20th September 1999

The Poisson-Furstenberg kernel of a bounded symmetric domain

Fabrice VANDEBROUCK

Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7 rue René Descartes, F-67084 Strasbourg Cedex, France. E-mail : vandebro@irma.u-strasbg.fr

1 Bounded symmetric domains and Jordan triple systems

[12] [7] [14] Let 𝔻 be a circled bounded homogeneous domain in a finite dimensional complex vector space 𝕍 = ℂ^{n}. Assume that 𝔻 is irreducible (i.e. not the direct product of two such domains). The classification of these domains is well known [1]: They fall into four infinite series: Type I_{p,q} (1 ≤ p ≤ q): (generalized) unit ball in the vector space 𝕍 of complex p × q-matrices. The domain 𝔻 = \{Z ∈ 𝕍, Id_{p} − Z^{t}\overline{Z} >> 0\} where X >> 0 means that X is a positive definite matrix; Type II_{n} (n ≥ 2): unit ball of alternating n × n matrices; Type III_{n} (n ≥ 1): unit ball of symmetric n × n matrices; Type IV_{n} (n ≠ 2): ”Lie ball” in ℂ^{n}; and two exceptional types: Type V: dimension 16; Type V I: dimension 27. (These domains are described in detail in section 13) We denote by G the identity component of the Lie group of holomorphic automorphisms of 𝔻. Then G is semi-simple and acts transitively on 𝔻. Let K be the isotropy group of the origin in G. Then K is a maximal compact subgroup of G and 𝔻 = G/K is an hermitian symmetric space of noncompact type. The Jordan triple system structure associated to 𝔻 is most naturally constructed using holomorphic vector fields on 𝔻. Every holomorphic function ξ : z → ξ(z) from 𝔻 to ℂ^{n} gives rise to a holomorphic vector field on 𝔻

ξ = ∑_{i=1}^{n} ξ_{i}(z)\frac{∂}{∂z_{i}}

such that

(ξf)(z) = Df(z).ξ(z)

for every holomorphic function f on 𝔻. The space of holomorphic vector fields on 𝔻 is a Lie algebra by the Poisson bracket:

[ξ, η] (z) = Dξ(z).η(z) − Dη(z).ξ(z).

Let 𝔤 (resp. 𝔨) denote the Lie algebra of G (resp. K). Every ξ ∈ 𝔤 is considered as a holomorphic vector field on 𝔻 by

ξ(z) = \frac{d}{dt}(exp tξ).z|_{t=0}

where exp is the exponential mapping 𝔤 → G. If the group G contains a real 1-parameter group g_{t} satisfying

\frac{d}{dt}g_{t}.z = ξ(g_{t}.z) ∀(t, z) ∈ ℝ × 𝔻,

the corresponding vector field ξ is called complete. The Lie algebra 𝔤 is the Lie algebra of all complete holomorphic vector fieds on 𝔻. It is a real Lie algebra. The symmetry −id_{𝔻} at the origin induces a Cartan decomposition 𝔤 = 𝔨 ⊕ 𝔭. The evaluation map ξ ⟶ ξ(0) is an isomorphism 𝔭 ⟶ 𝕍 of real vector spaces. For every v ∈ 𝕍, we denote by ξ_{v} the unique vector field in 𝔭 which takes the value v at the origin. Then v − ξ_{v}(z) is a homogeneous quadratic polynomial in z and is ℂ antilinear in v. We shall write

ξ_{v}(z) = v − Q(z)v

so that Q(z) : 𝕍 ⟶ 𝕍 is a complex linear map and Q : 𝕍 ⟶ End(𝕍) is a homogeneous quadratic polynomial map. Hence

Q(x, z) = Q(x + z) − Q(x) − Q(z) : 𝕍 ⟶ 𝕍

is bilinear and symmetric in x and z. For x, y, z ∈ 𝕍, we define

{x, y, z} = Q(x, z)y = D(x, y)z.

Example : [5] [3] [7] Type I_{p,q}. Vector fields in 𝔭 are given by

ξ_{V} (Z) = V − Z^{t}\overline{V} Z (V ∈ 𝕍),

and the triple product on 𝕍 is given by

\{X, Y, Z\} = X^{t}\overline{Y} Z + Z^{t}\overline{Y} X (X, Y, Z ∈ 𝕍).

The triple product {x, y, z} is complex bilinear and symmetric in x and z, complex antilinear in y, and satisfy the Jordan identity

{x, y, {u, v, z}} − {u, v, {x, y, z}} = {{x, y, u}, v, z} − {u, {v, x, y}, z} (x, y, z, u, v ∈ 𝕍).

The vector space 𝕍 endowed with this product is called the Jordan triple system associated to 𝔻. It is positive in the sense that

(u/v) = Trace D(u, v) (u, v ∈ 𝕍)

defines a positive definite inner product on 𝕍. The Bergman operator B(x, y) (x, y ∈ 𝕍) is defined by

B(x, y) = Id − D(x, y) + Q(x)Q(y).

If A is a linear operator on 𝕍, we denote by A^{∗} its adjoint operator w.r to (./.) defined by

(Au/v) = (u/A^{∗}v) (u, v ∈ 𝕍).

The Jordan identity implies that

D(u, v)^{∗} = D(v, u),  (Q(u)Q(v))^{∗} = Q(v)Q(u),  B(u, v)^{∗} = B(v, u) (u, v ∈ 𝕍).

An automorphism A : 𝕍 → 𝕍 is a linear isomorphism preserving the triple product

A{u, v, w} = {Au, Av, Aw} (u, v, w ∈ 𝕍).

The automorphisms of 𝕍 form a group, which is denoted Aut(𝕍), and whose identity component is exactly K. For u, v ∈ 𝕍, denote u□v the linear vector field z ⟶ {u, v, z}. The Lie algebra 𝔨 can be identify with the Lie algebra of linear applications A : 𝕍 → 𝕍 satisfying

[A, u□v] = (Au)□v + u□(Av) (u, v ∈ 𝕍).

The following formulas hold

[ξ_{u}, ξ_{v}] = u□v − v□u, [[ξ_{u}, ξ_{v}], ξ_{w}] = ξ_{{u,v,w}−{w,v,u}}.

In particular, holomorphic vector fiels of the form u□v − v□u (u, v ∈ 𝕍) belong to 𝔨.

2 Tripotents and Peirce decompositions

[7] [13] [10] An element e ∈ 𝕍 is called tripotent if Q(e)e = e. For such an element, we have the Peirce decomposition of 𝕍 relative to e: Let us put

𝕍_{j}(e) = {z ∈ 𝕍, D(e, e)z = jz} (j = 0, 1, 2),

then

𝕍 = 𝕍_{0}(e) ⊕ 𝕍_{1}(e) ⊕ 𝕍_{2}(e)

with the rules

{𝕍_{λ}(e), 𝕍_{μ}(e), 𝕍_{ν}(e)} ⊂ 𝕍_{λ−μ+ν}(e).

In particular, 𝕍_{0}(e),𝕍_{1}(e) and 𝕍_{2}(e) are Jordan triple subsystems of 𝕍 and 𝕍_{2}(e), endowed with

the product xy = \frac{1}{2}\{x, e, y\}, is a unital Jordan algebra with unit element e. The projectors onto 𝕍_{0}(e),𝕍_{1}(e) and 𝕍_{2}(e) are respectively

B(e, e),  D(e, e) − 2Q(e)^{2},  Q(e)^{2}.

The restriction of the operator Q(e) to 𝕍_{0}(e) ⊕ 𝕍_{1}(e) is zero and the restriction of Q(e) to 𝕍_{2}(e) is an involutive, complex antilinear automorphism of the Jordan algebra 𝕍_{2}(e). Denote by 𝔸(e) (respectively 𝔹(e)) its 1 (respectively −1) eigenspace, then 𝔸(e) is a Euclidean Jordan subalgebra of 𝕍_{2}(e), 𝔹(e) = i𝔸(e) and 𝕍_{2}(e) is the complexification of the real algebra 𝔸(e). Denote by 𝕐(e) the connected component of e in the set of invertible elements in 𝔸(e), then 𝕐(e) is a symmetric cone in the sense of [2]. Two tripotents e_{i}, e_{j} ∈ 𝕍 are said to be (strongly) orthogonal if D(e_{i}, e_{j}) = 0. We denote by 𝕄 the set of tripotents of 𝕍, and define an ordering on 𝕄 by

e_{i} < e_{j} if and only if e_{i} ≠ e_{j}, e_{i} − e_{j} ∈ 𝕄 and e_{i} and e_{j} − e_{i} are orthogonal.

In other words, e_{i} < e_{j} means that e_{j} = e_{i} + e_{k} where e_{k} is a non-zero tripotent orthogonal to e_{i}. One checks that this is indeed an ordering. A tripotent e is maximal with respect to this ordering if and only if the Peirce space 𝕍_{0}(e) = 0. This follows from the fact that the tripotents orthogonal to e are precisely the tripotent contained in 𝕍_{0}(e). A non-zero tripotent e is said to be primitive if it cannot be expressed as a sum of non-zero orthogonal tripotents. A non-zero tripotent e is primitive if and only if 𝕍_{2}(e) = ℂe. Finally, a Jordan frame of 𝕍 is a maximal sequence F = (e_{1}, e_{2}, ⋯ e_{r}) of mutually orthogonal tripotents. All frames of 𝕍 have the same number r of elements which is called the rank of 𝕍. Let F = (e_{1}, e_{2}, ⋯ e_{r}) be a fixed frame.

This means that the tripotents e_{1}, e_{2}, ⋯ e_{r} are primitive and E_{r} = e_{1} + e_{2} + ⋯ e_{r} is a maximal one. The operators D(e_{i}, e_{i}) 1 ≤ i ≤ r commute and are simultaneously diagonalizable, with eigenvalues in {0, 1, 2}. The only simultaneous eigenspaces which may be different from 0 are

𝕍_{i,j} = {z ∈ 𝕍, D(e_{k}, e_{k})z = (δ_{k,i} + δ_{k,j})z 1 ≤ k ≤ r}, for 0 ≤ i ≤ j ≤ r

(except 𝕍_{0,0} = {0}). We have 𝕍_{j,j} = ℂe_{j} (1 ≤ j ≤ r). All the 𝕍_{i,j} (1 ≤ i < j ≤ r) have the same dimension a. All the 𝕍_{0,j} (1 ≤ j ≤ r) have the same dimension b. The genus of 𝕍 is the number defined by q = 2 + a(r − 1) + b. The decomposition

𝕍 = ⨁_{0≤i≤j≤r} 𝕍_{i,j} ,

is called the Peirce decomposition of 𝕍 w.r. to the frame F. It satisfies to the Peirce rules

{𝕍_{i,j}, 𝕍_{j,k}, 𝕍_{k,l}} ⊂ 𝕍_{i,l}

will be called Peirce decomposition of 𝕍 w.r. to the frame F. The subspaces of the Peirce decomposition relative to the maximal tripotent E_{r} = e_{1} + e_{2} + ⋯ e_{r} are easily seen to be

𝕍_{1}(E_{r}) = ⨁_{1≤i≤r} 𝕍_{0,i}, 𝕍_{2}(E_{r}) = ⨁_{1≤i≤j≤r} 𝕍_{i,j}.

The Jordan triple system 𝕍 is said to be of tube type is 𝕍_{1}(E_{r}) = {0} (b = 0). In this case, 𝕍 = 𝕍_{2}(E_{r}) is the complexification of a Euclidean Jordan algebra 𝔸(E_{r}) with unit element E_{r} and genus 2 + a(r − 1).

