariant Levi algebra and 𝔱_{1} ⊆ 𝔱 a complement for the center of 𝔤 (cf. [12, Prop. II.11]). We set 𝔩 := 𝔱_{1} + 𝔰 and V := [𝔱, 𝔫]. Note that V is invariant under 𝔩 ([12, Prop. II.11]). So far we have only used that 𝔤 contains a compactly embedded Cartan algebra.

Since 𝔤 contains invariant cones, it has strong cone potential and therefore there exists a 𝔨-adapted positive system Δ^{+} and a linear functional ω ∈ 𝔷(𝔤)^{∗} such that

(5.2)  ω(i[\overline{X}_{α}, X_{α}]) > 0

whenever 0 ≠ X_{α} ∈ 𝔫_{C} ∩ 𝔤C^{α} (cf. Prop. III.15 and Th. III.39 in [12]). It follows in particular that 𝔤 and 𝔩 are quasihermitean (cf. Definition II.18). Now we define 𝔷_{1} := ker ω ∩ 𝔷, identify 𝔷 with IR × 𝔷_{1} via ω, and define Ω and q by

[v, w] = (Ω(v, w), q(v, w)),

where Ω(v, w) = ω([v, w]). Note that this works since [𝔫, 𝔫] ⊆ 𝔷(𝔤) by [12, Prop. II.10]. Now Ω is an 𝔩-invariant symplectic form on V and, in view of (5.2), Proposition II.23 shows that V is of convex type. Finally

𝔤 = V × IR × 𝔷_{1} × 𝔩

and all brackets are as in (5.1). This proves the theorem.

Remark V.2. A reductive Lie algebra 𝔤 containing invariant cones is simply a quasihermitean reductive Lie algebra which is not compact semisimple. Therefore Theorem V.1 provides an explicit description of all Lie algebras with invariant cones.

Corollary V.3. Let 𝔤 be a Lie algebra with invariant cones and 𝔤 = 𝔯× 𝔰 a Levi decomposition. Then all exceptional ideals of 𝔰 are ideals of 𝔤 and therefore split as ideal direct summands.

Proof. It follows from the classification of modules of convex type that an exceptional ideal 𝔞 of 𝔰 acts trivially on [𝔰, 𝔯] ⊆ [𝔱, 𝔫]. Therefore they act trivially on 𝔯 and this implies the assertion.

Remark V.4. Note that the representation of a Lie algebra with invariant cones as in Theorem V.1 depends on

• the choice of 𝔩, and

• the choice of a suitable decomposition, i.e., a non-zero linear functional on the center. To see that the choice of 𝔩 is inessential, we recall that all compactly

embedded Cartan subalgebras of 𝔤 are conjugate and that each compactly embedded Cartan algebra 𝔱 uniquely determines a 𝔱-invariant Levi complement 𝔰. Let 𝔩, 𝔩̃ ⊆ 𝔱+𝔰 be two different choices of an 𝔩-subalgebra. Let ψ: 𝔰+𝔱 → (𝔰+𝔱)/𝔷 denote the canonical quotient homomorphism. Then ψ |_{𝔩} and ψ |_{𝔩} are isomorphisms and we obtain an isomorphism

ψ̃:= (ψ |_{𝔩})^{−1} ∘ ψ |_{𝔩}: 𝔩 → 𝔩̃.

The corresponding modules are given by V = [𝔩, 𝔫] = [𝔱, 𝔫] = [𝔩̃, 𝔫] = Ṽ and one easily checks that if ρ resp. ρ̃ are the representations of 𝔩 resp. 𝔩̃ on V , then ρ̃ ∘ ψ̃ = ρ. Therefore the choice of 𝔩 is inessential and it determines the 𝔩-module V .

The following observations shed some light on the choice of the functional β ∈ 𝔷^{∗}. Here 𝔤 denotes a Lie algebra containing invariant cones and 𝔱 a fixed compactly embedded Cartan subalgebra.

