Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group

Wolfgang Bertram

Abstract. Using a geometric approach, we define and investigate the conformal group of a twisted complex symmetric space and, more generally, of a real form of such a space. In the non-degenerate case we characterize this group by a theorem generalizing the fundamental theorem of projective geometry.

0. Introduction

0.1. Symmetric spaces and circled spaces. A symmetric space may be defined as a manifold M with an affine connection ∇ such that the geodesic symmetry s_{x} with respect to any point x ∈ M is an affine diffeomorphism, i.e. it preserves the connection ∇ (cf. [KoNo69, Ch.XI.]). This is equivalent to requiring that, for all x ∈ M, the endomorphism −1_{x} := − id_{T_{x}M}, extends to an affine diffeomorphism s_{x} of M; by this we mean that s_{x} fixes x and has differential −1_{x} at x.

Now assume M to be a manifold with an affine connection ∇ and an invariant almost complex structure J , i.e. a tensor field of type (1,1) such that J_{x}^{2} = −1_{x} for all x ∈ M, and assume ∇ J = 0. Then we say that (M, J , ∇) is a circled space if J_{x} extends for all x ∈ M to an affine diffeomorphism j_{x} of M; as above, this means that j_{x} fixes x and has differential J_{x} at x. It follows immediately that j_{x}^{2} is an affine extension of J_{x}^{2} = −1_{x}, and therefore a circled space is a symmetric space with some additional structure. Note that our notion of a circled space is formally very close to the definition of a symmetric space, with the group ℤ/(2) = {id, s_{x}} replaced by the group ℤ/(4) = {id, j_{x}, j_{x}^{2} = s_{x}, j_{x}^{3}}. In fact, we then even have an affine action of the whole rotation group S^{1} for any point x (Prop.1.1.3). In [Lo72] a similar set-up is studied, with the action of an arbitrary compact group attached to any point of the space.

It is well-known that Hermitian symmetric spaces are circled in our sense. The pseudo-Hermitian symmetric spaces are also circled; more generally, we show (Th. 1.1.2) that the circled spaces are precisely the twisted complex symmetric spaces we have introduced in [Be97b]. The irreducible circled spaces can be classified, cf. [Be97b]. More generally, we have shown that the twisted complex symmetric spaces correspond bijectively to an algebraic category called Hermitian Jordan triple systems.

0.2. Real forms and dilation structures. In a second step we have introduced in [Be97b] a geometric category corresponding to the category of all Jordan triple systems. This is precisely the category of real forms of circled spaces, also called Makarevič spaces. We want to characterize such spaces geometrically in a similar way as the circled spaces. Since they are real forms of circled spaces, we have to select a structural feature of circled spaces which is invariant under all complex conjugations and contains all relevant geometric information. On an infinitesimal level, such an object is the structure tensor T which we have introduced in [Be97b]. On a local level, such an object is given by the dilation structure associated to a circled space: as an example, consider the unit disc D = {z ∈ ℂ | |z| < 1}. It is a Hermitian symmetric space, D = SU(1, 1)/ SO(2), the affine connection being the Levi-Civita connection of the Poincaré metric. Then multiplication J_{0} by i in the tangent space T_{0}D and on D itself are both given by the element z ↦ iz of SO(2). The complexified group SO(2, ℂ) acts naturally by the maps z ↦ wz with w ∈ ℂ^{∗}; in particular, its non-compact subgroup ℝ^{+} acts by positive real scalars. If the scalar is greater then one, this action can be seen as an action of a semigroup by dilations of D, and if it is smaller than one, it can be seen as the action of a semigroup by compressions of D. Further, any real form of D containing the origin 0 is stable under this action and inherits it therefore from D. This is the geometric picture that we generalize for arbitrary circled spaces. However, notions as dilation and compression should in general not be interpreted globally; in fact, the precise definition is made in terms of the vector field which generates the aforementioned semigroup-action (cf. Section 1.3). In the simple example given above, the vector field corresponding to the dilations is just the Euler operator; this fact is generalized in Th. 2.3.1. Finally one can show that dilation structures correspond bijectively to Makarevič spaces (Th. 1.3.5).

When passing from the “circular” real form SO(2) of SO(2, ℂ) to the “dilation real form” ℝ^{+} , one should note two things: first, in order to realize this change of real forms for the actions on the space, we need the fact that the almost complex structure J in question is actually integrable, i.e. D has actually the structure of a complex manifold inducing J . It is known that invariant almost complex structures on symmetric spaces are integrable (cf. [KoNo69]); for sake of completeness we include the simple proof in an Appendix (Prop. B.2). (We do not need the theorem of Newlander and Nirenberg (cf. [KoNo69]) in this work.) Second, the action of ℝ^{+} is not by automorphisms: it does not preserve the affine connection, and the corresponding vector field is not complete: if we dilate, we leave D after some time.

0.3. Structure theory. The main purpose of the present work is to develop a structure theory for Makarevič spaces which generalizes the well-developed structure theory of Hermitian symmetric spaces. Standard references for this topic are the book [Sa80] and the lecture notes [Lo77], where the relation with Jordan theory is emphasized. Remarkably enough, the concept of dilation structure allows to generalize most of the important features of Hermitian symmetric spaces without any use of the theory of semi-simple groups as in [Sa80] or an axiomatic theory of Jordan pairs as in [Lo77]. The results are not only more general, but also the methods are simpler and more geometric. We only need basic concepts of geometry and analysis.

More specifically, we generalize first the Borel- and the Harish-Chandra imbedding. The former can be seen as a realization of the c-dual symmetric space of the given one on the same underlying manifold; in the example of the unit disc, the c-dual space is the Riemann sphere in which D is imbedded. Of course, in our general situation this imbedding will no longer be global but only local (Section 2.1). For example, if M = G is a classical group, G and its c-dual space G_{ℂ}/G have a realization on the same manifold.

Next, the Harish-Chandra imbedding can be seen as a canonical realization of the symmetric space M in question as an open domain in a vector space V ; for example, if M is the unit disc, then V = ℂ. Again, this is generalized as a local realization (Section 2.2). Therefore we prefer the terminology canonical chart; it turns out later that this chart covers an open dense part of M. This chart has a nice interpretation in terms of the dilation structure: if φ_{t}, t ∈ ℝ is the dilation group w.r.t a point p ∈ M and ∇ is the affine connection of M, then lim_{t→∞}(φ_{t} · ∇) is a flat affine connection defined on a neighbourhood of p; it is precisely the flat connection induced by the canonical chart.

0.4. Conformal Lie algebra and Kantor-Koecher-Tits construction. The most important step in the structure theory of Hermitian symmetric spaces is the association of a

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 3

ℤ/(3)-graded Lie algebra usually denoted by 𝔭^{−} ⊕ 𝔨_{ℂ} ⊕ 𝔭^{+} to such a space. This is completely natural in the context of dilation structures: the (inner) conformal Lie algebra is the algebra generated by the affine vector fields together with the dilation vector field; the dilation vector field induces a derivation whose eigenspaces define the grading. As for Hermitian symmetric spaces in Harish-Chandra realization, one shows that in the canonical chart the conformal Lie algebra is represented by quadratic polynomial vector fields. The corresponding polynomials are explicitly given in terms of the associated Jordan triple product. Such a construction, starting from the axiomatic properties of Jordan structures, is known as Kantor-Koecher-Tits construction. Our approach can be seen as a geometric version of this construction.

0.5. Conformal group and the Liouville theorem. The next step is to associate a group to the conformal Lie algebra; this group is not an abstract group but a group coming in a certain realization, namely as a group acting by birational maps on the vector space V corresponding to the canonical chart. In the example of the unit disc it is the Möbius group acting by fractional linear transformations.

Our presentation of the conformal group has two sources: firstly, Koecher’s theory of groups of birational transformations ([Koe69a,b]), and secondly the Liouville-theorem for Jordan algebras ([Be96a]). Roughly speaking, Koecher defines the conformal group as the group of birational maps of V preserving the conformal Lie algebra. When one works over the base field of real or complex numbers, then one can actually avoid the assumption of rationality: any locally defined diffeomorphism preserving the conformal Lie algebra is automatically birational (Th.3.1.1). This simplifies our treatment compared with Koecher’s.

Our second motivation comes from the classical Liouville theorem on conformal maps of the Euclidean space ℝ^{3} . Once more the example of the unit disc is quite illuminating: as is wellknown, the Möbius group acts conformally on ℝ^{2} = ℂ, but this property does not characterize it; however, in dimension bigger than 2 the analogous statement holds – for n = 3 it is the Liouville theorem. Similarly, we prove (Th. 3.2.2) that the conformal group always satisfies a “Liouville-condition”, namely the differential of a conformal transformation at any point where it is non-singular belongs to a certain linear group, the structure group. The converse does not always hold; but if the Jordan triple system in question comes from a simple Jordan algebra not isomorphic to ℝ or ℂ, we have shown in [Be96a] that this property characterizes the conformal group. Even in the general case the theory becomes more transparent if one emphasizes the Liouville-condition from the very beginning (in [Koe69a, 4.Korollar 2] it appears at a later stage). The Liouville-condition can be formulated invariantly, without using the canonical chart: a local diffeomorphism satisfies the Liouville-condition if and only if it preserves a certain subbundle W of Hom(T M ⊗ T M, T M), the so-called structure bundle (Section 3.6).

0.6. The condition of generalized projectivity and the fundamental theorem. An important example of a circled space is the complex projective space ℂℙ^{n} = U(n+1)/(U(1)× U(n)). According to what has been explained in Section 0.2, its real form ℝℙ^{n} has a dilation structure: it is a Makarevič space. In this case the conformal group is the projective group; it acts by fractional transformations on the affine part ℝ^{n} of ℝℙ^{n}. The Liouville condition is empty: the structure group is Gl(n, ℝ), and no condition is imposed on the first differentials.

Now, it is well-known that for n > 1 the projective group is characterized by the fundamental theorem of projective geometry as the group preserving collinearity. By an elementary calculation one can show that this can be formulated as a condition on the second differential. We introduce a similar condition of generalized projectivity in the context of Makarevič spaces: a local diffeomorphism φ satisfies this condition iff it preserves the structure bundle W mentioned above, and in addition φ · ∇ − ∇ is a local section of W. Here ∇ is the flat connection of V or any other connection obtained by adding a section of W to ∇. One should note that the Liouville-condition appears here merely as an auxiliary condition guaranteering that the condition of generalized projectivity defines indeed a group. In the case of the original fundamental theorem this auxiliary condition does not appear since in this case the structure bundle is invariant under all diffeomorphisms (cf. Example 2.6.5). On the other hand, in the case of the Liouville-theorem the auxiliary condition alone is strong enough to imply the second-differential condition. In the general case, both conditions are significant.

We show that the conformal group always satisfies the condition of generalized projectivity (Th. 3.3.2). This fact plays an important role in the structure theorem (Th. 3.5.3) of the conformal group establishing an analogue of the Harish-Chandra decomposition for the complexification of Hermitian groups. Conversely, in the case that the Jordan triple system is simple and not isomorphic to ℝ or ℂ, we can characterize the conformal group by the condition of generalized projectivity (Th. 3.4.1). As in the case of the Liouville-theorem [Be96a], this theorem is reduced to a purely algebraic statement, namely we show that a certain continuation of the structure bundle, denoted by Hom_{s}(V, W), is reduced to zero (Prop. 2.6.4).

The original fundamental theorem is a special case of our theorem. This is all the more remarkable as another version of the Liouville-theorem for Jordan triple systems obtained by Gindikin and Kaneyuki ([GiKa96]) does not cover this case. Their notion of generalized conformal structure is different from the concepts presented here, and it remains an interesting open problem to examine the relation of these different structures. Further, one should mention two other approaches to the conformal group: for Jordan algebras, Koecher characterizes it as the group preserving an analogue of the classical cross-ratio ([Koe67]), and in the general case O. Loos ([Lo79]) describes it as the group of algebraic automorphisms of a certain algebraic variety corresponding to our notion of conformal completion (cf. Section 3.6).

0.7. Global Makarevič spaces. In a final chapter we use our knowledge of the conformal group in order to define global Makarevič spaces in the spirit of B.O. Makarevič’s original work [Ma73] (see also [Be96b,c]). We can not only generalize results which in [Be96b,c] were only proved for (semi-simple) Jordan-algebras, but give also more conceptual proofs.

The main result (Th. 4.4.1) describes the part of the global Makarevič space M which is covered by the canonical chart as the set of points z where a certain Bergman-operator B(z, z) is non-degenerate. We can interprete the function z ↦ B(z, z) globally as a section of a vector bundle related to the structure bundle mentioned above (Th. 4.3.1). This stage reached, most results from [Be96b,c] carry over to the more general framework given here; in particular, we can construct global straight and twisted (para-) complexifications of global Makarevič spaces. Thus one of the aims from [Be97b] is reached. We finish by giving an outlook what elements a further structure theory – in view of questions in harmonic analysis – should contain.

Contents of this paper:

1. Circled spaces and dilation structures

2. The conformal Lie algebra

3. Conformal group and fundamental theorem

4. Global Makarevič spaces

Appendix A: The canonical connection of a symmetric space Appendix B: Integrability of almost complex structures

This text is aimed both to readers coming from group theory and to readers coming from differential geometry. Although it would have been possible to formulate it entirely in a grouptheoretic language (e.g. as in [Sa80]), we felt that many aspects become clearer from the point of view of differential geometry. Unfortunately, the differential geometric viewpoint usually gets lost when in the literature a purely group theoretic set-up is chosen. Therefore we include in Appendix A a simple proposition (Prop. A.4) which makes it easy to pass from the group theoretical definition of a symmetric space to the differential geometric one and to use effectively concepts from differential geometry in the group theoretical set-up.