3 Generic norm, quasi-inverse and compactification

The generic norm of 𝕍 is defined by

Det B(x, y) = N (x, y)^{q}

where q is the genus of 𝕍. This is a K-invariant polynomial on 𝕍 × 𝕍 of bidegree (r, r) which has the expansion

N (x, y) = 1 − m_{1}(x, y) + ⋯ (−1)^{r}m_{r}(x, y)

where the m_{j}’s are polynomials, homogeneous of bidegree (j, j) and m_{j}(z, z) > 0 for all z ≠ 0. For each 1 ≤ i ≤ r, we have

m_{i}(x, y) = \overline{m_{i}(y, x)}, m_{i}(Q(u)x, z) = \overline{m_{i}(x, Q(u)y)} (u, x, y ∈ 𝕍).

Every z ∈ 𝕍 can be written

z = k ∑_{i=1}^{r} λ_{i}e_{i} (k ∈ K; λ_{i} ∈ ℝ 1 ≤ i ≤ r).

Then

m_{j}(z, z) = σ_{j}(λ_{1}^{2}, λ_{2}^{2}, ⋯ λ_{r}^{2})

where σ_{j} is the j-th elementary symmetric function. The r polynomials m_{j}(z, z) (1 ≤ j ≤ r) form algebraically independant generators of the algebra P(𝕍)^{K} of K-invariant polynomials on 𝕍. If B(x, y) (x, y ∈ 𝕍) is invertible (as a linear operator), the pair (x, y) is said to be invertible and the quasi-inverse x^{y} is then defined by

x^{y} = B(x, y)^{−1}(x − Q(x)y).

If x is close to the origin, then x^{y} = ∑_{n=1}^{∞} x^{(n,y)}, where

x^{(n,y)} = (\frac{1}{2}D(x, y))^{n−1} x (n > 0).

Example : Type I_{p,q}. The Bergman operator B(X, Y) is defined by

B(X, Y)Z = (Id − X^{t}Ȳ)Z(Id − Y^{t}X̄) (X, Y, Z ∈ 𝕍).

The generic norm is

N (X, Y) = Det(Id_{p} − X^{t}Ȳ) = 1 − Trace(X^{t}Ȳ) + ⋯ + (−1)^{r} Det(X^{t}Ȳ) (X, Y ∈ 𝕍),

and the quasi-inverse X^{Y} is given by

X^{Y} = (Id − X^{t}Ȳ)^{−1}X.

[7] [11] An equivalence relation R is defined in 𝕍 × 𝕍 as follows:

(x, y)R(x′, y′) if and only if (x, y − y′) is invertible and x′ = x^{y−y′}.

The equivalence class of (x, y) is denoted [x : y] and the quotient space is denoted by 𝕏. For each v ∈ 𝕍, let 𝕏_{v} = {[x : v]; x ∈ 𝕍}, then there exists on 𝕏 a unique stucture of smooth algebraic variety such that 𝕏_{v} is an open affine subvariety and [x : v] ⟶ x is an isomorphism of 𝕏_{v} onto 𝕍. The injection x ⟶ [x : 0] of 𝕍 in 𝕏 will be called the canonical compactification of 𝕍. Let G_{𝕏} be the connected component of the identity in the group Aut(𝕏) of all automorphisms of 𝕏. The Lie algebra of G_{𝕏} is 𝔤_{𝕏}. By restriction to 𝕍, we may consider 𝔤_{𝕏} as a Lie algebra of vector fields on 𝕍. Then we have

𝔤_{𝕏} = 𝔲^{+} ⊕ 𝔥 ⊕ 𝔲^{−}

where 𝔲^{+} consists of all vector fields in 𝔤_{𝕏} which, restricted to 𝕍, are constant, and 𝔲^{−} consists of all vector fields in 𝔤_{𝕏} which, restricted to 𝕍, are homogeneous polynomial functions of degree 2. The Lie algebra 𝔥 is the Lie algebra of linear applications A : 𝕍 ⟶ 𝕍 satisfying

[A, u□v] = (Au)□v − u□(A^{∗}v) (u, v ∈ 𝕍)

where A^{∗} denote the adjoint of A with respect to the inner product (./.). The Jordan identity implies that vector fields of the form u□v (u, v ∈ 𝕍) belong to 𝔥. Denote Θ the conjugaison on 𝔤_{𝕏} given by

Θ(h) = −h^{∗} (h ∈ 𝔥),  Θ(v) = Q(z)v (v ∈ 𝔲^{+}),  Θ(Q(z)v) = v (Q(z)v ∈ 𝔲^{−}).

Then the fixed point set of Θ on 𝔥 is 𝔨 and on 𝔲^{+} ⊕ 𝔲^{−} is i𝔭 = {v + Q(z)v, v ∈ 𝕍}. Therefore, the fixed point set of Θ on 𝔤_{𝕏} is the compact real form of 𝔤, 𝔤_{c} = 𝔨 ⊕ i𝔭. Denote G_{c} the connected group whose Lie algebra is 𝔤_{c}.

4 Polynomial and rational identities

The following identities hold in any Jordan triple system [7] [10] [11]: JP 1

{x, y, {u, v, z}} − {u, v, {x, y, z}} = {{x, y, u}, v, z} − {u, {v, x, y}, z}

JP 2

[D(x, y), D(u, v)] = D({x, y, u}, v) − D(u, {v, x, y})

JP 3

D(x, y)Q(x) = Q(x, Q(x)y)

JP 4

D(x, y)Q(x) = Q(x)D(y, x)

JP 5

Q(Q(x)y) = Q(x)Q(y)Q(x)

JP 6

Q(x, z)D(y, x) + Q(x)D(y, z) = Q(x, {x, y, z}) + Q(z, Q(x)y)

JP 7

D(x, y)D(z, y) = Q(x, z)Q(y) + D(x, Q(y)z)

JP 8

Q(x, z)D(y, x) = Q(Q(x)y, z) + D(z, y)Q(x)

JP 9

x^{y} = B(x, y)^{−1}(x − Q(x)y)

JP 10

x^{y} = x + Q(x)(y^{x})

JP 11

Q(y)(x^{Q(y)z}) = (Q(y)x)^{z}

JP 12

x^{y+z} = (x^{y})^{z}

JP 13

(x + y)^{z} = x^{y} + B(x, y)^{−1}(z^{y^{x}})

JP 14

B(x, y)Q(x^{y}) = Q(x^{y})B(y, x) = Q(x)

JP 15

D(z, x^{y})B(y, x) = D(z, x) − Q(y, z)Q(x)

JP 16

B(x, y)B(x^{y}, z) = B(x, y + z)

JP 17

B(x, y)^{−1} = B(x^{y}, −y) = B(−x, y^{x})

5 Root systems and the Lie algebra 𝔫

It is well known that the set of maximal tripotents in 𝕍 is exactly the Shilov boundary 𝕊 of the domain 𝔻. The group K acts transitively on 𝕊. Denote by L the stabilizer of E_{r}, then we have 𝕊 = K/L. In the subspace 𝔭, isomorphic to 𝕍, the vector space

𝔞 =< ξ_{e_{1}}, ξ_{e_{2}}, ⋯ ξ_{e_{r}} > |_{ℝ}

is a maximal abelian subalgebra of 𝔭, that is a Cartan subalgebra of the pair (𝔤, 𝔨). So the rank r of 𝕍 is the same as the real rank of the group G. We denote by 𝔽 the manifold of Jordan frames in 𝕍. The group K acts transitively on 𝔽. We denote by M the stabilizer of F. Then M is exactly the centralizer of 𝔞 in K. The manifold 𝔽 = K/M is also called the maximal Satake-Furstenberg boundary of the symmetric space G/K. Denote by A = exp 𝔞 the real torus whose Lie algebra is 𝔞. Define linear forms on 𝔞 by

α_{i}(ξ_{e_{j}}) = δ_{i,j} (1 ≤ i, j ≤ r).

A linear form α ≠ 0 on 𝔞 is said to be a (restricted) root of the pair (𝔤, 𝔞) if 𝔤^{α} ≠ {0} where

𝔤^{α} = {η ∈ 𝔤, ∀ξ ∈ 𝔞, [η, ξ] = α(η)ξ}.

Let Σ = Σ(𝔤, 𝔞) be the root system (i.e. the set of roots) of the pair (𝔤, 𝔞). The root decomposition of 𝔤 is given by

𝔤 = 𝔪 ⊕ 𝔞 ⊕ ∑_{α∈Σ} 𝔤^{α}.

Let Σ^{+} be the set of positive roots corresponding to the choice of the order in the frame F. Then

Σ^{+} = \begin{cases}\{2α_{j}, 1 ≤ j ≤ r α_{i} − α_{j}, α_{i} + α_{j}, 1 ≤ i < j ≤ r\} & tube type \\ \{2α_{j}, α_{j}, 1 ≤ j ≤ r α_{i} − α_{j}, α_{i} + α_{j}, 1 ≤ i < j ≤ r\} & non tube type. \end{cases}

One defines

𝔫 = ∑_{α∈Σ^{+}} 𝔤^{α}, N = exp 𝔫.

We give an explicit description of the Lie algebra 𝔫 in the language of Jordan triple systems.

Theorem 1 Let 𝕍_{i,j} = 𝔸_{i,j} ⊕ 𝔹_{i,j} (1 ≤ i ≤ j ≤ r) be the decomposition into real and imaginary part relative to the real form 𝔸(E_{r}) of 𝕍_{2}(E_{r}) then

𝔤^{2α_{j}} =< ξ_{ie_{j}} − ie_{j}□e_{j} > |_{ℝ}  (ie_{j} ∈ 𝔹_{j,j});
𝔤^{α_{i}−α_{j}} =< ξ_{v} + e_{i}□v − v□e_{i} v ∈ 𝔸_{i,j} > |_{ℝ};
𝔤^{α_{i}+α_{j}} =< ξ_{w} + e_{i}□w − w□e_{i} w ∈ 𝔹_{i,j} > |_{ℝ};
𝔤^{α_{j}} =< ξ_{u} + e_{j}□u − u□e_{j} u ∈ 𝕍_{0,j} > |_{ℝ}.

The (real) multiplicity of each 2α_{j} is 1. The multiplicity of each α_{i} − α_{j} and each α_{i} + α_{j} is a. The last case appears only in non tube type and then the multiplicity of each α_{i} is 2b.

Proof If h = ∑_{i=1}^{r} λ_{k}ξ_{e_{k}} belongs to 𝔞, it is easy to compute each of the brackets [h, .]. One

concludes using the point (9.18) of [7]. □

In this setting, one can easily describe 𝔫 ∩ 𝔭 and 𝔫 ∩ 𝔨. Define the linear form ρ on 𝔞 by

ρ(t_{1}, t_{2}, ⋯ t_{r}) = (b + 1) ∑_{i=1}^{r} t_{i} + a ∑_{i=1}^{r}(r − i)t_{i},

then ρ is the half sum of the positive restricted roots of the pair (𝔤, 𝔞). The Iwasawa decomposition of 𝔤 is 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫 and, at the group level, G = KAN.

6 Horocycles and Poisson-Furstenberg kernel

A horocycle in 𝔻 is by definition an orbit in 𝔻 of a subgroup of G conjugate to N. Horocycles are all closed manifolds in 𝔻 of codimension r. The horocycles which are orbits of the group N are called normal to the frame F. Every z ∈ 𝔻 can be uniquely written

z = n ∑_{i=1}^{r}(th t_{i})e_{i} (n ∈ N; t_{i} ∈ ℝ 1 ≤ i ≤ r)

and there exists a unique horocycle through z normal to F. The automorphism

a = exp t_{1}ξ_{e_{1}} exp t_{2}ξ_{e_{2}} ⋯ exp t_{r}ξ_{e_{r}} ∈ A carries the origin to the point ∑_{i=1}^{r}(th t_{i})e_{i}.

