Journal of Lie Theory Volume 11 (2001) 415–426 C ◯2001 Heldermann Verlag

Characterization of the Kantor–Koecher–Tits Algebra by a Generalized Ahlfors Operator

Wolfgang Bertram and Joachim Hilgert

Communicated by M. Cowling Dedicated to Jacques Faraut on his 60th birthday

Abstract. In the context of certain generalized conformal structures we define a first order differential operator S generalizing the classical Ahlfors operator. We prove its invariance under the corresponding conformal group and show that, under certain conditions, the Lie algebra of this group (which is also known as the “Kantor–Koecher–Tits algebra”) is precisely the space of solutions of the differential equation SX = 0.

0. Introduction

In his investigations on quasiconformal deformations, L. Ahlfors introduced the operator

SX = \frac{1}{2}(DX + (DX)^{t}) − \frac{1}{n} tr(DX)  (0.1)

(see [Ah74, Ah76]); it is a first order differential operator assigning to a vector field (“deformation”) X on ℝ^{n} a field of trace-free symmetric endomorphisms. He proved that this operator is invariant under the group of conformal transformations (which is the group generated by translations, similarities and the inversion) and that the solutions of the differential equation SX = 0 (“trivial deformations”) are precisely the vector fields of the form

X(x) = a + Bx + 2⟨c, x⟩x − ⟨x, x⟩c  (0.2)

with a, c ∈ ℝ^{n} and B a sum of a skew-symmetric matrix and a multiple of the identity. This space of quadratic vector fields is nothing but the Lie algebra of the conformal group.

The Ahlfors operator S has a natural generalization in Riemannian geometry (see, for instance, [Bra97, PØ96]). Let (M, g) be a Riemannian manifold and

∇ : Γ^{∞}(T M) → Γ^{∞}(End(T M)), X ↦ ∇X  (0.3)

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag be its associated Levi-Civita connection; it assigns to a vector field X the field of endomorphisms ∇X defined by (∇X)Y = ∇_{Y} X. Composing ∇ with a projection onto some subbundle of End(T M), we obtain what is nowadays often called a generalized gradient (usually in the more general context of connections on arbitrary bundles). In the case of a Riemannian manifold, the bundle End(T M) canonically decomposes into the sum of three subbundles, which are given in each tangent space T_{x}M respectively by skew-symmetric operators, multiples of the identity and symmetric trace-free operators (with respect to the metric g_{x}). Let p_{3} : End(T M) → End(T M) denote the projection onto the last subbundle. Then the Ahlfors operator is the generalized gradient S = p_{3} ∘ ∇.

In this paper we introduce a generalized gradient S on certain manifolds M having a “generalized conformal structure”. As in the classical case, the corresponding conformal group is much “bigger” than the automorphism group of the connection ∇ we use (Theorem 2.1). The operator S shares an essential feature with the classical Ahlfors operator. Under certain conditions, the equation SX = 0 characterizes the Lie algebra of the conformal group (Theorem 2.5). This Lie algebra is known as the Kantor–Koecher–Tits algebra in the context of Jordan algebras and, more generally, of Jordan triple systems. Our approach is placed in this context. The basic ingredients of this approach are reviewed in Section 1; let us just mention here that the manifolds M we are interested in are symmetric spaces and ∇ is the canonical connection of such a space (see [KN69] or [Lo69] for the general theory). Jordan theory comes in by defining the subbundle of End(T M) onto which we project. This subbundle is an additional structure on the symmetric space which cannot be recovered from the usual theory of symmetric spaces; it is an aspect of what one may call a “generalized conformal structure on M ” —in [Be01] and [Be00] the more neutral but not entirely satisfactory term “twist” is used; following a suggestion of the referee, we will say that M is a Jordan symmetric space. Most surprisingly, all classical and many of the exceptional symmetric spaces do admit such an additional structure and, as classification shows, it is in most cases unique. Thus there are many examples of the context in which we define our generalized Ahlfors operators; we mention some of them in Section 1.

The investigation of conformally invariant operators is a vast topic, see, for example, [CSS98] and [KR00]. In contrast to these approaches, our proof of conformal invariance is purely geometric and does not need any assumptions of irreducibility and the corresponding weight theory, thus giving a partial answer to a problem mentioned in the introduction of [KR00]. The fact that our operator is defined in geometric terms and not via representation theory also allows to give an algebraic formula for our operator (formula (2.7)) which is very similar to the classical formula (0.1).