Acknowledgements. I thank J. Hilgert and O. Loos for helpful discussions on the contents of this paper.

1. Circled spaces and dilation structures

1.0. Symmetric spaces. As usual in harmonic analyisis, we define a symmetric space to be a connected homogeneous space of the form M = G/H where H is an open subgroup of the group of fixed points of some non-trivial involution σ of G. We assume further that G acts effectively on M and consider 𝔤 as a subalgebra of the algebra 𝔛(M) of vector fields on M. The base point eH of M is denoted by o, and 𝔤 = 𝔥 ⊕ 𝔮 is the canonical decomposition of 𝔤 w.r.t. the differential of σ at the origin.

As explained in Appendix A, on a symmetric space M there is a canonical torsionfree affine connection ∇ having covariantly constant curvature: ∇R = 0. It is known that these properties in turn define locally affine symmetric spaces; cf. [KoNo69] or [Lo69]. The value of curvature tensor at the base point corresponds then to the triple Lie bracket in 𝔮 (App. A.5). Moreover, the geodesic symmetry s_{p} w.r.t. the point p ∈ M is an affine diffeomorphism (i.e. it preserves the affine connection) extending the tangent map −1_{p} := − id_{T_{p}M} in the following sense:

Definition 1.0.1. If M is a manifold, p ∈ M and a ∈ End(T_{p}M), a smooth map α is called an extension of a if it is locally defined on a neighbourhood of p ∈ M and if

(1) it fixes p: α(p) = p,

(2) its tangent map at p is a: T_{p}(α) = a.

If M has an affine connection ∇, an extension α is called affine if it preserves ∇.

Affine extensions, if they exist, are unique up to local equivalence (cf. e.g. [Lo69, p. 113]).

Proposition 1.0.2. If M is a symmetric space, p ∈ M and a ∈ End(T_{p}M), then a has an affine extension if and only if a is an automorphism of the curvature tensor R evaluated at p, i.e. iff

∀u, v, w ∈ T_{p}M : a R_{p}(u, v)w = R_{p}(au, av)aw.

Proof. Cf. [Lo69, Th. II.4.6].

Proposition 1.0.3. A manifold M with affine connection ∇ is locally affine symmetric if and only if for all p ∈ M the endomorphism −1_{p} = − id_{T_{p}M} has an affine extension.

Proof. As mentioned above, if (M, ∇) is affine symmetric, then the symmetry s_{p} is an affine extension of −1_{p}. Conversely, let s_{p} be an affine extension of −1_{p}. Since s_{p} is affine, it preserves the torsion T, the curvature R and all their covariant derivatives, and −1_{p}, being its derivative at p, preserves the value of all these tensors at p. In particular,

T_{p}(u, v) = (−1_{p}) T_{p}((−1_{p}u, −1_{p}v) = −T_{p}(u, v)

for all u, v ∈ T_{p}M, whence T = 0. In the same way we see that −1_{p} -invariant quadrilinear maps from T_{p}M to T_{p}M vanhish, whence ∇R = 0.

1.1. Locally and globally circled spaces. We consider a smooth manifold M. Recall that a tensor field J of type (1, 1) is a smooth field of endomorphisms J = (J_{p})_{p∈M}, J_{p} ∈ End(T_{p}M). If J_{p}^{2} = − id_{T_{p}M} for all p, then J is called an almost complex structure.

Definition 1.1.1. A locally (resp. globally) circled affine space or just a circled space is a triple (M, J , ∇), where J is an almost complex structure and ∇ an affine connection on a real manifold M such that (C1) J is invariant under ∇ (i.e. ∇ J = 0), (C2) J_{p} extends, for all p, to a local (resp. global) affine map j_{p}.

Recall that, by definition, (∇ J)(X, Y) = ∇_{X}(J Y) − J (∇_{X}Y), and condition (C1) can be written ∇_{X}(J Y) = J ∇_{X}Y for all X, Y ∈ 𝔛(M).

Theorem 1.1.2. The (locally) circled affine spaces are precisely the twisted complex (locally) symmetric spaces, i.e. the (locally) symmetric spaces with invariant almost complex structure J such that the curvature tensor R satisfies the relation R(J X, Y) = −R(X, J Y).

Proof. Let (M, J , ∇) be a circled space. Since j_{p} is an affine extension of J_{p}, j_{p}^{2} is an affine extension of J_{p}^{2} = −1_{p}. According to Prop. 1.0.3, (M, ∇) is thus a (locally) affine symmetric space. Now the condition ∇ J = 0 implies that

(1)  R(X, Y) J Z = J R(X, Y)Z.

The arguments used in the proof of Prop. 1.0.3 show that J_{p} is an automorphism of R_{p}. In the presence of (1) this is equivalent to

(2)  R_{p}(J_{p}^{−1} v, J_{p}^{−1} w, u) = R_{p}(v, w, u),

which in turn is equivalent to R_{p}(J_{p} v, w, u) = −R_{p}(v, J_{P} w, u) (u, v, w ∈ T_{p}M). This being true for all p, we have shown that J is a twisted almost complex structure for the symmetric space (M, ∇).

Conversely, given a symmetric space (locally isomorphic to M = G/H) with invariant twisted almost complex structure J , let ∇ be the canonical connection of M. The invariance of J under G_{o} implies the condition ∇ J = 0 (cf. Appendix A.8), and therefore (1) holds. By assumption, (2) holds, and together with (1) this implies that J_{p} is for all p an automorphism of R_{p}. By Prop. 1.0.2, J_{p} has an extension to an affine map of (M, ∇). Thus (M, ∇, J) is circled.

Since affine extensions are unique, it follows that for a circled space j_{p}^{2} = s_{p} is the symmetry w.r.t. p.

Proposition 1.1.3. If (M, J , ∇) is a circled space, then the operator e^{t J_{p}} has, for all p ∈ M and t ∈ ℝ, a unique affine extension.

Proof. It is easily verified that J_{p} is a derivation of R_{p} (cf. [Be97b, Prop.1.2.4]). Therefore e^{t J_{p}} is an automorphism of R_{p} for all t ∈ ℝ. According to Prop. 1.0.2, e^{t J_{p}} has an affine extension.

1.2. Infinitesimal version.

Definition 1.2.1. A vector field extension of a tangent vector v ∈ T_{p}M is by definition a vector field v ∈ 𝔛(M) such that v_{p} = v. A vector field-extension of an endomorphism a ∈ End(T_{p}M) is a vector field A such that A_{p} = 0 and

[v, A]_{p} = av

for all v ∈ T_{p}M and vector field extensions v of v.

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 7

Since A_{p} = 0 it is clear that [v, A]_{p} is independent of the chosen extension of v. If we introduce a local chart at p ∈ M, vector fields are identified with locally defined smooth maps of V = ℝ^{n}. The vector field v having constant value v ∈ V is a (chart-depending) vector field extension of v ∈ T_{p}M = V , and the relation

[v, A](p) = DA(0) · v

holds, where DA : V → End(V) is the ordinary total differential of A : V → V . Thus A is a vector field extension of a iff A(0) = 0 and DA(0) = a. It is easily seen that the local flow φ_{t} of a vector field X is an extension of e^{ta} ∈ Gl(T_{p}M) iff X a vector field extension of a ∈ End(T_{p}M).

Proposition 1.2.3. If (M, J , ∇) is a circled space, then J_{p} has for all p a unique vector field-extension to an affine vector field J^{p} of ∇.

Proof. This is the differentiated version of Prop. 1.1.3.

1.3. Dilation structure and real forms. We will use the fact that invariant (1,1)-tensor fields J with the property J^{2} = −1 or J^{2} = 1 on a symmetric space M = G/H are integrable (Appendix B). This means essentially that the Lie algebra

𝔤(J) = {X ∈ 𝔛(M)| ∀Y ∈ 𝔛(M) : [X, J Y] = J [X, Y]}

is stable under J and has a J -bilinear Lie-bracket (App. B, Prop. B.3). Since J is invariant under G, we have 𝔤 = Lie(G) ⊂ 𝔤(J).

Lemma 1.3.1. If (M, J , ∇) is a circled space and J^{p} denotes, for p ∈ M, the vector fieldextension of J_{p}, then the vector field

E^{p} := J^{−1} J^{p} ∈ 𝔤(J)

is a vector field extension of the operator 1_{p} = id_{T_{p}M}.

Proof. We remark first that (E^{p})_{p} = J_{p}^{−1}(J^{p})_{p} = 0. Next, since J^{p} ∈ 𝔤 ⊂ 𝔤(J) and since 𝔤(J) is stable under J (Prop. B.3), E^{p} = − J J^{p} belongs to 𝔤(J). Fix p ∈ M and choose a vector field extension v of v ∈ T_{p}M such that v ∈ 𝔤(J). This is possible: one may e.g. take the unique vector field X ∈ 𝔮, where 𝔤 = 𝔥 ⊕ 𝔮 is the decomposition of 𝔤 = Lie(G) w.r.t. the base point p, with X_{p} = v; then X ∈ 𝔤 ⊂ 𝔤(J). Now

[v, E^{p}]_{p} = −[v, J J^{p}]_{p} = − J_{p}[v, J^{p}]_{p} = − J_{p}^{2} v = v;

thus E^{p} is a vector field extension of 1_{p}.

We now fix an arbitrary point p in a circled space and choose it as a base point o in the corresponding symmetric space. The notation introduced in Section 1.0 is used. We are going to describe the Lie algebra 𝔤^{b} generated by 𝔤 and the vector field E^{o} defined in the preceding lemma. To this end let

𝔮̂:= ad(E^{o})𝔮 = {[E^{o}, X]| X ∈ 𝔮} ⊂ 𝔤(J)

and

𝔮^{b} := 𝔮 + 𝔮̂ ⊂ 𝔤(J).

Proposition 1.3.2. In the situation of the preceding lemma, the following holds: (D1) [[𝔮̂, 𝔮̂], 𝔮̂] ⊂ 𝔮̂ and [[𝔮^{b}, 𝔮^{b}], 𝔮^{b}] ⊂ 𝔮^{b} (i.e, 𝔮̂ and 𝔮^{b} are Lie triple-systems (LTS)). (D2) [[𝔮, 𝔮̂], 𝔮] ⊂ 𝔮̂. (D3) For all X ∈ 𝔮^{b}, ad(E^{o})^{2}X = X. (D4) For all X, Y ∈ 𝔮^{b}, ad(E^{o})[X, Y] = 0.

Proof. Since J defines on 𝔤(J) the structure of a complex Lie algebra, the inclusion

ι : 𝔤 → 𝔤(J)

has a unique ℂ-linear extension to a homomorphism

ι_{ℂ} : 𝔤_{ℂ} = 𝔤 ⊕ i𝔤 → 𝔤(J), (X + iY) ↦ X + J Y.

Note that ι_{ℂ}(i^{−1}J^{o}) = J^{−1} J^{o} = E^{o}, and since ι_{ℂ} is a homomorphism, the diagram

𝔤_{ℂ}  ⟶^{ι_{ℂ}}  𝔤(J)
− ad(iJ^{o}) ↓  ↓ ad(E^{o})
𝔤_{ℂ}  ⟶^{ι_{ℂ}}  𝔤(J)

commutes. Using that [J^{o}, 𝔮] = 𝔮, this implies 𝔮̂ = J [J^{o}, 𝔮] = J 𝔮 = ι_{ℂ}(i𝔮) and 𝔮^{b} = ι_{ℂ}(𝔮_{ℂ}). Since i𝔮 and 𝔮_{ℂ} are Lie triple systems, so are their images under ι_{ℂ}, proving (D1). The relation [[𝔮, i𝔮], 𝔮] = i[[𝔮, 𝔮], 𝔮] ⊂ i𝔮 yields (D2).

By assumption, ad(J^{o}) is an invariant twisted complex structure on 𝔮; therefore ad(J^{o}) : 𝔮_{ℂ} → 𝔮_{ℂ}, being its ℂ-linear continuation, is an invariant twisted complex structure on 𝔮_{ℂ}. Since (i ad(J^{o}))^{2} = i^{2}(ad(J^{o}))^{2} is the identity on 𝔮_{ℂ}, it follows that ad(E^{o})^{2} is the identity on 𝔮^{b}, whence (D3). Since, for all X ∈ 𝔥_{ℂ}, [iJ^{o}, X] = i[J^{0}, X] = 0, it follows that [E^{o}, Y] = 0 for all Y ∈ ι_{ℂ}(𝔥_{ℂ}), in particular for all [U, V], U, V ∈ 𝔮^{b} = ι_{ℂ}(𝔮_{ℂ}); whence (D4).

Note that (D3) and (D4) imply that [ad(E^{o})X, Y] + [X, ad(E^{o})Y] = ad(E^{o})[X, Y] = 0, i.e.

(D5)  ∀X, Y, Z ∈ 𝔮^{b} : [[ad(E^{o})X, Y], Z] = −[[X, ad(E^{o})Y], Z]

holds. Using the terminology introduced in [Be97b], (D3), (D4) and (D5) say that ad(E^{o}) is a twisted invariant polarization on the LTS 𝔮^{b}, and (D1) and (D2) say that 𝔮 has the essential properties of a para-real form of this polarized LTS. In fact, if the homomorphism ι_{ℂ} from the preceding proof is injective, then 𝔮^{b} is isomorphic to the polarized LTS 𝔮_{ℂ}; we return to this point in Section 1.4.

Definition 1.3.3. A dilation structure on a symmetric space M = G/H is given by a vector field extension E := E^{o} of 1_{o} := 1_{T_{o}M} having properties (D1) – (D4). Property (D4) allows to define a vector field extension E^{p} of 1_{p} for all p ∈ M in a G-invariant way which is uniquely determined by E^{o}; then (E^{p})_{p∈M} will also be called a dilation structure. The Lie algebra 𝔤^{b} := [𝔮^{b}, 𝔮^{b}] + 𝔮^{b} of vector fields generated by 𝔮 and E^{o} is called the associated inner conformal Lie algebra.

