THE HUA SYSTEM ON IRREDUCIBLE HERMITIAN SYMMETRIC SPACES OF NONTUBE TYPE

SPACES OF NONTUBE TYPE

DARIUSZ BURACZEWSKI

Abstract. Let G/K be an irreducible Hermitian symmetric space of noncompact type.

We study a G-invariant system of differential operators on G/K called the Hua system.

It was proved by K. Johnson and A. Kor´anyi that if G/K is a Hermitian symmetric space of tube type, then the space of Poisson-Szeg¨o integrals is precisely the space of zeros of the Hua system. N. Berline and M. Vergne raised the question about the nature of the common zeros of the Hua system for Hermitian symmetric spaces of nontube type. In this paper we show that these are exactly the pluriharmonic functions.

1. Introduction

Let G/K be an irreducible Hermitian symmetric space of noncompact type and let {E^j} be an orthonormal basis of p⁺. The Hua system, as defined in [JK], is

(1.1) H(F ) = X

j,k

EjE_k^∗F ⊗ [E^j, E_k^∗].

After a number of partial results, the earliest going back to Hua, the fundamental theorem concerning the Hua system, proved by K.Johnson and A. Kor´anyi is:

Theorem 1.2. (K. Johnson, A. Kor´anyi, 1980) A function F on a Hermitian symmetric space of tube type satisfies H(F ) = 0 if and only if it is the Poisson-Szeg¨o integral of a hyperfunction on the Shilov boundary of G/K.

Soon after the result of K. Johnson and A. Kor´anyi appeared, N. Berline and M. Vergne [BV] proved that (1.1) does not annihilate the Poisson-Szeg¨o kernel on type two Hermitian symmetric spaces and described a third order system that characterizes Poisson-Szeg¨o integrals for those spaces, and they raised the question about the nature of the common zeros of the Hua system for the type two Hermitian symmetric spaces, a question that has remained opened for twenty years.

The aim of this paper is to prove that the zeros of the Hua system on type two Hermitian symmetric spaces are the pluriharmonic functions (see section 4):

Theorem 1.3. Let F be a real valued function on a non-tube irreducible Hermitian sym- metric space. Then H(F ) = 0 if and only if F is pluriharmonic.

The author was partly supported by KBN grant 5P03A02821, Foundation for Polish Sciences, Subsidy 3/99, and by the European Commission via the TMR Network “Harmonic Analysis”, contract no. ERB FMRX–CT97–0159.

1

The system (1.1) annihilates holomorphic and antiholomorphic functions. The origin of (1.1) goes back to L. H. Hua [Hua], who in 1958 wrote a system that annihilates the Poisson-Szeg¨o kernel on some classical domains. His formula was not exactly the one above, but for classical tube domains the zeros of both systems are the same. Then A. Kor´anyi, E. Stein and J. Wolf obtained the formula for general tube domains and in an unpublished paper showed that the Poisson-Szeg¨o kernel is harmonic with respect to the system (see e.g. [JK]). The first results showing that differential equations actually characterize the class of Poisson-Szeg¨o integrals were obtained in special cases [KM], [J1], [J2]. Finally in 1980 K. Johnson and A. Kor´anyi proved Theorem (1.2).

For a particular case of functions having L² boundary values, the above theorem was proved in [BBDHPT]. The methods of [BBDHPT] make the utmost use of the strong growth restrictions and are not applicable here.

To treat zeros of H in full generality we combine two essential ingredients: the method of M.Lassalle [L] and the approach to pluriharmonic functions on symmetric Siegel domains developed in [DHMP], [BDH]. While the first one is based on the semi-simple group G, for the second one the use of the solvable Lie group S acting simply transitively on the corresponding Siegel domain seems indispensable. The reason is that the G-invariant operators don’t see pluriharmonicity while the S invariant do¹ The interplay between S- picture and G-picture is crucial for our story: the analysis is done on the group S and the special structure of S being the Iwasawa group is essential. That is why we describe so thoroughly both pictures: G/K and S and we pass from one to the other (sections 2-3).

