Journal of Lie Theory Volume 12 (2002) 217–243 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Poisson Kernels and Pluriharmonic H^{2}-Functions on Homogeneous Siegel Domains

Bartosz Trojan^{∗}

Communicated by J. Faraut

Abstract. In the paper we prove that a real function F defined on a homogeneous not necessarily symmetric Siegel domain satisfying an H^{2} condition is pluriharmonic if and only if HF = 0, LF = 0, LF = 0, where H, L, L are second order differential operators. This generalizes the result of [3] where symmetric domains were considered. Our approach to study non-symmetric case is based on T -algebras introduced by Vinberg in [11].

1. Introduction

This paper treats pluriharmonic functions on homogeneous Siegel domains. These are the functions locally characterized by the equations

∂_{z_{j}}∂_{\overline{z}_{k}}F = 0 for j, k = 1, …, n

n being the dimension of the underlying complex space. There are other local characterizations, one of them being Forelli’s theorem [5].

Here we are mostly interested in a global question i.e., we impose a growth condition, and we look for a characterization of pluriharmonic functions among the ones satisfying it. Similar problems have already been studied by various authors (for recent results see e.g. [1], [3], [2], [7], [8]), all of them being interested in symmetric domains while here we do not need symmetry at all.

Let Ω be an irreducible homogeneous cone, and let D be a corresponding homogeneous Siegel domain. We identify D with a solvable Lie group S that acts simply transitively on D as a group of biholomorphisms. We study S – invariant real elliptic degenerate second order operators on D, which annihilate holomorphic functions and, consequently, their real and imaginary part. Such operators will be called admissible. The particular interest in restricting our attention to second order degenerate elliptic operators is caused by the fact that for such operators there is a very well understood potential theory. Theory of

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag bounded functions harmonic with respect to an S –invariant operator L satisfying Hörmander condition was studied in [2], [4] and was based on an earlier work of H. Furstenberg, Y. Guivarc’h and A. Raugi. The basic result of this theory we use here is a description of bounded L–harmonic functions as Poisson integrals on a nilpotent subgroup N(L) of S. For an admissible L on a Siegel domain the boundary N(L) always contains a group N(Φ) which acts transitively on the Shilov boundary. In our case L is an arbitrary elliptic admissible operator and then we choose two other operators: H and L in the way that N(L+H) = N(Φ). Thus

∗ The author was partly supported by KBN grant 2P03A04316 and Foundation for Polish Sciences, Subsidy 3/99.

F (s) = ∫_{N(Φ)} f(s ∘ v)P (v)dv

where P is the Poisson kernel corresponding to L = L + H, and v ↦ s ∘ v is the action of S on the Bergman–Shilov boundary N(Φ). L is closely related to the tangential holomorphic structure of the Siegel domain of type two and it does not appear in the tube case. We show that three operators L, H and L are sufficient to characterize pluriharmonic functions F with (H^{2}) growth condition (Theorem 5.1)

sup_{z∈D} ∫_{N(Φ)} |F (w · z)|^{2}dw < ∞.  (H^{2})

For symmetric domains, this theorem was proved in [3]. Our strategy is to prove that the support of the integrated representation U_{f}^{λ} is included in \overline{Ω} ∪ −\overline{Ω}. For this we use the operator H which is basically the Laplace–Beltrami operator on a product of upper-planes. The proof exploits both the algebra of the underlying cone, and the Fourier analysis on N(Φ). The latter is pretty much the same as in [3]. Our main contribution here is in the algebraic part. (Section 2.)

Let V = ⨁_{1≤i≤jr} X_{ij} be the normal decomposition of the clan V , and c_{1}, …, c_{r} the corresponding system of simple idempotents. The authors of [3] used heavily the fact that for a symmetric cone X_{ij} ’s do not vanish. In this paper we have been able to overcome this difficulty by showing in fact that when Ω is irreducible there are enough of non-vanishing X_{ij} ’s.

Let S_{0} be a triangular group acting simply transitively on Ω. To push the argument through we study carefully the action of the S_{0} on V . Modulo a set of Lebesgue measure 0, V is the sum of the open orbits O_{η} of S_{0} where

η = ∑_{j=1}^{r} η_{j}c_{j}

for η ∈ {−1, 1}^{r}. The action of S_{0} on any of them is simply transitive and identifies O_{η} with S_{0} via the diffeomorphism (Theorem 2.9)

s ↦ s ∘ ∑_{j=1}^{r} η_{j}c_{j}.

The diagonalization of any non-degenerate element of V (Proposition 2.7) not only allows us to prove the main theorem, but also that the Fourier transform of P is smooth on the open orbits O_{η} (Theorem 4.5).

The author would like to express his gratitude to E. Damek, A. Hulanicki and R. Penney for valuable comments and suggestions. In particular, professor R. Penney strong belief, that the [3]–result should generalize beyond symmetric domains, encouraged the author a lot.

2. Homogeneous cones

Let Ω be a homogeneous cone in a vector space V . We are going to describe an algebraic structure of V . First, we state some definitions and facts.

Definition 2.1. A matrix algebra of rank r is an algebra U bigraded by subspaces U_{ij}, i, j = 1, …, r, such that U_{ij}U_{jk} ⊂ U_{ik}, and for j ≠ l, U_{ij}U_{lk} = 0.

Definition 2.2. An involution of a matrix algebra U is a linear mapping ^{⋆} of U onto itself that satisfies the following conditions:

i. x^{⋆⋆} = x;

ii. (xy)^{⋆} = y^{⋆}x^{⋆};

iii. U_{ij}^{⋆} = U_{ji};

for all x, y ∈ U.

Let U be an algebra with involution ^{⋆}. We define the subspace of Hermitian matrices in U

X = {x ∈ U| x^{⋆} = x}

and

T = ⨁_{1≤i≤j≤r} U_{ij},

the subalgebra of U consisting in upper triangular matrices.

Let U_{ii} = ℝc_{i}. We denote by ρ the unique isomorphism of U_{ii} onto the algebra of real numbers ℝ. For a matrix x ∈ U,

x = ∑_{i=1}^{r} x_{ii} + ∑_{i≠j} x_{ij},

we define its trace tr as follows

tr x = ∑_{i=1}^{r} n_{i}ρ(x_{ii}),  (1)

n_{i} = 1 + \frac{1}{2} ∑_{{j=1 \atop j≠i}}^{r} dim U_{ij}.  (2)

We need the following notation

[xy] = xy − yx,
[xyz] = x(yz) − (xy)z.

220 Trojan

Definition 2.3. A matrix algebra U with an involution ^{⋆} is called a T –algebra if the following conditions are satisfied:

i. for i = 1, …, r, U_{ii} = ℝc_{i};

ii. for x_{ij} ∈ U_{ij}, c_{i}x_{ij} = x_{ij}c_{j} = x_{ij}; iii. tr[xy] = 0; iv. tr[xyz] = 0; v. if x ≠ 0, then tr xx^{⋆} > 0; vi. for all t, u, w ∈ T , [tuw] = 0; vii. for all t, u ∈ T , [tuu^{⋆}] = 0.

For each matrix x ∈ U, we put

\overline{x} = \frac{1}{2} ∑_{i=1}^{r} x_{ii} + ∑_{i<j}^{r} x_{ij},

\underline{x} = \frac{1}{2} ∑_{i=1}^{r} x_{ii} + ∑_{j<i}^{r} x_{ij},

and define a bilinear operator △ by the formula

x△y = \overline{x}y + y\underline{x}.  (3)

Let

S_{0} = {t ∈ T | t_{ii} > 0, i = 1, …, r}.

The product in S_{0} is associative by property vi. Thus S_{0} is open in T and it is a connected Lie group. Its Lie algebra S_{0} can be identified with T with the bracket

[X, Y] = [XY].

Then we have S_{0} = N_{0} ⊕ A where N_{0} = ⨁_{1≤i<j≤r} U_{ij} and A = ⨁_{i=1}^{r} U_{ii}. Let N_{0} = exp N_{0}, A = exp A. Definition 2.4. An algebra L with linear form s and multiplication △ is called a clan if the following conditions hold:

i. the operator L(x) defined by L(x)y = x△y has only real eigenvalues; ii. [L(x), L(y)] = L(x△y − y△x); iii. s(x△y) = s(y△x); iv. if x ≠ 0, then s(x△x) > 0;

for all x, y ∈ L.

For every clan L with a unit element e there exists a normal decomposition (see [11]). This means that L has a direct sum decomposition

L = ⨁_{1≤i≤j≤r} L_{ij}

such that the subspaces L_{ij} are mutually orthogonal with respect to the scalar product

(x|y) = s(x△y) for x, y ∈ L,

and the following properties hold:

i. for each 1 ≤ i ≤ r there exists an idempotent c_{i} such that L_{ii} = ℝc_{i};

ii. for 1 ≤ i ≤ j ≤ k,

L_{ij}△L_{jk} ⊂ L_{ik}

and

L_{jk}△L_{ik} + L_{ik}△L_{jk} ⊂ L_{ij};

iii. for i ≤ j, k ≤ l, if j ≠ k and j ≠ l, then

L_{ij}△L_{kl} = 0;

iv. for i < j and x_{ij} ∈ L_{ij},

c_{i}△x_{ij} = \frac{1}{2}x_{ij},

x_{ij}△c_{i} = 0,

c_{j}△x_{ij} = \frac{1}{2}x_{ij},

x_{ij}△c_{j} = x_{ij}.