Rephrasing Prop. 1.3.2, we can say that a dilation structure is canonically associated to a circled space, as a sort of real version of the circled structure. It is therefore not surprising that this structure is inherited by any real form of a circled space. Recall that a real form is given by a conjugation, i.e. by an involutive automorphism τ with τ_{∗} J = − J .

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 9

Proposition 1.3.4. If τ is a conjugation of the circled space (M, J , ∇) such that τ(o) = o, then the vector field E^{o} is invariant under τ_{∗}, and it defines a dilation structure on the real form M^{τ} of M. Proof. We have τ_{∗}(J^{o}) = −J^{o} since τ_{∗}(J^{o}) is an affine vector field extending τ_{∗}(J_{o}) = − J_{o}. Using this, we get

τ_{∗}(E^{o}) = τ_{∗}(J^{−1} J^{o}) = − J^{−1} τ_{∗}(J^{o}) = E^{o}.

Thus E^{o} can be restricted to M^{τ} as a (locally defined) vector field, and 𝔮^{τ} + ad(E^{o})𝔮^{τ} is the real form (𝔮^{b})^{τ} of 𝔮^{b}; the properties (D1) – (D4) continue to hold for all X, Y, Z ∈ (𝔮^{b})^{τ}.

Recall from [Be97b] that a Makarevič space is by definition a real form of a twisted complex symmetric space, i.e. of a circled space. Thus Prop. 1.3.4 assigns canonically a dilation structure to any Makarevič space.

Theorem 1.3.5. The association of a dilation structure to a Makarevič space given by Prop. 1.3.4 is a bijection of the class of symmetric spaces with dilation structure and the class of Makarevič spaces. Proof. We have to define a twisted complexification of a space with dilation structure. Assume E^{o} is a dilation structure on M = G/H; as we have remarked after the proof of 1.3.2, ad(E^{o}) is an invariant twisted polarization on the LTS 𝔮^{b}, cf. [Be97b, Def. 1.4.1]. Therefore the formula

T̃(X, Y, Z) := \frac{1}{2}([[X, Y], Z] − ad(E^{o})[[X, ad(E^{o})Y], Z])

defines on 𝔮^{b} the structure of Jordan triple system (JTS) with the property

(!)  T̃(X, Y, Z) − T̃(Y, X, Z) = [[X, Y], Z],

cf. [Be97b, Section 1.4]. Because of (D1), (D2) and (D3), 𝔮 is stable under T̃. The restriction T of T̃ to a triple product on 𝔮 is again a JTS satisfying (!). In other words, T defines a Jordan-extension of the curvature tensor R, and according to [Be97b, Th.2.2.5] M is thus a Makarevič space.

We have to show that the construction just described is inverse to the construction from Prop. 1.3.4. This will follow from the following Lemma:

Lemma 1.3.6. Let T be the structure tensor of a circled space and E^{o} the associated dilation structure. Then, for X, Y, Z ∈ 𝔮, the relation

(∗)  T_{o}(X_{o}, Y_{o}, Z_{o}) = [[X, \frac{Y + ad(E^{o})Y}{2} ], Z]_{o},

holds, and this relation carries over to any real form. Proof. Recall that the structure tensor of a circled space is the tensor given by

T (X, Y, Z) := \frac{1}{2}(R(X, Y)Z − J R(X, J^{−1} Y)Z).

Therefore, for X, Y, Z ∈ 𝔮,

T_{o}(X_{o}, Y_{o}, Z_{o}) = \frac{1}{2}([[X, Y], Z] − ad(J^{o})[[X, ad(J^{o})^{−1}Y], Z])_{o}

= \frac{1}{2}([[X, Y], Z] − J^{−1} ad(E^{o})[[X, J ad(E^{o})^{−1}Y], Z])_{o}

= \frac{1}{2}([[X, Y], Z] − ad(E^{o})[[X, ad(E^{o})Y], Z])_{o}

= \frac{1}{2}([X, Y], Z]_{o} + [[X, ad(E^{o})Y], Z]_{o}),

where we have used that ad(E^{o}) = ad(J J^{o}) = J ∘ ad(J^{o}) = ad(J^{o})∘J , and that the Lie-bracket is J -bilinear. It is clear that both sides of this equation can be restricted to real forms.

We finish the proof of Th. 1.3.5. Comparing formula (∗) with the definition of T̃ above, restricted to 𝔮 and carried over to the tangent space T_{o}M via the evaluation-isomorphism, we see that both formulas coincide, and since the structure tensor determines the twisted complexification, both constructions are indeed inverse to each other.

Note that the vector field Y + ad(E^{o})Y appearing in formula (∗) vanishes at the origin: Y_{o} + [E^{o}, Y]_{o} = Y_{o} − Y_{o} = 0.

1.4. Faithful dilation structures. Let E^{o} be a dilation structure on a symmetric space M = G/H.

Definition 1.4.1. A dilation structure E^{o} is called faithful if 𝔮 ∩ 𝔮̂ = 0.

Lemma 1.4.2. The space 𝔮 ∩ 𝔮̂ is an ad(E^{o})-invariant ideal in 𝔮 with [𝔮 ∩ 𝔮̂, 𝔮 ∩ 𝔮̂] = 0.

Proof. It is clear that 𝔮 ∩ 𝔮̂ is ad(E^{o})-invariant, and from (D1), (D2) and (D4) it follows that it is an ideal of 𝔮. We now prove the last claim: since 𝔮∩𝔮̂ is ad(E^{o})-invariant, it is a direct sum of the ±1-eigenspaces of ad(E^{o}). But if [E^{o}, X] = X, then X_{o} = [E^{o}, X]_{o} = −[X, E^{o}]_{o} = −X_{o}, forcing X = 0; thus

𝔮 ∩ 𝔮̂ = {X ∈ 𝔮| [E^{o}, X] = −X}.

Therefore [X, Y] = −[[E_{o}, X], Y] = [X, [E_{o}, Y]] = −[X, Y] for X, Y ∈ 𝔮 ∩ 𝔮̂, proving our claim.

Proposition 1.4.3. If E^{o} is faithful, the formula

Θ(X + Y) := X − Y, X ∈ 𝔮, Y ∈ 𝔮̂

defines in involutive automorphism of the LTS 𝔮^{b} such that Θ ∘ ad(E^{o}) = − ad(E^{o}) ∘ Θ.

Proof. We assume first that E^{o} = − J J^{o} is the dilation structure of a circled space, and as in the proof of Prop. 1.3.2 we consider the inclusion ι : 𝔮 → 𝔤(J) and its complexification

ι_{ℂ} : 𝔮_{ℂ} = 𝔮 ⊕ i𝔮 → 𝔤(J), X + iY ↦ X + J Y.

Its kernel is

ker ι_{ℂ} = {X + i J X| X ∈ 𝔮, J X ∈ 𝔮} = {Z ∈ 𝔮_{ℂ}| i J Z ∈ 𝔮_{ℂ}}.

Thus

ker ι_{ℂ} = (𝔮 ∩ 𝔮̂)_{ℂ},

and we see that E^{o} is faithful iff ι_{ℂ} is injective (this explains the terminology). If this is the case, Θ is nothing but complex conjugation of 𝔮_{ℂ} w.r.t. 𝔮, which has clearly the properties stated.

Using Th. 1.3.5, the case of a general dilation structure is easily reduced to the circled case.

We will see later (Lemma 2.6.1) that non-degenerate structures are always faithful.

2. The conformal Lie algebra

We assume in this chapter a dilation structure E^{o} given on a symmetric space M = G/H. In this chapter we are interested in local properties; strictly speaking, we should assume M to be a germ of a symmetric space (cf. [Be97b]).

2.1. Local Borel-imbedding.

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 11

Proposition 2.1.1. The LTS 𝔮̂ = ad(E^{o})𝔮 is isomorphic to the dual of 𝔮 (i.e. its Lie triple bracket is isomorphic to the negative of the one in 𝔮). Consequently, the c-dual symmetric space of M has locally a canonical realization with base point o on the underlying manifold of M. Proof. From (D3) and (D4) we get for all X, Y, Z ∈ 𝔮,

[[ad(E^{o})X, ad(E^{o})Y], ad(E^{o})Z] = ad(E^{0})[[X, − ad(E^{o})^{2}Y], Z] = − ad(E^{o})[[X, Y], Z];

therefore ad(E^{o}) is an isomorphism from 𝔮 onto the LTS obtained from 𝔮̂ by taking the negative of the usual triple Lie bracket. The second statement is otained by integrating locally at o the vector fields X ∈ 𝔮̂.

If (M, J , ∇) is a circled space, the c-dual symmetric space of (M, ∇) admits, according to the preceding proposition, a local imbedding into M. This is a local generalization of the Borel-imbedding of a Hermitian symmetric space; it carries over to real forms.

2.2. Local Harish-Chandra imbedding.

Lemma 2.2.1. The inner conformal Lie algebra 𝔤^{b} (cf. Def. 1.3.4) is ℤ/(3)-graded. More precisely, 𝔤^{b} is stable under the derivation ad(E^{o}) and decomposes as a direct sum

𝔤^{b} = 𝔪^{−1} ⊕ 𝔪^{0} ⊕ 𝔪^{1}

of the eigenspaces 𝔪^{k} for the eigenvalues k = 1, 0, −1 of ad(E^{o}), and [𝔪^{k}, 𝔪^{l}] ⊂ 𝔪^{k+l}. In particular, 𝔪^{±1} are abelian and [[𝔪^{1}, 𝔪^{−1}], 𝔪^{±1}] ⊂ 𝔪^{±1} . Moreover,

𝔮^{b} = 𝔪^{−1} ⊕ 𝔪^{1}, [𝔮^{b}, 𝔮^{b}] = 𝔪^{0}.

Proof. By definition, 𝔤^{b} = [𝔮^{b}, 𝔮^{b}] + 𝔮^{b}; this is a Lie algebra since 𝔮^{b} is a Lie triple system. Further, 𝔮^{b} is stable under ad(E^{o}) whose square is, by (D3), the identity there; therefore 𝔮^{b} ⊂ 𝔪^{1} ⊕ 𝔪^{−1} . On the other hand, by (D4), [𝔮^{b}, 𝔮^{b}] ⊂ 𝔪^{0} , and we actually have equalities. This establishes the decomposition of 𝔤^{b} into eigenspaces.

Now, since ad(E^{o}) is a derivation, for all X ∈ 𝔪^{k}, Y ∈ 𝔪^{l}: ad(E^{o})[X, Y] = [ad(E^{o})X, Y]+ [X, ad(E^{o})Y] = [kX, Y] + [X, lY] = (k + l)[X, Y], proving the statement about the grading.

We will use the notation 𝔪^{±} := 𝔪^{±1} , 𝔥^{b} := 𝔪^{0} . Note that H_{o} = [H, E^{o}]_{o} = 0 for all H ∈ 𝔥^{b}, and also X_{o} = 0 for all X ∈ 𝔪^{+} since X_{o} = [E^{o}, X]_{o} = −[X, E^{o}]_{o} = −X_{o}.

Lemma 2.2.2. If we denote, for v ∈ T_{o}M, by X^{v} ∈ 𝔮 the unique vector field extension by an element of 𝔮, then

v := \frac{1}{2}(X^{v} − ad(E^{o})X^{v}) ∈ 𝔪^{−}

is a vector field extension of v. Proof. v_{o} = \frac{1}{2}(X_{o}^{v} + [X^{v}, E^{o}]_{o}) = \frac{1}{2}(X_{o}^{v} + 1_{o}X_{o}^{v}) = X_{o}^{v} = v.

Definition 2.2.3. The canonical chart associated to a dilation structure E^{o} on a symmetric space M = G/H is defined by assigning to v lying in some neighborhood of 0 in V := T_{o}M the value of the integral curve φ_{t}(o) of v for t = 1. Having fixed the base point o, we will identify an element v in the domain of the canonical chart with its image in M under this chart, and vector fields on the range of the canonical chart will be identified with smooth functions V → V ; thus the vector fields v ∈ 𝔪^{−} are just the constant vector fields.

The notion of the canonical chart can be seen as a local Harish-Chandra imbedding: in fact, integrating the vector fields X ∈ 𝔮 yields a local realization of M as an open domain in the vector space 𝔪^{−}. When M is a Hermitian symmetric space, this realization is known to be global and is precisely the well-known Harish-Chandra imbedding.

2.3. Geometric version of the Kantor-Koecher-Tits construction.

Theorem 2.3.1. Let E^{o} be a dilation structure on the symmetric space M and 𝔤^{b} the associated inner conformal Lie algebra. Then 𝔤^{b} is represented in the canonical chart as a Lie algebra of quadratic vector fields, i.e. by polynomial maps V → V of degree not bigger than two, such that the grading

𝔤^{b} = 𝔪^{−} ⊕ 𝔥^{b} ⊕ 𝔪^{+}

(cf. Lemma 2.2.1) coincides with the grading given by the degree. More precisely, if we define for v ∈ V a polynomial p_{v} by

p_{v}(x) := \frac{1}{2}T_{o}(x, v, x),

where T_{o} is the structure tensor defined by Eqn. (∗) in Lemma 1.3.6, then

(1) 𝔪^{−} = {v| v ∈ V },

(2) 𝔪^{+} = {p_{v}| v ∈ V },

(3) 𝔮 = {v − p_{v}| v ∈ V },

(4) 𝔮̂ = {v + p_{v}| v ∈ V },

(5) 𝔥^{b} = [𝔪^{+}, 𝔪^{−}] = Span{T (w, v)| v, w ∈ V }, where T (u, v)p := T (u, v, p),

(6) E^{o} is the vector field given by E^{o}(x) = x (Euler operator).