Lemma V.5. If Ω is an 𝔩-invariant symplectic structure on V such that (V, Ω) is a symplectic 𝔩-module of convex type, then there exists a 𝔨-adapted positive system Δ^{+} such that B is negative definite on VC^{α} if and only if α ∈ Δ_{r}^{+} (Definition II.18). Here the set Δ_{r}^{+} is uniquely determined by the isomorphy class of (V, Ω) as a symplectic 𝔩-module.

Proof. In view of Remark II.10, it is clear that the requirements on the set Δ_{r}^{+} determine the isomorphy class of (V, Ω) as a symplectic 𝔩-module uniquely.

Now assume that (V, Ω) is of convex type. Using Proposition II.23, we find X ∈ 𝔷(𝔨) such that the pseudohermitean form B is negative definite on V^{α} if and only if iα(X) > 0. Next we choose X′ ∈ 𝔷(𝔨) in such a way that α(X) ≠ 0 holds for all α ∈ Δ_{p} and that the signs of iα(X) and iα(X′) coincide for α ∈ Δ_{r}. Then Δ_{p}^{+} := {α ∈ Δ_{p}: iα(X′) > 0} is the system of positive noncompact roots for a 𝔨-adapted positive system Δ^{+} satisfying our requirements.

Proposition V.6. Let 𝔤 be a Lie algebra containing invariant cones and 𝔱 ⊆ 𝔤 a compactly embedded Cartan subalgebra. For a 𝔨-adapted positive system Δ^{+} we put

C_{min,𝔷} := cone\{i[\overline{X}_{α}, X_{α}]: α ∈ Δ_{r}^{+}, X_{α} ∈ 𝔤C^{α} ∩ 𝔫_{C}\}

and for β ∈ 𝔷^{∗} we write Ω_{β}(v, w) := β([v, w]) for the associated 𝔩-invariant skew-symmetric form on V := [𝔱, 𝔫]. Then the following assertions hold:

(i) (V, Ω_{β}) is an 𝔩-module of convex type if and only if β ∈ int C_{min,𝔷}^{⋆} for a 𝔨-adapted positive system.

(ii) Two symplectic 𝔩-modules of convex type (V, Ω_{β}) and (V, Ω_{γ}) are iso-

morphic if and only if the associated systems Δ_{r}^{+} coincide. (iii) If (V, Ω) is a symplectic 𝔩-module of convex type and Δ^{+} a 𝔨-adapted

positive system such that B is negative definite on VC^{α} if and only if α ∈ Δ_{r}^{+} , then there exists a β ∈ 𝔷^{∗} such that (V, Ω) ≅ (V, Ω_{β}) if and only if C_{min,𝔷} is a pointed cone.

Proof. (i) For X_{α} ∈ VC^{α} = 𝔤C^{α} ∩ 𝔫_{C} we have

iΩ_{β,C}(X_{α}, \overline{X}_{α}) = iβ([X_{α}, \overline{X}_{α}]) = −β(i[\overline{X}_{α}, X_{α}]).

We choose the 𝔨-adapted positive system Δ^{+} as in Lemma V.5. Then the pseudohermitean form B_{β} is negative definite on V^{α} if and only if α ∈ Δ_{r}^{+} . For α ∈ Δ_{r}^{+} we then have β(i[\overline{X}_{α}, X_{α}]) > 0 for X_{α} ≠ 0. Hence β ∈ int C_{min,𝔷}^{⋆}.

This proves the first half of (i). The other part has already been shown in the proof of Theorem V.1. (ii) If the symplectic 𝔤-modules (V, Ω_{β}) and (V, Ω_{γ}) are isomorphic, then the forms B_{β} and B_{γ} are positive definite on the same weight spaces V^{α}. As we have seen in (i), this property entails that β, γ ∈ C_{min,𝔷}^{⋆} for the same system Δ_{r}^{+} .