Define A(z, F) = (t_{1}, t_{2}, ⋯ t_{r}). The map z ⟶ A(z, F) is N-invariant and is called the complex distance from the origin to the horocycle through z normal to F. If μ ∈ L(ℝ^{r}, ℂ), F ∈ 𝔽, the function

e_{μ,F} : z ⟶ exp μ(A(z, F)) (z ∈ 𝔻)

is N-invariant and is an eigenfunction of each invariant differential operator. The joint eigenfunctions of the algebra D(𝔻)^{G} are the functions

f(z) = ∫_{𝔽} exp μ(A(z, F))dT (F) (z ∈ 𝔻)

where μ ∈ L(ℝ^{r}, ℂ) and T is a functional on 𝔽 [4]. A function f : 𝔻 ⟶ ℂ is said to be G-harmonic if Df = 0 for all G-invariant differential operators D on 𝔻 such that D1 = 0. For μ = 2ρ the function

z ⟶ P(z, F) = exp 2ρ(A(z, F))

is a N-invariant G-harmonic function called Poisson-Furstenberg kernel of 𝔻. It is a reproducing kernel for G-harmonic functions. It is well known that the Poisson kernel over the Shilov boundary can be expressed using the generic norm by [5]

P(z, w) = \frac{N (z, z)^{\frac{n}{r}}}{|N (z, w)|^{\frac{2n}{r}}} (z ∈ 𝔻 w ∈ 𝕊).

In the same way, the language of Jordan triple systems can be used to give explicit formulas for the joint eigenfunctions of D(𝔻)^{G} and for the Poisson-Furstenberg kernel. These formulas are given by r N-invariant functions called horocyclic equations. Example :

In the unit disk of ℂ, the horocycle through the point th t normal to the point 1 is the circle whose equation is

\frac{1 − zz̄}{|1 − z|^{2}} = e^{2t}.

7 Cayley transform

The partial Cayley transform defined by a tripotent e ∈ 𝕍 is [7]

γ_{e} = exp \frac{π}{4} (e + Q(z)e).

Since the vector field e + Q(z)e belongs to 𝔤_{c}, γ_{e} belongs to the compact group G_{c}. We have

γ_{e}^{8} = Id (γ_{e}^{4} = Id if, and only if, 𝕍 is of tube type)
γ_{0} = Id
γ_{e}^{−1} = γ_{−e}
kγ_{e}k^{−1} = γ_{ke} (k ∈ K)
γ_{c}γ_{d} = γ_{c+d} for orthogonal tripotents c and d.

Proposition 2 Let x = x_{2} ⊕ x_{1} ⊕ x_{0} be the components of a point x ∈ 𝕍 in the Peirce spaces with respect to the tripotent e. Then γ_{e}(x) ∈ 𝕍 if, and only if, e − x_{2} is invertible in the Jordan algebra 𝕍_{2}(e) and then,

γ_{e}(x) = (e − x_{2})^{−1}(e + x_{2}) ⊕ \sqrt{2}(e − x_{2})^{−1} ∘ x_{1} ⊕ x_{0} + P (x_{1})(e − x_{2})^{−1},
γ_{−e}(x) = (x_{2} + e)^{−1}(x_{2} − e) ⊕ \sqrt{2}(e + x_{2})^{−1} ∘ x_{1} ⊕ x_{0} − P (x_{1})(e + x_{2})^{−1}.

Here a^{−1} denotes the inverse of a in 𝕍_{2}(e), x ∘ y = {x, e, y} (x, y ∈ 𝕍) and P (x) = Q(x)Q(e).

Define f : 𝕍_{1}(e) × 𝕍_{1}(e) → 𝕍_{2}(e) by

(u, v) ⟶ {u, v, e} (u, v ∈ 𝕍_{1}(e)).

Then f is hermitian and positive definite in the sense that

f(u, v)^{∗} = f(v, u) in 𝕍_{2}(e)
f(u, u) ∈ 𝕐(e) (u ∈ 𝕍_{1}(e))
f(u, u) = 0 only for u = 0 in 𝕍_{1}(e).

We denote f(u) instead of \frac{1}{2}f(u, u) (u ∈ 𝕍_{1}(e)). The next proposition is the point 10.9 of [7]

Proposition 3 Let E be a maximal tripotent in 𝕍 (hence 𝕍_{0}(E) = 0). Then the image of 𝔻 under the Cayley transform γ_{E} is the Siegel domain of type two

𝕋̃(E) = {x_{2} ⊕ x_{1} ∈ 𝕍_{2}(E) ⊕ 𝕍_{1}(E), x_{2} + x_{2}^{∗} − 2f(x_{1}) > 0}

We have 𝕋̃(E) ∩ 𝔸(E) = 𝕐(E). In the special case where 𝕍 is a Jordan algebra with unit E (i.e. 𝕍_{1}(E) = 0), γ_{E}(𝔻) is the tube domain

𝕋(E) = 𝕐(E) ⊕ 𝔹(E)

over the cone 𝕐(E) ⊂ 𝔸(E).

In the following, we will use the Cayley transform with respect to the maximal tripotent E_{r} = e_{1}+e_{2}+⋯+e_{r} and f will be the application given by f(u, v) = {u, v, E_{r}} (u, v ∈ 𝕍_{1}(E_{r})). We will note 𝕐 (respectively 𝕋̃) instead of 𝕐(E_{r}) (respectively 𝕋̃(E_{r})). Denote G(𝕋̃) the connected component of the identity in the group of all analytic automorphisms of the domain 𝕋̃. We have G(𝕋̃) = γ_{E_{r}}Gγ_{E_{r}}^{−1}. Denote G(𝕐) the subgroup of G(𝕋̃) which stabilizes the cone 𝕐. The group G(𝕐) is a subgroup in the linear group of 𝕍. In the special case where 𝔻 is of tube type, G(𝕋̃) contains as a subgroup, the group of affine linear transformation x ⟶ Ax + b where A ∈ G(𝕐) and b ∈ 𝔹(E_{r}). We have G(𝕋̃) = {g ∈ G_{𝕏}, g(𝕋̃) ⊂ 𝕋̃}. The group G(𝕋̃) acts transitively on 𝕋̃. Denote by K(𝕋̃) the isotropy group of the tripotent E_{r} in G(𝕋̃), then K(𝕋̃) = γ_{E_{r}}Kγ_{E_{r}}^{−1}.

8 Cayley transform of vector fields

Denote by 𝔤(𝕋̃) the Lie algebra of the group G(𝕋̃). We have 𝔤(𝕋̃) = Ad γ_{E_{r}}(𝔤). We will use the two following lemmas.

Lemma 4 Let v ∈ 𝔸_{i,j} and w ∈ 𝔹_{i,j} (1 ≤ i < j ≤ r), then we have

v□e_{j} = e_{i}□v (v□e_{i} = e_{j}□v) and w□e_{j} = −e_{i}□w (w□e_{i} = −e_{j}□w).

Lemma 5 [7] lemmas (10.5) and (10.6). Let e_{j} be a minimal tripotent in the frame F = (e_{1}, e_{2}, ⋯ e_{r}). Then we have

(1) Ad γ_{e}.v = \frac{1}{\sqrt{2}}(v + v□e) (v ∈ 𝕍_{1}(e_{j})),

(2) Ad γ_{e_{j}}(ξ_{b} − b□e_{j}) = 2b (b ∈ 𝔹(e_{j})),

(3) Ad γ_{e_{j}}(ξ_{v} + e_{j}□v − v□e_{j}) = \sqrt{2}(v + e_{j}□v) (v ∈ 𝕍_{1}(e_{j})),

(4) Ad γ_{e_{j}}(ξ_{v} + Δ) = (ξ_{v} + Δ) (v ∈ 𝕍_{0}(e_{j}), Δ ∈ 𝔨 tel que Δ(e_{j}) = 0).

Proposition 6 Let e_{i} and e_{j} (i < j) two orthogonal tripotents in the frame F = (e_{1}, e_{2}, ⋯ e_{r}). The generators of the Lie algebra 𝔫 are transformed according to the following rules:

(1) Ad γ_{e_{k}}(τ_{j,j}) = τ_{j,j} (k ≠ j)
(2) Ad γ_{e_{j}}(τ_{j,j}) = 2ie_{j}
(3) Ad γ_{e_{i}+e_{j}}(τ_{i,j,v}^{−}) = 2e_{i}□v
(4) Ad γ_{e_{i}+e_{j}}(τ_{i,j,w}^{+}) = 2w
(5) Ad γ_{e_{k}}(τ_{i,j,.}^{±}) = τ_{i,j,.}^{±} (k ≠ i, j)
(6) Ad γ_{e_{j}}(τ_{0,j,u}) = \sqrt{2}(u + E_{r}□u)
(7) Ad γ_{e_{k}}(τ_{0,j,u}) = τ_{0,j,u} (k ≠ j).

We also have

(8) Ad γ_{e_{i}−e_{j}}(τ_{i,j,v}^{−}) = 2v.

Proof Identities (1), (5) and (7) follow from (4) in lemma 5. The identity (2) follows from (2) in the same lemma. Then, we have

Ad γ_{e_{j}}(τ_{0,j,u}) = \sqrt{2}(u + e_{j}□u),
Ad γ_{e_{i}}(τ_{i,j,v}^{−}) = \sqrt{2}(v + e_{i}□v),
Ad γ_{e_{i}}(τ_{i,j,w}^{+}) = \sqrt{2}(w + e_{i}□w),

which follow from (3). For u ∈ 𝕍_{0,j}, e_{j}□u = E_{r}□u and (6) follows. Using lemma 4, we also have

\sqrt{2}(v + e_{i}□v) = \sqrt{2}(v + v□e_{j}) (v ∈ 𝔸_{i,j}),
\sqrt{2}(w + e_{i}□w) = \sqrt{2}(w − w□e_{j}) (w ∈ 𝔹_{i,j}).

Using the first relation of lemma 5,

\sqrt{2}(v + v□e_{j}) = 2 Ad γ_{e_{j}}.v,
\sqrt{2}(w + w□(−e_{j})) = 2 Ad γ_{−e_{j}}.w,

from which we deduce

Ad γ_{e_{i}}(τ_{i,j,v}^{−}) = 2 Ad γ_{e_{j}}.v,
Ad γ_{e_{i}}(τ_{i,j,w}^{+}) = 2 Ad γ_{−e_{j}}.w.

Recall that γ_{c}γ_{d} = γ_{c+d} for orthogonal tripotents c and d. Then we easily see (8) and (4). Last, using the first relation of lemma 5, we have

\sqrt{2}(v + v□(−e_{j})) = 2 Ad γ_{−e_{j}}.v,

so

2v = \sqrt{2} Ad γ_{e_{j}}.v − \sqrt{2} Ad γ_{e_{j}}.(v□e_{j}),
\sqrt{2} Ad γ_{e_{j}}.(v□e_{j}) = v + e_{i}□v − 2v = e_{i}□v − v,

and

Ad γ_{e_{j}}(Ad γ_{e_{i}}(τ_{i,j,v}^{−}) = \sqrt{2} Ad γ_{e_{j}}.v + \sqrt{2} Ad γ_{e_{j}}(e_{i}□v)
= v + e_{i}□v + e_{i}□v − v
= 2e_{i}□v.

This is the relation (3). □

We now use the following lemma.

Lemma 7 [7] lemmas (10.5) and (10.6). Let e be a minimal tripotent. Let v ∈ 𝕍_{0}(e) and Δ ∈ 𝔨 such that Δ(e) = 0, then

(1) Ad γ_{e}.v = v
(2) Ad γ_{e}.Δ = Δ.

The Cayley transform with respect to the maximal tripotent E_{r} is the commutative product of the partial Cayley transforms γ_{e_{j}} (1 ≤ j ≤ r). Using the previous lemma and the results of the proposition, we obtain

Proposition 8 The Lie algebra 𝔫(𝕋̃) is generated by the vector fields

Ad γ_{E_{r}}(τ_{j,j}) = 2ie_{j} (1 ≤ j ≤ r),
Ad γ_{E_{r}}(τ_{i,j,v}^{−}) = 2e_{i}□v (v ∈ 𝔸_{i,j}, 1 ≤ i < j ≤ r),
Ad γ_{E_{r}}(τ_{i,j,w}^{+}) = 2w (w ∈ 𝔹_{i,j}, 1 ≤ i < j ≤ r),
Ad γ_{E_{r}}(τ_{0,j,u}) = \sqrt{2}(u + E_{r}□u) (u ∈ 𝕍_{0,j}, 1 ≤ j ≤ r).