Acknowledgement: The second author would like express his gratitude to the Institut Elie Cartan in Nancy, where part of this work was done, for its hospitality.

1. Preliminaries

1.1. Symmetric spaces. A symmetric space is a manifold M together with a complete torsion free affine connection ∇ such that the curvature R is covariantly constant: ∇R = 0. If M is connected, then M can be written as a homogeneous symmetric space: M = G/H is homogeneous under the action of a Lie group G such that H is an open subgroup of the group G^{σ} of fixed points of some nontrivial involution σ of G (see [Lo69] or [KN69]). Let 𝔤 = 𝔥_{p} ⊕ 𝔮_{p} be the decomposition of the Lie algebra 𝔤 of G (assumed to act effectively on M) under the differential of the involution σ_{p} corresponding to the point p ∈ M. In this group theoretic set-up, the affine connection can then be recovered by the formula

(∇_{X}Y)_{p} = [L(X_{p}), Y]_{p}  (p ∈ M)  (1.1)

where for a tangent vector v ∈ T_{p}M, L(v) denotes the unique vector field Z ∈ 𝔮_{p} such that Z_{p} = v (see [Be01, Section I.2]).

1.2. The Jordan tensor. We assume that (M, ∇, T) is a Jordan symmetric space: (M, ∇) is a symmetric space and T is a Jordan extension of the curvature R, that is, a covariantly constant tensor field of the same type as R (that is, of type (3, 1)) such that for all u, v, w, x, y ∈ T_{p}M, p ∈ M, the following identities hold

(JT1) (symmetry) T_{p}(u, v, w) = T_{p}(w, v, u), (JT2) [T_{p}(u, v), T_{p}(x, y)] = T_{p}(T_{p}(u, v)x, y) − T_{p}(x, T_{p}(v, u)y),

where T_{p}(a, b)c := T_{p}(a, b, c),

and such that T is related to R by

R(X, Y) = −(T (X, Y) − T (Y, X)).  (1.2)

The identities (JT1) and (JT2) mean that, for each p ∈ M, T_{p} is a Jordan triple system on the tangent space T_{p}M. It follows from the identity (JT2) that the subspace of End(T_{p}M) spanned by the operators T_{p}(a, b), a, b ∈ T_{p}M, is a Lie algebra, called the (inner) structure algebra of T_{p} and denoted by 𝔰𝔱𝔯(T_{p}). In most parts of the following text, the Jordan triple system T_{p} will be assumed to be nondegenerate in the sense that the trace form

β_{p}(u, v) := tr T_{p}(u, v),  (1.3)

is nondegenerate. Then (see [Be01, Lemma V.2.4]) β_{p} is symmetric—and thus β is a pseudometric tensor field on M —and it is associative in the sense that the transpose of T_{p}(u, v) with respect to β_{p} is given by

T_{p}(u, v)^{⊤} = T_{p}(v, u).  (1.4)

1.3. Construction of Jordan symmetric spaces. The nondegenerate Jordan symmetric spaces are all obtained by the following construction (which is described in detail in [Be01]). One starts with a nondegenerate Jordan triple system T_{0} on a finite-dimensional vector space V . With T_{0} one associates a Kantor–Koecher–Tits algebra 𝔠𝔬(T) = 𝔠𝔬(T_{0}), that is the Lie algebra of quadratic vector fields of the form

ξ(x) = v + Hx + P (x)w,  (1.5)

for v, w ∈ V and H ∈ 𝔰𝔱𝔯(T_{0}), where

P (x)w = \frac{1}{2}T_{0}(x, w, x).

We will use the notation v for the constant vector field having value v on V and A(w) for the homogeneous quadratic vector field given by

(A(w))(x) = P (x)w  (1.6)

so that ξ = v + H + A(w). Any local diffeomorphism φ of V preserving this Lie algebra (in the sense that the push-forward φ^{∗}ξ of every element ξ ∈ 𝔠𝔬(T) coincides on its domain of definition with an element of 𝔠𝔬(T)) is in fact given by a birational formula, and hence extends to a birational map of V . The birational maps so obtained form a group Co(T) = Co(T_{0}), called the conformal group or Kantor–Koecher–Tits group. It is isomorphic to the adjoint group of 𝔠𝔬(T), hence is a Lie group. Moreover, the adjoint representation is faithful, and therefore 𝔠𝔬(T) is the Lie algebra of Co(T). The group of translations τ_{v}, v ∈ V , is a subgroup of Co(T), as well as the structure group, which is the linear group Str(T) := Str(T_{0}) defined by

Str(T_{0}) := {g ∈ GL(V)| (∀u, v, w ∈ V) gT_{0}(u, v, w) = T_{0}(gu, (g^{⊤})^{−1}v, gw)}.