To see the converse, we note that the positive system Δ_{r}^{+} fixes the signs of B on the weight spaces VC^{α}, hence, in view of Remark II.10, it also determines the isomorphy class of the symplectic 𝔤-module (V, Ω_{β}). It follows in particular that (V, Ω_{β}) ≅ (V, Ω_{β′}). (iii) It is clear that (V, Ω) ≅ (V, Ω_{β}) if and only if β ∈ int C_{min,𝔷}^{⋆}. Such a functional β exists if and only if the cone C_{min,𝔷} is pointed.

Remark V.7. In general there can be more symplectic structures on V turning it into a symplectic module of convex type as one can obtain by forms of the type Ω_{β}, β ∈ int C_{min,𝔷}^{⋆}. A typical example can be obtained as follows. Let (V, Ω) = (IR^{2}, Ω_{0})^{2} as module of 𝔰𝔩(2, IR)^{2} , where Ω_{0} is any symplectic form on IR^{2} . We form the associated Lie algebra 𝔤 = V × IR × 𝔩 as in Theorem V.1 with 𝔷_{1} = {0}. Then, as we have seen in Remark II.10, the 𝔩-module V permits 4 different classes of symplectic structures turning it into an 𝔩-module of convex type. Accordingly there exist 4 different systems Δ_{r}^{+} for 𝔤, but for only two of them the cone C_{min,𝔷} is pointed.

Remark V.8. If we consider only those non-reductive Lie algebras with invariant cones where dim 𝔷 = 1 and 𝔩 is given, then their isomorphy classes correspond to isomorphy classes of symplectic 𝔩-modules of convex type. Therefore the open problem mentioned in Remark II.10 is directly related to the classification of this type of Lie algebras. In view of Lemma V.5 we know at least that we can parametrize the isomorphy classes by a certain set of positive systems Δ_{r}^{+} .

Relations to Siegel domains

Definition V.9. Let (V, A, I) be a symplectic vector space endowed with a complex structure I such that IA(v, w) := A(v, Iw) is a positive definite symmetric bilinear form on V .

Let further U be a real vector space, C ⊆ U an open convex cone which contains no non-trivial affine subspaces, e ∈ int C^{⋆}, and Â: V × V → U a skewsymmetric bilinear map such that the forms A_{u}(v, w) := ⟨u, Â(v, w)⟩ satisfy the condition A_{e} = A and IA_{u} is positive definite for u ∈ int C^{⋆}. Note that this condition is equivalent to Â(v, Iv) ∈ \overline{C} \ \{0\} for v ∈ V \ \{0\}.

With the data (V, Â, U, C) given, we define the Siegel domain

S := S(V, Â, U, C) := {(u, v) ∈ U_{C} × V: Im u − Â(v, Iv) ∈ C}.

Note that this is a convex domain in the complex vector space U_{C} × V (cf. [21, p.128]). Accordingly we say that (V, Â, U, C) is a Siegel data. We write

Sp(V, Â) := {g ∈ End_{IR}(V): (∀v, w ∈ V)Â(g.v, g.w) = Â(v, w)}.

Then Sp(V, Â) is a reductive quasihermitean real algebraic group and the corresponding symmetric domain is the generalized Siegel space 𝔖(V, Â, C) consisting of all complex structures J on V such that JA_{u} is positive definite for all u ∈ int C^{⋆}.

We recall that 𝔖(V, Â, C) does not depend on the cone C (cf. [21]). One only needs the assumption that there exists a cone C satisfying the requirements. Written as a condition on the U -valued map Â, this means that

(C1) Â(v, Iv) ≠ 0 for v ∈ V \ {0},

(C2) D_{A}:= cone{Â(v, Iv): v ∈ V } ⊆ U is pointed, and (C3) A_{e} = A for an element e ∈ int(D_{A})^{⋆}.