9 The group N(𝕐) in G(𝕋̃)

Denote 𝔫(𝕐) the Lie subalgebra in 𝔫(𝕋̃) generated by the vector fields

2e_{i}□v (v ∈ 𝔸_{i,j}, 1 ≤ i < j ≤ r).

It is a Lie algebra of linear vector fields, and this is a nilpotent subalgebra of 𝔥 [6] [15]. Denote N(𝕐) the subgroup of G(𝕋̃) with Lie algebra 𝔫(𝕐). It is a linear subgroup of G(𝕋̃). Let us remark that

{𝕍_{2}(E_{r}), 𝕍_{2}(E_{r}), 𝕍_{i}(E_{r})} ⊂ 𝕍_{i}(E_{r}),

so the group N(𝕐) stabilizes both spaces 𝕍_{2}(E_{r}) and 𝕍_{1}(E_{r}).

Lemma 9 The elements of the group N(𝕐) commute with the conjugation Q(E_{r}) in 𝕍_{2}(E_{r}).

Proof Using the fact that Q(E_{r}) is an automorphism of 𝕍_{2}(E_{r}), we show that any generator

δ = e_{i}□v (v ∈ 𝔸_{i,j}, 1 ≤ i < j ≤ r)

of the Lie algebra 𝔫(𝕐) satisfies

Q(E_{r})(δ.z) = δ(Q(E_{r})z) (z ∈ 𝕍_{2}(E_{r})).
□

Proposition 10 The group N(𝕐) is a subgroup of G(𝕐).

Proof Let y ∈ 𝕐 and Δ ∈ N(𝕐). The automorphism Δ is an automorphism of the tube, so we have Δ.y + (Δ.y)^{∗} ∈ 𝕐. But (Δ.y)^{∗} = Δ.y^{∗} = Δ.y, so 2Δ.y ∈ 𝕐 and Δ.y ∈ 𝕐. □

The two groups G(𝕐) et N(𝕐) are isomorphic under Θ with the groups which appear in the proposition V I.3.3 of [2]. We have Θ(e_{i}□v) = −v□e_{i} and N(𝕐) is the conjugate of the group N in [2]. In the same way as in this book, we can show the following results:

Proposition 11 Every x ∈ 𝕐 can be uniquely written

x = n ∑_{i=1}^{r} e^{2t_{i}}e_{i}  (n ∈ N(𝕐);  t_{i} ∈ ℝ, 1 ≤ i ≤ r).

The r functions Δ_{k}^{∗} (1 ≤ k ≤ r) defined on the cone 𝕐 by

Δ_{k}^{∗}(y) = m_{k}(y^{−1}, E_{k})^{−1} (y ∈ 𝕐)

are N(𝕐)-invariant. Moreover, if

x = ∑_{i=1}^{r} e^{2t_{i}}e_{i},

Δ_{k}^{∗}(x) = exp (2 ∑_{i=1}^{k} t_{i}) (1 ≤ k ≤ r).

We shall use the following lemma.

Lemma 12 The elements of the group N(𝕐) are automorphisms of the hermitian form f in the sense that

f(Δx, Δy) = Δf(x, y) (x, y ∈ 𝕍_{1}(E_{r}), Δ ∈ T(𝕐))

Proof Using JP1 and JP8, one can show that every generator

δ = e_{i}□v (v ∈ 𝔸_{i,j}, 1 ≤ i < j ≤ r)

of the Lie algebra 𝔫(𝕐) satisfies

δf(x, y) = f(δx, y) + f(x, δy) (x, y ∈ 𝕍_{1}(e)).
□

10 Horocyclic coordinates

In the special case of tube type, the Lie algebra 𝔫(𝕋) is only generated by the vector fields

Ad γ_{E_{r}}(τ_{j,j}) = 2ie_{j} (1 ≤ j ≤ r),
Ad γ_{E_{r}}(τ_{i,j,v}^{−}) = 2e_{i}□v (v ∈ 𝔸_{i,j}, 1 ≤ i < j ≤ r),
Ad γ_{E_{r}}(τ_{i,j,w}^{+}) = 2w (w ∈ 𝔹_{i,j}, 1 ≤ i < j ≤ r).

Let us remark that the all group of translations in the direction 𝔹(E_{r}) is generated by the translations

t_{b} (b ∈ 𝔹_{j,j}, 1 ≤ j ≤ r) and
t_{w} (w ∈ 𝔹_{i,j}, 1 ≤ i < j ≤ r)

Hence, the group N(𝕋) is the group of all affine transformations of the tube 𝕋 generated by the transformations of the group N(𝕐) and by the translations in the direction t_{ih} (h ∈ 𝔸(E_{r})). The space 𝔹(E_{r}) is invariant under N(𝕐) ; so we have the following result:

Proposition 13 In the special case of tube type, the group N(𝕋) is the group of all affine transformations of the form

x ⟶ Δx + ih (Δ ∈ N(𝕐), h ∈ 𝔸(E_{r})).

Let us see the general case. We study integral curves of the vector field

\sqrt{2}(u + E_{r}□u) (u ∈ 𝕍_{0,j}, 1 ≤ j ≤ r).

If x = x_{2} ⊕x_{1} ∈ 𝕍_{2}(E_{r})⊕𝕍_{1}(E_{r}) is the Peirce decomposition of the point x w.r. to the maximal tripotent E_{r}, we solve the differential equation

\frac{d}{dt}η(t) = \sqrt{2}(u + \{E_{r}, u, η(t)\})

η(0) = x_{2} ⊕ x_{1}.

The solution of the homogeneous associated equation is

η(t) = [Exp t\sqrt{2}(E_{r}□u)].K

then, we have

K′(t) = \sqrt{2}[Exp(−t\sqrt{2}(E_{r}□u)].u
K′(t) = \sqrt{2}[u − t\sqrt{2}\{E_{r}, u, u\}] and
K(t) = η(0) + t\sqrt{2}u − t^{2}\{E_{r}, u, u\}.

It follows that

η(t) = [Exp t\sqrt{2}(E_{r}□u)].(x_{2} − t^{2}\{E_{r}, u, u\} ⊕ x_{1} + \sqrt{2}tu).

Using {𝕍_{2}(e_{j}), 𝕍_{1}(e_{j}), 𝕍_{2}(e_{j})} = 0, we obtain

η(t) = x_{2} + 2t^{2}f(u) + t\sqrt{2}f(x_{1}, u) ⊕ x_{1} + \sqrt{2}tu.

With t = 1 and \sqrt{2}u replaced by u, we obtain the following lemma. Lemma 14 The elements of N(𝕋̃) generated by holomorphic vector fields of the form

u + E_{r}□u (u ∈ 𝕍_{0,j}, 1 ≤ j ≤ r)

are of the form

x ⟶ μ_{U}(x) + ih (U ∈ 𝕍_{1}(E_{r}), h ∈ 𝔸(E_{r}))

where

μ_{U} : x = x_{2} ⊕ x_{1} ⟶ x_{2} + f(U) + f(x_{1}, U) ⊕ x_{1} + U.

The product μ_{U_{2}} ∘ μ_{U_{1}} of two such automorphisms μ_{U_{1}} et μ_{U_{2}} is exactly μ_{U_{1}+U_{2}} + U_{1,2} where

U_{1,2} = \frac{1}{2}[f(U_{1}, U_{2}\} − f(U_{2}, U_{1})] ∈ i𝔸(E_{r}).

Using also lemma 12, we obtain Proposition 15 The group N(𝕋̃) is the group of all transformations

x = x_{2} ⊕ x_{1} ⟶ Δx_{2} + f(u) + f(Δx_{1}, u) + ih ⊕ Δx_{1} + u

where

Δ ∈ N(𝕐), h ∈ 𝔸(E_{r}), u ∈ 𝕍_{1}(E_{r}).

Finally, we have the Jordan decomposition for 𝕋̃. Proposition 16 Every x = (x_{2}, x_{1}) ∈ 𝕋̃ can be written in an unique way

(x_{2}, x_{1}) = (Δ ∑_{i=1}^{r} e^{2t_{i}}e_{i} + f(u) + ih, u),

where Δ ∈ N(𝕐), h ∈ 𝔸(E_{r}) and u ∈ 𝕍_{1}(E_{r}). Proof It is enough to write

x_{2} = \frac{1}{2}[x_{2} + x_{2}^{∗} − f(x_{1}, x_{1})] + f(x_{1}) + \frac{1}{2}[x_{2} − x_{2}^{∗}],

where \frac{1}{2}[x_{2} + x_{2}^{∗} − f(x_{1}, x_{1})] belongs to the cone 𝕐. □

Example : [8]. Type I_{m,n} (m ≤ n). The domain is of tube type if and only if n = m. For any element W ∈ 𝕄_{m,n}(ℂ), denote W = (W_{1}, W_{2}) (W_{1} ∈ 𝕄_{m,m}(ℂ), W_{2} ∈ 𝕄_{m,n−m}(ℂ)) its Peirce decomposition with respect to the maximal tripotent E_{m} = (Id_{m}, 0). A matrix H ∈ 𝕄_{m,m}(ℂ) belongs to 𝔸(E_{m}) if and only if it is hermitian (H^{∗} = H). We consider the frame F = (e_{1,1}, e_{2,2}, ⋯ e_{m,m}) of 𝕄_{m,n}(ℂ). The Cayley transform with respect to E_{m} = (Id_{m}, 0) given by

Z_{1} = (Id_{m} − W_{1})^{−1}(Id_{m} + W_{1})
Z_{2} = \sqrt{2}(Id_{m} − W_{1})^{−1}W_{2}

transforms the domain 𝔻 = \{W ∈ 𝕄_{m,n}(ℂ), Id_{m} −W^{t}\overline{W} >> 0\} into the unbounded realisation 𝕋̃ = \{Z = (Z_{1}, Z_{2}) ∈ 𝕄_{m,n}(ℂ), \frac{1}{2}(Z_{1} +^{t} \overline{Z_{1}}) − Z_{2}^{t}\overline{Z_{2}} >> 0\}. The elements of N(𝕐) are of the form (Z_{1}, Z2) ⟶ (AZ_{1}^{t}\overline{A}, AZ_{2}), where A ∈ 𝕄_{m,m}(ℂ) is an upper unipotent triangular matrix. See page 111 in [2]. The elements Φ_{U} of N(𝕋̃) induced by the holomorphic vector field U + Id_{m}□U (U ∈ 𝕍_{1}(E_{m})) is of the form

(Z_{1}, Z_{2}) ⟶ (Z_{1} + \frac{1}{2}\{Id_{m}, U, U\} + \{Id_{m}, U, Z_{2}\}, Z_{2} + U).

Denote B ∈ 𝕄_{m,n−m}(ℂ) the matrix whose embedding into 𝕄_{m,n}(ℂ) is the matrix −U. Let us remark that

\{Id_{m}, U, U\} = B^{t}\overline{B} ∈ 𝕄_{m,m}(ℂ)
\{Id_{m}, U, Z_{2}\} = −Z_{2}^{t}\overline{B} ∈ 𝕄_{m,m}(ℂ).

With these notations, the automorphism Φ_{U} can be written

(Z_{1}, Z_{2}) ⟶ (Z_{1} + \frac{1}{2}B^{t}\overline{B} − Z_{2}^{t}\overline{B}, Z_{2} − B);

in this example, the group N(𝕋̃) is the group of all transformations of the form

(Z_{1}, Z_{2}) ⟶ (AZ_{1}^{t}\overline{A} + \frac{1}{2}B^{t}\overline{B} − AZ_{2}^{t}\overline{B} + iH, AZ_{2} − B),

with A ∈ 𝕄_{m,m}(ℂ) (upper unipotent triangular matrix), B ∈ 𝕄_{m,n−m}(ℂ), and H ∈ 𝕄_{m,m}(ℂ) (hermitian). These are the horocyclic coordinates given by Lu Qikeng in [8].