Let Q be the subgroup of Co(T) of all elements φ ∈ Co(T) which are defined at 0 and satisfy φ(0) = 0. It is known that Co(T) is semisimple and that Q is a maximal parabolic subgroup of Co(T) (see [Lo71]). The map

V → V^{c} := Co(T)/Q, v ↦ τ_{v}Q  (1.7)

is an imbedding with open dense image, called the conformal compactification of V .

The Kantor–Koecher–Tits algebra carries an involution Θ interchanging constant and homogeneous quadratic vector fields. It is given by the formulas

Θ(v) = −A(v), Θ(T_{0}(u, v)) = −T_{0}(v, u), Θ(H) = −H^{⊤} (H ∈ 𝔰𝔱𝔯(T_{0})).

(1.8)

This involution induces an involution Θ on the adjoint group of 𝔠𝔬(T) and hence on the conformal group Co(T). Let G = Co(T)_{o}^{Θ} be the identity component of the fixed point group of this involution. Then

M = G.0 ⊂ V^{c}  (1.9)

is an open orbit carrying the structure of a homogeneous symmetric space G/H. Here H is the automorphism group of T_{0} , and therefore T_{0} extends to a Ginvariant tensor field T on M which is indeed a Jordan extension of the curvature tensor. The space M has a natural chart on a neighborhood of the base point 0, namely the “vectorialization” V ∩ M; we call the associated coordinates Jordan coordinates. In Jordan coordinates, the geodesic symmetry with respect to the base point is simply given by − id_{V} , and the extension L(v) at the origin is given by

L(v) = v − A(v), v ∈ V = T_{0}M,  (1.10)

see [Be01, Theorem VII.2.4].

1.4. Jordan algebras and spaces of the first kind. To a unital Jordan algebra (V, ·) one associates the Jordan triple system T_{0} given by

T_{0}(u, v, w) = 2(u(vw) − v(uw) + (uv)w).  (1.11)

If e ∈ V is the unit element, then this Jordan triple system has the special property that P (e) = id_{V} . More generally, we say that a Jordan triple system T_{0} is of the first kind if it has invertible elements, that is, elements x ∈ V such that P (x) is invertible. All such Jordan triple systems are of the form T_{0}^{(α)} , where T_{0} is associated with the Jordan algebra as above, α ∈ GL(V) satisfies

αT_{0}(u, αv, w) = T (αu, v, αw)  (1.12)

for all u, v, w ∈ V , and

T_{0}^{(α)}(u, v, w) = T_{0}(u, αv, w)  (1.13)

is the “α-modification of T_{0} ” (see [Be01, Section XI.1]).

1.5. Examples. Classification shows that all classical and many exceptional symmetric spaces admit a (generically unique) Jordan extension of the curvature. The corresponding list is therefore very long (see [Be01]). Here we just mention some particularly interesting examples (all of them, with the exception of GL(n, ℝ), are Riemannian); for the geometric realization of these data we refer to [Be01].

(1) Classical conformal space: M = S^{n} = SO(n + 1)/ SO(n), conformal group: SO(n+1, 1), structure group: SO(n)×ℝ^{∗}, Jordan triple system: ℝ^{n} with T_{0}(u, v, w) = ⟨u, v⟩w + ⟨w, v⟩u − ⟨u, w⟩v (thus the conformal Lie algebra is indeed given by (0.2)).

(2) Hyperbolic spaces: M = H^{n} = SO(n, 1)/ SO(n): conformal group and structure group as in (1), Jordan triple system: same space as in (1) with the negative of the product given there.

(3) Grassmannians: M = O(p+q)/(O(p)×O(q)), conformal group: PGL(p+ q, ℝ), structure group P(GL(p, ℝ)×GL(q, ℝ)), Jordan triple system V = M(p, q; ℝ) (real p × q matrices) with T_{0}(X, Y, Z) = −(XY^{t}Z + ZY^{t}X). Here the P means that one considers the respective projective group where the multiples of the identity are factored out. The cases p = 1 (projective space) and p = q (see (4)) play a somewhat special role. For ℝ replaced by ℂ or ℍ there are similar formulae.