It is clear that the requirements from above imply (C1)-(C3). Suppose, conversely, that (C1)-(C3) are satisfied. Then IA_{u} is positive definite for all u ∈ int(D_{A})^{⋆}, hence in particular for some pointed generating cone C^{⋆} ⊆ (D_{A})^{⋆}. Then C := int(C^{⋆})^{⋆} satisfies the requirements.

Theorem V.10. The tuple (V, Â, U) is a Siegel data if and only if the associated generalized Heisenberg algebra

𝔫 = V × U with [(v, u), (v′, u′)] = (0, Â(v, v′))

occurs as the nilradical of a Lie algebra containing invariant cones. Proof. Suppose that (V, Â, U) is a Siegel data and that C ⊆ U is an appropriate open pointed convex cone. We define

𝔤 := 𝔫× 𝔰𝔭(V, Â),

where 𝔰𝔭(V, Â) acts canonically on V and trivially on U. Then it is easy to see that 𝔤 is a Lie algebra whose nilradical is 𝔫. We claim that 𝔤 contains invariant cones. In view of Theorem V.1, it suffices to show that the symplectic 𝔰𝔭(V, Â)- module (V, A_{e}) for an element e ∈ C is of convex type. Since the inclusion 𝔰𝔭(V, Â) → 𝔰𝔭(V, A) is an (H_{2})-homomorphism, this follows from Theorem IV.6.

Suppose, conversely, that 𝔤 = 𝔫× 𝔩 is a Lie algebra containing invariant cones (cf. Theorem V.1). We choose a compactly embedded Cartan algebra 𝔱 ⊆ 𝔤 and a 𝔨-adapted positive system of roots such that the cone C_{min} is pointed (cf. [12, III]). Let V^{+} := ∑_{α∈Δ_{r}^{+}} 𝔤C^{α}. Then the IR-linear isomorphism V → V^{+} can be used to obtain a complex structure I on V . For X, Y ∈ V^{+} we then have

Â(X + \overline{X}, I(X + \overline{X})) = [X + \overline{X}, iX − i\overline{X}] = 2[i\overline{X}, X] ∈ C_{min,𝔷},

where C_{min,𝔷} := C_{min}∩𝔷. Since 𝔤 has cone potential, Â(v, Iv) ≠ 0 for 0 ≠ v ∈ V , i.e., (C1) is satisfied. Since D_{A} = C_{min,𝔷} is pointed, the condition (C2) is also satisfied. Fixing e ∈ int C^{⋆}, we therefore obtain a symplectic structure A_{e} on V such that IA_{e} is positive definite and therefore (C3) holds. This completes the proof.

In view of Theorem V.10, the groups (V ⊕ U)× 𝔰𝔭(V, Â) are the prototypes of non-reductive Lie algebras with invariant cones since for a given generalized Heisenberg algebra 𝔫 = V ⊕ U all algebras 𝔫× 𝔩 which contain invariant cones arise by certain (H_{1})-homomorphisms κ: 𝔩 → 𝔰𝔭(V, Â) satisfying the condition that κ^{−1}(W_{max}) is generating (cf. Theorem V.1).

Parabolic subgroups and Lie algebras with invariant cones

We have already seen that the nilpotent radicals which occur in Lie algebras with invariant cone are precisely those generalized Heisenberg algebras which arise as Siegel data. The connection between non-reductive Lie algebras with invariant cones and symmetric domains is much deeper as we will explain in this subsection.

Let (𝔤, H_{0}) be a simple Lie algebra of hermitean type and

κ: 𝔰𝔩(2, IR) → 𝔤

an (H_{1})-homomorphism.

We put

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

We write 𝔟_{κ} = (U + V)× 𝔷_{𝔤}(H_{κ}), where

U = 𝔤(ad H_{κ}; 2) and V = 𝔤(ad H_{κ}; 1).

Then 𝔟_{κ} is a maximal parabolic subalgebra of 𝔤 (cf. [21]) and 𝔫 := U + V is a generalized Heisenberg algebra since U is central in 𝔫 and [V, V] ⊆ U.