11 N-invariant functions

Let z = (z_{2}, z_{1}) a point in 𝕍 given in the Peirce decomposition of 𝕍 with respect to the maximal tripotent E_{r}, 𝕍 = 𝕍_{2}(E_{r})⊕𝕍_{1}(E_{r}). Since the Cayley transform is one to one from 𝔻 onto 𝕋̃, the point z = (z_{2}, z_{1}) belongs to 𝔻 if, and only if, the point (E_{r} −z_{2})^{−1}(E_{r} +z_{2})⊕\sqrt{2}(E_{r} −z_{2})^{−1} ∘z_{1}

belongs to 𝕋̃. The Cayley transform of the point ∑_{i=1}^{r}(th t_{i})e_{i} is the point ∑_{i=1}^{r} e^{2t_{i}}e_{i}. So the

point z = (z_{2}, z_{1}) ∈ 𝔻 belongs to the N orbit of the point ∑_{i=1}^{r}(th t_{i})e_{i} if and only if the point

(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) ⊕ \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1} ∈ 𝕋̃ belongs to the N(𝕋̃) orbit of the point ∑_{i=1}^{r} e^{2t_{i}}e_{i}.

Let us recall the proposition 16: the point ((E_{r} − z_{2})^{−1}(E_{r} + z_{2}), \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}) in 𝕋̃ can be uniquely written

((E_{r} − z_{2})^{−1}(E_{r} + z_{2}), \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}) = (Δ ∑_{i=1}^{r} e^{2t_{i}}e_{i} + f(u) + ih, u)

where Δ ∈ N(𝕐), h ∈ 𝔸(E_{r}) and u ∈ 𝕍_{1}(E_{r}). In this proof, the point Δ ∑_{i=1}^{r} e^{2t_{i}}e_{i} which is

used is the point

\frac{1}{2}[(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) + ((E_{r} − z_{2})^{−1}(E_{r} + z_{2}))^{∗} − f(\sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}, \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})].

So the point in 𝕋̃

((E_{r} − z_{2})^{−1}(E_{r} + z_{2}), \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})

belongs to the N(𝕋̃) orbit of the point ∑_{i=1}^{r} e^{2t_{i}}e_{i} if, and only if, the point of the cone

\frac{1}{2}[(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) + ((E_{r} − z_{2})^{−1}(E_{r} + z_{2}))^{∗} − f(\sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}, \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})]

belongs to the N(𝕐) orbit of this same point ∑_{i=1}^{r} e^{2t_{i}}e_{i} (Horocycle of the cone).

Definition 17 For any point z = (z_{2}, z_{1}) ∈ 𝔻, given in the Peirce decomposition of 𝕍 with respect to the maximal tripotent E_{r}, denote H(z, E_{r}) the point

\frac{1}{2}[(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) + (E_{r} − z_{2})^{−1}(E_{r} + z_{2})^{∗} − f(\sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}, \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})]

The point H(z, E_{r}) belongs to the cone 𝕐 = 𝕐(E_{r}), in the Euclidean Jordan algebra 𝔸(E_{r}).

Example : In the case of the unit disk 𝔻 in ℂ, consider a point e in the boundary of 𝔻. Then F=(e) is a frame in ℂ. The inverse of a point x ∈ ℂ is x^{−1} = e/x and his conjugate is x^{∗} = e^{2}x̄. If z ∈ 𝔻, the point H(z, e) is

\frac{1}{2} (\frac{(e + z)e}{e − z} + \frac{(ē + z̄)e}{ē − z̄}) = \frac{(1 − zz̄)e}{|1 − zē|^{2}}.

The equation for the horocycle H(t, e) through the point (th t)e and normal to the frame e is

\frac{1 − zz̄}{|1 − zē|^{2}} = exp 2t.

Definition 18 For any 1 ≤ k ≤ r et for any z ∈ 𝔻, define

Δ̃_{k}(z, E_{k}) = Δ_{k}^{∗}(H(z, E_{r}))

where E_{k} denote the tripotent e_{1} + e_{2} + ⋯ e_{k}.

Using proposition 11 and the previous remarks, we obtain Proposition 19 The r equations

Δ̃_{k}(z, E_{k}) = exp (2 ∑_{i=1}^{k} t_{i}) (1 ≤ k ≤ r)

are the equations of the horocycle through the point

∑_{i=1}^{r}(th t_{i})e_{i}

normal to the frame F = (e_{1}, e_{2}, ⋯ e_{r}).

Corollary 20 For each 1 ≤ k ≤ r, the function

z ⟶ Δ̃_{k}(z, E_{k}) (z ∈ 𝔻)

is N-invariant in the sense that

Δ̃_{k}(nz, E_{k}) = Δ̃_{k}(z, E_{k}) (z ∈ 𝔻, n ∈ N)

Moreover, if

z = ∑_{i=1}^{r}(th t_{i})e_{i}

we have

Δ̃_{k}(z, E_{k}) = exp (2 ∑_{i=1}^{k} t_{i}) .

12 Expressions in the language of Jordan triple systems

We give, for the functions Δ̃_{k}, expressions which use uniquely the language of Jordan triple systems, and without using the Peirce decomposition of 𝕍 with respect to the tripotent E_{r}. Let z = (z_{2}, z_{1}) a point in 𝔻 given in the Peirce decomposition of 𝕍 with respect to the maximal tripotent E_{r}. The point H(z, E_{r}) is the point of the cone 𝕐 defined by

\frac{1}{2}[(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) + ((E_{r} − z_{2})^{−1}(E_{r} + z_{2}))^{∗} − f(\sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}, \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})].

Lemma 21 With the previous notations, the following formula holds:

\frac{1}{2} [(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) + ((E_{r} − z_{2})^{−1}(E_{r} + z_{2}))^{∗}] = Q(E_{r})^{2}[z^{E_{r}} + E_{r}^{z}]

Proof Remark that

(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) + E_{r} = 2(E_{r} − z_{2})^{−1} = 2(E_{r} + z_{2}^{E_{r}}),

where z_{2}^{E_{r}} is the quasi-inverse of z_{2} with respect to E_{r}. So

(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) = E_{r} + 2z_{2}^{E_{r}},

and

\frac{1}{2} [(E_{r} − z_{2})^{−1}(E_{r} + z_{2}) + ((E_{r} − z_{2})^{−1}(E_{r} + z_{2}))^{∗}]

is equal to

E_{r} + z_{2}^{E_{r}} + (z_{2}^{E_{r}})^{∗}.

The orthogonal projection of 𝕍 onto 𝕍_{2}(E_{r}) is Q(E_{r})^{2}. Using the identity JP11, we have

(Q(E_{r})z)^{E_{r}} = Q(E_{r}) (z^{E_{r}}) .

So, we have, using also JP10,

E_{r} + z_{2}^{E_{r}} + (z_{2}^{E_{r}})^{∗} = Q(E_{r})^{2}[E_{r} + z^{E_{r}} + Q(E_{r})z^{E_{r}}] = Q(E_{r})^{2}[z^{E_{r}} + E_{r}^{z}].
□

In the same way, we show that

\frac{1}{2}[f(\sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}, \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})]

can be expressed without using the Peirce decomposition of z.

Lemma 22 With the same notations, we have

z_{1} + {z_{1}, E_{r}, z_{2}^{E_{r}}} = [Id − Q(E_{r})^{2}]z^{E_{r}}.

Proof Using the identities JP13 and JP17, we have

(z_{1} + z_{2})^{E_{r}} = z_{2}^{E_{r}} + B(z_{2}, E_{r})^{−1}z_{1}^{E_{r}^{z_{2}}}
= z_{2}^{E_{r}} + B(z_{2}^{E_{r}}, −E_{r})z_{1}^{E_{r}^{z_{2}}}.

Remark that for any (x, y) ∈ 𝕍_{1}(E_{r}) × 𝕍_{2}(E_{r}), we have x^{y} = x. It is true for x near the origin because in this case,

x^{y} = ∑_{n=0}^{∞} x^{(n,y)}

and all the products {𝕍_{2}, 𝕍_{1}, 𝕍_{2}} vanish. By analytic continuation, this remark is true everywhere in 𝕍_{1}(E_{r}). In particular, since z_{1} ∈ 𝕍_{1}(E_{r}) and since E_{r}^{z_{2}} is in 𝕍_{2}(E_{r}), we have

z_{1}^{E_{r}^{z_{2}}} = z_{1}.

So

z^{E_{r}} = z_{2}^{E_{r}} + z_{1} + D(z_{2}^{E_{r}}, E_{r})z_{1} + Q(z_{2}^{E_{r}})Q(E_{r})z_{1}.

Let us remark now that Q(E_{r})z_{1} = 0 since the conjugaison Q(E_{r}) is equal to 0 in restriction to 𝕍_{1}(E_{r}). So

z^{E_{r}} = z_{2}^{E_{r}} + z_{1} + D(z_{2}^{E_{r}}, E_{r})z_{1}.

In this equality,

z_{2}^{E_{r}} ∈ 𝕍_{2}(E_{r}) and z_{1} + D(z_{2}^{E_{r}}, E_{r})z_{1} ∈ 𝕍_{1}(E_{r}).

The orthogonal projection onto 𝕍_{1}(E_{r}) is D(E_{r}, E_{r}) − 2Q(E_{r})^{2}, so

[D(E_{r}, E_{r}) − 2Q(E_{r})^{2}]z^{E_{r}} = z_{1} + {z_{2}^{E_{r}}, E_{r}, z_{1}}.

Finally, let us remark that, since E_{r} is a maximal tripotent, the orthogonal projection B(E_{r}, E_{r}) onto 𝕍_{0}(E_{r}) is 0. This implies

D(E_{r}, E_{r}) − 2Q(E_{r})^{2} = Id − Q(E_{r})^{2}

and the lemma follows. □

Lemma 23 The following formula holds:

\frac{1}{2}[f(\sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}, \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})] = Q(E_{r})^{2} \{(Id − Q(E_{r})^{2})z^{E_{r}}, z^{E_{r}}, E_{r}\}

Proof We have

\frac{1}{2}[f(\sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1}, \sqrt{2}(E_{r} − z_{2})^{−1} ∘ z_{1})]

= {(E_{r} − z_{2})^{−1} ∘ z_{1}, (E_{r} − z_{2})^{−1} ∘ z_{1}, E_{r}}
= {{z_{1}, E_{r}, E_{r} + z_{2}^{E_{r}}}, {z_{1}, E_{r}, E_{r} + z_{2}^{E_{r}}}, E_{r}}
= {z_{1} + {z_{1}, E_{r}, z_{2}^{E_{r}}}, z_{1} + {z_{1}, E_{r}, z_{2}^{E_{r}}}, E_{r}}
= {(Id − Q(E_{r})^{2})z^{E_{r}}, (Id − Q(E_{r})^{2})z^{E_{r}}, E_{r}}

using the previous lemma. The rules {𝕍_{1}, 𝕍_{1}, 𝕍_{2}} ⊂ 𝕍_{2} and {𝕍_{1}, 𝕍_{2}, 𝕍_{2}} = 0 implie that it can also be written

Q(E_{r})^{2} {(Id − Q(E_{r})^{2})z^{E_{r}}, z^{E_{r}}, E_{r}} .
□

We obtain

Proposition 24 For any z ∈ 𝔻, we define the point H(z, E_{r}) ∈ 𝕐, by

H(z, E_{r}) = Q(E_{r})^{2} [z^{E_{r}} + E_{r}^{z} − {(Id − Q(E_{r})^{2})z^{E_{r}}, z^{E_{r}}, E_{r}}]

Then the horocycle equations in 𝔻 are given by the r functions

Δ̃_{k}(z, E_{k}) = m_{k}(H(z, E_{r})^{−1}, E_{k})^{−1} (1 ≤ k ≤ r)

where H(z, E_{r})^{−1} is the inverse of H(z, E_{r}) in the Jordan algebra 𝕍_{2}(E_{r}).