(4) Group case M = GL(n, ℝ), conformal group: P(GL(2n, ℝ)), structure group: P(GL(n, ℝ) × GL(n, ℝ)), Jordan triple system V = M(n, n, ℝ) with

T_{0}(X, Y, Z) = XY Z + ZY X.

(5) Group case M = U(n), conformal group SU(n, n), structure group GL(n, ℂ)/S^{1} , Jordan triple system V = Herm(n, ℂ) (Hermitian matrices) with

T_{0}(X, Y, Z) = −(XY Z + ZY X).

(6) Lagrangian Grassmannian M = U(n)/O(n), conformal group PO(n, n), structure group GL(n, ℝ), Jordan triple system Sym(n, ℝ) (symmetric matrices) with T_{0}(X, Y, Z) = −(XY Z + ZY X).

(7) Symmetric cones (see [FK94]): M isomorphic to the symmetric cone Ω = G/K associated with a Euclidean Jordan algebra V , G an open subgroup of the structure group, conformal group isomorphic to the automorphism group of the tube over Ω, Jordan triple system associated with V via formula (1.11) (notice that we use here the bounded realization M = (−e + Ω) ∩ (e − Ω) of Ω, see [Be01]).

(8) Bounded symmetric domains (see [Lo77]): M = G/K is a Hermitian symmetric space in its Harish-Chandra realization in 𝔭_{ℂ} ≅ ℂ^{n}, conformal group G_{ℂ}, structure group K_{ℂ}, Jordan triple system on 𝔭_{ℂ} as defined in [Lo77].

The spaces of the first kind among the above examples are precisely (1), (2), (3) p = q, (4), (5), (6), (7), and the spaces of tube type from (8).

2. The generalized Ahlfors operator

Keeping the notation of the preceding section, we now decompose the bundle End(T V^{c}) = (T V^{c})^{∗} ⊗ (T V^{c}) into three Co(T)-invariant subbundles. Observe first that the space End(V) decomposes as

End(V) = {X ∈ 𝔰𝔱𝔯(T)| tr(X) = 0} ⊕ ℝ id_{V} ⊕ 𝔰𝔱𝔯(T)^{⊥},  (2.1)

where the orthocomplement is taken with respect to the usual trace form. Clearly this decomposition is invariant under Str(T), and therefore the three pieces can be transported to all tangent spaces of V^{c} in a well-defined way, giving rise to a Co(T)-invariant decomposition of End(T V^{c}). We denote by p_{i} : End(T V^{c}) → End(T V^{c}), i = 1, 2, 3, the corresponding projections. (Here, p_{2} is essentially the trace.)

The generalized Ahlfors operator is the map S assigning to a vector field X the section of End(T V^{c}) given by

SX := p_{3}(∇X).  (2.2)

Theorem 2.1. The Ahlfors operator S : Γ^{∞}(T M) → Γ^{∞}(End(T M)) extends to a conformally invariant differential operator S : Γ^{∞}(T V^{c}) → Γ^{∞}(End(T V^{c})).

Proof. Let g ∈ Co(T). We denote by g_{∗} the push-forward of tensor fields and differential operators and by g^{∗} = (g^{−1})_{∗} the canonical pull-back. We have to show that g_{∗}S and S coincide on the open dense set g(M) ∩ M ⊂ V^{c}. Clearly, since the projections p_{i} are Co(T)-invariant, we have on this set

g_{∗}S − S = g_{∗}(p_{3} ∘ ∇) − p_{3} ∘ ∇ = p_{3} ∘ (g_{∗}∇ − ∇).

The condition p_{3} ∘ (g_{∗}∇ − ∇) = 0 means that, for all p ∈ M ∩ g(M) and vector fields X, (g_{∗}∇ − ∇)X_{p} ∈ 𝔰𝔱𝔯(T_{p}). To prove this, note first that the difference of two connections is a tensor field of type (2, 1), i.e. an algebra structure on each tangent space. If both connections are torsion free this algebra structure is commutative. The important fact, to be proved next, is that for all connections involved in our context the differences are fields of Jordan algebras which are derived from our Jordan triple system T_{0} by fixing the middle argument. To make this precise, we introduce the following terminology.

Definition 2.2.