Proof. (1) is true by definition of the canonical chart. In order to prove the other relations, we use formula (2) from Appendix A for the Lie bracket of two vector fields X, Y , considered as smooth functions on V , which implies that for any vector field Y and all v ∈ V ,

ad(v)^{k}Y (p) = D^{k}Y (p) · (⊗^{k}v),

where D^{k}Y : V → Hom(S^{k}V, V) is the ordinary k -th total differential of Y : V → V . Now, it follows from Lemma 2.2.1 that [𝔪^{−}, [𝔪^{−}, [𝔪^{−}, 𝔤^{b}]]] = 0, and therefore, for all Y ∈ 𝔤^{b} and p in the domain of the canonical chart,

D^{3}Y (p) = 0.

Integration shows that Y is quadratic as claimed.

Let us prove (6): E^{o} is a linear vector field since [v, [v, E^{o}]] = −[v, v] = 0 and (E^{o})_{o} = 0. Since [v, E^{o}] = v_{o} = v, E^{o} = id_{V} .

Now, the solutions of the Euler differential equation

[E^{o}, X](p) = DX(p) · p − X(p) = k X(p)

for k ∈ ℕ_{0} are precisely the homogeneous polynomials of degree k+1; therefore 𝔥^{b} is represented by linear vector fields and 𝔪^{+} by homogenous quadratic ones.

It remains to calculate the second differential of the vector field X^{w} ∈ 𝔮 with X^{w}(o) = w; we use that v = \frac{1}{2}(X^{v} − [E^{o}, X^{v}]) and the fact that 𝔪^{±} are abelian:

D^{2}X^{w}(o) · v ⊗ v = [v, [v, X^{w}]]_{o}

= \frac{1}{4}[X^{v} − ad(E^{o})X^{v}, [X_{v} − ad(E^{o})X^{v}, X^{w}]]_{o}

= \frac{1}{2}([X^{v}, [X^{v} − ad(E^{o})X^{v}, X^{w}]])_{o}

= \frac{1}{2}([X^{v}, [X^{v}, X^{w} + ad(E^{o})X^{w}]])_{o}

= −T (v, w, v)

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 13

when comparing with Eqn. (∗) from Lemma 1.3.6. Since D^{2}w = 0, we have 2D^{2}(X^{w}) = D^{2}(X^{w} + ad(E^{o})X^{w}); the vector field X^{w} + ad(E^{o})X^{w} being homogeneous quadratic is given by the formula

(X^{w} + ad(E^{o})X^{w})(x) = \frac{1}{2}D^{2}(X^{w} + ad(E^{o})X^{w})(0)(x, x) = −\frac{1}{2}T (x, w, x) = −p_{w}(x).

This proves (2) and gives the formula

(X^{w})(z) = w − \frac{1}{2}T (z, w, z),

proving (3). Finally, [E^{o}, v − p_{v}] = −v − p_{v}, proving (4).

We prove (5): Clearly, [𝔮^{b}, 𝔮^{b}] = [𝔪^{+}, 𝔪^{−}], and we have explicitly

[p_{v}, w](p) = −Dp_{v}(p)w = −T (w, v, p) = −T (w, v)p;

this implies the claim.

Proposition 2.3.2. Let assumptions and notation be as in the preceding theorem. Then the dilation structure is faithful if and only if the map p : v ↦ p_{v} is injective, and in this case the involution Θ : 𝔤^{b} → 𝔤^{b} defined in Prop. 1.4.3 is given by

Θ(v + H + p_{w}) = −w + Θ(H) − p_{v}.

Proof. Comparing (3) and (4) of the theorem, we see that

𝔮 ∩ 𝔮̂ = {v| v ∈ V, p_{v} = 0}.

This space is reduced to zero iff p is injective. If this is the case, then Θ exists and reverses the grading since Θ ∘ ad(E) = − ad(E) ∘ Θ. In particular, 𝔥^{b} is Θ-stable, and on 𝔪^{−} ⊕ 𝔪^{+} , Θ is uniquely determined by the conditions Θ(X) = X for X ∈ 𝔮 and Θ(X) = −X for X ∈ 𝔮̂ which are clearly verified for the formula given in the claim.

Note that, under the assumptions of the proposition, for all u, v, w ∈ V , the vector field [[u, Θv], w] is the constant vector field with value T (u, v, w). A relation of this kind is used in [Sa80, p.55] in order to define the triple product T associated to a Hermitian symmetric space.

2.4. The structure group and its Lie algebra. Recall that, if X is a vector field on V (considered as a smooth function V → V ) and φ a (locally defined) diffeomorphism of V , then the natural forward transport of X is given by

(φ_{∗}X)(p) = (Dφ^{−1}(p))^{−1} · X(φ^{−1}(p)).

Lemma 2.4.1. A locally defined diffeomorphism g of M satisfies the conditions

(1)  g(o) = o, g_{∗}(𝔪^{−}) = 𝔪^{−}, g_{∗}(𝔪^{+}) = 𝔪^{+}

if and only if it is represented in the canonical chart by an invertible linear map g : V → V such that there exists an element g^{♯} ∈ Gl(V) with

(2)  ∀u, v, w ∈ V : gT (u, v, w) = T (gu, g^{♯}v, gw).

Proof. Let g satisfy (1). Then, for all v ∈ V , (g_{∗}v)(p) = (Dg^{−1}(p))^{−1}v has to be a constant vector field, i.e., Dg^{−1} is a constant function. Integrating, we obtain that g^{−1} is an affine map of V ; it preserves the base point and is hence linear. Conversely, the invertible linear maps preserve 𝔪^{−}.

Using the description of 𝔪^{+} given by Th.2.3.1 (2), we see that the condition g_{∗}(𝔪^{+}) = 𝔪^{+} means that, for every v ∈ V , there exists w ∈ V such that g_{∗}(p_{v}) = p_{w}. If we denote by V_{0} the kernel of the linear map p : V → 𝔪^{+}, v ↦ p_{v}, and require w to be in a fixed complementary subspace V_{1} of V_{0} , then the choice of w becomes unique. Now we define g^{♯} to be arbitrary invertible on V_{0} and let g^{♯}(v) := w for v ∈ V_{1} . Then clearly the condition

(3)  g_{∗}(p_{v}) = p_{g^{♯}v}

holds for all v ∈ V , and since p is linear, so is g^{♯}. Finally, since

g_{∗}(p_{v})(p) = g(p_{v})(g^{−1}(x)) = \frac{1}{2}gT (g^{−1}x, v, g^{−1}x),

and T is symmetric in the outer variables, (3) is equivalent to (2). Moreover, all arguments can be reversed, and we thus see that (2) implies that g_{∗}(𝔪^{+}) = 𝔪^{+} .

Definition 2.4.2. The group

Str(T) = {g ∈ Gl(V)| ∃g^{♯} ∈ Gl(V) : ∀u, v, w ∈ V : gT (u, v, w) = T (gu, g^{♯}v, gw)}

described in the previous lemma is called the structure group of the JTS T, resp. of the dilation structure E^{o} which is equivalent to T. Its closure in End(V) is called the structure monoid.

Proposition 2.4.3. If T is faithful and Θ denotes the involution of 𝔤^{b} described in Prop. 2.3.2, then for all g ∈ Str(T) the element g^{♯} is unique, it belongs again to Str(T), and ♯ : g ↦ g^{♯} is an involutive automorphism of Str(T), determined by the condition

(g^{♯})_{∗} = Θ ∘ g_{∗} ∘ Θ.

Proof. Everything follows by comparing Eqn. (3) above with

g_{∗}p_{v} = g_{∗}(Θ(−v)) = Θ ∘ (Θ ∘ g_{∗} ∘ Θ)(−v) = p_{(Θ∘g_{∗}∘Θ)(v)}

and noticing that g ↦ g_{∗} is injective.

Lemma 2.4.4. The structure group is a closed subgroup of Gl(V) whose Lie algebra is

𝔰𝔱𝔯(T) = {X ∈ 𝔤𝔩(V)| [X, 𝔪^{+}] ⊂ 𝔪^{+}}
= {X ∈ 𝔤𝔩(V)| ∃X^{♯} ∈ 𝔤𝔩(V) : ∀u, v, w ∈ V :
XT (u, v, w) = T (Xu, v, w) + T (u, X^{♯}v, w) + T (u, v, Xw)}.

It contains the algebra Der(T) of derivations of T as a subalgebra. The endomorphisms T (u, v) (u, v ∈ V ) belong to 𝔰𝔱𝔯(T), and one may choose T (u, v)^{♯} = −T (v, u). Proof. Clearly Str(T) is an algebraic, hence closed subgroup of Gl(V). Differentiating, resp. integrating, we see that for X ∈ 𝔤𝔩(V) the conditions: ∀t : (exp tX)_{∗}𝔪^{+} ⊂ 𝔪^{+} and [X, 𝔪^{+}] ⊂ 𝔪^{+} are equivalent, proving the first equation. If we define for X ∈ 𝔰𝔱𝔯(T) the element X^{♯} similarly as in the proof of 2.4.1 by [X, p_{v}] = p_{X^{♯}v}, we get the second equality (note that this condition does not define X^{♯} uniquely). The derivations of T arise precisely if X = X^{♯}.

Finally, it is clear that 𝔥^{b} = [𝔪^{+}, 𝔪^{−}] satisfies the first condition. In Th. 1.3.5 we have proved that T is a Jordan triple product, and the defining identity (JT2) of a Jordan triple product (cf. [Be97b, Appendix A]) says precisely that the condition of the second description holds with T (u, v)^{♯} = −T (v, u).

The reader may notice that we have not used in this chapter the condition (D2) up to Prop. 2.4.3, but it is essential for the last statement we have proved.

Definition 2.4.5. The Lie algebra 𝔰𝔱𝔯(T) is called the structure algebra of T, and its subalgebra 𝔥^{b} spanned by the T (u, v), u, v ∈ V is called the inner structure algebra.

2.5. The conformal Lie algebra: general properties.

Definition 2.5.1. The Lie algebra

𝔠𝔬(T) := 𝔪^{−} ⊕ 𝔰𝔱𝔯(T) ⊕ 𝔪^{+}

is called the conformal Lie algebra of T, and its subalgebra 𝔤^{b} = 𝔪^{−} ⊕ 𝔥^{b} ⊕ 𝔪^{+} is called the inner conformal Lie algebra of T.

Definition 2.5.2. A (possibly only locally defined) vector field X on V is called T-conformal if it is of class C^{3} and, for all p ∈ V where X is defined, (L) the first differential DX(p) belongs to 𝔰𝔱𝔯(T), (P) the second differential D^{2}X(p) : V ⊗ V → V belongs to the space

W := {T (·, v, ·)| v ∈ V } ⊂ Hom(V ⊗ V, V).

Proposition 2.5.3. The Lie algebra 𝔠𝔬(T) is a Lie algebra of T -conformal vector fields.

Proof. (L) and (P) are trivial for the vector fields belonging to 𝔪^{−} or 𝔰𝔱𝔯(V). Let us assume that X = p_{v} ∈ 𝔪^{+} . Then DX(p) = T (p, v) ∈ 𝔰𝔱𝔯(T) and D^{2}X(p) = T (·, v, ·) ∈ W.

We will prove in Section 2.6 that in the non-degenerate case the converse also holds. The next theorem is due to Koecher ([Koe69b, Satz 3.1.]). For the sake of completeness, we give a (simplified) proof.

Theorem 2.5.4. Every derivation of the conformal Lie algebra is an inner derivation, and the adjoint representation

ad : 𝔠𝔬(T) → Der(𝔠𝔬(T)), X ↦ ad(X)

is an isomorphism onto.

Proof. Let is first show that ad is injective, i.e. the center of 𝔠𝔬(T) is trivial. Note that, for any X ∈ 𝔛(V),

\begin{aligned}[X, v](p) & = −DX(p) · v, \\ [X, E](p) & = X(p) − DX(p) · p, \end{aligned}

where E = E^{o}, and thus X(p) = [X, E](p) + [X, p](p). Therefore, if X centralizes 𝔠𝔬(T), then X = 0; in particular, the center of 𝔠𝔬(T) is reduced to zero.

Let δ be a derivation of 𝔠𝔬(T). We prove that δ is inner. W.l.o.g. we may assume that δE ∈ 𝔰𝔱𝔯(T): In fact, writing δE = S + Z with S ∈ 𝔰𝔱𝔯(T) and Z ∈ 𝔪^{+} ⊕ 𝔪^{−}, we have (δ + ad([E, Z]))E = S + Z + [[E, Z], E] = S + Z − Z = S ∈ 𝔰𝔱𝔯(T).

We claim that then δE = 0: In fact, for all v ∈ 𝔪^{−},

−δv = δ[E, v] = [δE, v] + [E, δv].

We evaluate at 0 and use that [E, Z](0) = −[Z, E](0) = −Z(0) for all Z ∈ 𝔠𝔬(T). We obtain [δE, v](0) = 0. But [δE, v] ∈ [𝔰𝔱𝔯(T), 𝔪^{−}] ⊂ 𝔪^{−}, and therefore [δE, v] = 0, whence δE = 0.

This implies [δ, ad(E)] = ad(δE) = 0; i.e. δ preserves the eigenspaces of ad(E). Now, any derivation δ preserving the grading is uniquely determined by its restriction to 𝔪^{−}: δX for X ∈ 𝔰𝔱𝔯(T) is uniquely determined by the condition δ[X, v] = [δX, v] + [X, δv], and similarly δ[v, p_{w}] = [δv, p_{w}] + [v, δp_{w}]. The same arguments hold for the Lie algebra Pol(V, V) of all polynomial vector fields on V . In fact, ad(H) with H ∈ 𝔤𝔩(V) given by H(v) = −(δv)(0) is the unique extension of δ|_{𝔪^{−}} to a derivation of Pol(V) preserving the grading. It follows that ad(H) : 𝔠𝔬(T) → ad(H)(𝔠𝔬(T)) and δ : 𝔠𝔬(T) → δ(𝔠𝔬(T)) are the same, and thus 𝔠𝔬(T) is stable under ad(H) and δ = ad(H). Moreover [H, 𝔪^{+}] ⊂ 𝔪^{+} since ad(H) respects the grading. But according to Lemma 2.4.4, this means precisely that H ∈ 𝔰𝔱𝔯(T), und thus δ = ad(H) is an inner derivation.