M.Lassalle [L] reproved theorem (1.2) introducing new methods and, at the same time, cutting down the number of equations. We adopt his method to reduce the problem to bounded functions. Namely, we prove that a Hua harmonic function is G-harmonic (section 5)². To do so we use only a part of the system, the “strongly diagonal operators”

(see section 4). These r equations (r being the rank) correspond to the system H^h of M. Lassalle. Next we use Harish-Chandra theorem [HC] in order to expand f in terms of its projections on spaces of K-finite functions f_δ. Each of these functions is Hua harmonic, hence G-harmonic. A K-finite and G-harmonic function can be written as a Poisson integral of a continuous bounded function defined on the maximal Furstenberg boundary, therefore all functions fδ are bounded.

After restricting to bounded functions we transfer our problem to Siegel domains. For this we pass to realization of G/K as a Siegel domain ^cD in C^m, which is described in section (3.3) following the Kor´anyi-Wolf theory [KW]. However, for our purposes, we have to transform it further on in order to write strongly diagonal operators on the solvable Lie group S, the one that acts simply transitively on ^cD (section 6). This gives an extra advantage: the whole system may be replaced by strongly diagonal operators. In fact, we prove (see section 4):

1Bounded pluriharmonic functions are Poisson integrals i.e they are annihilated by all the G-invariant operators, but there are many G-harmonic functions that are not pluriharmonic.

2The fact that for type two domains Hua harmonicity implies G-harmonicity was mentioned without proof in [JK].

Theorem 1.4. Let F be a bounded real valued function on a non-tube Hermitian irre- ducible symmetric space. If F is annihilated by the Laplace-Beltrami operator and the strongly diagonal Hua operators then F is pluriharmonic.

Hence only r +1 operators are needed which is considerably less than in either Johnson- Kor´anyi’s or Lassalle’s proof.

The rest of the proof uses S (section 6). First we show that a bounded function annihi- lated by the strongly diagonal Hua operators is the Poisson-Szeg¨o integral (see [BBDHPT], [DHP]). Then, we notice that the Laplace-Beltrami operator ∆_T for the corresponding tube domain T_Ω is a linear combination of the above r + 1 operators. Combining these two facts with Theorem (1.2) applied to TΩ we obtain some more equations (see section 6).

Now the strategy is to single out operators whose common zeros have to be plurihar- monic. These operators, however, cannot be obtained directly as linear combinations of the ones studied so far. The method we apply is the induction on the rank of the domain.

The crucial observation is that S is a semi-direct product S = S^r−1Sr,

where S^r−1 is the group acting simply transitively on the Siegel domain of rank r − 1 and Sr the group acting simply transitively on the Siegel half plane D^r (biholomorphic to the complex ball). Since a part of the equations that we have at our disposal are on Sr, we restrict the function F to Sr and apply some Fourier analysis methods on the Heisenberg group (section 7). The induction produces equations to which we can apply the results of [BDH] and conclude that F is pluriharmonic. Since this method requires that we deal with bounded functions not L², the analytic part here is somewhat more delicate than in [BBDHPT].

The author would like to express his deep gratitude to Ewa Damek for her numerous ideas, suggestions and corrections incorporated in the paper. Indeed, some parts of this paper are in fact a joint work. Also the author wishes to thank Aline Bonami, Jacques Faraut and Andrzej Hulanicki for their valuable comments.

2. Preliminaries on Hermitian symmetric spaces

Let G/K be an irreducible Hermitian symmetric space of noncompact type and G the connected component of its isometry group. G is a centerless semisimple Lie group, and K is its maximal compact subgroup. We need some standard notation concerning semisimple Lie groups and algebras. For more details we refer to [H1], [Kn] or [K].

2.1. Root space decomposition. Let g and k be the Lie algebras of respectively G and K, and let θ : g 7→ g be the Cartan involution on g which is identity on k. If p is the -1 eigenspace of θ, we get the Cartan decomposition g = k ⊕p, where [k, p] = p and [p, p] = k.

Let g^C (k^C, p^C resp.) denote the complexification of g (k, p resp.). We extend θ to be a complex linear involution on g^C. G^C is the adjoint group of g^C with K^C the analytic subgroup corresponding to k^C.

u = k ⊕ ip is a compact real form of g^C. Denote by τ the conjugation operator on g^C with respect to u. If B is the Killing form on g^C, then the bilinear form defined by B_τ(X, Y ) = −B(X, τY ) is positive definite. σ = τθ = θτ is the conjugation of g^C with respect to g. (Usually we shall write E instead of σE.)