The number r is invariant under isomorphism and is called the rank of the clan L or the rank of the associated cone Ω(L).

Let us define the subspaces X_{ij} for 1 ≤ i < j ≤ r by

X_{ij} = X ∩ (U_{ij} + U_{ji}) ,
X_{ii} = U_{ii}.

Then we can state the following

Theorem 2.5. ([11]) The subspace of Hermitian matrices X in a T –algebra U with the multiplication given by the formula (3) and a linear form tr defined by (1) is a clan with a unit element. Moreover, the decomposition X = ⨁_{1≤i≤j≤r} X_{ij} is a normal decomposition. Conversely, let L be a clan with a unit element. Then there is a unique T –algebra U such that L is isomorphic to the clan X.

Let Ω(X) be the homogeneous cone associated with the clan X. Then

Ω(X) = {ss^{⋆}| s ∈ S_{0}}

and the mapping s ↦ ss^{⋆} is one-to-one. The group S_{0} acts simply transitively on Ω(X) by

t ∘ (ss^{⋆}) = (ts)(s^{⋆}t^{⋆}).  (4)

The transformation (4) corresponding to t is the restriction to Ω(X) of a linear transformation π(t) ∈ GL(X). For t = exp y, y ∈ T , the differential dπ(y) of π(t) is given by

dπ(y)x = yx + xy^{⋆} = L(y + y^{⋆})x

for x ∈ X.

Let G be the identity component of the group G(Ω) of all transformations in GL(X) which leave Ω(X) invariant. Then S_{0} is a maximal triangular subgroup of G.

Let Ω′(X) denote the open dual homogeneous cone defined by

Ω′(X) = \{x′ ∈ X | (x|x′) > 0, ∀x ∈ \overline{Ω(X)} − \{0\}\}.

It was proved in [11] that

Ω′(X) = Ω^{⋆}(X)

and the group S_{0}^{⋆} acts simply transitively on Ω′(X). Thus

S′_{0} = S_{0}^{⋆}.

Here Ω^{⋆}(X) and S_{0}^{⋆} denote the images under the involution ^{⋆} of Ω(X) and S_{0}, respectively. We shall denote the open dual cone to Ω by Ω^{⋆}.

In the T –algebra U we consider the subspaces

X^{k} = X ∩ ⨁_{i,j=1}^{k} U_{ij}

for k = 1, …, r. With every element x ∈ X,

x = ∑_{i=1}^{r} x_{ii} + ∑_{i<j}^{r} x_{ij} + ∑_{i<j}^{r} x_{ij}^{⋆},

we associate a sequence of matrices x^{k} ∈ X^{k}, k = 1, …, r, as follows

x^{r} = x,  (5)
x^{k−1} = ∑_{i,j=1}^{k−1} (ρ(x_{kk}^{k})x_{ij}^{k} − x_{ik}^{k}x_{kj}^{k}) .  (6)

We put

p_{k}(x) = ρ(x_{kk}^{k}) for k = 1, …, r.

We define

J = {x ∈ X | p_{k}(x) ≠ 0, k = 1, …, r}.

Since p_{k} are non-zero polynomials, the set J^{c} is closed in X and has measure 0 in X.

The following Lemma 2.6 and Proposition 2.7 are based on [11] Lemma 3 and Proposition 2.

Lemma 2.6. For a_{1}, …, a_{r} ∈ ℝ, we put a = ∑_{i=1}^{r} a_{i}c_{i}. Let

x = t (∑_{i=1}^{r} a_{i}c_{i}) t^{⋆}

for t ∈ S_{0}. Then

x_{ij}^{k} = ( ∏_{s=k+1}^{r} p_{s}(x)) ∑_{p=1}^{k} a_{p}t_{ip}t_{jp}^{⋆}  (7)

for i, j = 1, …, k and k = 1, …, r.

Proof. Since for every k = 1, …, r and 1 ≤ i, j ≤ k

(a_{k}t_{ik}c_{k})t_{jk}^{⋆} = t_{ik}(a_{k}c_{k}t_{jk}^{⋆}),

we have [tat^{⋆}] = 0.

The proof of the formula (7) is inductive. For k = r we simply get

x_{ij}^{r} = ∑_{p=1}^{k} t_{ip}a_{p}t_{jp}^{⋆}.

Let us assume that (7) holds for some k, 1 ≤ k ≤ r. Then

x_{ik}^{k} = ( ∏_{s=k+1}^{r} p_{s}(x)) ∑_{p=1}^{k} a_{p}t_{ip}t_{kp}^{⋆} = ( ∏_{s=k+1}^{r} p_{s}(x)) a_{k}ρ(t_{kk})t_{ik},

x_{kj}^{k} = ( ∏_{s=k+1}^{r} p_{s}(x)) ∑_{p=1}^{k} a_{p}t_{kp}t_{jp}^{⋆} = ( ∏_{s=k+1}^{r} p_{s}(x)) a_{k}ρ(t_{kk})t_{jk}^{⋆},

and

ρ(x_{kk}^{k}) = ( ∏_{s=k+1}^{r} p_{s}(x)) ρ (∑_{p=1}^{k} a_{p}t_{kp}t_{kp}^{⋆}) = ( ∏_{s=k+1}^{r} p_{s}(x)) ρ(t_{kk})^{2}a_{k}.

For i, j = 1, …, k − 1, we have

x_{ij}^{k−1} = ρ(x_{kk}^{k})x_{ij}^{k} − x_{ik}^{k}x_{kj}^{k}

= (∏_{s=k}^{r} p_{s}(x)) ∑_{p=1}^{k} a_{p}t_{ip}t_{jp}^{⋆} − ( ∏_{s=k+1}^{r} p_{s}(x))^{2} ρ(t_{kk})^{2}a_{k}^{2}t_{ik}t_{jk}^{⋆}

= (∏_{s=k}^{r} p_{s}(x)) (∑_{p=1}^{k} a_{p}t_{ip}t_{jp}^{⋆} − a_{k}t_{ik}t_{jk}^{⋆}) = (∏_{s=k}^{r} p_{s}(x)) ∑_{p=1}^{k−1} a_{p}t_{ip}t_{jp}^{⋆}.

which finishes the proof.

224 Trojan Proposition 2.7. If x ∈ X, then the element x belongs to J if and only if there are t ∈ S_{0} and η ∈ {−1, 1}^{r} such that

x = t (∑_{i=1}^{r} η_{i}c_{i}) t^{⋆}.  (8)

Moreover, the operator t and the sequence η are unique. Proof. First, we show that every x of the form (8) belongs to J. Let

x = t (∑_{i=1}^{r} η_{i}c_{i}) t^{⋆}

for t ∈ S_{0} and η ∈ {−1, 1}^{r}. By Lemma 2.6,

p_{i}(x) = ( ∏_{s=i+1}^{r} p_{s}(x)) ρ(t_{ii})^{2}η_{i} ≠ 0 for i = 1, …, r,  (9)

and so x ∈ J. Moreover, for i = 1, …, r

η_{i} = sign ∏_{s=i}^{r} p_{s}(x),  (10)
t_{ii} = \sqrt{|\frac{p_{i}(x)}{∏_{s=i+1}^{r} p_{s}(x)}|}c_{i}.  (11)

Further, by Lemma 2.6, we have for 1 ≤ i < j ≤ r

x_{ij}^{j} = ( ∏_{s=j+1}^{r} p_{s}(x)) η_{j}ρ(t_{jj})t_{ij}.

By (9)–(11), we get

t_{ij} = \frac{η_{j}η_{j+1}x_{ij}^{j}}{\sqrt{|∏_{s=j}^{r} p_{s}(x)|}} for 1 ≤ i < j < r,  (12)

and

t_{jr} = \frac{η_{r}x_{jr}}{\sqrt{|p_{r}(x)|}} for j = 1, …, r.  (13)

Thus t and η are determined by x.

Conversely, if x ∈ J, then we define t ∈ S_{0} and η ∈ {−1, 1}^{r} by formulas (10)–(13). Let

y = t (∑_{i=1}^{r} η_{i}c_{i}) t^{⋆}.

Then we have for 1 ≤ i ≤ j ≤ r

y_{ij} = ∑_{p=j}^{r} η_{p}t_{ip}t_{jp}^{⋆} = η_{j}ρ(t_{jj})t_{ij} + ∑_{p=j+1}^{r} η_{p}t_{ip}t_{jp}^{⋆}.