The following facts can be found in [21, p.113]. We have

𝔷_{𝔤}(H_{κ}) = 𝔤_{κ}^{(1)} ⊕ 𝔤_{κ}^{(2)},

where 𝔤_{κ}^{(1)} = 𝔩_{2} ⊕ 𝔤_{κ} and 𝔤_{κ}^{(2)} = IRH_{κ} ⊕ 𝔤_{κ}^{(2)}′, where 𝔩_{2} is a compact ideal in 𝔷_{𝔤}(H_{κ}) which is maximal with respect to the property of being compactly embedded in 𝔤, 𝔤_{κ} is simple hermitean, and the commutator algebra of 𝔤_{κ}^{(2)} is either {0} or simple and non-compact.

Let A denote the symplectic structure on V given by the element X_{κ} ∈ U by

A(v, v′) = −\frac{1}{4}⟨X_{κ}, [v, v′]⟩.

Then we obtain an (H_{2})-homomorphism ρ_{V}: 𝔤_{κ}^{(1)} → 𝔰𝔭(V, A) with respect to \frac{1}{2}I ∈ 𝔰𝔭(V, A), and [𝔤_{κ}^{(1)}, U] = \{0\}.

We identify U with U^{∗} via the scalar product given by

⟨X, Y ⟩ := −B(X, θY),

where θ is a Cartan involution of 𝔤 and B is the Cartan-Killing form. Then the orbit of X_{κ} under the group G_{κ}^{(2)} := ⟨exp 𝔤_{κ}^{(2)}⟩ in U is an irreducible self-dual homogeneous cone, denoted C, and we have a skew-symmetric mapping

Â: V × V → U, (X, Y) ↦ −\frac{1}{4}[X, Y]

satisfying IA_{u} = I(u ∘ Â) >> 0 for every u ∈ C.

Proposition V.11. The Lie algebra 𝔭_{κ} := 𝔫× 𝔤_{κ}^{(1)} contains invariant cones, 𝔫 is its nilradical, and 𝔤_{κ}^{(1)} contains one hermitean factor. Proof. If V = {0}, then 𝔭_{κ} is reductive quasihermitean and there is nothing to prove. Assume that V ≠ {0}.

We know already that 𝔤_{κ}^{(1)} is reductive and quasihermitean with one hermitean factor, and since [𝔤_{κ}^{(1)}, U] = {0}, we have ρ_{V} (𝔤_{κ}^{(1)}) ⊆ 𝔰𝔭(V, Â).

From the positive definiteness of the form IA_{u} for all u ∈ C, it follows that (V, Â, U, C) is a Siegel data. Moreover 𝔤_{κ}^{(1)} acts effectively with V_{fix} = {0} (cf. [Sa80, p.112]). In view of Theorem V.1, it remains to show that V is a symplectic 𝔤_{κ}^{(1)} -module of convex type which follows from the observation that ρ_{V} is an (H_{2})-homomorphism (Theorem V.10).

The preceding result shows how Lie algebras containing invariant cones arise as subalgebras of maximal parabolic subalgebras of hermitean simple Lie algebras. The size of 𝔤_{κ}^{(1)} in 𝔷_{𝔥}(H_{κ}) depends on the rank of the homomorphism κ (cf. [21]). If the rank is minimal, then 𝔤_{κ}^{(2)} = IRH_{κ} ([21, p.113]), so that 𝔷_{𝔤}(H_{κ}) = IRH_{κ} ⊕ 𝔤_{κ}^{(1)} and therefore 𝔟_{κ} = 𝔭_{κ} × IRH_{κ}. Moreover, the inspection of the restricted root system of 𝔤 shows that in this case U = IRX_{κ}, hence 𝔫 is a Heisenberg algebra. Moreover, the representation ρ_{V} of 𝔤_{κ}^{(1)} is irreducible over IR ([21, p.112]). Therefore the 𝔤_{κ}^{(1)} -module V is contained in the list of representations in Theorem III.15.