Remark : In the special case of tube domains, we have 𝕍 = 𝕍_{2}(E_{r}) and Q(E_{r})^{2} = Id. In these easy cases, H(z, E_{r}) ∈ 𝕐 reduces to

H(z, E_{r}) = z^{E_{r}} + E_{r}^{z}.

13 Explicit computations in the case of Jordan algebras

Let 𝕍 be a Jordan triple system of tube type and of rank r and let 𝔻 be the bounded symmetric domain associated.

Lemma 25 With the usual notations, denote

m_{k}^{∗} = m_{r−k} (m_{0} = 1) and E_{k}^{∗} = E_{r} − E_{k}.

Then we have for each 1 ≤ k ≤ r,

m_{k}([z^{E_{r}} + E_{r}^{z}]^{−1}, E_{k})^{−1} = \frac{det[z^{E_{r}} + E_{r}^{z}]}{m_{k}^{∗}([z^{E_{r}} + E_{r}^{z}], E_{k}^{∗})},

where det(y) = m_{r}(y, E_{r}) is the determinant of the Jordan algebra 𝕍_{2}(E_{r}). In other words, for each 1 ≤ k ≤ r,

Δ̃_{k}(z, E_{k}) = \frac{det[z^{E_{r}} + E_{r}^{z}]}{m_{k}^{∗}([z^{E_{r}} + E_{r}^{z}], E_{k}^{∗})}.

Proof For y ∈ 𝕐, we have

Δ_{k}(y^{−1})^{−1} = \frac{det(y)}{Δ_{r−k}^{∗}(y)}

where the notations are the same as in [2]. □ Proposition 26 The following identity holds in the Jordan triple system of tube type 𝕍:

Q(z^{E_{r}} + E_{r}^{z})Q(E_{r}) = B(z, E_{r})^{−1}B(z, z)B(E_{r}, z)^{−1} (z ∈ 𝔻).

In particular,

det(z^{E_{r}} + E_{r}^{z}) = \frac{N (z, z)}{|N (z, E_{r})|^{2}} (z ∈ 𝔻).

We prove the following lemma. Lemma 27 With the same notations, the following identity holds:

Q(z^{E_{r}} + E_{r}^{z})Q(E_{r})B(E_{r}, z) = B(z^{E_{r}}, z − E_{r}).

Proof Applying E_{r}^{z} = z^{E_{r}} + E_{r} + Q(E_{r})z^{E_{r}}, we obtain Q(z^{E_{r}} + E_{r} + Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z)

= Q(z^{E_{r}})Q(E_{r})B(E_{r}, z) + Q(E_{r})^{2}B(E_{r}, z)
+ Q(Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z) + Q(z^{E_{r}}, E_{r})Q(E_{r})B(E_{r}, z)
+ Q(z^{E_{r}}, Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z) + Q(E_{r}, Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z).

We have:

(1) Q(E_{r})^{2}B(E_{r}, z) = B(E_{r}, z)

since Q(E_{r})^{2} = Id for tube type,

(2) Q(Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z) = Q(E_{r})Q(z^{E_{r}})Q(E_{r})^{2}B(E_{r}, z)
= Q(E_{r})Q(z^{E_{r}})B(E_{r}, z)
= Q(E_{r})Q(z)

using JP5 and JP14,

(3) Q(z^{E_{r}}, E_{r})Q(E_{r})B(E_{r}, z) = (D(z^{E_{r}}, E_{r})D(E_{r}, E_{r}) − D(z^{E_{r}}, Q(E_{r})E_{r})) B(E_{r}, z)
= D(z^{E_{r}}, E_{r})B(E_{r}, z)

using JP7 and the fact that D(E_{r}, E_{r}) = 2Id for tube type,

(4) Q(z^{E_{r}}, Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z)
= (Q(E_{r}, z^{E_{r}})D(z^{E_{r}}, E_{r}) − D(z^{E_{r}}, z^{E_{r}})Q(E_{r})) Q(E_{r})B(E_{r}, z)  by JP 8
= Q(E_{r}, z^{E_{r}})Q(E_{r})D(E_{r}, z^{E_{r}})B(E_{r}, z^{E_{r}}) − D(z^{E_{r}}, z^{E_{r}})B(E_{r}, z)  by JP 4
= Q(E_{r}, z^{E_{r}})Q(E_{r})D(E_{r}, z) − Q(E_{r}, z^{E_{r}})Q(E_{r})Q(E_{r}, E_{r})Q(z)  by JP15
− D(z^{E_{r}}, z) + Q(E_{r}, z^{E_{r}})Q(z)  by JP15
= Q(E_{r}, z^{E_{r}})Q(E_{r})D(E_{r}, z) − 2Q(E_{r}, z^{E_{r}})Q(z)
− D(z^{E_{r}}, z) + Q(E_{r}, z^{E_{r}})Q(z).

Finally,

(5) Q(E_{r}, Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z) = D(E_{r}, z^{E_{r}})B(E_{r}, z)
= D(E_{r}, z) − 2Q(E_{r})Q(z)

using JP3 and JP15. Therefore, the quantity

Q(z^{E_{r}} + E_{r} + Q(E_{r})z^{E_{r}})Q(E_{r})B(E_{r}, z)

is equal to

Q(z^{E_{r}})Q(E_{r})B(E_{r}, z) + B(E_{r}, z) + Q(E_{r})Q(z)
+D(z^{E_{r}}, E_{r})B(E_{r}, z) + Q(E_{r}, z^{E_{r}})Q(E_{r})D(E_{r}, z)
−Q(E_{r}, z^{E_{r}})Q(z) − D(z^{E_{r}}, z) + D(E_{r}, z) + 2Q(E_{r})Q(z).

Applying the expression of B(E_{r}, z), we obtain, using Q(E_{r})^{2} = Id,

Q(z^{E_{r}})Q(E_{r}) − Q(z^{E_{r}})Q(E_{r})D(E_{r}, z) + Q(z^{E_{r}})Q(z)
+Id − D(E_{r}, z) + Q(E_{r})Q(z) + Q(E_{r})Q(z) + D(z^{E_{r}}, E_{r}) − D(z^{E_{r}}, E_{r})D(E_{r}, z)
+D(z^{E_{r}}, E_{r})Q(E_{r})Q(z) + Q(E_{r}, z^{E_{r}})Q(E_{r})D(E_{r}, z)
−Q(E_{r}, z^{E_{r}})Q(z) − D(z^{E_{r}}, z) + D(E_{r}, z) + 2Q(E_{r})Q(z)
+Q(z^{E_{r}})Q(z, E_{r}) − Q(z^{E_{r}})Q(z, E_{r}).

Remark that the two last terms appear. Hence

Q(z^{E_{r}})Q(E_{r}) + Q(z^{E_{r}})Q(z) + Id + D(z^{E_{r}}, E_{r}) − D(z^{E_{r}}, z) − Q(z^{E_{r}})Q(z, E_{r})

is equal to B(z^{E_{r}}, z − E_{r}). Moreover, the four terms −D(E_{r}, z), 2Q(E_{r})Q(z), D(E_{r}, z) and −2Q(E_{r})Q(z) vanish. Therefore, it is enough to show that

− Q(y)Q(E_{r})D(E_{r}, z) − D(y, E_{r})D(E_{r}, z) + D(y, E_{r})Q(E_{r})Q(z)
+ Q(E_{r}, y)Q(E_{r})D(E_{r}, z) − Q(E_{r}, y)Q(z) + Q(y)Q(z, E_{r})

vanish. This will give the result with y = z^{E_{r}}. Using JP6 and D(E_{r}, E_{r}) = 2Id, we have

Q(E_{r})D(E_{r}, z) = 2Q(E_{r}, z) + Q(z, E_{r}) − 2Q(E_{r}, z) = Q(z, E_{r}).

In the same way, we have

D(y, E_{r})Q(E_{r}) = Q(y, E_{r}), D(y, E_{r}) = Q(y, E_{r})Q(E_{r})

and the lemma follows. □

Proof (of the proposition) According to the previous lemma, we have

Q(z^{E_{r}} + E_{r}^{z})Q(E_{r}) = B(z^{E_{r}}, z − E_{r})B(E_{r}, z)^{−1}.

Using JP16, we have

B(z, E_{r})B(z^{E_{r}}, z − E_{r}) = B(z, E_{r} + z − E_{r})

hence

B(z, E_{r})^{−1}B(z, z) = B(z^{E_{r}}, z − E_{r})

and the first part of the proposition follows. For the second part, recall that for x ∈ 𝕐(E_{r}) [2]

det(x)^{q} = Det Q(x)Q(E_{r})

where q = 2 + a(r − 1) is the genus of 𝕍. □

Using the second part of this proposition, we obtain Proposition 28 Horocyclic equations in 𝕍 are given by

Δ̃_{k}(z, E_{k}) = \frac{N (z, z)}{|N (z, E_{r})|^{2}m_{k}^{∗}([z^{E_{r}} + E_{r}^{z}], E_{k}^{∗})} (z ∈ 𝔻, 1 ≤ k ≤ r).

In particular, for k = r,

Δ̃_{r}(z, E_{r}) = \frac{N (z, z)}{|N (z, E_{r})|^{2}}

like in the usual Poisson kernel.

The case k = r is also true for non tube type domains. The most important result is the following.

Proposition 29 For (x, y, z) ∈ 𝕍 and each 1 ≤ k ≤ r, let us define

Λ_{k}(z, y, x) = |N (z, y)|^{2}m_{k}^{∗}([z^{y} + y^{z}], x) (m_{r}^{∗} = 1).

Then for each 1 ≤ k ≤ r, Λ_{k} is polynomial with respect to the variables z, y, x and Λ_{k}(z, E_{r}, E_{r}− E_{k}) depends uniquely of the point z and of the tripotent E_{k}. Let

Λ̃_{k}(z, E_{k}) = Λ_{k}(z, E_{r}, E_{r} − E_{k})

and

Δ̃_{k}(z, E_{k}) = \frac{N (z, z)}{Λ̃_{k}(z, E_{k})}.

Then the horocyclic equations for tube domains are

Δ̃_{k}(z, E_{k}) = exp (2 ∑_{i=1}^{k} t_{i}) (1 ≤ k ≤ r).

For k = r, (and it’s also true for non tube type domains) we have

Δ̃_{r}(z, E_{r}) = \frac{N (z, z)}{|N (z, E_{r})|^{2}}

like in the usual Poisson kernel.

In particular, for each 1 ≤ k ≤ r, Δ̃_{k} is a rational map with numerator equal to the generic norm N (z, z). We shall give a proof of the first point of this proposition using the classification. Let us give two examples. Example : Type IV_{n} (Lie balls) 𝕍 = ℂ^{n} = ℝ^{n} ⊗_{ℂ} ℂ where ℝ^{n} is endowed with its euclidian structure < ., . >. For x, y, z ∈ 𝕍, denote

q(z) =< z, z >= ∑_{i=1}^{n} z_{i}^{2} q(x, y) = 2 < x, y >= 2 ∑_{i=1}^{n} x_{i}y_{i}.

then the triple product is given by

{x, y, z} = q(x, ȳ)z + q(z, ȳ)x − 2 < x, z > ȳ.

It is a Jordan triple system of type tube and of rank two. The generic norm is given by

N (x, y) = 1 − q(x, ȳ) + q(x)\overline{q(y)}.

We have

𝔻 = \{z ∈ 𝕍, 2 − q(z, z̄) > 0, 1 − q(z, z̄) + q(z)\overline{q(z)} > 0\},
𝕄_{1} = {z ∈ 𝕍, q(z, z̄) = 1, q(z) = 0},
𝕄_{2} = \{z ∈ 𝕍, q(z, z̄) = 2, q(z, z̄) − q(z)\overline{q(z)} = 1\}.