By (10)–(12) and (5), we may write

\begin{matrix}y_{ij} = & \frac{x_{ij}^{j}}{∏_{s=j+1}^{r} p_{s}(x)} + η_{j+1}t_{i,j+1}t_{j,j+1}^{⋆} + ∑_{p=j+2}^{r} η_{p}t_{ip}t_{jp}^{⋆} \\ = & \frac{x_{ij}^{j}}{∏_{s=j+1}^{r} p_{s}(x)} + \frac{x_{i,j+1}^{j+1} (x_{j,j+1}^{j+1})^{⋆}}{∏_{s=j+1}^{r} p_{s}(x)} + ∑_{p=j+2}^{r} t_{ip}η_{p}t_{jp}^{⋆} \\ = & \frac{x_{ij}^{j+1}}{∏_{s=j+2}^{r} p_{s}(x)} + ∑_{p=j+2}^{r} t_{ip}η_{p}t_{jp}^{⋆}. \end{matrix}

This argument allows us to raise by one the upper index in the first term. We may repeat the argument till we reach r and obtain the equation

y_{ij} = x_{ij}^{r} = x_{ij}.

In the same way we can prove

y_{ii} = ∑_{p=i}^{r} η_{p}t_{ip}t_{ip}^{⋆} = x_{ii}.

Let x be any element of J. Using Proposition 2.7, we can write

x = t (∑_{i=1}^{r} η_{i}c_{i}) t^{⋆}.

Since the action by t on X is a bounded linear operator, we have

t ∘ c_{k} = tc_{k}t^{⋆}.

Thus

s ∘ (tc_{k}t^{⋆}) = (st)c_{k}(st)^{⋆}.

Now, by linearity and continuity, the action of the group S_{0} on X restricted to the set J can be written in the following form

s ∘ x = (st) (∑_{i=1}^{r} η_{i}c_{i}) (t^{⋆}s^{⋆})

for s ∈ S_{0}.

For a sequence η ∈ {−1, 1}^{r}, we denote by O_{η} the orbit of S_{0} in V passing through ∑_{i=1}^{r} η_{i}c_{i}. The following lemma is an immediate consequence of Proposition 2.7.

Lemma 2.8. The group S_{0} acts on O_{η} simply transitively. Moreover,

i. O_{η} is open in V ;

ii. For η ≠ η′, O_{η} ∩ O_{η′} = Ø;

iii. J = ⋃_{η∈{−1,1}^{r}} O_{η}.

Theorem 2.9. The mapping x ↦ t(x) restricted to O_{η} is a diffeomorphism.

Proof. Clearly, the action of S_{0} on X is C^{∞}. Let x ∈ O_{η}. By (10)–(13), the mapping x ↦ t(x) restricted to O_{η} is C^{∞}. Moreover,

x ↦ t(x) ↦ t(x) ∘ (∑_{i=1}^{r} η_{i}c_{i}) = x,

which finishes the proof.

A homogeneous cone Ω in a vector space V is said to be irreducible if there are no non-trivial subspaces V′, V′′ and homogeneous cones Ω′ ⊂ V′, Ω′′ ⊂ V′′ such that V is the direct sum of V′ and V′′, and Ω = Ω′ + Ω′′. We state an equivalent condition for irreducibility.

Proposition 2.10. Let Ω be a homogeneous cone of rank r > 1 in L with a normal decomposition {L_{ij}}_{1≤i≤j≤r}. Then Ω is irreducible if and only if for each non-constant sequence η ∈ {−1, 1}^{r} there are 1 ≤ p < q ≤ r such that dim L_{pq} > 0 and η_{p}η_{q} = −1.

Proof. Let Ω be irreducible. Let η ∈ {−1, 1} be a non-constant sequence such that for all p and q, 1 ≤ p < q ≤ r, η_{p}η_{q} = −1 implies dim L_{pq} = 0. We introduce a partition of {1, …, r}:

P = {i| η_{i} = 1}, Q = {j| η_{j} = −1},

and define

L′ = ⨁_{{i,j∈P \atop i≤j}} L_{ij}, L′′ = ⨁_{{i,j∈Q \atop i≤j}} L_{ij}.

Then L = L′ ⊕ L′′. By properties of η, we get

L′△L′′ = 0.

Moreover, L′ and L′′ are subalgebras of L with unit elements e′ = ∑_{i∈P} c_{i} and e′′ = ∑_{i∈Q} c_{i}. Let Ω′, Ω′′ be homogeneous cones in L′, L′′, respectively. Then the Lie group S_{0} has a decomposition

S_{0} = S′_{0}S′′_{0},

which implies Ω = Ω′ ⊕ Ω′′.

Assume that for all η there exist p and q such that η_{p}η_{q} = −1 and dim L_{pq} > 0. Suppose Ω is not irreducible. Then there are non-empty subspaces L′ and L′′ and homogeneous cones Ω′ ⊂ L′, Ω′′ ⊂ L′′ such that

L = L′ ⊕ L′′, Ω = Ω′ ⊕ Ω′′.

Thus the Lie group S_{0} has a decomposition S_{0} = S′_{0}S′′_{0}. Hence, L′ and L′′ are subalgebras with unit elements e′ and e′′, respectively. We define a partition of {1, …, r}

P = {i| c_{i}△e′ = c_{i}}, Q = {j| c_{j}△e′′ = c_{j}}.

Then L_{pq} = {0} for p ∈ P and q ∈ Q. Taking

η_{i} = \{ \begin{matrix}−1 & for & i ∈ P, \\ 1 & for & i ∈ Q, \end{matrix}

for 1 ≤ i ≤ r, we get a contradiction.

Let Ω be an irreducible, open homogeneous cone of rank r > 1 in a matrix T –algebra U. For y ∈ N_{0}, ξ ∈ V and k = 1, …, r, we define

W_{k}(ξ, y) = 2π(ξ|y ∘ c_{k}).

Let us denote by π_{k}, π^{k} the projections

π_{k} : U ↦ ⨁_{1≤i≤j≤k} U_{ij}, π^{k} : U ↦ ⨁_{i=1}^{k−1} U_{ik}.

We may write

W_{k}(ξ, y) = W_{k}(π_{k}(ξ), e + π^{k}(y)),  (14)

since π_{k}(U) is a subalgebra of U and

y ∘ c_{k} = (e + π^{k}(y)) ∘ c_{k}, (ξ|y ∘ c_{k}) = (π_{k}(ξ)|(e + π^{k}(y)) ∘ c_{k}),

for y ∈ N_{0}, ξ ∈ X.

Theorem 2.11. Let Ω be an irreducible, open homogeneous cone of rank r > 1 in a matrix T –algebra U. Then for every ξ ∈ J^{⋆} and ξ ∉ \overline{Ω^{⋆}} ∪ −\overline{Ω^{⋆}} there exist i ∈ {1, …, r} and y_{1}, y_{2} ∈ N_{0} such that

W_{i}(ξ, y_{1}) > 0, W_{i}(ξ, y_{2}) < 0.

Proof. The proof is inductive over the rank of the cone. Direct calculation shows that for cones of rank r = 2 the theorem is true. Now assume that the theorem holds for cones of rank ≤ r. Let Ω be a homogeneous cone of rank r + 1. π_{r}(Ω) is a homogeneous cone in π_{r}(U), not necessarily irreducible. Therefore, there are non-zero subalgebras {U^{i}}_{i=1}^{k} of π_{r}(U) and homogeneous irreducible cones Ω_{1}, …, Ω_{k} such that

Ω_{i} ⊂ X^{i}, π_{r}(Ω) = ⨁_{i=1}^{k} Ω_{i}, π_{r}(U) = ⨁_{i=1}^{k} U^{i}.

For i ∈ {1, …, k}, we put I_{i} = {j| c_{j} ∈ U_{i}}. Let P_{i}, Q_{i} denote the projections

P_{i} : U ↦ U^{i}, Q_{i} : ⨁_{j=1}^{r} U_{j,r+1} ↦ ⨁_{j∈I_{i}} U_{j,r+1}.

Since Ω is irreducible, dim Q_{i}(π^{r+1}(U)) > 0 for every i = 1, …, k.

We fix ξ ∈ J^{⋆} and ξ ∉ \overline{Ω^{⋆}} ∪ −\overline{Ω^{⋆}}. Assume that there is i such that

P_{i}(ξ) ∉ \overline{Ω_{i}^{⋆}} ∪ −\overline{Ω_{i}^{⋆}}.

By the induction hypothesis, there exist j ∈ I_{i} and t_{1}, t_{2} ∈ P_{i}(N_{0}) such that

(P_{i}(ξ)|t_{1} ∘ c_{j}) > 0, (P_{i}(ξ)|t_{2} ∘ c_{j}) < 0.

Taking y_{l} = e + π^{j}(t_{l}) ∈ N_{0} for l = 1, 2, we have t_{l} ∘ c_{j} = y_{l} ∘ c_{j} ∈ X_{i}, and so

(ξ|y_{l} ∘ c_{j}) = (P_{i}(ξ)|t_{l} ∘ c_{j}).

Hence, the conclusion follows.

Assume now that for all i ∈ \{1, …, k\}, P_{i}(ξ) ∈ \overline{Ω_{i}^{⋆}} ∪ −\overline{Ω_{i}^{⋆}}. Since ξ ∈ J^{⋆},

P_{i}(ξ) ∈ Ω_{i}^{⋆} ∪ −Ω_{i}^{⋆}.