(i) For a Jordan triple system T_{0} on a vector space V , let

W_{0} := {T_{0}(·, v, ·)| v ∈ V } ⊂ Hom(V ⊗ V, V)

be the space of algebra structures obtained by fixing the middle argument of T_{0} . (If T_{0} is nondegenerate so that β_{0} induces an isomorphism V → V^{∗}, v ↦ v^{∗}, it follows immediately from the definitions that the map V^{∗} → W_{0} , v^{∗} ↦ T_{0}(·, v, ·) is an isomorphism of vector spaces with inverse W_{0} → V^{∗}, c ↦ (x ↦ tr c(x, ·).) The space W_{0} is Str(T)- invariant and hence gives rise to a Co(T)-invariant subbundle W of Hom(T V^{c} ⊗ T V^{c}, T V^{c}), called the structure bundle.

(ii) We say that two affine connections ∇^{1} and ∇^{2} on the open set U ⊂ V^{c} are conformally equivalent if their difference C := ∇^{2} − ∇^{1} is a section of the structure bundle W over U.

Note that, if c = T_{0}(·, v, ·) ∈ W_{0} , then c(w, ·) = T_{0}(w, v) ∈ 𝔰𝔱𝔯(T_{0}); therefore, if ∇^{1} and ∇^{2} are conformally equivalent, then ((∇^{1} − ∇^{2})X)_{p} ∈ 𝔰𝔱𝔯(T_{p}) for all vector fields X and points p where things are defined. Thus the theorem will be proved if we can show that g_{∗}∇ and ∇ are conformally equivalent. More generally, we will prove

Theorem 2.3. Let ∇^{0} be the canonical flat connection on the vector part V ⊂ V^{c}. Then

(i) ∇^{0} and ∇ are conformally equivalent. More precisely, for C := ∇ − ∇^{0} we have the formula

C_{x}(u, v) = T_{0}(u, (id −P (x))^{−1}x, v), u, v ∈ V, x ∈ M ∩ V.

(ii) For any g ∈ Co(T), the connections ∇, g_{∗}∇ and g_{∗}∇^{0} are all conformally equivalent.

Proof. (i) We show first that C_{0} = 0: in fact, for u ∈ V = T_{0}M, taking account of Equation (1.10), we get from Equation (1.1) (∇_{u}v)_{0} = [L(u), v]_{0} = [u − A(u), v]_{0} = 0.

Suppose that φ: M → M is an affine diffeomorphism with respect to ∇ such that x := φ(0) ∈ V . We show that

C_{x}(u, v) = ((D^{2}φ)(0))(Dφ(0)^{−1}u, Dφ(0)^{−1}v).  (2.3)

In fact, since φ^{∗}∇ = ∇, we have

φ^{∗}C − C = (φ^{∗}∇ − ∇) − (φ^{∗}∇^{0} − ∇^{0}) = (∇^{0} − φ^{∗}∇^{0}).

An elementary computation shows that the effect of φ on the canonical flat connection ∇^{0} of a vector space is given by

(φ^{∗}∇^{0} − ∇^{0})_{x}(u, v) = (Dφ(x))^{−1} · ((D^{2}φ)(x))(u ⊗ v)

(see [Be01, Appendix I.B]). Therefore

(φ^{∗}C − C)_{0}(u, v) = −(Dφ(0))^{−1} · ((D^{2}φ)(0))(u ⊗ v).

On the other hand, since C_{0} = 0, the definition of φ^{∗} gives

(φ^{∗}C − C)_{0}(u, v) = (Dφ(0))^{−1}C_{φ(0)}(Dφ(0)u, Dφ(0)v).

Comparison of the last two equations yields (2.3). Next we have to calculate the second differential D^{2}φ(0). We claim that

((Dφ)(0))^{−1} · D^{2}(0)(φ)(u ⊗ v) = T_{0}(u, φ^{−1}(0), v).  (2.4)

In fact, we may decompose φ = vgn with g = Dφ(0), v = τ_{φ(0)} and n = Θ(τ_{φ^{−1}(0)}) = exp(−A(φ^{−1}(0))) (see [Be01, Theorem VIII.2.3(5)]). Then D^{2}φ(0) = g ∘ D^{2}n(0) and

D^{2}n(0) = −D^{2}(A(−φ^{−1}(0)))(0) = T_{0}(·, φ^{−1}(0), ·)