It is instructive to have a closer look at this situation. The situation is relatively simple for the class of those hermitean algebras arising via skewhermitean forms.

Example V.12. Let IK = IR,C, IH and V a IK-left vector space endowed with a non-degenerate skewhermitean form ⟨·, ·⟩, i.e.,

\overline{⟨v, w⟩} = −⟨w, v⟩, λ⟨v, w⟩ = ⟨λv, w⟩ and ⟨v, λw⟩ = ⟨v, w⟩\overline{λ}.

The basic bulding block for such vector spaces is IK^{2} endowed with the form ⟨(x, y), (x′, y′)⟩ := x\overline{y′} − y\overline{x′}. We write 𝔰𝔲(V) for the Lie algebra of the group SU(V) := U(V) ∩ Sl(V), where U(V) denotes the group of all IK-linear isometries.

Up to equivalence, we have the following cases:

(1) IK = IR, V = IR^{2n}, 𝔰𝔲(V) = 𝔰𝔭(n, IR).

(2) IK =C, V =C^{p+q}, 𝔰𝔲(V) = 𝔰𝔲(p, q).
(3) IK = IH, V = IH^{n}, 𝔰𝔲(V) = 𝔰𝔬^{∗}(2n).

Note that this covers all simple hermitean Lie algebras up to 𝔰𝔬(n, 2) and the two exceptional ones.

Let v_{0} ∈ V be a non-zero isotropic vector. We choose v_{1} ∈ V with ⟨v_{1}, v_{0}⟩ = 1. Set V′ := v_{0}^{⊥} ∩ v_{1}^{⊥}. Then V ≅ IKv_{0} ⊕ IKv_{1} ⊕ V′ ≅ IK^{2} ⊕ V′. For IK^{2} , the real rank of 𝔰𝔲(IK^{2}) is 1 and the corresponding (H_{1})-homomorphism κ: 𝔰𝔩(2, IR) → 𝔰𝔲(IK^{2}) is the natural inclusion. We have in particular

H_{κ} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

Using the embedding 𝔰𝔲(IK^{2}) → 𝔰𝔲(V) obtained by splitting the IK^{2} -factor, we see that

𝔷_{𝔰𝔲(V)}(H_{κ}) = IRH_{κ} ⊕ 𝔰𝔲(V′), 𝔭_{κ} ≅ (V′ + IRX_{κ})× 𝔰𝔲(V′)

and

𝔟_{κ} = {X ∈ 𝔰𝔲(V): X.v_{0} ∈ IKv_{0}}.

Example V.13. For the exceptional hermitean algebra 𝔤 = 𝔢_{(6,−14)} one finds for κ of rank 1 that 𝔤_{κ}^{(1)} ≅ 𝔰𝔲(5, 1) and that V is a real module of dimension 20, hence a real form of ⋀^{3}(C^{6}).

Example V.14. For the exceptional hermitean algebra 𝔤 = 𝔢_{(7,−25)} one finds for κ of rank 1 that 𝔤_{κ}^{(1)} ≅ 𝔰𝔬(2, 10) and that V is a real module of dimension 32, hence a factor of the corresponding spin representation.

Example V.15. For 𝔤 = 𝔰𝔬(2, n) we obtain 𝔤_{κ}^{(1)} ≅ 𝔰𝔩(2, IR)⊕𝔰𝔬(n−2) and that V is a real module of dimension 2(n−2). More precisely, it is the tensor product IR^{2} ⊗ IR^{n−2} , where the factors are endowed with the natural representations.

VI. More on the structure of Lie algebras with invariant cones

The following theorem was the original motivation for us to consider the structure of a Lie algebra with invariant cones from the point of view presented in Section II.