Choose a Cartan subalgebra h in k. Then h^C is a Cartan subalgebra of g^C. Define

∆ to be the system of roots of g^C with respect to h^C. Any root space g^α is contained either in k^C or in p^C. In the first case α is called compact (α ∈ C) and in the second case noncompact (α ∈ Q). Clearly ∆ is a disjoint union of C and Q. For a fixed α ∈ ∆ there are eHα ∈ ih, eEα ∈ g^α, eE−α∈ g^−α such that

α(H) = B(H, eHα), for every H ∈ h^C, [ eE_α, eE_−α] = eH_α,

τ eEα = − eE−α, B( eEα, eE−α) = 1, (2.1)

([H1], page 220). In particular, eHα, eEα and eE−α span a subalgebra of g^C, isomorphic to sl(2, C).

We define a Hermitian product on (h^C)^∗ by

(2.2) hα, βi = B( eHα, eHβ) = α( eHβ) = β( eHα), for α, β ∈ (h^C)^∗.

Let

(2.3) cα =

s 2

α( eHα) =

s 2

hα, αi .

c_α is well defined because α( eH_α) = B( eH_α, eH_α) > 0. Clearly c_α = c_−α. Using c_α’s we introduce a second normalization:

Eα = cαEeα, H_α = c²_αHe_α. (2.4)

By (2.1) these vectors satisfy the following relations:

[Eα, E−α] = Hα, τ Eα = −E^−α, B(Eα, E−α) = 2

α( eHα) = 2 hα, αi, α(H_α) = 2.

(2.5)

If c is the center of k, then there exists an element Z ∈ c such that (adZ)² = −1 on p^C ([Kn], Theorem 7.117). Let p⁺be the (i)–eigenspace of adZ and p⁻be the (−i)–eigenspace of adZ. Then p⁺ and p⁻ are Abelian Lie subalgebras invariant under the action of k^C,

and [p⁺, p⁻] ⊂ k^C ([H1], page 313). Moreover, there is an ordering of ∆ decomposing Q so that Q = Q⁺∪ Q⁻, and

p⁺= X

α∈Q⁺

g^α = X

α∈Q⁺

CEα, p⁻= X

α∈Q⁺

g^−α = X

α∈Q⁺

CE−α. (2.6)

Indeed, we can select an ordering on ∆ as follows: for two roots α, β we say that α is bigger than β if and only if −i(α − β)(Z) > 0. Q⁺ is referred as the set of positive noncompact roots, while Q⁻ is called the set of negative noncompact roots.

For α ∈ Q⁺, let

X_α = E_α+ E_−α, Yα = i(Eα− E^−α).

(2.7)

Then the set {X^α, Yα}^α∈Q+ spans p.

The restriction of adZ to p gives the complex structure on p, which will be denote by J . Thus we have

J X^α = Yα, J Y^α = −X^α,

E_α = 1

2(X_α− iYα), E−α = 1

2(Xα+ iYα).

(2.8)

Similarly to (2.7), for every positive compact root α, take Xα, Yα ∈ k to be Xα = Eα− E^−α,

Yα = i(Eα+ E−α).

(2.9)

2.2. Some algebraic preliminaries. In this subsection we are going to introduce some further algebraic properties of g^C, which will be needed later.

First our goal is to describe the restricted root system for G/K. Two roots α, β ∈ ∆ are called strongly orthogonal if neither α + β nor α − β are roots. One can easily check that strong orthogonality implies orthogonality with respect to the form (2.2). Let (2.10) Γ = {γ1, . . . , γ_r} ⊂ Q⁺

(r = rank G/K) be a maximal set of strongly orthogonal positive noncompact roots.

Then

a=X

γ∈Γ

RXγ, (2.11)

is a maximal Abelian subalgebra of p. (For a construction of Γ we refer to [H1] pages 385-387.)