(for the last equality see [Be01, Prop. VIII.2.2 and its proof]). After multiplication by g^{−1} this gives (2.4). Putting (2.3) and (2.4) together, we have

\begin{aligned}C_{φ(0)}(u, v) & = Dφ(0)T_{0}((Dφ(0))^{−1}u, φ^{−1}(0), (Dφ(0))^{−1}v) \\ & = T_{0}(u, Θ(Dφ(0))φ^{−1}(0), v), \end{aligned}  (2.5)

where the last equality is due to the fact that Dφ(0) belongs to the structure group of T. Finally, we claim that, with x = φ(0) in some neighborhood of the origin, the equality

Θ(Dφ(0))φ^{−1}(0) = (id −P (x))^{−1}x  (2.6)

holds. In fact, we may choose φ = exp(v − A(v)); then x = Exp(v) is given by x = tanh(v) in the sense of [Be01, Section X.4]. Here Exp denotes the exponential map of the symmetric space. On the other hand, Θ(Dφ(0)) = (Dφ^{−1}(0))^{−1} (see [Be01, VIII.2.3 (5)]. Note that φ^{−1} = (− id) ∘ φ ∘ (− id), and thus Dφ^{−1} = Dφ. Now [Be01, Prop. X.4.2] states that

Θ(Dφ(0))φ^{−1}(0) = −(Dφ^{−1}(0))^{−1}φ(0) = cosh^{2}(v) tanh(v).

Define the power series tanh^{2}(v) to be ∑ a_{k}P (v)^{k}, where ∑ a_{k}z^{2k} is the usual power series of tanh^{2}(z). By power associativity, the operator tanh^{2}(v) coincides with P (tanh v) on the Jordan span of v. Then the identity cosh^{2} z = (1 − tanh^{2} z)^{−1} for complex z yields (again by power associativity) the identity

cosh^{2}(v)w = (id − tanh^{2}(v))^{−1}w

for all w in the Jordan span of v, and in particular for w = tanh(v) = x. Thus we get

cosh^{2}(v) tanh(v) = (id −P (tanh v))^{−1} tanh(v).

Together with the preceding equation, this yields (2.6). Finally, the claim follows by combining (2.5) and (2.6).

(ii) Note first that conformal equivalence is indeed an equivalence relation (assuming that all objects are defined on Zariski-dense subsets). Now, it is proved in [Be01, Theorem VIII.1.11] that g_{∗}∇^{0} and ∇^{0} are conformally equivalent. By (i), ∇^{0} and ∇ are conformally equivalent. These two facts together imply that

g_{∗}∇ − ∇ = g_{∗}(∇^{0} + C) − (∇^{0} + C) = g_{∗}∇^{0} − ∇^{0} + g_{∗}C − C

is again a section of the structure bundle, whence the claim.

Corollary 2.4. In Jordan coordinates, the Ahlfors operator is the constant operator (p_{3})_{0} ∘ D obtained by composing (p_{3})_{0} : End(V) → 𝔰𝔱𝔯(T_{0})^{⊥} with the ordinary first differential, i.e.

(SX)(p) = (p_{3})_{0}(DX)(p).

Proof. Since p_{3} is invariant under the conformal group, and thus in particular under translations, it is represented in the chart V by the constant map (p_{3})_{0} . Next, SX = p_{3}∇(X) = p_{3}(∇^{0} + C)(X) = p_{3}∇^{0}X since p_{3}C(X) = 0 by Theorem 2.3 (i). Now the claim follows since ∇^{0} is nothing but the ordinary first differential: (∇_{Y}^{0} X)(p) = DX(p) · Y (p).

More generally, the preceding argument shows that the Ahlfors operators which one can associate to any two conformally equivalent connections are the same.

Theorem 2.5. Let M be the Jordan symmetric space associated with a Jordan triple system T of the first kind obtained from a simple Jordan algebra which is not isomorphic to ℝ or ℂ (or a direct sum of such Jordan triple systems), and let S = p_{3} ∘ ∇ be the corresponding Ahlfors operator. Then a (possibly only locally defined) vector field X is conformal (i.e., it is the restriction of an element of the conformal Lie algebra 𝔠𝔬(T)) if and only if SX = 0.