2.6. Non-degenerate spaces and algebraic version of the fundamental theorem. Recall that a JTS T is called non-degenerate if the trace-form (x|y) := Tr T (x, y) is nondegenerate.

Lemma 2.6.1. Let T be a non-degenerate JTS. Then T is faithful; for all g ∈ Str(T), g^{♯} = (g^{∗})^{−1} , and for all X ∈ 𝔰𝔱𝔯(T), X^{♯} = −X^{∗}, where the adjoint is taken w.r.t. the trace-form.

Proof. T is non-degenerate iff the map

∗ : V → V^{∗}, v ↦ v^{∗} := (x ↦ Tr T (x, v))

is injective. But ^{∗} is a composition of

A : V ↦ W ⊂ Hom(S^{2}V, V), v ↦ T (·, v, ·)

and

κ : Hom(S^{2}V, V) → V^{∗}, B ↦ (x ↦ Tr(B(x, ·))).

Therefore A has to be injective, i.e. T is faithful. Further, by definition of ♯, A has the equivariance property g · A(v) = A(g^{♯}v), and κ is natural, i.e. κ(g · B) = (g^{∗})^{−1}κ(B). When we identify V and V^{∗} via ^{∗}, this proves the claim. Similarly for X^{♯}.

Theorem 2.6.2. If T is a non-degenerate JTS containing no ideal isomorphic to ℝ or ℂ, then 𝔠𝔬(T) is precisely the space of conformal vector fields on V ; i.e. any locally defined vector field of class C^{3} having properties (L) and (P) (cf. Def. 2.5.2) is actually polynomial and belongs to 𝔠𝔬(T).

Proof. Because of Prop. 2.5.3 we only have to show that a conformal vector field ξ belongs to 𝔠𝔬(T). By definition, the second differential

D^{2}ξ : V → W ⊂ Hom(S^{2}V, V), x ↦ D^{2}ξ(x)

is of class C^{1} . The third differential D^{3}ξ = D(D^{2}ξ) takes therefore values in Hom(V, W). On the other hand, D^{3}ξ(x) is a totally symmetric V -valued trilinear map. Therefore D^{3}ξ takes values in

Hom_{s}(V, W) := Hom(V, W) ∩ Hom(S^{3}V, V) = {α : S^{3}V → V | ∀x ∈ V : α(x, ·, ·) ∈ W }.

If we can show that Hom_{s}(V, W) = 0, then D^{3}ξ = 0 and we can integrate. We then find that ξ is in fact quadratic, given by

ξ(x) = ξ(0) + Dξ(0) · x + \frac{1}{2}D^{2}ξ(0) · (x, x).

Since Dξ(0) ∈ 𝔰𝔱𝔯(T) and D^{2}ξ(0) ∈ W by conformality, ξ is of the form given in Th. 2.3.1, i.e. ξ ∈ 𝔠𝔬(T). We collect these observations:

Proposition 2.6.3. If Hom_{s}(V, W) = 0, the Lie algebra 𝔠𝔬(T) is equal to the Lie algebra of conformal vector fields.

(The converse also holds, as is easily seen.) In order to prove the theorem we now show:

Proposition 2.6.4. Under the assumptions of the theorem, Hom_{s}(V, W) = 0.

Proof. Since T is non-degenerate, the map A : V → W, v ↦ T (·, v, ·) is bijective (Prop. 2.6.1). Therefore

Hom_{s}(V, W) → End(V), α ↦ β := A^{−1} ∘ α

is injective, i.e., we define β by the relation α(u) · v ⊗ w = T (v, βu, w). By the assumption on α, this expression is totally symmetric in u, v, w. In particular, the relation

(1)  T (v, βu) = T (u, βv)

holds for all u, v ∈ V . When applying ♯ to both sides, the relation T (a, b)^{♯} = −T (b, a) (cf. Lemma 2.4.4) yields T (βu, v) = T (βv, u). Using the notaton R(a, b) = T (a, b) − T (b, a), we get R(v, βu) = R(u, βv), i.e.

(2)  R(v, βu) = −R(βv, u)

holds for all u, v ∈ V . We claim that this implies the relation

(3)  R(v, u) ∘ β = −β ∘ R(v, u)

for all u, v ∈ V . In order to prove this, we note that for all x, y ∈ V , (y|βx) = tr T (y, βx) = tr T (βy, x) = (βy|x), i.e. β is self-adjoint. Using this and the relation

(R(x, y)u|v) = (R(u, v)x|y),

which holds for all x, y, u, v ∈ V (cf. [Hel62, p.68]), we get from (2)

(R(u, v)βx|y) = (R(βx, y)u|v)
= −(R(x, βy)u|v)
= −(R(u, v)x|βy)
= −(βR(u, v)x|y).

Since the trace-form is non-degenerate, (3) follows by comparing the first and the last expression. From (3) we deduce that for all a, b, u, v ∈ V ,

(4)  [R(a, b), R(u, v)] ∘ β = β ∘ [R(a, b), R(u, v)].

Thus equations (3) and (4) together imply that for all H ∈ [𝔥, 𝔥] ⊂ 𝔥,

(5)  H ∘ β = β ∘ H = 0.

Thus β(V) ⊂ V is a submodule on which [𝔥, 𝔥] acts trivially.

Let us assume now that V is simple and not isomorphic to ℝ or ℂ. Then R ≠ 0 and [𝔥, 𝔥] ≠ 0. It is known (cf. [N85, Th.1.11]) that three cases may arise:

(a) V is an irreducible [𝔥, 𝔥]-module of dimension bigger than one;

(b) V is the direct sum of two irreducible [𝔥, 𝔥]-modules of dimension bigger than one;

(c) V is the direct sum of a trivial one-dimensional and an irreducible non-trivial [𝔥, 𝔥]- module.

In the first two cases the only submodule on which [𝔥, 𝔥] acts trivially is zero, whence β = 0. In the third case β has to be a multiple of the projection onto the trivial one-dimensional module. This case arises precisely if T comes from a simple Jordan algebra (over 𝕂 = ℝ or ℂ) with product (x, y) ↦ xy via the formula T (u, v, w) = u(vw) − v(uw) + (uv)w, and the trivial submodule is the space 𝕂e of multiples of the unit element e. Thus β is given by β(x) = λ(x|e)e with λ ∈ 𝕂, and

T (u, βv) = λ(v|e)T (u, e) = λ(v|e)L(u),

where L(u)x = ux. If λ ≠ 0, then condition (1) with v = e implies that L(u) = (u|e)L(e) = (u|e) id_{V} for all u ∈ V . This implies that 𝔥 = 0; this case corresponds to the simple Jordan algebra V = 𝕂 and is excluded. Therefore λ = 0 and β = 0.

Thus the proposition and the theorem are proved in the case that T is simple. The general non-degenerate case is reduced to the simple case as in [Be96a].

Note, by the way, that in case (c) there always exists a modification of T ([Be97b, Lemma A.6]) leading to case (a) or (b). Since the space W is independent of modifications, this reduces case (c) to the former cases. Also, case (c) is covered by [Be96a, Th. 1.3.1].

Example 2.6.5. If V = ℝ^{n}, then

T (u, v, w) := uv^{t}w + wv^{t}u

defines a JTS on V that is related to the real projective space M = SO(n + 1)/ ± SO(n) with twisted complexification the complex projective space U(n + 1)/(U(1) × U(n). It is an easy exercise to prove Prop. 2.6.4 in this case by hand. The resulting special case of Th. 2.6.2 is the algebraic version of the fundamental theorem of projective geometry (real case).

3. Conformal group and the fundamental theorem

3.1. Definition of the conformal group. The conformal group is a Lie group with Lie algebra 𝔠𝔬(T), coming in a certain realization, namely as a group of birational maps of the vector space V . Since birational maps are not defined everywhere, we start with local diffeomorphisms φ of V , i.e. diffeomorphisms defined on some domain U ⊂ V .

Theorem 3.1.1. Let φ : U → φ(U) ⊂ V be a local (C^{1} -regular) diffeomorphism of V such that

φ_{∗}(𝔠𝔬(T)) = 𝔠𝔬(T),

i.e. if for all X ∈ 𝔠𝔬(T), the vector field φ_{∗}(X) coincides on φ(U) with an element of 𝔠𝔬(T). Then φ has a unique extension to a birational map φ̃ of V such that φ̃_{∗}(𝔠𝔬(T)) = 𝔠𝔬(T). The birational maps thus obtained form a group.

Proof. Recall the formula

(φ_{∗}X)(p) = (Dφ^{−1}(p))^{−1} · X(φ^{−1}(p)),

where D denotes the ordinary total differential on a vector space. We remark first that, since (φ^{−1})_{∗} = (φ_{∗})^{−1} =: φ^{∗}, the conditions φ_{∗}(𝔠𝔬(T)) = 𝔠𝔬(T) and φ^{∗}(𝔠𝔬(T)) = 𝔠𝔬(T) are equivalent.

Let X = v with v ∈ V (constant vector field). By assumption, φ^{∗}v coincides on U with an element of 𝔠𝔬(T). Since 𝔠𝔬(T) is a Lie algebra of (quadratic) polynomial vector fields (Th.

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 19

2.3.1), the map U → V , p ↦ (φ^{∗}v)(p) = (Dφ(p))^{−1}v is (quadratic) polynomial for all v ∈ V , and thus

d_{φ} : U → Gl(V) ⊂ End(V), p ↦ d_{φ}(p) := (Dφ(p))^{−1}

is (quadratic) polynomial. Now consider the Euler operator X = E^{o} ∈ 𝔠𝔬(T). The same arguments as before show that

n_{φ} : U → V, p ↦ (φ^{∗}E^{o})(p) = (Dφ(p))^{−1} · φ(p)

is (quadratic) polynomial. Therefore

φ : U → V, p ↦ φ(p) = d_{φ}(p)^{−1} · n_{φ}(p)

is rational. We continue n_{φ} and d_{φ} to polynomials on V , and define

φ̃(x) := d_{φ}(x)^{−1} · n_{φ}(x)

for all x with det d_{φ}(x) ≠ 0 (since det d_{φ} is a non-zero polynomial, φ̃ is defined almost everywhere). Then φ̃ is a birational map, its inverse given by φ̃^{−1} . Let us prove that φ̃_{∗}(𝔠𝔬(T)) = 𝔠𝔬(T). For all X ∈ 𝔠𝔬(T),

((φ̃)^{∗}X)(p) = (Dφ̃(p))^{−1}X(φ̃(p)) = d_{φ}(p)X(φ̃(p))

is rational in p and coincides for p ∈ U with the polynomial Y := φ^{∗}X ∈ 𝔠𝔬(T), whence ((φ̃)^{∗}X)(p) = φ^{∗}X(p) everywhere. In other words, (φ̃)^{∗}(𝔠𝔬(T)) ⊂ 𝔠𝔬(T). The same being true for φ^{−1} , we actually have equality.

If φ and ψ with φ_{∗}(𝔠𝔬(T)) = 𝔠𝔬(T) and φ_{∗}(𝔠𝔬(T)) = 𝔠𝔬(T) are given, then (although φ and ψ may be not composable) φ̃ and ψ̃ are always composable on a dense open set, and

(φ̃ ∘ ψ̃)_{∗}(𝔠𝔬(T)) = φ̃_{∗}(ψ̃_{∗}(𝔠𝔬(T))) = 𝔠𝔬(T).

This, and the remark made above that (φ̃)^{−1} = φ̃^{−1} , prove the last claim.

Definition 3.1.2. The group of birational maps of V defined in the preceding theorem is denoted by Co(T) and is called the conformal group of T. For any g ∈ Co(T), the quadratic polynomial

d_{g} : V → End(V), x ↦ (Dg(x))^{−1}

is called its denominator, and the quadratic polynomial

n_{g} : V → V, x ↦ (Dg(x))^{−1} · g(x) = d_{g}(x) · g(x)

is called its numerator (cf. the preceding proof).

The following result is due to Koecher ([Koe69, Satz 2.2]). In order to introduce some relevant notation, we include its proof.

Theorem 3.1.3. The representation

∗ : Co(T) → Aut(𝔠𝔬(T)), g ↦ g_{∗}

is injective, and its image Co_{∗}(T) is an open subgroup of Aut(𝔠𝔬(T)); in particular, Co(T) has the structure of a Lie group with Lie algebra 𝔠𝔬(T).

Proof. We have seen above that g(p) = d_{g}(p)^{−1}n_{g}(p) with d_{g}(p)v = (g^{∗}v)(p) and n_{g}(p) = (g^{∗}E)(p). This defines an inverse of ∗ from Co_{∗}(T) onto Co(T); thus ∗ is injective.

In order to prove that Co_{∗}(T) is open in Aut(𝔠𝔬(T)) we have to find enough elements in Co(T). Let X ∈ 𝔠𝔬(T) let φ_{t} be the local flow of X. If X ∈ 𝔪^{−} or X ∈ 𝔰𝔱𝔯(T), then the flows are global on V ; we write exp(X) := φ_{1} . Thus

exp(𝔪^{−}) = t_{V} := {t_{v}|t_{v}(x) = x + v, x, v ∈ V }, exp(𝔰𝔱𝔯(T)) ⊂ Str(T).