Take h⁻ to be the real span of the elements iHγ, and h⁺ to be the orthogonal comple- ment of h⁻ in h via the Killing form B:

h⁻ =X

γ∈Γ

RiHγ, h= h⁻⊕^Bh⁺. (2.12)

Let α, β ∈ ∆. Denote α ∼ β if and only if α|^h− = β|^h−. Define:

(2.13)

C₀ = { α ∈ C: α ∼ 0},

Ci = { α ∈ C: α ∼ −¹₂γi} for i = 1, . . . , r, Cij = { α ∈ C: α ∼ ¹₂(γj − γⁱ)} for 1 ≤ i < j ≤ r, Qi = { α ∈ Q: α ∼ ¹2γi} for i = 1, . . . , r, Qij = { α ∈ Q: α ∼ ¹2(γi+ γj)} for 1 ≤ i < j ≤ r.

Then the map α 7→ γⁱ+ α is a bijection of Ci onto Qi and Cij onto Qij. It is also known that ∆⁺ is the disjoint union of the sets C0, Ci, Cij, Γ, Qi, Qij, and Q⁺ is the disjoint union of the sets Γ, Qi, Qij ([H3], pages 457-460).

We shall call G/K a tube space if all sets Q_i are empty. Otherwise G/K is a nontube space.

We introduce numbers Mα,β, which will be helpful in next sections in computing some brackets relations. For α, β ∈ ∆ define M^α,β by:

(2.14) [Eα, Eβ] = Mα,βEα+β if α + β ∈ ∆, Mα,β = 0 if α + β 6∈ ∆.

(some properties of these numbers are described in [H1], pages 146-152).

We may assume ([H1], Theorem V.5.5) that

(2.15) Mα,β = −M^−α,−β.

Proposition 2.16. Fix k between 1 and r.

a) If α = γj ∈ Γ then α(H^γk) = 2δjk.

b) If α ∈ Q^k or α ∈ Q^jk, and β = α − γ^k, then α(Hγk) = 1 and hα, αi = hβ, βi = hγ^k, γki.

c) For the rest of positive roots α: α(Hγk) = 0.

Proof. a) and c) are obvious. We prove b) for α ∈ Q^jk. The second case is similar.

Notice that by (2.2) and (2.4)

(2.17) α(H_γk) = hα, αi

hγ^k, γkiγ_k(H_α),

therefore it is enough to compute α(Hγk) and γk(Hα). For this purpose we shall look at corresponding roots series. The γk-series containing α is {α − γ^k, α}, because α − 2γ^k ∼

γj−3γk

2 , which by the above remark implies that α − 2γk is not a root. If α − γk is a root, then α + γ_k cannot belong to ∆ ([H3], Lemma V.4.4), as well. Hence α(H_γk) = 1,

which is immediate consequence of Theorem V.5.3 in [H3]. Similarly we get γk(Hα) = 1.

Therefore hα, αi = hγk, γ_ki. ¤

Corollary 2.18. If α ∈ Q^k or α ∈ Q^jk, β = α − γ^k then Mγk,β = ±1.

Proof. We have just proved that cα = cβ and then the corollary follows from Lemma 5.V.2

in [H1]. ¤

3. Irreducible symmetric Siegel domains

In this chapter we introduce symmetric Siegel domains and following A. Kor´anyi and J. Wolf [KW], we describe realization of a Hermitian symmetric space as a Siegel domain.

3.1. Preliminaries on irreducible symmetric cones. Let Ω be an irreducible sym- metric cone in an Euclidean space. Our aim is to describe a solvable group S0 acting simply transitively on Ω. We are going to use heavily the language of Jordan algebras so we recall briefly some basic facts which will be needed later. The reader is referred to the book of J. Faraut and A. Koranyi [FK] for more details.

A finite dimensional algebra V with a scalar product h·, ·i is an Euclidean Jordan algebra, if for all elements x, y and z in V :

xy = yx, x(x²y) = x²(xy), hxy, zi = hy, xzi.

We denote by L(x) the self-adjoint endomorphism of V given by the multiplication by x, i.e. L(x)y = xy. For an irreducible symmetric cone Ω contained in a linear space V of the same dimension, the space V can be made a simple real Euclidean Jordan algebra with unit element e, so that

Ω = int {x² : x ∈ V }.