Let (ξ|c_{r+1}) > 0. The case (ξ|c_{r+1}) < 0 is similar. For y ∈ N_{0}, we can write

\begin{aligned}\frac{1}{2π} W_{r+1}(ξ, y) & = (ξ|(e + π^{r+1}(y)) ∘ c_{r+1}) = (ξ|(e + ∑_{i=1}^{k} Q_{i}(y)) ∘ c_{r+1}) \\ & = ∑_{i=1}^{k}(P_{i}(ξ)|Q_{i}(y)Q_{i}(y)^{⋆}) + ∑_{i=1}^{k}(ξ|Q_{i}(y) + Q_{i}(y)^{⋆}) + (ξ|c_{r+1}), \end{aligned}

since Q_{i}(y)Q_{i}(y)^{⋆} ∈ P_{i}(U). Let us consider

F (λ, y) = ∑_{i=1}^{k} λ_{i}^{2}(P_{i}(ξ)|Q_{i}(y)Q_{i}(y)^{⋆}) + ∑_{i=1}^{k} λ_{i}(ξ|Q_{i}(y) + Q_{i}(y)^{⋆}) + (ξ|c_{r+1})

for y ∈ N_{0} and λ ∈ ℝ^{k}. Suppose that F does not change the sign i.e.,

F (λ, y) ≥ 0.  (15)

Then, for all i = 1, …, k, P_{i}(ξ) ∈ Ω_{i}^{⋆}, since P_{i}(ξ) ∈ −Ω_{i}^{⋆} implies (P_{i}(ξ)|Q_{i}(y)Q_{i}(y)^{⋆}) < 0. Thus for every y ∈ N_{0} and all i = 1, …, r

W_{i}(ξ, y) ≥ 0.  (16)

Since ξ ∉ \overline{Ω^{⋆}} ∪ −\overline{Ω^{⋆}}, there exists x ∈ Ω such that (ξ|x) < 0. Writing

x = t ∘ (∑_{i=1}^{r+1} a_{i}c_{i})

for t ∈ N_{0} and a_{i} > 0, i = 1, …, r + 1, we get

2π(ξ|x) = ∑_{i=1}^{r} a_{i}W_{i}(ξ, t) + 2πa_{r+1}(ξ|t ∘ c_{r+1}) < 0

i.e., by (16),

\frac{1}{2π} W_{r+1}(ξ, t) = (ξ|t ∘ c_{r+1}) < 0.

But

F ((1, …, 1) , t) = \frac{1}{2π} W_{r+1}(ξ, t) < 0,

which contradicts (15).

3. The Siegel domains of type II

Identification with a solvable Lie group. Let Ω be an open homogeneous irreducible cone in a real vector space V . We may assume that V is a clan with a unit element. Let U be the matrix T –algebra such that the homogeneous cone Ω(X) is isomorphic to Ω (see [11] Theorem 4) i.e., there is an isomorphism of clans with a unit element σ : V ↦ X such that σ(Ω) = Ω(X). We identify V with X.

Let V^{ℂ} = V + iV be the complexification of V . We extend the action of S_{0} to V^{ℂ}.

In addition to V^{ℂ}, suppose that we are given a complex vector space Z. Let Φ : Z ×Z ↦ V^{ℂ} be a Hermitian symmetric sesquilinear mapping. We assume that Φ is Ω–positive i.e., Φ(ζ, ζ) ∈ \overline{Ω} for all ζ ∈ Z and Φ(ζ, ζ) = 0 only if ζ = 0.

The Siegel domain of type II associated with the cone Ω is defined as

D = {(ζ, z) ∈ Z × V^{ℂ} | ℑz − Φ(ζ, ζ) ∈ Ω}.

There is a representation σ : S_{0} ↦ GL(Z) such that

gΦ(ζ, ω) = Φ(σ(g)ζ, σ(g)ω).

Therefore, the transformation (ζ, z) ↦ (σ(g)ζ, σ(g)ω) is a biholomorphic automorphism of D. The elements ζ ∈ Z, x ∈ V and g ∈ S_{0} acts on D in the following way

ζ · (ω, z) = (ζ + ω, z + 2iΦ(ω, ζ) + iΦ(ζ, ζ)),
x · (ω, z) = (ω, z + x),
g · (ω, z) = (σ(g)ω, g ∘ z).

The first two actions generate a two-step nilpotent group N(Φ) (or Abelian if Z = 0) of biholomorphic automorphisms of D. The multiplication in N(Φ) is given by

(ζ, z)(ζ′, z′) = (ζ + ζ′, z + z′ + 2ℑΦ(ζ, ζ′)).

All three actions generate a solvable Lie group S = N(Φ)S_{0}, N(Φ) being a normal subgroup of S. For s ∈ S, we use the notation s = (ζ, x)ya with ζ ∈ Z, x ∈ V , y ∈ N_{0} and a ∈ A.

The action of σ(A) is diagonalizable i.e.,

Z = ⨁_{j=1}^{r} Z_{j}  (17)

with σ(H)ζ = \frac{λ_{i}(H)}{2}ζ for ζ ∈ Z_{j} where λ_{1}, …, λ_{r} is dual basis to c_{1}, …, c_{r} (see e.g. [3]).

Given λ ∈ V^{⋆} let
H_{λ}(ζ, ω) = 4(λ|Φ(ζ, ω)).

For λ ∈ Ω^{⋆}, the Hermitian form H_{λ} is not degenerate. If λ = ∑_{j=1}^{r} a_{j}c_{j}, a_{j} ∈ ℝ, the form H_{λ} decomposes nicely as

H_{λ}(ζ, ω) = ∑_{j=1}^{r} a_{j}H_{c_{j}}(ζ_{j}, ω_{j})

where ζ = ∑_{j=1}^{r} ζ_{j}, ω = ∑_{j=1}^{r} ω_{j}, ζ_{j}, ω_{j} ∈ Z_{j}. For j ≠ k, we have

\begin{aligned}H_{c_{j}}(ζ, ω_{k}) & = 4(c_{j}|t_{jj} ∘ Φ(ζ, ω_{k})) = 4(c_{j}|Φ(σ(t_{jj})ζ, ω_{k})) + 4(c_{j}|Φ(ζ, σ(t_{jj})ω_{k})) \\ & = 4 ∑_{l=1}^{r}(c_{j}|Φ(\frac{λ_{l}(t_{jj})}{2}ζ_{l}, ω_{k})) + 4(c_{j}|Φ(ζ, \frac{λ_{k}(t_{jj})}{2}ω_{k})) = \frac{1}{2}H_{c_{j}}(ζ_{j}, ω_{k}), \end{aligned}

and so H_{c_{j}}(ζ, ω_{k}) = 0.

For i = 1, …, r, we define

d_{i} = dim Z_{i} ≥ 0, m_{i} = n_{i} + \frac{d_{i}}{2} .

The Lie algebra S of S has the following decomposition

S = ⨁_{j=1}^{r} Z_{j} ⊕ ⨁_{1≤i≤j≤r} V_{ij} ⊕ ⨁_{1≤i<j≤r} N_{ij} ⊕ ⨁_{i=1}^{r} A_{i}

where ⨁_{1≤i≤j≤r} V_{ij} is the normal decomposition of the clan V . For 1 ≤ i < j ≤ r, we choose an orthogonal basis {e_{ij}^{α}} of the subspace V_{ij} such that

e_{ij}^{α}△e_{ij}^{β} = δ_{αβ}c_{i}.

Then

(e_{ij}^{α}|e_{ij}^{β}) = tr(e_{ij}^{α}△e_{ij}^{β}) = δ_{αβ}n_{i}.

We identify e_{ii}^{α} = c_{i}. Let {t_{ij}^{α}} be the corresponding basis for U_{ij} i.e.,

t_{ij}^{α} = \overline{e_{ij}^{α}}, t_{ii} = e_{ii}^{α}.

For 1 ≤ k ≤ r, 1 ≤ i < j ≤ r, 1 ≤ α ≤ dim V_{ij} = dim N_{ij}, we define the left-invariant vector fields on S: X_{k} ∈ V_{kk}, H_{k} ∈ A_{k}, X_{ij}^{α} ∈ V_{ij}, Y_{ij}^{α} ∈ N_{ij} by identifying at the identity element with \frac{c_{k}}{\sqrt{m_{k}}}, \frac{t_{kk}}{\sqrt{m_{k}}}, \frac{e_{ij}^{α}}{\sqrt{m_{i}}}, \frac{t_{ij}^{α}}{\sqrt{m_{i}}}, respectively.

In Z we choose coordinates compatible with the decomposition (17). Let {e_{jα}} be a basis of Z_{j} such that H_{c_{j}}(e_{jα}, e_{jβ}) = δ_{αβ}. Then

H_{c_{j}}(ζ, ω) = ∑_{α}^{d_{j}} ζ_{jα}\overline{ω}_{jα}

where ζ = ∑_{α=1}^{d_{j}} ζ_{jα}e_{jα} and ω = ∑_{α=1}^{d_{j}} ω_{jα}e_{jα}.