The manifold 𝔽 of Jordan frame is diffeomorphic to 𝕄_{1} × Δ (where Δ is the open unit disk in ℂ) under [12]

g̃(u, σ) = (u, σū) (u ∈ 𝕄_{1}, σ ∈ Δ).

Fixing u ∈ 𝕄_{1} and σ ∈ Δ, then F = (u, σū) is a Jordan frame in 𝕍. Denote v the maximal tripotent u + σū, then

Δ̃_{2}(z, v) = \frac{1 − q(z, z̄) + q(z)\overline{q(z)}}{|1 − q(z, v̄) + q(z)\overline{q(v)}|^{2}} (z ∈ 𝔻).

Now, we have

Δ̃_{1}(z, u) = \frac{1 − q(z, z̄) + q(z)\overline{q(z)}}{Λ_{1}(z, v, σū)}

and the proposition says that Λ_{1}(z, v, σū) depends only on the minimal tripotent u. But

Λ_{1}(z, v, σū) = |N (z, v)|^{2}q([z^{v} + v^{z}], σ̄u).

For (x, y) ∈ 𝕍^{2}, the quasi-inverse is given by

x^{y} = \frac{x − q(x)y}{N (x, y)} (x, y ∈ 𝕍)

so

Λ_{1}(z, v, σū) = q([N (v, z)(z − q(z)v̄) + N (z, v)(v − q(v)z̄)], σ̄u).

The quantity

N (v, z)(z − q(z)v̄) + N (z, v)(v − q(v)z̄)

is equal to

(1 − q(v, z̄) + q(v)\overline{q(z)})(z − q(z)v̄) + (1 − q(z, v̄) + q(z)\overline{q(v)})(v − q(v)z̄)
= z − q(u, z̄)z − σq(ū, z̄)z + σ\overline{q(z)}z − q(z)ū − σ̄q(z)u + q(z)q(u, z̄)ū
+ σ̄q(z)q(u, z̄)u + σq(z)q(ū, z̄)ū + σσ̄q(z)q(ū, z̄)u − σq(z)\overline{q(z)}ū − σσ̄q(z)\overline{q(z)}u
+ u + σu − q(z, ū)u − σ̄q(z, u)u − σq(z, ū)ū − σ̄σq(z, u)ūσ̄q(z)u
+ σσ̄q(z)ū − σz̄ + σq(x, ū) + σσ̄q(z, u)z̄ − σσ̄q(z)z̄.

Using σσ̄ = 1, q(u, ū) = 1 and q(u) = 0, we obtain Λ_{1}(z, v, σū) =

σ̄q(z, u) − σ̄q(u, z̄)q(z, u) − q(ū, z̄)q(z, u) + \overline{q(z)}q(z, u) − σ̄q(z) + σ̄q(z)q(u, z̄)
+ q(z)q(ū, z̄) − q(z)\overline{q(z)} + 1 − q(z, ū) − σ̄q(z, u) + σ̄q(z) − q(z̄, u)
+ q(z, ū)q(z̄, u) + σ̄q(z, u)q(z̄, u) − σ̄q(z)q(z̄, u).

The terms with σ̄ vanish and Λ_{1}(z, v, σū) is only a polynomial depending only on the point z and on the minimal tripotent u:

Λ̃_{1}(z, u) = |1 − q(z, ū)|^{2} − |q(z) − q(z, u)|^{2}.

Finally, we have

Δ̃_{1}(z, u) = \frac{1 − q(z, z̄) + q(z)\overline{q(z)}}{|1 − q(z, ū)|^{2} − |q(z) − q(z, u)|^{2}} .

Example : Type 𝕍 = I_{2,2} endowed with a frame (U, V). As this system is isomorphic to IV_{4}, we have

Δ̃_{2}(Z, U + V) = \frac{Det(Id − Z^{t}\overline{Z})}{| Det(Id − Z^{t}\overline{U + V})|^{2}} ,

Δ̃_{1}(Z, U) = \frac{Det(Id − Z^{t}\overline{Z})}{|1 − Trace Z^{t}\overline{U}|^{2} − | Det(Z) − Σ_{2}(Z, U)|^{2}} ,

where Σ_{2}(X, Y) = Det(X + Y) − Det(X) − Det(Y).

The first point of proposition 29 is a consequence of the following

Proposition 30 For (x, y, z) ∈ 𝕍^{3}, the product

|N (z, y)|^{2} N ([z^{y} + y^{z}], x)

is polynomial with respect to the variables x, y, z.

In fact, Λ_{k}(z, y, x) is the homogeneous component of degree r − k in the variable x in this product. This result also hold in the non tube type case. Proof Type 𝕍 = I_{p,q}. Let Z, Y, X ∈ 𝕍, we have to compute

Det(Id − (Z^{Y} + Y^{Z})^{t}\overline{X})

which is equal to

Det(Id − [(Id − Z^{t}\overline{Y})^{−1}Z + (Id − Y^{t}\overline{Z})^{−1}Y]^{t}\overline{X}).

We obtain

Det(Id − Z^{t}\overline{Y})^{−1} Det (Id − Z^{t}\overline{Y} − Z^{t}\overline{X} − (Id − Z^{t}\overline{Y})(Id − Y^{t}\overline{Z})^{−1}Y^{t}\overline{X}) .

If A, B, C, D, with D invertible, are four matrices, remark that

(\begin{matrix}D & 0 \\ B & Id \end{matrix}) (\begin{matrix}Id & D^{−1}C \\ 0 & A − BD^{−1}C \end{matrix}) = (\begin{matrix}D & C \\ B & A \end{matrix}) .

So

Det(D) Det(A − BD^{−1}C) = Det (\begin{matrix}D & C \\ B & A \end{matrix}) .

Aplying this result with

A = Id − Z^{t}\overline{Y} − Z^{t}\overline{X},  B = Id − Z^{t}\overline{Y} ,  C = Y^{t}\overline{X},  D = Id − Y^{t}\overline{Z},

we obtain

Det (Id − Z^{t}\overline{Y} − Z^{t}\overline{X} − (Id − Z^{t}\overline{Y})(Id − Y^{t}\overline{Z})^{−1}Y^{t}\overline{X}) = \frac{Det (\begin{matrix}Id − Y^{t}\overline{Z} & Y^{t}\overline{X} \\ Id − Z^{t}\overline{Y} & Id − Z^{t}\overline{Y} − Z^{t}\overline{X} \end{matrix})}{Det(Id − Y^{t}\overline{Z})}.

So

Det(Id − [(Id − Z^{t}\overline{Y})^{−1}Z + (Id − Y^{t}\overline{Z})^{−1}Y]^{t}\overline{X}) = \frac{Det (\begin{matrix}Id − Y^{t}\overline{Z} & Y^{t}\overline{X} \\ Id − Z^{t}\overline{Y} & Id − Z^{t}\overline{Y} − Z^{t}\overline{X} \end{matrix})}{Det(Id − Z^{t}\overline{Y}) Det(Id − Y^{t}\overline{Z})}.

The numerator of this fraction is a polynomial in the variable X, Y, Z, so the result is proved in this case.

The same proof give the result in the cases II et III using pfaffian instead of determinant.

Type 𝕍 = IV_{n}. For (z, y, x) ∈ 𝕍^{3}, we have

|N (z, y)|^{2}N ([z^{y} + y^{z}], x) = |N (z, y)|^{2} (1 − q(z^{y} + y^{z}, x̄) + q(z^{y} + y^{z})\overline{q(x)}) .

The quasi-inverse is given by

z^{y} = \frac{z − q(z)ȳ}{N (z, y)}, y^{z} = \frac{y − q(y)z̄}{N (y, z)},

hence, by linearity,

|N (z, y)|^{2}q(z^{y} + y^{z}, x̄)

is a polynomial in the variable x, y, z. To prove the result for

|N (z, y)|^{2}q(z^{y} + y^{z})\overline{q(x)},

we can compute q(z^{y} + y^{z}). We have

q(z^{y} + y^{z}) = q(z^{y}) + q(z^{y}, y^{z}) + q(y^{z}),

and

|N (z, y)|^{2}q(z^{y}, y^{z})

is polynomial in z, y by linearity. By symmetry, it is sufficient to check that so is

|N (z, y)|^{2}q(z^{y}).

We have

q(z^{y}) = \frac{q(z − q(z)ȳ)}{N (z, y)^{2}},

so

|N (z, y)|^{2}q(z^{y}) = N (y, z)\frac{q(z − q(z)ȳ)}{N (z, y)}.

But

q(z − q(z)ȳ) = q(z) − q(z)q(z, ȳ) + q(z)^{2}\overline{q(y)} = q(z)N (z, y),

so the result is also proved. The product

|N (z, y)|^{2}N ([z^{y} + y^{z}], x)

is a polynomial with respect to the variable x, y, z.

Type V I. [9] (Exceptional domain of dimension 27). Let 𝕎 = ℍ_{3}(O_{ℂ}), the space of 3 × 3 matrices with entries in the space O_{ℂ} of octonions over ℂ, which are hermitian with respect to the Cayley conjugaison x → x̃. An element a ∈ 𝕎 is of the form JX 1

A = (\begin{matrix}α_{1} & a_{3} & ã_{2} \\ ã_{3} & α_{2} & a_{1} \\ a_{2} & ã_{1} & α_{3} \end{matrix} )

where α_{1}, α_{2}, α_{3} ∈ ℂ and a_{1}, a_{2}, a_{3} ∈ O_{ℂ}. Instead of JX1, one often writes JX 2

A = ∑_{j=1}^{3} α_{j}e_{j} + ∑_{j=1}^{3} F_{j}(a_{j})

The determinant of A ∈ 𝕎, written of the form JX1 is defined by JX 3

det A = α_{1}α_{2}α_{3} − ∑_{j=1}^{3} α_{j}n(a_{j}) + a_{1}(a_{2}a_{3}) + (ã_{3}ã_{2})ã_{1}

where n(u) = uũ denotes the Cayley norm of an octonion u ∈ O_{ℂ}. The above relation is the convenient generalization of the ”Sarrus rule” for ordinary 3×3 matrices. The (complex) scalar product on 𝕎 is the symmetric non singular bilinear form defined by JX 4

(A : B) = ∑_{j=1}^{3} α_{j}β_{j} + ∑_{i+1}^{3}(a_{j} : b_{j}) (A, B ∈ 𝕎);

here (x : y) = xỹ + yx̃ (x, y ∈ O_{ℂ}) is the symmetric bilinear form on O_{ℂ} associated to the Cayley norm n; A and B are of the form JX1. The adjoint of an element A is A^{#} defined by JX 5

A^{#} = ∑_{i=1}^{3}(α_{j}α_{k} − n(a_{j}))E_{i} + ∑_{i=1}^{3} F̃_{i}(a_{j}a_{k} − α_{i}α̃_{i});

in the right hand side of this formula, (i, j, k) is the even permutation of (1, 2, 3) determined by i, and F̃_{i} denotes the transposed F_{i}(c̃) of F_{i}(c). The product A × B of A, B ∈ 𝕎 is defined by JX 6

A × B = (A + B)^{#} − A^{#} − B^{#} (A, B ∈ 𝕎),

The trilinear form T : 𝕎 × 𝕎 × 𝕎 → 𝕎 defined by JX 7

T (A, B, C) = (A × B : C) (A, B, C ∈ 𝕎),

is then symmetric and one has JX 8

det A = \frac{1}{6}T (A, A, A) = \frac{1}{3}(A^{#} : A) (A ∈ 𝕎).

We are now able to define the Jordan triple structure of 𝕎. The quadratic operator is defined by JX 9

Q(A)B = (A : \overline{B})A − A^{#} × \overline{B} (A, B ∈ 𝕎)

where \overline{B} is the conjugate of B. The hermitian product in 𝕎 is defined by

(A, B) = (A : \overline{B}) (A, B ∈ 𝕎).