Now consider X ∈ 𝔪^{+} . Since X vanishes of order 2 at 0, there is a neighbourhood U of 0 in V such that φ_{t}(x) is defined for x ∈ U and t = 1 (cf. [Be96, annexe (A2)]). We let

exp X : U → V, x ↦ φ_{1}(x).

Since X ∈ 𝔠𝔬(T), it follows that (φ_{t})_{∗} preserves 𝔠𝔬(T) in the sense of Th. 3.1.1; in particular exp(X) ∈ Co(T). For simplicity of notation, we identify exp(X) with its rational continuation guaranteed by Th. 3.1.1. Then the relation

∀Y ∈ 𝔠𝔬(T) : (exp X)_{∗}Y = e^{ad(X)}Y

holds. Summing up, e^{ad(X)} belongs to Co_{∗}(T) for all X ∈ 𝔪^{±} and X ∈ 𝔰𝔱𝔯(T); thus the identity component of the group Int(𝔠𝔬(T)) of inner automorphisms, generated by e^{ad(𝔠𝔬(T))} , belongs to Co_{∗}(T). But this is equal to the identity component of Aut(𝔠𝔬(T)) because ad(𝔠𝔬(T)) = Der(𝔠𝔬(T)) (Th. 2.5.3). Thus Co_{∗}(T) is open in Aut(𝔠𝔬(T)).

In [Koe69, Satz 2.1], Koecher gives a further description of the open subgroup Co_{∗}(T) ⊂ Aut(𝔠𝔬(T)) which he calls the group of essential automorphisms.

3.2. The Liouville-condition.

Definition 3.2.1. We say that a locally defined diffeomorphism g of V satisfies the Liouvillecondition if, for all x where g is defined,

(L′)  Dg(x) ∈ Str(T).

Of course, this terminology is motivated by the classical Liouville-theorem concerning the case where Str(T) is the scalar-extended orthogonal group (cf. [Be96a]).

Theorem 3.2.2. All elements of Co(T) satisfy the Liouville-condition.

Proof. Given x ∈ V with det d_{g}(x) ≠ 0, we let y := g(x) and g′ := t_{−y} ∘ g ∘ t_{x}. Then g′ ∈ Co(T), g′(0) = 0 and Dg′(0) = Dg(x). Replacing g by g′, we may assume that x = 0 and g(0) = 0.

We claim that then g_{∗}(𝔪^{+}) ⊂ 𝔪^{+} . In order to prove this, note that 𝔪^{+} is the subalgebra of elements of 𝔠𝔬(T) vanishing of order 2 at the origin. Differentiating the expression (g_{∗}X)(p) = Dg(g^{−1}(x)) · X(g^{−1}(p)), we obtain

(1)  \begin{aligned}D(g_{∗}X)(p) & = (D^{2}g(g^{−1}(p)) · X(g^{−1}(p))) ∘ Dg^{−1}(p) \\ & + Dg(g^{−1}(p)) ∘ DX(g^{−1}(p)) ∘ Dg^{−1}(p). \end{aligned}

If g(0) = 0, X(0) = 0 and DX(0) = 0, then the right-hand side vanishes at the origin. Also, (g_{∗}X)(0) = 0, and thus g_{∗}(𝔪^{+}) ⊂ 𝔪^{+} .

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 21

Next we want to show that for all X ∈ 𝔪^{+} , g_{∗}X = (Dg(0))_{∗}X. In order to prove this, we differentiate the expression (1) for D(g_{∗}X)(p) and evaluate at 0. Then, under our assumptions on g and X, only one term remains, namely

D^{2}(g_{∗}X)(0) = Dg(0) ∘ D^{2}X(0) ∘ Dg(0)^{−1} ⊗ Dg(0)^{−1}.

If we replace in this expression g by Dg(0), the right-hand side does not change; therefore

D^{2}(g_{∗}X)(0) = D^{2}((Dg(0))_{∗}X)(0).

Since both g_{∗}X and (Dg(0))_{∗}X are are homogeneous quadratic vector fields, it follows that g_{∗}X = (Dg(0))_{∗}X.

We have shown that 𝔪^{+} = g_{∗}(𝔪^{+}) = (Dg(0))_{∗}(𝔪^{+}). According to Prop. 2.4.1 and Def. 2.4.2, Dg(0) belongs thus to the structure group.

Corollary 3.2.3. For all g ∈ Co(T), the image of the polynomial d_{g} is contained in the structure monoid. Proof. This is an immediate consequence of the definition of the structure monoid as the closure of the structure group in End(V).

3.3. The condition of generalized projectivity. Recall from Def. 2.5.2 the definition of the subspace W ⊂ Hom(S^{2}V, V).

Definition 3.3.1. A locally defined diffeomorphism g of V satisfies the condition of generalized projectivity if it satisfies the Liouville-condition (L′), and in addition for all x where g is defined,

(P′).  (Dg(x))^{−1} ∘ D^{2}g(x) ∈ W.

We give a more conceptual definition of condition (P′): (P′) is equivalent to requiring that

(P′′)  (g^{−1} · ∇ − ∇)_{x} ∈ W,

where ∇ is the canonical flat connection of V (cf. Appendix A.9, Eq. (7)). In other words, g^{−1} · ∇ − ∇ is a section of the subbundle W of Hom(S^{2}T V, V) with constant fiber W_{p} = W. The Liouville-condition (L′) means just that g preserves W. Therefore, if both g and h satisfy (L′) and (P′), then

(gh)^{−1}∇ − ∇ = h^{−1}(g^{−1}∇ − ∇) + g^{−1}∇ − ∇

is again a section of W. We have thus shown that the condition of generalized projectivity defines a pseudogroup of diffeomorphisms.

Theorem 3.3.2. All elements of Co(T) satisfy the condition of generalized projectivity. Proof. Let g ∈ Co(T) and x ∈ V be given. Since (L′) holds by Th. 3.2.2, only (P′) remains to be verified. As in the proof of Th. 3.2.2, we may assume w.l.o.g that x = 0 and g(x) = x; composing with Dg(0)^{−1} , we see that we may further assume that Dg(0) = id_{V} .

We specialize Eqn. (1) in the proof of 3.2.2 to the Euler operator X = E:

D(g_{∗}E)(p) = (D^{2}g(g^{−1}(p)) · g^{−1}(p)) ∘ Dg^{−1}(p) + id_{V} .

Thus, under our assumptions on g, D(g_{∗}E)(0) = id_{V} , and differentiating further and evaluating at 0, we find that

D^{2}(g_{∗}E)(0) = (D^{2}g)(0).

Since g_{∗}E ∈ 𝔠𝔬(T) by definition of Co(T), it follows from Prop. 2.5.3 that (D^{2}g)(0) ∈ W.

3.4. The fundamental theorem.

Theorem 3.4.1. If T is non-degenerate and contains no ideal isomorphic to ℝ or ℂ, then any local diffeomorphism (of class C^{4} ) satisfying (L′) and (P′) is birational, and its birational continuation belongs to Co(T). In other words, Co(T) is exactly the group satisfying the condition of generalized projectvity.

Proof. Let g be a local diffeomorphism satisfying (L′) and (P′). According to Th. 2.6.2, 𝔠𝔬(T) can under the given hypothesis be characterized as the Lie algebra of vector fields X (of class C^{3} ) satisfying the properties

∀p ∈ V : (L) DX(p) ∈ 𝔰𝔱𝔯(T), (P) D^{2}X(p) ∈ W.

Thus in order to prove the theorem it suffices to prove the following:

(1) If g satisfies (L′) and X satisfies (L), then g_{∗}X satisfies (L).

(2) If X satisfies (L) and (P) and g satisfies (L′) and (P′), then g_{∗}X satisfies (P). In fact, from (1) and (2) it follows that, if g satisfies (L′) and (P′), then it preserves the

conformal Lie algebra which is characterized by (L) and (P) (Th. 2.6.1), whence g ∈ Co(T).

Now, (1) is proved in [Be96a, Lemme 2.1.2]. Condition (2) can be proved similarly, however the calculations become quite messy. Therefore we indicate a more conceptual proof: Conditions (L) and (P) are the infinitesimal versions of (L′) and (P′); using notation defined in Def. 3.3.1 they can be rephrased as

X · C^{∞}(V, W) ⊂ C^{∞}(V, W)

and (cf. Appendix A.7 for notation)

X · ∇ ∈ C^{∞}(V, W).

But then

(g_{∗}X) · ∇ = g · (X · (g^{−1} · ∇)) ∈ C^{∞}(V, W),

and thus g_{∗}X satisfies (P), and (2) is proved. Observe finally that for these arguments it suffices to assume C^{4} -regularity for g.

Example 3.4.2. (Projective Space) If W corresponds to the triple product T defined in Ex. 2.6.5, then it is a classical result of H. Weyl that two torsionfree affine connections ∇, ∇′ have the same geodesics (as unparametrized lines) iff ∇ − ∇′ is a section of W. Then a local diffeomorphism preserves straight lines iff it satisfies condition (P′′).

3.5. The structure of the conformal group. We return to the general case.

Proposition 3.5.1. An element g ∈ Co(T) is uniquely determined by its 2-jet at one point p ∈ V with d_{g}(p) ≠ 0.

Proof. Let g_{1}, g_{2} ∈ Co(T) have the same 2-jet at a point p ∈ V . Then g := g_{1}^{−1}g_{2} has the 2-jet of the identity at p. Replacing g by t_{−g(p)} ∘ g ∘ t_{p}, we may further assume that p = 0. We thus have to show that the only element g ∈ Co(T) with g(0) = 0, Dg(0) = id_{V} and D^{2}g(0) = 0 is the identity. The arguments proving Th. 3.3.2 show that, under these conditions on g, g_{∗}E = E. Then, for all v ∈ V ,

[E, g_{∗}v] = [g_{∗}E, g_{∗}v] = g_{∗}[E, v] = −g_{∗}v.

However, the only solutions of the differential equation [E, X] = −X are the constant vector fields, therefore (g_{∗}v)(p) = (Dg^{−1}(p))^{−1}v is constant; this means that Dg^{−1} : V → End(V) is constant and thus g^{−1} is linear. Since Dg(0) = id_{V} by assumption, this implies that g = id_{V} .

Proposition 3.5.2. Let Co′(T) := {g ∈ Co(T)| d_{g}(0) ≠ 0}. Then the map

κ : Co′(T) → V × Str(T) × W, g ↦ (g(0), Dg(0), Dg(0)^{−1} ∘ D^{2}g(0))

is bijective. Proof. The map is well-defined since g is regular at 0 by definition of Co′(T), and Dg(0) and Dg(0)^{−1}D^{2}g(0) lie in the correct spaces according to Th. 3.2.1 and 3.3.1. Since these data describe precisely the 2-jet of g, the preceding proposition now implies that κ is injective.

In order to prove that κ is surjective, we have to calculate the 2-jet of exp X for X ∈ 𝔪^{+} (defined in the proof of Th. 3.1.3). It is easily seen that (exp X)(0) = 0 and (D exp X)(0) = id_{V} (cf. [Be96, Appendix A2]). Therefore, as we have seen in the proof of Th.3.3.2,

D^{2}(exp X)(0) = D^{2}((exp X)_{∗}E)(0),

and we can calculate

D^{2}(exp X)(0) = D^{2}(e^{ad(X)}E)(0)
= D^{2}(E + [X, E])(0)
= −D^{2}(X)(0)

since [X, E] = −X for all X ∈ 𝔪^{+} und thus ad(X)^{k}E = 0 for all k > 1.

Now, given v, w ∈ V , g ∈ Str(T), we let g := t_{v} ∘ g ∘ exp(−p_{w}) ∈ Co(T) and prove that κ(g) = (v, g, 2T (·, w, ·)): In fact, g(0) = v, Dg(0) = g and Dg(0)^{−1} ∘ D^{2}g(0) = (D^{2} exp(−p_{w}))(0) = D^{2}(p_{w})(0) = 2T (·, w, ·).

We collect the preceding results. By

Co(T)_{0} := {g ∈ Co′(T)| d_{g}(0) ≠ 0, g(0) = 0}

we denote the “stabilizer of the base point”, and by

Co(T)_{00} := {g ∈ Co(T)_{0}| Dg(0) = id_{V} }

the kernel of the first isotropy representation g ↦ Dg(0) of Co(T)_{0} .

Theorem 3.5.3. Co(T) is a Lie group with Lie algebra 𝔠𝔬(T). It is generated by exp(𝔪^{−}), exp(𝔪^{+}) and Str(T). More precisely:

(1) Co(T)_{00} = exp(𝔪^{+}).

(2) Co(T)_{0} = Str(T) exp(𝔪^{+}); this is a semidirect product.

(3) Co′(T) = exp(𝔪^{−}) Str(T) exp(𝔪^{+}); more precisely: every element φ of the open dense set Co′(T) ⊂ Co(T) has a unique decomposition

φ = t_{g(0)} Dg(0) exp(p_{w}), t_{g(0)} ∈ exp(𝔪^{−}), Dg(0) ∈ Str(V), p_{w} ∈ 𝔪^{+}.

(4) Co(T) = exp(𝔪^{−}) Str(T) exp(𝔪^{+}) exp(𝔪^{−}) (the corresponding decomposition is not unique).

(5) If T is faithful, then Co(T) carries an involution Θ such that, for all φ = t_{v}g exp(p_{w}) ∈

Co(T)′,

Θ(t_{v}g exp(p_{w}))^{−1} = t_{w}(g^{♯})^{−1} exp(p_{v}).

Proof. Claims (1), (2) and (3) are a direct consequence of Prop. 3.5.2. We prove (4): If g ∈ Co(T), we choose a point p with d_{g}(p) ≠ 0; we let g′ := g ∘ t_{p}; then d′_{g}(0) ≠ 0, and we can apply (3).