Let ζ_{jα} = x_{jα} + iy_{jα} and let X_{jα}, Y_{jα} be the left-invariant vector fields on S corresponding to \frac{∂_{x_{jα}}}{\sqrt{m_{j}}} and \frac{∂_{y_{jα}}}{\sqrt{m_{j}}} , respectively.

Then

X_{k}, X_{ij}^{α}, H_{k}, Y_{ij}^{α}, X_{jα}, Y_{jα}  (18)

form a basis for S.

Admissible operators. Let T be the tangent bundle for the complex domain D and let T^{ℂ} be the complexified tangent bundle. We extend the complex structure J and Bergman metric g from T to T^{ℂ} by complex linearity. The space of smooth sections of T, T^{ℂ} will be denoted by Γ(T), Γ(T^{ℂ}), respectively. We extend the corresponding Riemannian connection ∇ from Γ(T) to Γ(T^{ℂ}) by complex linearity.

For Z, W ∈ Γ(T^{ℂ}), we define

△(Z, W) = Z\overline{W} − ∇_{Z} \overline{W} .

In T^{ℂ} we introduce a Hermitian scalar product

(Z, W) = \frac{1}{2}g(Z, \overline{W}).

Assume that we are given a system z_{1}, …, z_{m} of coordinates in D such that g(∂_{z_{i}}, ∂_{z_{j}}) = δ_{ij} at the point (0, ie). Let E_{1}, …, E_{m} be the unique S –invariant orthonormal frame such that E_{j}(0, ie) = ∂_{z_{j}}. Since for every j, k ∈ {1, …, m}

∇_{∂_{z_{j}}} ∂_{\overline{z}_{k}} = ∇_{∂_{\overline{z}_{k}}} ∂_{z_{k}} = 0,

a simple calculation proves that

△(E_{j}, E_{k}) = △_{j,k} = ∑_{p,q} b_{pq}^{jk}(z)∂_{z_{p}}∂_{\overline{z}_{q}}

and

∂_{z_{j}}∂_{\overline{z}_{k}} = ∑_{p,q} c_{pq}^{jk}(z)△_{p,q}

for some smooth functions b_{pq}^{jk}(z), c_{pq}^{jk}(z), and

△_{j,k}(0, ie) = ∂_{z_{j}}∂_{\overline{z}_{k}}.  (19)

This shows that a second order real operator annihilating holomorphic functions can be written as L = ∑_{j,k} a_{j,k}(z)∂_{z_{j}}∂_{\overline{z}_{k}}, or L = ∑_{j,k} b_{j,k}(z)△_{j,k} with a_{j,k}(z) = \overline{a_{k,j}(z)}, b_{j,k}(z) = \overline{b_{k,j}(z)}, respectively. Finally, △_{j,k} are unique S – invariant operators with the property (19). This implies that if on top of the above assumptions we add S –invariance then L = ∑_{j,k} b_{j,k}△_{j,k} for b_{j,k} ∈ C with the property b_{j,k} = \overline{b_{k,j}}. Such operators will be called admissible (see [4], [3]).

For our purpose it will be much more convenient to consider admissible operators as operators on the group S. To do that we identify D with S by

θ : S ↦ θ(s) = s ∘ ie ∈ D,

and we transport both the Bergman metric g and the complex structure J from D to S. Although, we follows closely the calculations of [3], we keep most of them, but not all because of normalizations specific to the non-symmetric situation.

In coordinates

(ζ, z) = ( ∑_{j,α}(x_{jα} + iy_{jα})e_{jα}, ∑_{{i≤j \atop α}}(x_{ij}^{α} + iy_{ij}^{α})e_{ij}^{α})

the differential dθ of θ becomes

\begin{matrix}dθ(X_{j}) = \frac{1}{\sqrt{m_{j}}} ∂_{x_{j}}, & dθ(X_{ij}^{α}) = \frac{1}{\sqrt{m_{i}}} ∂_{x_{ij}^{α}}, \\ dθ(H_{j}) = \frac{1}{\sqrt{m_{j}}} ∂_{y_{jj}}, & dθ(Y_{ij}^{α}) = \frac{1}{\sqrt{m_{i}}} ∂_{y_{ij}^{α}}, \\ dθ(X_{jα}) = \frac{1}{\sqrt{m_{j}}} ∂_{x_{jα}}, & dθ(Y_{jα}) = \frac{1}{\sqrt{m_{j}}} ∂_{y_{jα}}. \end{matrix}

This implies the following identities

\begin{matrix}J (X_{j}) = H_{j}, & J (X_{ij}) = Y_{ij}, \\ J (H_{j}) = −X_{j}, & J (Y_{ij}) = −X_{ij}, \\ J (X_{jα}) = Y_{jα}, & J (Y_{jα}) = −X_{jα}. \end{matrix}

We need some commutation relations. First, we notice that the adjoint action of A preserves all the subspaces V_{ij}, N_{ij}. More precisely, if H ∈ A, then

[H, X] = \frac{λ_{i}(H) + λ_{j}(H)}{2}X for X ∈ V_{ij},  (20)
[H, Y] = \frac{λ_{i}(H) − λ_{j}(H)}{2}Y for Y ∈ N_{ij},  (21)

where λ_{1}, …, λ_{r} denote the dual basis to c_{1}, …, c_{r}. Next, for i < j, we have

[Y_{ij}^{α}, X_{ij}^{α}] = \frac{1}{\sqrt{m_{i}}} X_{i},  (22)
[Y_{ij}^{α}, X_{j}] = \frac{1}{\sqrt{m_{j}}} X_{ij}^{α},  (23)

since [Y_{ij}^{α}, X_{ij}^{α}] is identified in e with

\frac{1}{m_{i}} (t_{ij}^{α}e_{ij}^{α} + e_{ij}^{α}t_{ij}^{α}^{⋆}) = \frac{1}{m_{i}} e_{ij}^{α}△e_{ij}^{α} = \frac{1}{m_{i}} c_{i},

and [Y_{ij}^{α}, X_{j}] with

\frac{1}{\sqrt{m_{j}m_{i}}} (t_{ij}^{α}c_{j} + c_{j}t_{ij}^{α}^{⋆}) = \frac{1}{\sqrt{m_{j}m_{i}}} e_{ij}^{α}.

For every j = 1, …, r with d_{j} > 0, the subgroup Z_{j} ⊕ V_{jj} is a Heisenberg group in which multiplication is

(ζ, x)(ω, y) = (ζ + ω, x + y + \frac{1}{2} ∑_{α=1}^{d_{j}} ζ_{jα}\overline{ω}_{jα}).

Therefore, [Y_{jα}, X_{jα}] = \frac{1}{\sqrt{m_{j}}} X_{j} and for α ≠ β

[Y_{jα}, X_{jβ}] = [X_{jα}, X_{jβ}] = [Y_{jα}, Y_{jβ}] = 0.

Then

Lemma 3.1. The basis X_{j}, H_{j}, X_{ij}^{α}, Y_{ij}^{α}, X_{jα}, Y_{jα} is orthonormal with respect to the Riemannian structure g on S.

Proof. The Riemannian structure on S derived from the Bergman metric on D is given by the formula (see [6])

g(X, Y) = \frac{1}{2}β([J X, Y])

where for X ∈ S

β(X) = Tr(ad_{X} −J ad_{X}).

Using (20), (22) and (2), for every j ∈ {1, …, r}, we get

β(X_{j}) = 2\sqrt{m_{j}}

and β ≡ 0 on ⨁_{j=1}^{r} Z_{j} ⨁_{i<j} V_{ij} ⊕ ⨁_{ij} N_{ij}. Thus, the lengths of the vectors are

g(X_{j}, X_{j}) = g(H_{j}, H_{j}) = \frac{1}{2}β([H_{j}, X_{j}]) = \frac{1}{2\sqrt{m_{j}}} β(X_{j}) = 1,

g(X_{ij}^{α}, X_{ij}^{α}) = g(Y_{ij}^{α}, Y_{ij}^{α}) = \frac{1}{2}β([Y_{ij}^{α}, X_{ij}^{α}]) = \frac{1}{2\sqrt{m_{i}}} β(X_{i}) = 1,

g(X_{jα}, X_{jα}) = g(Y_{jα}, Y_{jα}) = \frac{1}{2}β([Y_{jα}, X_{jα}]) = \frac{1}{2\sqrt{m_{j}}} β(X_{j}) = 1.

Orthogonality of the basis follows from the fact that if S_{λ}, S_{η} are root spaces corresponding to roots λ and η, respectively, then [S_{λ}, S_{η}] ⊂ S_{λ+η}.

The Riemannian form g and the bracket in S determine the invariant Riemannian connection ∇ in S.

Lemma 3.2. The Riemannian connection is given by the formulas

∇_{X_{j}}X_{j} = \frac{1}{\sqrt{m_{j}}} H_{j}, ∇_{H_{j}}H_{j} = 0,

∇_{X_{ij}^{α}}X_{ij}^{α} = \frac{1}{2} (\frac{1}{\sqrt{m_{i}}} H_{i} + \frac{1}{\sqrt{m_{j}}} H_{j}) , ∇_{Y_{ij}^{α}}Y_{ij}^{α} = \frac{1}{2} (\frac{1}{\sqrt{m_{i}}} H_{i} − \frac{1}{\sqrt{m_{j}}} H_{j}) ,

∇_{X_{jα}}X_{jα} = \frac{1}{2\sqrt{m_{j}}} H_{j}, ∇_{Y_{jα}}Y_{jα} = \frac{1}{2\sqrt{m_{j}}} H_{j}.