Let G0 be the connected component of the group of all transformations in GL(V ) which leave Ω invariant, and let g0 be its Lie algebra. Then g0 is a subspace of the space of endomorphisms of V which contains all L(x) for all x ∈ V , as well as all x²y for x, y ∈ V , where

(3.1) x2y = L(xy) + [L(x), L(y)]

(see [FK] for these properties).

We fix a Jordan frame {c¹, . . . , cr} in V , that is, a complete system of orthogonal primitive idempotents:

c²_i = ci,

cicj = 0 if i 6= j, c₁+ ... + c_r = e

and none of the c1, ..., cr is a sum of two non-zero idempotents. Let us recall that the length r is independent of the choice of the Jordan frame. It is called the rank of V . To have an example in mind, one may think of the space V of the symmetric r × r matrices endowed with the symmetrized product of matrices ¹₂(xy + yx). Then the corresponding cone is the set of symmetric positive definite r × r matrices, the set of diagonal matrices with all entries equal to 0 except for one equal to 1 being a Jordan frame.

The Peirce decomposition of V related to the Jordan frame {c¹, . . . , cr} ([FK], Theorem IV.2.1) may be written as

(3.2) V = M

1≤i≤j≤r

Vij.

It is given by the common diagonalization of the self-adjoint endomorphism L(c_j) with respect to their only eigenvalues 0, ¹₂, 1. In particular Vjj = Rcj is the eigenspace of L(cj) related to 1, and, for i < j, Vij is the intersection of the eigenspaces of L(ci) and L(cj) related to ¹₂. All Vij, for i < j, have the same dimension d.

For each i < j, we fix once for all an orthonormal basis of Vij, which we note {e^αij}, with 1 ≤ α ≤ d. To simplify the notation, we write e^αii = ci (α taking only the value 1).

Then the system {e^αij}, for i ≤ j and 1 ≤ α ≤ dim V^ij, is an orthonormal basis of V . Let us denote by a₀ the Abelian subalgebra of g₀ consisting of elements H = L(a), where

a = Xr

j=1

ajcj ∈M

i

Vii.

We set λj the linear form on a0 given by λj(H) = aj. It is clear that the Peirce decom- position gives also a simultaneous diagonalization of all H ∈ a⁰, namely

(3.3) Hx = L(a)x = λ_i(H) + λ_j(H)

2 x, x ∈ V^ij.

Let A0 = exp a0. Then A0 is an Abelian group, and this is the Abelian group in the Iwasawa decomposition of G0. We now describe the nilpotent part N0. Its Lie algebra n0

is the space of elements X ∈ g⁰ such that, for all i ≤ j, XVij ⊂ M

k≥l (k,l)>(i,j)

Vkl,

where the pairs ordered lexicographically. Once n0 is defined, we define s0 as the direct sum n0⊕a⁰. The groups S0and N0 are then obtained by taking the exponentials. It follows from the definition of n0 that the matrices of elements of s0 and S0, in the orthonormal basis {e^αij}, are upper-triangular.

The solvable group S₀ acts simply transitively on Ω. This may be found in [FK] Chapter VI, as well as the precise description of n₀ which will be needed later. One has

(3.4) n₀ = M

i<j≤r

n_ij,

where

(3.5) n_ij = {z²ci : z ∈ V^ij}.

This decomposition corresponds to a diagonalization of the adjoint action of a0 since

(3.6) [H, X] = λj(H) − λⁱ(H)

2 X, X ∈ n^ij.

Finally, let V^C= V + iV be the complexification of V . We extend the action of G to V^C in the obvious way.

3.2. Irreducible symmetric Siegel domains. Suppose that we are given a complex vector space Z and a Hermitian symmetric bilinear mapping

Φ : Z × Z 7→ V^C. We assume that

Φ(ζ, ζ) ∈ Ω, ζ ∈ Z , and Φ(ζ, ζ) = 0 implies ζ = 0.

The Siegel domain associated with these data is defined as

(3.7) ^cD = {(ζ, z) ∈ Z × V^C: ℑz − Φ(ζ, ζ) ∈ Ω}³.