We prove (5): The involutive automorphism Θ of 𝔠𝔬(T) described in Th. 2.3.2 and Lemma 2.4.4 induces by conjugation an involution of the group Aut(𝔠𝔬(T)) which will be denoted by Θ. We have to show that the subgroup Co_{∗}(T) ⊂ Aut(𝔠𝔬(T)) is stable under Θ. But this is easily checked on the generators of Co(T) exhibited in Part (4): Θ(t_{v}) = Θ(exp v) = exp(Θ(v)) = exp(p_{−v}) and Θ(g) = g^{♯} for g ∈ Str(T) (cf. Prop. 2.4.3). This also proves the last equation.

3.6. Conformal completion and global version of the fundamental theorem. We denote by P := Str(T) exp(𝔪^{+}) the stabilizer Co(T)_{0} and by 𝔭 := 𝔰𝔱𝔯(T) ⊕ 𝔪^{+} its Lie algebra. Then P is closed in Co′(T) and in Co(T).

Definition 3.6.1. The space V^{c} := Co(T)/P as well as the map

V → V^{c}, v ↦ t_{v}P

are called the conformal completion of V (w.r.t T).

Proposition 3.6.2. The conformal completion is an imbedding with open dense image.

Proof. The map V → V^{c} is injective since P ∩ t_{V} = {id_{V} }. Its image is open dense since it is the image of the open dense set Co′(T) ⊂ Co(T) under the surjective submersion π : Co(T) → V^{c}, g ↦ gP.

It is clear that 𝔠𝔬(T) is realized as an algebra of vector fields on V^{c} and that Co(T) is precisely the group normalizing this algebra. It suffices to assume that diffeomorphisms normalizing 𝔠𝔬(T) are only locally defined in order to conclude a global continuation onto V^{c}; this explains the terminology. We now give a global description of the conditions (L′) and (P′).

Proposition 3.6.3. The subbundle W of Hom(S^{2}T V, T V) with constant fiber W_{p} = W (where W is as in Def. 2.5.2) has a unique extension to a Co(T)-invariant subbundle (again denoted by W) of the bundle Hom(S^{2}T (V^{c}), T (V^{c})).

Proof. We only have to check that W = W_{0} is invariant under the natural action of the stabilizer P. But this is clear since, for all p ∈ P, Dp(0) belongs to Str(T) which is precisely the linear group preserving W.

Definition 3.6.4. The subbundle W of Hom(S^{2}T (V^{c}), T (V^{c})) is called the structure bundle associated to T.

As we have seen in the proof of the fundamental theorem (3.4.1), the Liouville-condition is equivalent to saying that a local diffeomorphism φ preserves the structure bundle, and the condition of generalized projectivity that g · ∇ − ∇ is a local section of the structure bundle, where ∇ is the flat connection of V (which does not extend to a global connection on V^{c}). The interested reader may easily rephrase Th. 3.4.1 in this context, using the relevant definitions from Appendix A.

4. Global Makarevič spaces

4.1. The global Makarevič space associated to a local one. Recall that a germ of a Makarevič space (M^{o}, T_{o}) is given by the germ M^{o} of a symmetric space (equivalent to a LTS 𝔮) and a Jordan-extension T = T_{o} of the LTS 𝔮 (or, equivalently, a dilation structure E^{o} at o). Using the conformal group Co(T) of T, we now construct a certain globally symmetric space M whose germ at o is given by M^{o}.

Proposition 4.1.1.

(1) Let G be the analytic subgroup of Co(T) with Lie algebra 𝔤 = Der(T) ⊕ 𝔮 ⊂ 𝔠𝔬(T). Then

the orbit

M := M_{T} := G · o ⊂ V^{c}

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 25

is open in V^{c}; it is a globally symmetric space whose associated LTS is 𝔮. The geodesic symmetry s_{0} w.r.t. the base point o is given in the canonical chart by − id_{V} .

(2) If T is faithful, then G = (Co(T)^{Θ})_{o} is the identity component of the fixed point group of the involution Θ defined in Th.3.5.3 (5). In particular, it is closed in Co(T).

Proof. (1) Using the description of 𝔮 from Th.2.3.1 (3), we see that evaluation T_{o}M → T_{o}(V^{c}) = V , X ↦ X(0) is surjective, and therefore M is open in V^{c}. Further, ((− id)_{∗}X)(p) = −X(−p) defines an involution on 𝔤 which is the identity on Der(T) (linear vector fields, cf. Lemma 2.4.4) and minus the identity on 𝔮 (immediate verification). This involution lifts to the involution (− id)_{∗}g = (− id) ∘ g ∘ (− id) of G, and the stabilizer G_{0} is then an open subgroup of the fixed point group G^{(− id)_{∗}}. The geodesic symmetry w.r.t. the base point 0 is then calculated as s_{0}(g · 0) = ((− id)_{∗}g)(0) = −g(0).

(2) If T is faithful, then 𝔤 = 𝔠𝔬(T)^{Θ} (cf. Prop.2.3.2 and Lemma 2.4.4), and the statements follow immediately.

The space M = M_{T} defined in the preceding proposition will be called the global Makarevič space associated to the germ (M^{o}, T_{o}). In order to avoid technical complications, we will in the rest of this section assume that T is faithful. Recall that in the non-degenerate case this is always the case (Lemma 2.6.1).

4.2. The Borel-imbedding generalized. We assume that M is defined as in Prop. 4.1.1 (2). Let α be an element of the structure group of T with Θ(α) = α^{−1} , i.e.

α ∈ {g ∈ Gl(V)| ∀u, v, w ∈ V : gT (u, gv, w) = T (gu, v, gw)}.

Let α_{∗} be conjugation by α in Co(T). Then Θα_{∗} is an involution of Co(T): in fact,

(Θα_{∗})^{2} = Θα_{∗}Θα_{∗} = ΘΘ(Θ(α))_{∗}α_{∗} = id_{Co(T)} .

We denote by G^{(α)} := (Co(T)^{Θα_{∗}})_{o} the identity component of the corresponding fixed point group.

Proposition 4.2.1. The orbit

M^{(α)} := G^{(α)} · o ⊂ V^{c}

is open in V^{c}; it is the global Makarevič space associated to the JTS

T^{(α)}(u, v, w) = T (u, αv, w).

The spaces M^{(α)} and M^{(−α)} are c-dual symmetric spaces.

Proof. The Lie algebra of G^{(α)} is 𝔤^{(α)} = 𝔠𝔬(T)^{Θα_{∗}}. It is stable under (− id)_{∗}; the corresponding decomposition is

𝔤^{(α)} = 𝔥^{(α)} ⊕ 𝔮^{(α)}

with 𝔥^{(α)} = 𝔰𝔱𝔯(T)^{Θα_{∗}}. In order to determine 𝔮^{(α)} , we use Prop. 2.3.2 and find that

Θα_{∗}(v + p_{w}) = Θ(αv + p_{α^{♯}w}) = −α^{−1}w − p_{αv}.

Therefore

𝔮^{(α)} = {v − p_{αv}| v ∈ V }.

Thus the evaluation map 𝔮^{(α)} → T_{o}(V^{c}) = V , X ↦ X(0) is surjective, and as in the proof of Prop. 4.1.1 it follows that M^{(α)} is an open orbit in V^{c} having the structure of a symmetric space with geodesic symmetry − id_{V} at the origin.

It remains to be shown that T^{(α)} is a Jordan-extension of the Lie triple product R^{(α)} of the LTS 𝔮^{(α)} . In order to prove this, we calculate

R^{(α)}(u, v)w = [[u − p_{αu}, v − p_{αv}], w − p_{αw}]_{0}
= −[(T (u, αv) − T (v, αu)), w − p_{αw}]_{0}
= (T (u, αv, w) − T (v, αu, w)) = (T^{α}(u, v, w) − T^{α}(v, u, w)),

thus T^{α} is a Jordan-extension of the LTS 𝔮^{(α)} .

Finally, since T^{−α} = −T^{α} and R^{−T} = −R^{T} , the Lie triple product of 𝔮^{(α)} is isomorphic to the negative of the one in 𝔮^{(−α)} , and this means precisely that both are c-dual.

In particular, the proposition shows that both M = M^{(id_{V} )} and its c-dual space M^{(− id_{V} )} have a global realization on V^{c}. This is the global version of Prop. 2.1.1 (there 𝔮^{(− id_{V} )} was denoted by 𝔮̂). The more general statement given here means that all para-real forms of the para-Hermitian complexification M_{phℂ} have a global realization on V^{c} (cf. [Be97b, Appendix B, Cor. B.2]).

Remark 4.2.2. In case T comes from a unital Jordan algebra (x, y) ↦ xy via the formula T (x, y, z) = x(yz) − y(xz) + (xy)z, the involution Θ is given by j_{∗}, where j is the Jordaninversion j(x) = x^{−1} in the algebra, and the definition of the spaces M^{(α)} given here coincides with the one given in [Be96b]. We call such spaces of the first kind. Koecher characterizes them intrinsically by the property that Θ is an inner automorphism of the conformal Lie algebra ([Koe69b, III.4.3]). Many spaces which are not of the first kind can be realized as “subspaces” of spaces of the first kind, cf. [Be96c, Section 2.3] and [Be97a, Section 3.2].

4.3. Bundles over V^{c} with global sections. We continue to assume that T is faithful and use the notation G for the identity component of Co(T)^{Θ} and M = G · o ⊂ V^{c}. Recall that the structure tensor T is a G-invariant tensor field on M = G/H (in fact, H = Str(T)^{Θ} = Aut(T_{o})). If we consider T in the form

A : T M → W, A_{p}(v) = T_{p}(·, v, ·),

then A_{p} : T_{p}M → W_{p} is bijective since this is true for T_{o} by faithfulness. Therefore

A^{−1} : W → T M, (A^{−1})_{p} := (A_{p})^{−1}

is a well-defined G-invariant section of Hom(W, T M) over M. We are going to derive an explicit formula for this section and show that this defines a global section over V^{c}. Note that, over the open dense subset V ⊂ V^{c}, we can trivialize the bundles T (V^{c}), W and Hom(W, T (V^{c})) by identifying all fibers via the translation group t_{V} with the fiber at 0. Then a section of Hom(W, T (V^{c})) is described by a function V → Hom(W, V) and by a function V → End(V) if we identify V with W via A_{o}.

We define the Bergman-operator of T as the polynomial B : V × V → End(V) given by

B(u, v) := id_{V} −T (u, v) + \frac{1}{4}P (u)P (v); P (z) := T (z, ·, z).

Theorem 4.3.1.

(1) The section A^{−1} has a unique extension to a G-invariant and analytic section F of the bundle Hom(W, T (V^{c})) over V^{c}.

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 27

(2) If we identify sections of Hom(W, T (V^{c})) over V with functions V → End(V) as explained

above, then F is given by the formula

F (z) = B(z, z).

Proof. (1) For any g ∈ Co(T) and x ∈ V^{c} we denote by (T_{x}g)_{∗} : W_{x} → W_{gx} the natural map induced from the tangent map T_{x}g : T_{x}(V^{c}) → T_{gx}(V^{c}) via the formula

(T_{x}g)_{∗}m = T_{x}g ∘ m ∘ ((T_{x}g)^{−1} ⊗ (T_{x}g)^{−1}).

Then, for all x = g · o ∈ V^{c} (g ∈ Co(T)) we define

F_{x} := T_{o}g ∘ d_{Θ(g)^{−1}g}(0) ∘ A_{o}^{−1} ∘ (T_{o}g)_{∗}^{−1} : W_{x} → T_{x}(V^{c}),

(where d_{φ} is the denominator of φ, cf. Def. 3.1.2):

W_{x}  ⟶^{F_{x}}  T_{x}(V^{c})
(T_{o}g)_{∗} ↑  ↑ T_{o}g
W_{o}  ⟶^{d_{Θ(g)^{−1}g}(0)∘A_{o}^{−1}}  T_{o}M

Let us show that F_{x} is well-defined: replace g in the above definition by gp with p ∈ P, then

F_{gp·o} = T_{o}(gp) ∘ d_{Θ(gp)^{−1}gp}(0) ∘ A_{o}^{−1} ∘ (T_{o}(gp))_{∗}^{−1}
= T_{o}g ∘ T_{o}p ∘ (T_{o}p)^{−1} ∘ d_{Θ(g)^{−1}g}(0) ∘ d_{Θ(p)^{−1}}(0) ∘ A_{o}^{−1} ∘ (T_{o}p)_{∗}^{−1} ∘ (T_{o}g)_{∗}^{−1}
= F_{g·o}.

In the last line we used that, due to the fact that Θ(P) acts affinely on V ,

d_{Θ(p)^{−1}}(0) ∘ A_{o}^{−1} = (T_{o}(Θ(p)^{−1}))^{−1} ∘ A_{o}^{−1} = Θ(T_{o}p) ∘ A_{o}^{−1} = A_{o}^{−1} ∘ (T_{o}p)_{∗}.

Having shown that F_{x} is well-defined, it es immediately verified that F_{x} = A_{x}^{−1} for x ∈ M: just write x = g · o with g ∈ Co(T)^{Θ} ; one gets that F_{x} = T_{o}g ∘ A_{o}^{−1} ∘ (T_{o}g)_{∗}^{−1} is the natural push-forward of A_{o}^{−1} by g, whence F_{x} = A_{x}^{−1} . That F is an analytic section follows from the claim to be proved next.

(2) Identifying V and W, the definition of F_{g·o} in the proof of (1) reduces to

F_{g·o} := T_{o}g ∘ d_{Θ(g)^{−1}g}(0) ∘ (T_{o}g)_{∗}^{−1}.

For z ∈ V we choose g = t_{z}, and because of the trivialization

F_{z} = d_{Θ(t_{−z})t_{z}}(0).