Proof. Let X be defined on the open set U ⊂ V^{c}. By possibly translating and shrinking U, we may assume that U ⊂ V . By definition of S, the condition (SX)_{p} = 0 for all p ∈ U is equivalent to

(∇X)_{p} ∈ 𝔰𝔱𝔯(T_{p})

for all p ∈ U. By the preceding corollary, this amounts to

DX(p) ∈ 𝔰𝔱𝔯(T_{0})

for all p ∈ U. This is the condition of 𝔰𝔱𝔯(T_{0})-conformality from [Be96], and according to [Be96, Theorem 1.3.2] (see also [Be00] and [Be01, Chapter IX]) it describes precisely the conformal Lie algebra. (Note that 𝔰𝔱𝔯(T) = 𝔰𝔱𝔯(T^{(α)}) for all α-modifications of T, allowing us to assume without loss of generality that T is already the Jordan triple system associated with a Jordan algebra.)

Remark 2.6. The proof of Theorem 2.3 shows that the condition SX = 0 is equivalent to 𝔰𝔱𝔯(T_{0})-conformality in the sense of [Be96]. Thus the conclusion of the theorem holds whenever the condition Hom_{s}(V, 𝔰𝔱𝔯(T_{0})) = 0 from [Be96] is verified. In the literature one can find the remark (see, for example, [GK98, Go87]) that this condition holds not only in the Jordan algebra case but more generally for a simple Jordan triple system which is not isomorphic to a modification of the projective Jordan triple system M(1, n; ℝ). For the latter case it certainly does not hold since then 𝔰𝔱𝔯(T_{0}) is the full general Lie algebra, and hence our Ahlfors operator is zero.

An explicit formula for the Ahlfors operator. Finally, by means of the Jordan tensor T and by using some results of K. Meyberg on traces in Jordan triple systems, we will give an “explicit formula” for our Ahlfors operator in the spirit of formula (0.1). To this end, given a nondegenerate Jordan triple system T_{0} on V , let

T̃_{0} : V ⊗ V^{∗} → End(V), v ⊗ w^{∗} ↦ T_{0}(v, w)

(with w^{∗} defined using the trace form). Using the isomorphism V ⊗ V^{∗} → End(V), v ⊗ φ ↦ (x ↦ φ(x)v), we may and will consider T̃_{0} as a linear map T̃_{0} ∈ End(End(V)). The important point is that this map is Str(T)-equivariant. Using this, Schur’s lemma and some trace calculations, one gets the following result (see [Mey84,Mey93]).

Proposition 2.7. Let End(V) = E_{1} ⊕ E_{2} ⊕ E_{3} be the decomposition given by Equation (2.1). Then T̃_{0} preserves the subspaces E_{i}, i = 1, 2, 3. Moreover,

(i) the restriction of T̃_{0} to E_{3} is zero, (ii) T̃_{0}(id_{V}) = id_{V} , (iii) tr ∘T̃_{0} = tr, (iv) if the derived algebra 𝔰𝔱𝔯(T_{0})′ is simple, then the restriction of T̃_{0} to E_{1}

acts by the scalar (dim 𝔰𝔱𝔯(T_{0}) − 1)/(dim V − 1).

Since we have always (p_{1})_{0}A = A − n^{−1} tr(A) id_{V} , we deduce that under the assumption of (iv), (p_{3})_{0} is given by

(p_{3})_{0}(A) = A − \frac{s − 1}{n − 1}T̃_{0}(A) + \frac{s − n}{n(n − 1)} tr(A) id_{V} ,

where n = dim V , s = dim 𝔰𝔱𝔯(T_{0}). An explicit formula for the Ahlfors operator in Jordan coordinates is now deduced from Corollary 2.4 by letting A = DX

SX = DX − \frac{s − 1}{n − 1}T̃_{0}(DX) + \frac{s − n}{n(n − 1)} tr(DX) id_{V} .  (2.7)

We conclude by remarking that the assumption of (iv) is satisfied for all simple Jordan triple systems T except for the case V = M(p, q; 𝔽) (p, q > 1), for which 𝔰𝔱𝔯(T)′ is a direct product 𝔰𝔩(p, 𝔽)×𝔰𝔩(𝔮, 𝔽) (𝔽 = ℝ, ℂ, ℍ). The conclusion of (iv) still holds if p = q but is no longer true if p ≠ q.

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Wolfgang Bertram Institut Elie Cartan Université Nancy I B.P. 239 F-54506 Vandœuvre lès Nancy, Cedex France email: bertram@iecn.u-nancy.fr Joachim Hilgert Institut für Mathematik TU Clausthal Erzstr. 1 D-38678 Clausthal-Zellerfeld Germany email: hilgert@math.tu-clausthal.de

Received April 25, 2000 and in final form December 4, 2000