It is called of tube type, if Z is reduced to {0}. Otherwise, it is called of type II.

There is a representation σ : S0 ∋ s → σ(s) ∈ GL(Z) such that

(3.8) sΦ(ζ, w) = Φ(σ(s)ζ, σ(s)w),

and such that all automorphisms σ(s), for s ∈ A⁰, admit a joint diagonalization (see [KW]). To reduce notations, we shall as well denote by σ the corresponding representation of the algebra s0. For X ∈ s⁰, (3.8) implies that

(3.9) XΦ(ζ, w) = Φ(σ(X)ζ, w) + Φ(ζ, σ(X)w).

As an easy consequence, one can prove that the only possible eigenvalues for σ(H), with H ∈ a⁰ are λj(H)/2, for j = 1, . . . , r. So we may write

(3.10) Z =

Mr j=1

Z^j with the property that

(3.11) σ(H)ζ = λj(H)

2 ζ, ζ ∈ Zj.

Moreover, all the spaces Zj have the same dimension. A proof of these two facts may be found in [DHMP]. We call χ the dimension of Z^j for j = 1, . . . , r. Let us remark, using (3.9) and (3.11), that for ζ, w ∈ Z^j, we have L(cj)Φ(ζ, w) = Φ(ζ, w). Therefore,

3We denote a Siegel domain by^cD to be consistent with A. Kor´anyi and J. Wolf notation needed in the next subsections.

Φ(ζ, w) = Qj(ζ, w)cj, for ζ, w ∈ Z^j. Moreover, hc^j, Φ(ζ, ζ)i > 0 for ζ ∈ Z^j and so the Hermitian form Q_j is positive definite on cz_j.

The representation σ allows to consider S₀ as a group of holomorphic automorphisms of ^cD. More generally, the elements ζ ∈ Z, x ∈ V and s ∈ S⁰ act on ^cD in the following way:

ζ · (w, z) = (ζ + w, z + 2iΦ(w, ζ) + iΦ(ζ, ζ)), x · (w, z) = (w, z + x),

s · (w, z) = (σ(s)w, sz).

(3.12)

We call N (Φ) the group corresponding to the first to actions, that is N (Φ) = Z × V with the product

(3.13) (ζ, x)(ζ^′, x^′) = (ζ + ζ^′, x + x^′ + 2ℑΦ(ζ, ζ^′)).

All three actions generate a solvable Lie group

(3.14) S = N (Φ)S0 = N (Φ)N0A0 = N A0,

which identifies with a group of holomorphic automorphisms acting simply transitively on^cD. The group N(Φ), that is two-step nilpotent, is a normal subgroup of S. The Lie algebra s of S admits the decomposition

(3.15) s= n(Φ) ⊕ s⁰ =

à M

1≤j≤r

Z^j

!

⊕

à M

1≤i≤j≤r

Vij

!

⊕

à M

1≤i<j≤r

n_ij

!

⊕ a⁰. Moreover, by (3.3), (3.6) and (3.11), one knows the adjoint action of elements H ∈ a⁰:

(3.16)

[H, X] = ^λj(H)₂ X, for X ∈ Z^j, [H, X] = ^λi(H)+λ₂ ^j(H) X, for X ∈ Vij, [H, X] = ^λj(H)−λ₂ ^i(H) X, for X ∈ nij.

Since S acts simply transitively on the domain ^cD, we may identify S and ^cD. More precisely, we define

(3.17) θ : S ∋ s → θ(s) = s · e ∈ ^cD,

where e is the point (0, ie) in ^cD. The Lie algebra s is then identified with the tangent space of^cD at e using the differential dθ^e. We identify e with the unit element of S. We then transport both the Bergman metric g and the complex structure J from ^cD to S, where they become left-invariant tensor fields on S. We still write J for the complex structure on S.

3.3. Realization of Hermitian symmetric space as a Siegel domain. The goal of this subsection is to describe connections between Hermitian symmetric spaces and Siegel domains. For a space G/K, which is supposed to be in the Harish-Chandra realization, we find a biholomorphically equivalent domain, equipped with the structure of Siegel

domains. We follow closely [KW] and [K], but at the end we shall need a little more, namely we are interesting in full description of a basis of s in terms of the Lie algebra g^C. First we recall the Harish-Chandra realization. Let us denote the analytic subgroups of G^C corresponding to subalgebras p⁺, p⁻ by P⁺ and P⁻, respectively. They are Abelian.