Using the definition of the denominators in the proof of Th. 3.1.1 and that Θ(t_{v}) = exp(p_{−v}) (Th. 3.5.3 (5)), we calculate (cf. [Be96c, Prop. 1.2.1])

(d_{Θ(t_{−x})t_{y}}(0))v = (t_{y}^{∗}Θ(t_{−x})^{∗}v)(0) = (Θ(t_{−x}^{∗})^{∗}v)(y)

= (exp(p_{x})^{∗}v)(y) = (e^{ad(p_{x})}v)(y)

= (v + [p_{x}, v] + \frac{1}{2}[p_{x}, [p_{x}, v]])(y)

= v − T (v, x, y) − \frac{1}{2}[T (v, x), p_{x}](y)

= v − T (y, x)v + \frac{1}{2}p_{T (x,v,x)}(y)

= (id_{V} −T (y, x) + \frac{1}{4}T (y, T (x, v, x), y) = B(x, y)v,

whence F (z) = B(z, z).

4.4. Global version of the Harish-Chandra imbedding. If n = dim V , consider the G-invariant section

Λ^{n}F : Λ^{n}W → Λ^{n}(T V^{c})

of the line bundle (Λ^{n}W)^{∗} ⊗ Λ^{n}(T V^{c}), where F is as in the preceding theorem. Then F_{x} : W_{x} ↦ T_{x}(V^{c}) is bijective iff (Λ^{n}F)_{x} is non-zero.

Theorem 4.4.1. The G-orbit G · x of a point x ∈ V^{c} is open if and only if

(i)  (Λ^{n}F)_{x} ≠ 0,

and M is the connected component of o of the set

N := {x ∈ V^{c}| (Λ^{n}F)_{x} ≠ 0}.

If x ∈ V , then (i) is equivalent to

(ii)  Det B(x, x) ≠ 0.

Proof. Clearly (ii) is the trivialized version of (i) (Th. 4.3.1 (2)).

Since the section Λ^{n}A^{−1} is non-trivial, analytic and G-invariant, N is an open dense Ginvariant set of V^{c}, and every open G-orbit must belong to this set. We have to prove that conversely every point of this set belongs to an open G-orbit. This is proved as in [Be96c, Th.2.1.1]. We recall briefly the main steps of the proof:

Note first that x ∈ N iff x = g · o with g ∈ Co(V) s.t. Det d_{Θ(g)^{−1}g}(0) ≠ 0 (see the definition of F_{go} in the proof of Th.4.3.1). Given such an element g, let α := Θ(g)^{−1}g. Then α ∈ Co′(V) is an element with Θ(α) = α^{−1} . Replacing g by gp with p ∈ P suitable, we can assume that α ∈ Str(T). Then

g^{−1}(G · x) = Co(V)_{o}^{g_{∗}^{−1}Θg_{∗}} · o = Co(V)_{o}^{Θ(Θ(g)^{−1})_{∗}g_{∗}} · o = Co(V)_{o}^{Θα_{∗}} · o.

But since α is a symmetric element of the structure group, this is just the modified space M^{(α)} which is open in V^{c} by Prop. 4.2.1. Therefore also G · x is open in V^{c}.

If follows that the connected components of N are precisely the open G-orbits, and M is the connected component of o.

Corollary 4.4.2. All open G-orbits are symmetric spaces, and they are modifications of M in the sense of Prop. 4.2.1.

Proof. In the preceding proof we have remarked that G · x ≅ M^{(α)} .

4.5. Global complexifications. As in [Be96b, Section 2.3] it is proved that the space M from Prop. 4.1.1 (2) admits global straight and twisted (para-) complexifications: The conformal completion of the space V_{ℂ} equipped with the ℂ-trilinear triple product T_{ℂ},

(V_{ℂ})^{c} = Co(T_{ℂ})/P_{ℂ},

is a natural complexification of V^{c} (the conjugation is the usual conjugation τ of V_{ℂ} w.r.t V , considered as element of Co(V_{ℂ})). If Θ denotes also the holomorphic extension of Θ to the group of holomorphic conformal maps of V_{ℂ}, then

M_{ℂ} := Co(T_{ℂ})^{Θ} · o ⊂ (V_{ℂ})^{c}

is a global straight complexification of M, and the space

M_{hℂ} := Co(T_{ℂ})^{Θτ_{∗}} · o ⊂ (V_{ℂ})^{c}

is a global twisted complexification of M. Further, as in [Be96c, 2.1. Ex.4] it is seen that M has global imbedding into the space

M_{phℂ} := Co(T)/ Str(T) ⊂ V^{c} × V^{c}

which is its global para-complexification.

4.6. Outlook. The structure theory presented here has been developed with applications to harmonic analysis in mind. The next step should be a geometric treatment of the Peircedecomposition (root theory) and an explicit description of differential operators in terms of the Weyl-algebra of the canonical chart. In the special case of symmetric cones the Jordan-theoretic approach has proved to give more explicit results than the general theory of symmetric spaces (cf. [FK94]), and it is our impression that it is precisely the supplementary information given by a twist which allows to get these explicit results. Therefore one may hope to obtain results of a similar kind for the much bigger class of spaces treated here.

Appendix A: The canonical connection of a symmetric space

A.1. We recall the concept of an affine connection ∇ on the tangent bundle T M of a manifold M (cf. [Hel62]): ∇ is a ℝ-bilinear map

∇ : 𝔛(M) × 𝔛(M) → 𝔛(M), (X, Y) ↦ ∇_{X}Y

(where 𝔛(M) denotes the space of vector fields on M) which is function-linear in the first variable and satisfies the Leibnitz-rule ∇_{X}(fY) = (Xf)Y + f∇_{X}Y (f ∈ C^{∞}(M)) w.r.t. the second variable. We will only be interested in torsionfree connections, i.e. connections satisfying

∀X, Y ∈ 𝔛(M) : ∇_{X}Y − ∇_{Y} X = [X, Y].

A.2. The difference ∇ − ∇′ of two affine connections ∇, ∇′ is a (2, 1)-tensor field (i.e. a section of Hom(T M ⊗T M, T M)), and the difference of two torsionfree connections is a symmetric (2, 1)-tensor field (i.e. a section of Hom(S^{2}T M, T M).

A.3. We identify vector fields on a vector space V ≅ ℝ^{n} canonically with smooth functions V → V . Then the formula

(1)  (∇_{X}Y)(p) := (DY)(p) · X(p)

defines a torsionfree affine connection on V , where DY : V → End(V) is the ordinary total differential. Thus the commutator of two vector fields is given by

(2)  [X, Y](p) = (∇_{X}Y − ∇_{Y} X)(p) = (DY)(p) · X(p) − (DX)(p) · Y (p).

We call ∇ the canonical flat connection of V . Its construction is a special case of

Proposition A.4. Let M = G/H be a symmetric space (in the group theoretical sense). For p ∈ M denote by 𝔤 = 𝔥^{p} ⊕ 𝔮^{p} the associated decomposition of the Lie algebra 𝔤 of G and let

l_{p} : T_{p}M → 𝔮^{p} ⊂ 𝔛(M)

be the inverse of the bijective evaluation map ev_{p} : 𝔮^{p} → T_{p}M, X ↦ X_{p}. Then the formula

(∇_{X}Y)_{p} := [l_{p}(X_{p}), Y]_{p}, (X, Y ∈ 𝔛(M))

defines a torsionfree connection on M. Proof. The defining relations of a connection are immediately verified. In order to prove that the torsion tensor T or(X, Y) = ∇_{X}Y − ∇_{Y} X − [X, Y] vanishes at an arbitrary point p ∈ M, we use the tensor nature of T or: w.l.o.g. we may assume that X = l_{p}X_{p}, Y = l_{p}Y_{p}. Inserting this in the formulas for T or and ∇, we get T or_{p} = 0 since ev_{p}([𝔮^{p}, 𝔮^{p}]) ⊂ ev_{p}(𝔥^{p}) = 0.

A.5. We call the connection ∇ just defined the canonical connection of M = G/H. One can prove that

(R(X, Y)Z)_{p} = [[l_{p}X_{p}, l_{p}Y_{p}], l_{p}Z_{p}]_{p},

R(X, Y)Z = (∇_{[X,Y]} − ([∇_{X}, ∇_{Y}])Z is the negative of the curvature tensor (we have put in the change of sign in order to have this equality).

A.6. We denote by g_{∗}X the natural forward transport of a vector field X by a diffeomorphism g. Then, for any connection ∇, the formula

(3)  (g · ∇)_{X}Y := g_{∗}(∇_{g_{∗}^{−1}X}g_{∗}^{−1}Y)

defines a connection g · ∇; if g · ∇ = ∇, then g is called affine w.r.t. ∇. It is not hard to verify that G is a group of affine transformations of the connection ∇ defined in Prop. A.4.

A.7. As an infinitesimal version of (3), the formula

(4)  (Z · ∇)_{X}Y := [Z, ∇_{X}Y] − ∇_{[Z,X]}Y − ∇_{X}[Z, Y]

defines a tensor field of type (2,1) (symmetric if ∇ is torsionfree), and Z is called an affine vector field w.r.t. ∇ if Z · ∇ = 0. It is easily seen that, in the situation of Prop. A.4, 𝔤 is a Lie algebra of affine vector fields.

A.8. In the situation of Prop. A.4, using the action of the Lie algebra 𝔛(M) on the tensor algebra by the usual Lie derivative, we can describe the covariant derivative ∇_{X}S of a tensor field S by the formula

(5)  (∇_{X}S)_{p} = ((l_{p}X_{p}) · S)_{p}.

In particular, if 𝔤 · S = 0, then ∇S = 0, i.e. G-invariant tensor fields are covariantly constant. For instance, if ∇ is as in A.4, then G_{o} · ∇ = ∇; therefore G_{o} · R = R, 𝔤 · R = 0 and hence ∇R = 0.

A.9. Let ∇ be the flat connection of a vector space V (cf. A.3). Then, if g is a (local) diffeomorphism of V and X a vector field,

(6) (g_{∗}X)(p) = (Dg^{−1}(p))^{−1} · X(g^{−1}(p)); (g^{∗}X)(p) := ((g^{−1})^{∗}X)(p) = (Dg(p))^{−1} · X(g(p)).

and, for all u, v ∈ V ,

(7)  (g^{−1} · ∇ − ∇)_{p}(u ⊗ v) = (Dg(p))^{−1} · (D^{2}g)(p)(u ⊗ v),

where D^{2}g : V → Hom(S^{2}V, V) is the ordinary second differential. (In order to verify this formula, note that g^{−1} · ∇ − ∇ is a tensor field of type (2,1); hence it suffices to determine its values on the constant vector fields.)

Appendix B: Integrability of almost complex structures

B.1. Recall that an almost complex structure (resp. polarization) is a tensor field J of type (1,1) such that J_{p}^{2} = −1_{p} (resp. J_{p}^{2} = 1_{p}) for all p ∈ M. It is called integrable if its torsion tensor

N(X, Y) := [J X, J Y] + J^{2}[X, Y] − J [X, J Y] − J [J X, Y]

vanishes.

Complexifications of Symmetric Spaces and Jordan theory Part II: Integrability and Conformal Group 31

Proposition B.2. Let J be a G-invariant almost complex structure or polarization on a symmetric space M = G/H. Then J is integrable.

Proof. We use the canonical connection ∇ of M defined in Prop. A.4. Since [X, Y] = ∇_{X}Y − ∇_{Y} X for all X, Y ∈ 𝔛(M), we get

N(X, Y) = ∇_{J X} J Y −∇_{J Y} J X+J^{2}(∇_{X}Y −∇_{Y} X)−J (∇_{X} J Y −∇_{J Y} X−∇_{J X}Y +∇_{Y} J X).

As remarked above (A.8), G · J = J implies ∇ J = 0; this means that ∇_{Z} J S = J ∇_{Z}S for all Z, S ∈ 𝔛(M). Using this property, all terms in the above expression of N(X, Y) cancel out.

B.3. We consider the invariance group G(J) of J ; this is the group of diffeomorphisms g of M such that for all Y ∈ 𝔛(M), g_{∗}(J Y) = J g_{∗}Y holds. We call G(J) also the group of almost (para-) holomorphic transformations of (M, J). The corresponding infinitesimal object is the Lie algebra

𝔤(J) := {X ∈ 𝔛(M)| ∀Y ∈ 𝔛(M) : [X, J Y] = J [X, Y]}.

We remark that for all X, Y ∈ 𝔤(J),

N(X, Y) = [J X, J Y] − J^{2}[X, Y],

and therefore, if J is integrable, then we have for all X, Y ∈ 𝔤(J),

(1)  [J X, J Y] = J^{2}[X, Y].

Proposition B.4. Let J be an integrable almost complex structure or a polarization on a manifold M and assume that 𝔤(J) contains for any point p ∈ M a local basis of 𝔛(M) around p. Then 𝔤(J) is stable under the map X ↦ J X. In particular, if J^{2} = − id_{𝔛(M)} , then 𝔤(J) is a complex Lie algebra with complex structure J .

Proof. Let X ∈ 𝔤(J) and Y ∈ 𝔛(M). On a neighbourhood of a point p ∈ M we can write Y = ∑_{i=1}^{m} f_{i}Y_{i} with smooth functions f_{i} and Y_{i} ∈ 𝔤(J). Without loss of generality we may assume that m = 1 and write Y = fỸ. Then using (1), locally,

[J X, J Y] = [J X, f J Ỹ] = df(J X) · J Ỹ + f[J X, J Ỹ]
= J (df(J X) · Ỹ + [J X, Ỹ])
= J [J X, Y].

Using a partition of unity, we get the same relation for general Y ∈ 𝔛(M), whence J X ∈ 𝔤(J), proving the claim. Now (i) is an immediate consequence.

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