Proof. Let X ∈ S. Using the usual formulas for the Riemannian connection, we obtain

g(∇_{X}X, W) = g([W, X], X)

for W ∈ S. The proof follows directly by (20)–(22).

By Lemma 3.2, we get

Theorem 3.3. Let

△_{j} = △(X_{j} + iH_{j}, X_{j} + iH_{j}), △_{ij}^{α} = △(X_{ij}^{α} + iY_{ij}^{α}, X_{ij}^{α} + iY_{ij}^{α}),
L_{j}^{α} = △(X_{jα} + iY_{jα}, X_{jα} + iY_{jα}).

Then

△_{j} = X_{j}^{2} + H_{j}^{2} − \frac{1}{\sqrt{m_{j}}} H_{j}, △_{ij}^{α} = (X_{ij}^{α})^{2} + (Y_{ij}^{α})^{2} − \frac{1}{\sqrt{m_{i}}} H_{i},

L_{j}^{α} = (X_{jα})^{2} + (Y_{jα})^{2} − \frac{1}{\sqrt{m_{j}}} H_{j}.

The partial Fourier transform. We present some basic facts about Fourier analysis on N(Φ). All what we need has been elaborated in [9].

Let (·, ·) be the Hermitian scalar product in which the basis {e_{jα}} is orthonormal. For λ ∈ V , we define a Hermitian transformation M_{λ} : Z ↦ Z by

(M_{λ}ζ, ω) = H_{λ}(ζ, ω)  (24)

where ζ, ω ∈ Z, and consider the set

Λ = {λ ∈ V^{⋆}| Det M_{λ} ≠ 0} = {λ ∈ V^{⋆}| H_{λ} is not degenerate}.

The set Λ^{c} is closed set of measure 0, since we have H_{λ}(ζ, ζ) > 0 for λ ∈ Ω^{⋆}, and Det M_{λ} is a non-zero polynomial of λ. The set Λ carries the Plancherel measure (see [9]) ρ(λ)dλ = | Det M_{λ}|dλ. For every λ ∈ Λ, we define a complex structure J_{λ} which corresponds to λ and determines the representation space H_{λ}. Let |M_{λ}| be the positive Hermitian transformation such that |M_{λ}|^{2} = M_{λ}^{2}. Then

J_{λ} = i|M_{λ}|^{−1}M_{λ}.

Let B_{λ} = ℑH_{λ}. We define a realization of the unitary irreducible representation U^{λ} (the Fock representation) associated with λ ∈ Λ. Let H_{λ} be the set of all C^{∞}(Z) functions F which are holomorphic with respect to the complex structure J_{λ} and such that

∫_{Z} |F (ζ)|^{2}ρ(λ)e^{−πB_{λ}(J_{λ}ζ,ζ)}dζ < ∞.

The appropriate scalar product in H_{λ} and the representation U^{λ} are defined by

(F_{1}, F_{2})_{λ} = ∫_{Z} F_{1}(ζ)\overline{F_{2}(ζ)}e^{−πB_{λ}(J_{λ}ζ,ζ)}ρ(λ)dζ

and

U_{(ζ,x)}^{λ}F (ω) = e^{−2π(λ|x)− \frac{π}{2} |ζ|^{2}+πω\overline{ζ}}F (ω − ζ)

with ω\overline{ζ} = B_{λ}(J_{λ}ω, ζ) + iB_{λ}(ω, ζ), |ζ|^{2} = ζ\overline{ζ}.

We will define an orthonormal basis of H_{λ} for λ ∈ Ω^{⋆} ∪ −Ω^{⋆}. First, we notice that for λ ∈ Ω^{⋆} ∪ −Ω^{⋆}, the complex structure J_{λ} has the form

J_{λ}ζ = \{ \begin{matrix}J ζ & for & λ ∈ Ω^{⋆} \\ −J ζ & for & λ ∈ −Ω^{⋆}. \end{matrix}

Hence, inside Ω^{⋆} ∪ −Ω^{⋆}, the action of s^{⋆} ∈ S_{0}^{⋆} does not change the complex structure. By (24) we have M_{s^{⋆}∘λ} = σ(s)^{⋆}M_{λ}σ(s), and so

ρ(s^{⋆} ∘ λ) = | Det M_{s^{⋆}∘λ}| = ρ(λ) Det σ(s).

For s ∈ S_{0} and ξ ∈ H_{λ}, we put

s · ξ(ζ) = ξ(s · ζ).  (25)

Therefore,

(s · ξ, s · η)_{s^{⋆}∘λ} = ∫_{Z} ξ(s · ζ)\overline{η(s · ζ)}e^{−πB_{s^{⋆}λ}(J_{s^{⋆}∘λ}ζ,ζ)}ρ(s^{⋆} ∘ λ)dζ

= ∫_{Z} ξ(ζ)\overline{η(ζ)}e^{−πB_{λ}(J_{λ}ζ,ζ)} Det σ(s^{−1})ρ(s^{⋆}λ)dζ = (ξ, η)_{λ} (26)

i.e., the action (25) is an isometry. Moreover,

U_{(ζ,x)}^{s^{⋆}∘λ}s · ξ(ω) = U_{(σ(s)ζ,s∘x)}^{λ}ξ(s · ω).

Hence,

(U_{(ζ,x)}^{s^{⋆}∘λ}s · ξ, s · η)_{s^{⋆}∘λ} = (U_{(σ(s)ζ,s∘x)}^{λ}ξ, η)_{λ}.  (27)
For a multi-index γ_{j} = (γ_{j1}, …, γ_{jd_{j}}), we define

ξ_{γ_{j}}(ζ) = \frac{π^{\frac{|γ_{j} |}{2}}}{\sqrt{γ_{j}!}} ∏_{α} ζ_{jα}^{γ_{jα}}

where |γ_{j}| = γ_{j1} + ⋯ + γ_{jd_{j}}, γ_{j}! = γ_{j1}! ⋯ γ_{jd_{j}}!. Then the polynomials

ξ_{γ}(ζ) = ∏_{j=1}^{r} ξ_{γ_{j}}(ζ)

form an orthonormal basis of H_{e}. Let λ ∈ Ω^{⋆} ∪ −Ω^{⋆}. By Proposition 2.7, there is s(λ) ∈ N_{0} such that λ = s^{⋆}(λ) ∘ η with η = −e or η = e. Then putting ξ_{γ}^{λ}(ζ) = s(λ) · ξ(ζ), we get an orthonormal basis of H_{λ}.

4. Poisson kernels and regularity of their Fourier transform

Poisson kernels. Let L be a second order real elliptic S –invariant differential operator which annihilates holomorphic functions. We write

L = ∑_{p,q} c_{pq}△(Z_{p}, Z_{q})

where {c_{pq}} is a Hermitian positive-definite matrix, and

Z_{p} ∈ {X_{ij}^{α} + iY_{ij}^{α}, X_{i} + iH_{i}, X_{jα} + iY_{jα}}.

Let π_{A} denote the canonical homomorphism π_{A} : S ↦ S/N(Φ)N_{0}. Let Y be the first order part of π_{A}(L). Then

Y = ∑_{i=1}^{r} b_{j}H_{j}

with b_{j} < 0 (see [4]).

We define two subalgebras of N (Φ) ⊕ N_{0}

N_{1}(L) = N (Φ) ⊕ ⨁_{{(λ_{i}−λ_{j})(Y )<0 \atop i<j}} N_{ij}, N_{0}(L) = ⨁_{{(λ_{i}−λ_{j})(Y )≥0 \atop i<j}} N_{ij},

and two subgroups N_{1}(L) = exp N_{1}(L), N_{0}(L) = exp N_{0}(L). Then N(Φ)N_{0} = N_{1}(L)N_{0}(L) in the sense that

N_{1}(L) × N_{0}(L) ∋ (x, y) ↦ xy ∈ N(Φ)N_{0}

is a diffeomorphism (see e.g. [2]). Let π_{1} : S ↦ N_{1}(L) be given by π_{1}(xya) = x for x ∈ N_{1}(L), y ∈ N_{0}(L), and a ∈ A. The space H_{L} of bounded L–harmonic functions on S is characterized in the following way.

Theorem 4.1. (see e.g. [2]) There is a unique positive bounded smooth function ν on N_{1}(L) with ∫_{N_{1}(L)} ν(x)dx = 1 such that the bounded L–harmonic functions F on S are in one-to-one correspondence with functions f in L^{∞}(N_{1}(L)) via the Poisson integral

F (s) = ∫_{N_{1}(L)} f(π_{1}(sx))v(x)dx.

N_{1}(L) is the maximal boundary for L.