The exponential map from p^± to P^± is biholomorphic and P^±is biholomorphically equiv- alent with Cⁿ for some n.

The mapping (p1, k, p2) 7→ p¹kp2 is a diffeomorphism of P⁺× K^C× P⁻ onto an open submanifold of G^C containing G. For g ∈ G let p⁺(g) denote the unique element in p⁺ such that g ∈ exp(p⁺(g))K^CP⁻. One can show that p+(g) = p+(gk) and p+ is a diffeomorphism of G/K onto a domain D ⊂ p⁺ = Cⁿ. G acts biholomorphically on D by g · p⁺(eg) = p⁺(geg). Let o = p⁺(e), then D is the G-orbit of o and the group K is the stabilizer of the point o. This is the Harish–Chandra embedding and in fact realizes G/K as a bounded symmetric domain (we refer for more details to [H1] or [Kn]).

From now we shall assume that D is the above realization. Put X^Γ = P

γ∈ΓXγ, E_Γ=P

γ∈ΓE_γ and define an element of G^C called the Cayley transform:

(3.18) c = expπ

4iXΓ, Let

cG = Ad(c)G,

cK = Ad(c)K,

cg = Ad(c)g,

ck = Ad(c)k.

For g ∈ G, c exp(p⁺(g)) ∈ P⁺K^cP⁻ and so the mapping p+(g) 7→ p⁺(c exp p+(g)) defines a biholomorphism of D onto a domain ^cD ⊂ p⁺ ([KW]). Clearly, ^cD is the orbit of the point c · o = iE^γ under the action of the group^cG, and^cK is the isotropy group of iEγ.

A simple computation proves the following lemma Lemma 3.19 ([K], Lemma IV.1.1). Let γ ∈ Γ. Then

Ad(c) · Xγ = X_γ, Ad(c) · Y^γ = Hγ, Ad(c) · H^γ = −Y^γ. Furthermore Ad(c) acts trivially on h⁺.

It can be shown that c⁸ = I and Ad(c⁴) preserves k and p. We decompose both Lie algebras:

− p^T, p2 is the (±1)-eigenspace of Ad(c⁴) in p,

− k^T, q2 is the (±1)-eigenspace of Ad(c⁴) in k,

− gT = k_T ⊕ pT.

If gT = g, then the space G/K is of tube type, otherwise it is of nontube type. In the obvious way we introduce Lie algebras p^C_T, p⁺_T, etc., and analytic subgroups of G^C: G_T, KT, etc. We denote by p^j, p^jk, p^2j subspaces corresponding to restricted roots γj, ^γj+γ₂ ^k,

γj

2. Similarly define subspaces of k^C: k^kj, k^j2, m^C which are restricted root spaces of ^γj−γ₂ ^k,

γj

2 and 0, respectively. Notice that

p⁺_T = ³M p^j´

⊕³ M

j<k

p^jk´ ,

k^C_T = m^C⊕ (h⁻)^C⊕³ M

j6=k

k^jk

´ . Furthermore,

p⁺₂ = M p^j2, q⁺₂ = M

k^j2. Next we introduce

n^j = ^cg∩ p^j,

n^jk = ^cg∩ p^jk for j < k, n^kj_K = ^cg∩ k^kj for j > k,

n^j2 = ^cg∩ (p^j2 ⊕ k^j2).

Moreover, let

n⁺_T =³M n^j

´

⊕³ M

j<k

n^jk

´ , n⁺₂ =M

j

n^j2, n_K =M

j>k

n^kj_K. (3.20)

One can easily see that n⁺_T = ^cg∩ p⁺T and is a real form of p⁺_T. Similarly n⁺₂ is a real form of q⁺₂ ⊕ p⁺2. One can prove that Ad(c²) preserves kT and pT. We define:

l, qT is the (±1)-eigenspace of Ad(c) in k^T.