It is easily checked that the above operations in 𝕎 are compatible with the conjugation, in the sense that

det \overline{A} = \overline{det A},  (\overline{A} : \overline{B}) = \overline{(A : B)},  \overline{A}^{#} = \overline{A^{#}},  \overline{A} × \overline{B} = \overline{A × B}

The rank is r = 3. The generic norm is

JX 10

N (A, B) = 1 − m_{1}(A, B) + m_{2}(A, B) − m_{3}(A, B), (A, B ∈ 𝕎)

where

m_{1}(A, B) = (A : \overline{B})
m_{2}(A, B) = (A^{#} : \overline{B}^{#})
m_{3}(A, B) = det A \overline{det B}.

The quasi-inverse A^{B} is then given by

JX 11

N (A, B)A^{B} = A − A^{#} × \overline{B} + det A \overline{B}^{#}.

A frame of 𝕎 is given by the tree matrices e_{1}, e_{2}, e_{3}

e_{1} = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})  e_{2} = (\begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix})  e_{3} = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) .

Define the subspaces

F_{1} = F_{1}(O_{ℂ}), F_{2} = F_{2}(O_{ℂ}), and F_{3} = F_{3}(O_{ℂ}),

then the Peirce decomposition for the tripotent e_{1} is

𝕎_{2}(e_{1}) = ℂe_{1},  𝕎_{1}(e_{1}) = F_{2} ⊕ F_{3},  𝕎_{0}(e_{1}) = ℂe_{2} ⊕ ℂe_{3} ⊕ F_{3}.

The triple subsystem 𝕎_{0}(e_{1}) is isomorphic to the classical system of type IV_{10}. The subsystem 𝕎_{1}(e_{1}) is isomorphic to the Jordan triple system of type V. The subspaces of the simultaneous Peirce decomposition with respect to the frame (e_{1}, e_{2}, e_{3}) are

𝕎_{i,i} = ℂe_{i},  𝕎_{i,j} = F_{k}  (1 ≤ i, j, k ≤ 3, i < j, i, j ≠ k).

The numerical invariants are a = 8, b = 0. It’s a tube domain. The following identities hold

JX 12

A^{##} = (det A)A

JX 13

det(A^{#}) = (det A)^{2}

JX 14

A^{#} × (A × B) = (det A)B + (A^{#} : B)A

JX 15

A × (A^{#} × C) = (det A)C + (A : C)A^{#}

JX 16

A^{#} × B^{#} + (A × B)^{#} = (A : B^{#})A + (B : A^{#})B

In view of JX10, we have

N (z^{y} + y^{z}, x) = 1 − (z^{y} + y^{z}, x) + ((z^{y} + y^{z})^{#}, x^{#}) − det(z^{y} + y^{z})\overline{det x}.

So, it is sufficient to check that

C_{1}(z, y) = |N (z, y)|^{2}(z^{y} + y^{z}),
C_{2}(z, y) = |N (z, y)|^{2}(z^{y} + y^{z})^{#},
C_{3}(z, y) = |N (z, y)|^{2} det(z^{y} + y^{z}),

are polynomial in z, y. As N (z, y)z^{y} is polynomial, it is clear that C_{1}(z, y) is polynomial

C_{1}(z, y) = U(z, y)N (y, z) + N (z, y)U(y, z)

where

U(z, y) = z − z^{#} × \overline{y} + (det z)\overline{y}^{#}

is the numerator of z^{y} and U(y, z) is the numerator of y^{z}. For the two other terms, let us prove the following lemmas.

Lemma 31 We have

U(z, y)^{#} = N (z, y)(z^{#} − (det z)\overline{y}).

This means also that

N (z, y)(z^{y})^{#} = z^{#} − (det z)\overline{y}.

Proof We have

U(z, y)^{#} = z^{#} + (z^{#} × \overline{y})^{#} + (det z)^{2}\overline{y}^{##} − z × (z^{#} × \overline{y})
+ (det z)(z × \overline{y}^{#}) − (det z)(z^{#} × \overline{y}) × \overline{y}^{#}.

Using JX15, we have

z × (z^{#} × \overline{y}) = (det z)\overline{y} + (z : \overline{y})z^{#}.

Using JX12, we have (det z)z = z^{##} and using JX16 with z^{#} and \overline{y}, we have

(z^{#} × \overline{y})^{#} + (det z)(z × \overline{y}^{#}) = (z^{#} : \overline{y}^{#})z^{#} + (\overline{y} : (det z)z)\overline{y}.

Finally, using JX14 with \overline{y} and z^{#}, we have

(z^{#} × \overline{y}) × \overline{y}^{#} = \overline{y}^{#} × (\overline{y} × z^{#}) = (det \overline{y})z^{#} + (\overline{y}^{#} : z^{#})\overline{y}.

At last, we obtain

U(z, y)^{#} = (1 − (z : \overline{y}) + (z^{#} : \overline{y}^{#}) − det z det \overline{y}) (z^{#} − (det z)\overline{y}) ,

that is

U(z, y)^{#} = N (z, y) (z^{#} − (det z)\overline{y}) .
□

Lemma 32 We have

det(U(z, y)) = N (z, y)^{2}(det z).

This also means that

N (z, y) det(z^{y}) = det(z).

Proof We have

3 det(U(z, y)) = (U(z, y)^{#} : U(z, y)) = N (z, y)(V (z, y) : U(z, y))

where

V (z, y) = z^{#} − (det z)\overline{y}.

So, it is sufficient to compute the product

(z^{#} − (det z)\overline{y} : z − z^{#} × \overline{y} + (det z)\overline{y}^{#}) .

We have

(z^{#} : z) = 3 det z,
(z^{#} : z^{#} × \overline{y}) = 2(z^{##} : \overline{y}) = 2(det z)(z : \overline{y}),
(\overline{y} : z^{#} × \overline{y}) = 2(\overline{y}^{#} : z^{#}),

so

(V (z, y) : U(z, y)) = 3(det z)N (z, y),

and

det(U(z, y)) = N (z, y)^{2}(det z).
□

Let us prove, now, that the two terms

C_{2}(z, y) = |N (z, y)|^{2}(z^{y} + y^{z})^{#},
C_{3}(z, y) = |N (z, y)|^{2} det(z^{y} + y^{z}),

are polynomial with respect to z and y. We have

(z^{y} + y^{z})^{#} = (z^{y})^{#} + (y^{z})^{#} + z^{y} × y^{z}.

Using lemma 31 and JX11, we obtain

(z^{y})^{#} = \frac{U(z, y)^{#}}{N (z, y)^{2}} = \frac{V (z, y)}{N (z, y)},

(y^{z})^{#} = \frac{U(y, z)^{#}}{N (y, z)^{2}} = \frac{V (y, z)}{N (y, z)},

so

C_{2}(z, y) = V (z, y)N (y, z) + N (z, y)V (y, z) + U(z, y) × U(y, z).

For the third term, write

det(z^{y} + y^{z}) = det(z^{y}) + det(y^{z}) + ((z^{y})^{#} : y^{z}) + ((y^{z})^{#} : z^{y}) .

Using lemma 31 and lemma 32, we obtain

C_{3}(z, y) = (det z)N (y, z) + N (z, y)(det y) + (V (z, y) : U(y, z)) + (V (y, z) : U(z, y)).

So the product

|N (z, y)|^{2}N (z^{y} + y^{z}, x)

is again polynomial with respect to the variables x, y, z. Type V The result is also true because the type V is imbedded in the type V I such that the restriction of the generic norm of V I to V is exactly the generic norm of V. [7] (4.17). □

14 Eigenfunctions of invariant differential operators. The Poisson-Furstenberg kernel

For (s_{1}, s_{2}, ⋯ s_{r}) ∈ ℂ^{r}, the N invariant function

Δ̃(z, F) = Δ̃_{1}(z, E_{1})^{s_{1}−s_{2}} Δ̃_{2}(z, E_{2})^{s_{2}−s_{3}} ⋯ Δ̃_{r}(z, E_{r})^{s_{r}}

is an eigenfunction of each invariant differential operator on 𝔻, whose value at the point

∑_{i=1}^{r}(th t_{i})e_{i} is exp(2s_{1}t_{1}) exp(2s_{2}t_{2}) ⋯ exp(2s_{r}t_{r}).

The G-invariant laplacian L is the most important G-invariant operator. If {c_{1}, c_{2}, ⋯ c_{n}} is an orthonormal basis of 𝕍 with respect to (., .), then

L = ∑_{i=1}^{n}(B(z, z)c_{j}, c_{i})\frac{∂^{2}}{∂z_{i}∂\overline{z_{j}}}

The expression of the G-invariant laplacian is N-invariant coordinates is given by ([4] page 261):

Δ_{N}(L) = ∑_{i=1}^{r} \frac{d}{dt_{i}^{2}} − 2(b + 1) ∑_{i=1}^{r} \frac{d}{dt_{i}} − 2a ∑_{i=1}^{r}(r − i)\frac{d}{dt_{i}} .

It is easy to show that the function Δ̃(z, F) is an eigenfunction of the G-invariant laplacian by computing

Δ_{N}(L) exp (2 ∑_{i=1}^{r} s_{i}t_{i})

= (∑_{i=1}^{r} 4s_{i}^{2} − 4(b + 1)s_{i} − 4a(r − i)s_{i}) exp (2 ∑_{i=1}^{r} s_{i}t_{i})

= (∑_{i=1}^{r} 2s_{i}(2s_{i} − 2(b + 1) − 2a(r − i))) exp (2 ∑_{i=1}^{r} s_{i}t_{i}) .

Hence

L_{𝔻} Δ̃_{1}(z, E_{1})^{s_{1}−s_{2}} Δ̃_{2}(z, E_{2})^{s_{2}−s_{3}} ⋯ Δ̃_{r}(z, E_{r})^{s_{r}} =

(∑_{i=1}^{r} 2s_{i}(2s_{i} − 2(b + 1) − 2a(r − i))) Δ̃_{1}(z, E_{1})^{s_{1}−s_{2}} Δ̃_{2}(z, E_{2})^{s_{2}−s_{3}} ⋯ Δ̃_{r}(z, E_{r})^{s_{r}}.

A function f is said to be weakly G-harmonic if L_{𝔻}.f = 0. Note that there is two weakly G-harmonic function:

(1) The case

s_{i} = \frac{n}{r} (1 ≤ i ≤ r)

which give the usual Poisson kernel over the Shilov boundary,

P(z, E_{r}) = Δ̃_{r}(z, E_{r})^{\frac{n}{r}} .

In fact

\frac{2n}{r} = 2(b + 1) + a(r − 1)

so

∑_{i=1}^{r} 2s_{i}(2s_{i} − 2(b + 1) − 2a(r − i)) = \frac{2n}{r}a ∑_{i=1}^{r}(2i − r − 1) = 0.

(2) The case

s_{i} = (b + 1) + a(r − i) (1 ≤ i ≤ r)

which give the N-invariant function

P(z, F) = Δ̃_{1}(z, E_{1})^{a}Δ̃_{2}(z, E_{2})^{a} ⋯ Δ̃_{r−1}(z, E_{r−1})^{a}Δ̃_{r}(z, E_{r})^{b+1}.

If z = ∑_{i=1}^{r}(th t_{i})e_{i}, then

P(z, F) = ∏_{k=1}^{r−1} exp (2a ∑_{i=1}^{k} t_{i}) exp (2(b + 1) ∑_{i=1}^{r} t_{i})

= exp (2((b + 1) ∑_{i=1}^{r} t_{i} + a ∑_{i=1}^{r}(r − i)t_{i}))

= exp (2ρ(t_{1}, t_{2}, ⋯ t_{r})) .

So it is clear that

z ⟶ P(z, F)

is the Poisson-Furstenberg kernel of the domain 𝔻 with respect to the frame F = (e_{1}, e_{2}, ⋯ e_{r}).

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