As a straightforward generalization of Lemma 2.1 from [3], we get the following

Lemma 4.2. ([3]) There exist positive numbers γ_{1}, …, γ_{r−1} such that, if Y is the A component of the first order part of

L = L + ∑_{i=1}^{r} γ_{i}△_{i},

then (λ_{i} − λ_{j})(Y) ≥ 0 for all i < j.

We fix such an operator L. Let F be a function on D annihilated by L which satisfies

sup_{s∈S_{0}} ∫_{N(Φ)} |F ((ζ, x)s)|^{2}dζdx < ∞.  (H^{2})

By Lemma 4.2, N_{1}(L) = N(Φ), and so there exists f ∈ L^{2}(N(Φ)) such that

F (s) = ∫_{N(Φ)} f(π_{1}(su))ν(u)du = Det s^{−1} ∫_{N(Φ)} f(u)ν(s^{−1} · u)du

where s ∈ S and Det s is the determinant of the adjoint action (ζ, x) ↦ s(ζ, x)s^{−1} on N(Φ). Let

P_{ya}(ζ, x) = P ((ζ, x)ya) = Det(ya)^{−1}ν̆((ya)^{−1} · (ζ, x))  (28)

where ν̆(ζ, x) = ν((ζ, x)^{−1}). Then

LP ((ζ, x)ya) = 0.  (29)

For a function F on S and a fixed s ∈ S, we define the function F_{s} on N(Φ) by putting F_{s}(ζ, x) = F ((ζ, x)s). Then F_{s}(ζ, x) = f ∗ P_{s}(ζ, x). For F satisfying (H^{2}), the operator U_{F_{s}}^{λ} given by

(U_{F_{s}}^{λ}ξ, η)_{λ} = ∫_{N(Φ)} F_{s}(ζ, x)(U_{(ζ,x)}^{λ}ξ, η)_{λ}dζdx

is defined for almost every λ, and it is a Hilbert-Schmidt operator on H_{λ}.

Let D be a left-invariant differential operator on S. By Harnack’s inequality,

|DP ((ζ, x)ya)| ≤ cP ((ζ, x)ya)  (30)

with a constant c = c(D) independent of (ζ, x)ya ∈ S. Hence,

DF ((ζ, x)ya) = f ∗ DP ((ζ, x)ya),  (31)

and so DF satisfies (H^{2}). Thus

U_{(DF)_{s}}^{λ} = U_{f}^{λ}U_{(DP)_{s}}^{λ}.  (32)

Moreover, applying the differential operator D to the variable s, we get

D(U_{F_{s}}^{λ}ξ, η)_{λ} = (U_{(DF)_{s}}^{λ}ξ, η)_{λ}.  (33)

In particular, by (29), L(U_{P_{s}}^{λ}ξ, η)_{λ} = 0. Hence, by (29), s ↦ (U_{P_{s}}^{λ}ξ, η) is real analytic for almost every λ ∈ Λ and all ξ, η ∈ H_{λ}.

Admissible operators on the Fourier transform side. Let X̃_{jα}, Ỹ_{jα}, X̃_{ij}^{α}, X̃_{j} be a basis of the Lie algebra of N(Φ) corresponding to the vectors X_{jα}, Y_{jα}, X_{ij}^{α}, X_{j} in S parallel to N(Φ). For a function F on S and any X ∈ N(Φ), we have

(XF)((ζ, x)ya) = Ad_{ya}(X̃)F_{ya}(ζ, x)  (34)

with (ζ, x) ∈ N(Φ) and ya ∈ S_{0}. Then 238 Trojan Proposition 4.3. The Fourier transform U_{P_{s}}^{λ} satisfies

(U_{(△_{j}P)_{ya}}^{λ}ξ, η)_{λ} = (−4π^{2}(λ| Ad_{ya} X̃_{j})^{2} + H_{i}^{2} − \frac{1}{\sqrt{m_{j}}} H_{j}) (U_{P_{ya}}^{λ}ξ, η)_{λ},  (35)
(U_{(△_{ij}^{α}P)_{ya}}^{λ}ξ, η)_{λ} = (−4π^{2}(λ| Ad_{ya} X̃_{ij}^{α})^{2} + (Y_{ij}^{α})^{2} − \frac{1}{\sqrt{m_{i}}} H_{i}) (U_{P_{ya}}^{λ}ξ, η)_{λ}, (36)

for all ξ, η ∈ H^{λ}, ya ∈ S_{0}, and almost every λ ∈ Λ. Proof. Since

U_{exp(−t Ad_{ya} X̃_{j})}^{λ} = e^{2πit(λ| Ad_{ya}(X̃_{j}))}I,

so, by (30) and (34), we have

U_{(X_{j}P)_{ya}}^{λ} = \frac{d}{dt} ∫_{N(Φ)} P_{ya}((ζ, x) exp(t Ad_{ya}(X̃_{j})))U_{(ζ,x)}^{λ}dζdx|_{t=0}
= 2πi(λ| Ad_{ya}(X̃_{j}))U_{P_{ya}}^{λ}.

Using again (30) and (33) for H_{j}^{2} − \frac{1}{\sqrt{m_{j}}} H_{j}, we get (35). The proof of (36) is similar.

For λ ∈ Ω^{⋆} ∪ −Ω^{⋆}, we define
Φ_{α,β}^{λ}(ζ, x) = (U_{(ζ,x)}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ}.

Let j ∈ {1, …, r} be such that d_{j} > 0. We consider the left invariant operator L_{j} on S given by

L_{j} = ∑_{α=1}^{d_{j}} (X_{jα})^{2} + (Y_{jα})^{2} ,

and the corresponding operator on N(Φ)

L_{j} = ∑_{α=1}^{d_{j}} \binom{}{X_{jα}}^{2} + \binom{}{Y_{jα}}^{2} .

We state the following Lemma 4.4. Let λ ∈ Ω^{⋆} ∪ −Ω^{⋆} and y(λ) ∈ N_{0} be such that λ = y^{⋆}(λ) ∘ λ^{0} with λ^{0} = ∑_{j=1}^{r} λ_{j}^{0}c_{j}, λ_{1}^{0} …, λ_{r}^{0} ∈ ℝ. Then

(U_{(L_{j}P)_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = −2πa_{j}|λ_{j}^{0}|\frac{2|α_{j}| + d_{j}}{m_{j}} (U_{P_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ},  (37)
(U_{(X_{j}^{2}P)_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = −4π^{2}(λ_{j}^{0})^{2}a_{j}^{2}(U_{P_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ}.  (38)

Trojan 239

Proof. By (27) and (34), we have

\begin{aligned}(U_{(L_{j}P)_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} & = a_{j} ∫_{N(Φ)} L̃_{j}P_{y(λ)^{−1}a}(ζ, x)Φ_{α,β}^{λ^{0}}(y · ζ, y ∘ x)dζdx \\ & = a_{j}(L̃_{j}P_{y(λ)^{−1}a}, y · Φ_{α,β}^{λ^{0}}) \end{aligned}

where

L̃_{j} = ∑_{α}(Ad_{y(λ)^{−1}} X̃_{jα})^{2} + (Ad_{y(λ)^{−1}} Ỹ_{jα})^{2}.

Given a function g on N(Φ) let

y(λ) ∘ g(ζ, x) = g(σ(y(λ)) · ζ, y(λ) ∘ x).

For any left-invariant vector field X on S, we have

Ad_{y(λ)^{−1}a} X̃ (y(λ) · Φ_{α,β}^{λ^{0}}) = y(λ) · X̃Φ_{α,β}^{λ^{0}}.

Hence,

L̃_{j} (y(λ) · Φ_{α,β}^{λ^{0}}) = y(λ) · L_{j}Φ_{α,β}^{λ^{0}} = −2πa_{j}|λ_{j}^{0}|\frac{2|α_{j}| + d_{j}}{m_{j}} y(λ) · Φ_{α,β}^{λ^{0}},

since, by [10] Section 2.1,

L_{j}Φ_{α,β}^{λ^{0}} = −2π|λ_{j}^{0}|\frac{2|α_{j}| + d_{j}}{m_{j}} Φ_{α,β}^{λ^{0}}

where the factor \frac{1}{m_{j}} follows from our normalization (18). Moreover, Φ_{α,β}^{λ^{0}} and all its left-invariant derivatives are bounded functions (see [3]). Thus we have

\begin{aligned}(L̃_{j}P_{y(λ)^{−1}a}, y(λ) · Φ_{α,β}^{λ^{0}}) & = (P_{y(λ)^{−1}a}, L̃_{j}(y(λ) · Φ_{α,β}^{λ^{0}})) \\ & = −2πa_{j}|λ_{j}^{0}|\frac{2|α_{j}| + d_{j}}{m_{j}} (P_{y(λ)^{−1}a}, y(λ) · Φ_{α,β}^{λ^{0}}) \end{aligned}

which proves (37). The proof of (38) is similar. Fourier transform of the Poisson kernel. Theorem 4.5. For a fixed s ∈ S_{0} the partial Fourier transform

J^{⋆} ∋ λ ↦ P_{s}(λ̂)

is smooth and bounded on J^{⋆}. Proof. By (35) and (36),

0 = L̂P (λ, s) = LP_{s}(λ̂)

where L is an elliptic operator with analytic coefficients applied to the variable s. Hence, for a fixed λ ∈ V the function

s ↦ P_{s}(λ̂)  (39)

is real analytic.