Theorem VI.1. Let 𝔤 be a Lie algebra with invariant cones, 𝔱 ⊆ 𝔤 a compactly embedded Cartan algebra, and 𝔤 = 𝔯× 𝔰 a 𝔱-invariant Levi decomposition. Let α, β ∈ Δ with 𝔤C^{α} ⊆ 𝔯_{C}, 𝔤C^{β} ⊆ 𝔰_{C}, and X_{β} ∈ 𝔤C^{β} with

β([X_{β}, \overline{X}_{β}]) = 2.

Then

α([X_{β}, \overline{X}_{β}]) ∈ \{−1, 0, 1\}.

Proof. Let 𝔩 be the intersection of 𝔤 with the smallest conjugation invariant subalgebra containing 𝔤C^{β}. Then 𝔩 ≅ 𝔰𝔩(2, IR) ([12, Prop. II.8]). Therefore Proposition II.28 applies to the module 𝔪 = [𝔱, 𝔯] of convex type (Theorem V.1) and it follows that every non-trivial simple 𝔩-submodule is of dimension at most 2. Now α is a 𝔱_{C}-weight of 𝔪_{C} and therefore the restriction of these weights to 𝔩_{C} ∩ 𝔱_{C} are the corresponding weights for 𝔩. But 𝔪 contains only two types of simple 𝔩-submodules: one-dimensional trivial modules and two-dimensional modules. Therefore α([X_{β}, \overline{X}_{β}]) ∈ \{−1, 0, 1\}.

Theorem VI.1 has some important consequences for the structure of Lie algebras with invariant cones.

Corollary VI.2. Let 𝔤 be a Lie algebra with invariant cones, Δ^{+} a 𝔨adapted positive system of roots, 𝔤 = 𝔯× 𝔰 a 𝔱-invariant Levi decomposition, and α, β ∈ Δ_{p}^{+} such that 𝔤C^{α} ⊆ 𝔯_{C} and 𝔤C^{β} ⊆ 𝔰_{C}. Then the following assertions hold:

(i) [𝔤C^{α}, 𝔤C^{β}] = {0}.

(ii) [𝔤C^{−α}, 𝔤C^{−β}] = {0}.

(iii) [𝔤C^{α}, 𝔤C^{−β}] = {0} or α − β ∈ −Δ_{p}^{+} .

(iv) The subalgebra 𝔭C^{+} := ∑_{α∈Δ_{p}^{+}} 𝔤C^{α} is abelian.

Proof. (i) Let X_{β} ∈ 𝔤C^{β} with β([X_{β}, \overline{X}_{β}]) = 2. Then

(6.1)  α([X_{β}, \overline{X}_{β}]) ∈ \{0, 1\}

since α is a positive non-compact root and Δ^{+} is 𝔨-adapted which implies that C_{min} ⊆ C_{max} (cf. [12, Th. III.20]).

Suppose that {0} ≠ [𝔤C^{α}, 𝔤C^{β}] ⊆ 𝔤C^{α+β}. Then α + β ∈ Δ_{p}^{+} and (α + β)([X_{β}, \overline{X}_{β}]) ≥ 2. This contradicts Theorem VI.1. (ii) This follows from (i) by interchanging Δ^{+} and −Δ^{+} . (iii) If

{0} ≠ [𝔤C^{α}, 𝔤C^{−β}] ⊆ 𝔤C^{α−β} ⊆ 𝔯_{C},

then α − β ∈ Δ_{p} and (α − β)([X_{β}, \overline{X}_{β}]) ≤ −1, so that α − β ∈ −Δ_{p}^{+} follows from (6.1). (iv) It follows from (i) that root spaces in 𝔯_{C} and those in 𝔰_{C} commute. On the other hand those in 𝔯_{C} commute by [12, Prop. II.10] and those in 𝔰_{C} by [7, Lemma 7.7].