Let λ ∈ J^{⋆}. There exist t^{⋆}(λ) ∈ S_{0}^{⋆} and ϵ ∈ {−1, 1}^{r} such that λ = t^{⋆}(λ)∘η with η = ∑_{i=1}^{r} ϵ_{i}c_{i}. Since

P_{s}(t^{−1}(λ) ∘ η) = Det t(λ)P_{t(λ)s}(η),

we have P_{s}(λ̂) = P_{t(λ)s}(η̂). The conclusion follows now by Theorem 2.9.

5. Pluriharmonic H^{2} functions

Now we are ready to prove our main theorem.

Theorem 5.1. Let D be a homogeneous Siegel domain, and let F be a real function on D such that

sup_{s∈S_{0}} ∫_{N(Φ)} |F ((ζ, x)s)|^{2}dζdx < ∞.

Assume that F is annihilated by an elliptic admissible operator L. Let

H = ∑_{i=1}^{r} γ_{i}△_{i}, γ_{i} > 0,  (40)
L = ∑_{{1≤i≤r \atop α}} γ_{i}L_{i}^{α},  (41)

be such that the maximal boundary for L = L + H is the Shilov boundary. If HF = 0 and LF = 0, then F is the real part of a holomorphic H^{2} function.

For a tube domain, the condition LF = 0 is void.

Proof. First, we show that U_{F_{s}}^{λ} = 0 for every λ ∈ \overline{Ω^{⋆}} ∪ −\overline{Ω^{⋆}} and every s ∈ S_{0}.

Let P be the Poisson kernel for L defined by (28). There exists f ∈ L^{2}(N(Φ)) such that

F_{s}(ζ, x) = f ∗ P_{s}(ζ, x).

Thus for almost all λ ∈ Λ U_{F_{s}}^{λ} = U_{f}^{λ}U_{P_{s}}^{λ}, and, by (39), the mapping

s ↦ (U_{F_{s}}^{λ}ξ, η)_{λ}  (42)

is real analytic. By Proposition 2.10 and the formula (31),

(U_{(HF)_{ya}}^{λ}ξ, η)_{λ} = ∑_{i=1}^{r} γ_{i} (−4π^{2}(λ| Ad_{ya} X̃_{i})^{2} + H_{i}^{2} − \frac{1}{\sqrt{m_{i}}} H_{i}) (U_{F_{ya}}^{λ}ξ, η)_{λ}.

Thus

∑_{i=1}^{r} γ_{i} (−4π^{2}(λ| Ad_{ya} X̃_{i})^{2} + H_{i}^{2} − \frac{1}{\sqrt{m_{i}}} H_{i}) (U_{F_{ya}}^{λ}ξ, η)_{λ} = 0.

Writing H_{i} in coordinates, we get

H_{i}((ζ, x)ya) = \frac{a_{i}}{\sqrt{m_{i}}} ∂_{a_{i}}.  (43)

Hence (U_{F_{ya}}^{λ}ξ, η)_{λ} satisfies the following differential equation

∑_{i=1}^{r} \frac{γ_{i}}{m_{i}} a_{i}^{2}(−W_{i}(λ, y)^{2} + ∂_{a_{i}}^{2})(U_{F_{ya}}^{λ}ξ, η)_{λ} = 0.

Therefore (see e.g. [3]),

(U_{F_{ya}}^{λ}ξ, η)_{λ} = c(λ, y)e^{− ∑_{i=1}^{r} a_{i}|W_{i}(λ,y)|}

with c(λ, y) = lim_{a_{j}→0}(U_{F_{ya}}^{λ}ξ, η)_{λ} for j = 1, …, r. Since P_{ya} is an approximate identity for a → 0, we get c(λ, y) = (U_{f}^{λ}ξ, η)_{λ}, and so

(U_{F_{ya}}^{λ}ξ, η) = (U_{f}^{λ}ξ, η)_{λ}e^{− ∑_{i=1}^{r} a_{i}|W_{i}(λ,y)|}.  (44)

Let λ ∈ J^{⋆}, λ ∉ \overline{Ω^{⋆}} ∪ −\overline{Ω^{⋆}} be such that (U_{f}^{λ}ξ, η)_{λ} ≠ 0. Then, by Theorem 2.11, there is 1 ≤ i ≤ r such that W_{i}(λ, y) changes sign. Therefore, (U_{F_{ya}}^{λ}ξ, η)_{λ} cannot be smooth as a function of ya at the points y for which W_{i}(λ, y) = 0. This contradicts (42).

Thus for λ ∉ \overline{Ω^{⋆}} ∪ −\overline{Ω^{⋆}} (U_{f}^{λ}ξ, η)_{λ} = 0.

We fix λ ∈ Ω^{⋆} ∪ −Ω^{⋆}. By (40) and (41),

∑_{{1≤i≤r \atop α}} γ_{i}(U_{(L_{i}^{α}F)_{ya}}^{λ}ξ, η)_{λ} = 0, ∑_{i=1}^{r} γ_{i}(U_{(△_{i}F)_{ya}}^{λ}ξ, η)_{λ} = 0,

for every ξ, η ∈ H_{λ}. Let y(λ) ∈ N_{0} and λ_{1}^{0} …, λ_{r}^{0} ∈ ℝ be such that

λ = y^{⋆}(λ) ∘ λ^{0}

with λ^{0} = ∑_{j=1}^{r} λ_{j}^{0}c_{j}. Then, by Lemma 4.4,

∑_{i=1}^{r} γ_{i} (−2πa_{i}|λ_{i}^{0}|\frac{2|α_{i}| + d_{i}}{m_{i}} − \frac{d_{i}}{\sqrt{m_{i}}} H_{i}) (U_{F_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = 0,

∑_{i=1}^{r} γ_{i} (−4π^{2}(λ_{i}^{0})^{2}a_{i}^{2} + H_{i}^{2} − \frac{1}{\sqrt{m_{i}}} H_{i}) (U_{F_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = 0.

Writing H_{i} in coordinates (43), we get

∑_{i=1}^{r} γ_{i} (−2πa_{i}|λ_{i}^{0}|\frac{2|α_{i}| + d_{i}}{m_{i}} − d_{i}a_{i}∂_{a_{i}}) (U_{F_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = 0,  (45)

∑_{i=1}^{r} γ_{i} (−4π^{2}|λ_{i}^{0}|^{2}a_{i}^{2} + ∂_{a_{i}}^{2}) (U_{F_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = 0.  (46)

Solving (46) (see e.g. [3]), we get

(U_{F_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = c(λ, y)e^{−2π ∑_{i=1}^{r} a_{i}|λ_{i}^{0}|}.

Then plugging it into the first equation, we obtain (U_{F_{y(λ)^{−1}a}}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = 0 for all β and α ≠ 0. Again, since P_{ya} is an approximate identity for a → 0, we get (U_{f}^{λ}ξ_{α}^{λ}, ξ_{β}^{λ})_{λ} = 0. To finish the proof of Theorem 5.1, we use the following

Lemma 5.2. (see e.g. [3])

U λ ; P̆_{ya}ξ_{0} = { e^{−2π(λ|ya∘e)}ξ_{0} ifλ ∈ Ω^{⋆} e^{2π(λ|ya∘e)}ξ_{0} ifλ ∈ −Ω^{⋆}.  (47)

By (47) and the Fourier inversion formula (see [9]),

\begin{aligned}F ((ζ, x)ya) & = ∫_{Ω^{⋆}∪−Ω^{⋆}} Tr(U_{(−ζ,−x)}^{λ}U_{f∗P_{ya}}^{λ})ρ(λ)dλ \\ & = ∫_{Ω^{⋆}∪−Ω^{⋆}}(U_{f∗P_{ya}}^{λ}ξ_{0}, U_{(ζ,x)}^{λ}ξ_{0})_{λ}ρ(λ)dλ. \end{aligned}

We define a function G on S by the formula

G((ζ, x)ya) = ∫_{Ω^{⋆}} e^{−2π(λ|ya∘e)}(U_{f}^{λ}ξ_{0}, U_{(ζ,x)}^{λ}ξ_{0})_{λ}ρ(λ)dλ.

Then G is a holomorphic H^{2} function (see e.g. [3]). Moreover,

\overline{G}((ζ, x)ya) = ∫_{−Ω^{⋆}} e^{2π(λ|ya∘e)}(U_{f}^{λ}ξ_{0}, U_{(ζ,x)}^{λ}ξ_{0})_{λ}ρ(λ)dλ.

Hence, by Lemma 5.2,

F ((ζ, x)ya) = G((ζ, x)ya) + \overline{G}((ζ, x)ya),

and the conclusion follows.

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Bartosz Trojan Instytut Matematyczny Uniwersytet Wrocławski Plac Grunwaldzki 2/4 50-384 Wrocław Poland trojan@math.uni.wroc.pl

Received November 27, 2000 and in final form June 12, 2001