Corollary VI.3. We keep the assumptions of Corollary VI.2 and set Δ_{r} := {α ∈ Δ: 𝔤C^{α} ⊆ 𝔯_{C}} (cf. Definition II.18) and 𝔪C^{+} := ⨁_{α∈Δ_{r}^{+}} 𝔤C^{α}. Then the mapping

𝔪C^{+} → 𝔪 := [𝔱, 𝔯], X ↦ X + \overline{X}

is a bijection which induces a complex structure on 𝔪 given by

I.(X + \overline{X}) = i(X − \overline{X}).

Let 𝔰 = (𝔨 ∩ 𝔰) + 𝔭 be a Cartan decomposition of 𝔰. Then the complex structure on 𝔪 is invariant under 𝔨 and the restrictions of the operators ad X, X ∈ 𝔭 are antilinear, i.e.,

ad X |_{𝔪} ∘ I = −I ∘ ad X |_{𝔪}

for all X ∈ 𝔭. Proof. The invariance under 𝔨 follows from that fact that the subalgebra 𝔪C^{+} = ∑_{α∈Δ_{r}^{+}} 𝔤C^{α} is invariant under 𝔨 and this is the i-eigenspace for I on 𝔪_{C}.

The (−i)-eigenspace is 𝔪C^{−} := \overline{𝔪C^{+}} which is also 𝔨-invariant.

For X ∈ 𝔤C^{α} ⊆ 𝔰_{C}, α ∈ Δ_{p}^{+} , it follows from Corollary VI.2(iii) that ad X maps 𝔪C^{+} into 𝔪C^{−} and vice versa. This means that ad X anticommutes with I on 𝔪_{C}. The same holds for \overline{X} ∈ 𝔤C^{−α}. Then ad(X + \overline{X}) also anticommutes with I on 𝔪 and this proves the assertion.

Corollary VI.4. We set Δ_{p,s} := {α ∈ Δ: 𝔤C^{α} ⊆ 𝔰_{C}} and 𝔭_{s}^{±} := ∑_{α∈Δ_{p,s}^{+}} 𝔤C^{α}. Then

𝔤_{C} = 𝔭_{s}^{−} ⊕ 𝔪C^{−} ⊕ 𝔨_{C} ⊕ 𝔪C^{+} ⊕ 𝔭_{s}^{+}

is a 5-grading of 𝔤_{C}.

Proof. In view of Corollary VI.2, this is an elementary verification. The crucial point is that [𝔭_{s}^{+}, 𝔪C^{−}] ⊆ 𝔪C^{+} and [𝔭_{s}^{−}, 𝔪C^{+}] ⊆ 𝔪C^{−}.

Remark VI.5. Let us say that a Lie algebra 𝔤 is admissible if 𝔤 ⊕ IR contains invariant cones. If 𝔤 has this property but does not itself contain invariant cones, then it must be compact semisimple ([12, Th. III.39]). Hence Theorem VI.1 and all its consequences also remains true for this slightly extended class of Lie algebras since it is trivially true for compact semisimple Lie algebras.

Example VI.6. An instructive example which illustrates the above results is the Lie algebra 𝔤 = 𝔥_{𝔫} × 𝔰𝔭(n, IR), where 𝔥_{n} is the (2n + 1)-dimensional Heisenberg algebra. Here the compact roots Δ_{k} are the roots of the compact Lie algebra 𝔲(n), hence

Δ_{k}^{+} = {ε_{k} − ε_{j}: 1 ≤ k < j ≤ n − 1}

and the non-compact positive roots of 𝔰𝔭(n, IR) are given by

Δ_{p,s}^{+} = {ε_{k} + ε_{j}: 1 ≤ k, j ≤ n}.

In 𝔤 we also have positive roots corresponding to the root spaces contained in (𝔥_{n})_{C} which are given by

Δ_{r}^{+} = {ε_{i} : 1 ≤ j ≤ n}.

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Mathematisches Institut Universität Erlangen-Nürnberg Bismarckstrasse 1 \frac{1}{2} D-91054 Erlangen Germany Email: neeb@mi.uni-erlangen.de

Received March 20, 1994 and in final form January 25, 1994