Take k^∗_T = l ⊕ iq^T, then k^∗_T = ^cg∩ k^CT ([K], Lemma IV.2.6). By K_T^∗ and L we denote corresponding Lie groups.

Now we are ready to describe the domain ^cD. The image of the point iE^γ under the action of K_T^∗ is a self dual cone in in⁺_T, with the group L as a stabilizer of iE_γ ([K],

Theorem IV.2.10). We shall denote this cone by Ω. Define a function Φ : p⁺₂ × p⁺2 7→ p⁺T

by

Φ(X, Y ) = 1

2adX(adY )^∗EΓ,

where (adY )^∗ is the adjoint operator of adY with respect to B_τ. It can be shown ([KW], lemma 6.4) that Φ satisfies all asumptions listed in the previous subsection and

(3.21) ^cD = ^cG · iE^Γ = {X + iY + Z : X, Y ∈ n⁺T, Z ∈ p⁺2, Y − Φ(Z, Z) ∈ Ω}.

([KW], Theorem 6.8). Therefore^cD is a Siegel domain of type I or II.

Now we want to recognize the group S in this picture. The mapping iqT 7→ n⁺T given by

iqT ∋ X 7→ [X, E^Γ]

is a bijection ([KV], lemma 2.5). Let L be the inverse map, then we define multiplication in n⁺_T by:

XY = adL(X)adL(Y )EΓ.

One can easily prove that the above definition coincides with the one given in [FK], page 49, and multiplication so defined gives the structure of Jordan algebra in n⁺_T.

Take ci = Eγi, then L(ci) = ₂¹Hγi and the set {c¹, . . . , cr} is a Jordan frame. The Peirce decomposition (3.2) with respect to this frame is given by the decomposition (3.20) of n⁺_T with Vj = n^j and Vjk = n^jk. Then a0 is spanned by vectors Hγi and the Gauss decomposition (3.4) of n0 = nK coincides with (3.20) for nij = n^ij_K. Therefore, the solvable part of Iwasawa decomposition of ^cg with respect to a0 is

s= n⁺₂ ⊕ n⁺T ⊕ n^K⊕ a⁰,

and this is exactly the same decomposition as (3.15). The group S corresponding to s acts simply transitively on the domain^cD.

3.4. An orthonormal basis for a Siegel domain of type II. Now we describe an orthonormal basis of s for the nontube case corresponding to the decomposition (3.15).

This will be the same basis as in [DHMP], [DHP], [BBDHPT]. We begin with finding a basis of n^ij. Take α ∈ Q^ij and put

(3.22) α = γe i+ γj − α,

then by [L] (page 141) eα ∈ Q^ij. The classification theorem for Hermitian symmetric spaces says that in the nontube case dimension of each space Qij is even. Since α 6= ^γi+γ₂ ^j if and only if eα 6= ^γi+γ₂ ^j, it follows that α 6= eα. By Q_ij we shall denote the subset of Qij

such that from each pair of roots α and eα exactly one is contained in Q_ij. Define

β = α − γⁱ = γj− eα, θ = α − γj = γ_i− eα, (3.23)

Then β and θ are positive compact roots. Applying Lemma V.5.1 in [H1] and Corollary (2.18) we obtain:

ε = Mγi,β = Mβ,−α = M−α,γi = −M^−β,α = −M^α,−γi = −M^−γi,−β, δ = Mγj,θ = Mθ,−α= M−α,γj = −M^−θ,α = −M^α,−γj = −M^−γj,−θ, ρ = M−γj,β = M_β,eα= Mα,−γ_e j = −M−β,−eα = −M−eα,γj = −M^γj,−β, σ = M−γi,θ = M_θ,eα = Mα,−γ_e i = −M−θ,−eα = −M−eα,γi = −M^γi,−θ, (3.24)

for some ε, σ, ρ, δ ∈ {−1, 1}. We have α + eα − γⁱ− γ^j = 0, thus it follows from Lemma V.5.3 [H1] that ερ = δσ, which implies

(3.25) εδ = (δσρ⁻¹)δ = δ²σρ⁻¹ = σρ.

From (3.18) and (3.24) we obtain Ad(c⁻¹)Eα = 1

2Eα−1

2ερE_−eα− 1

2i(εEβ + δEθ), Ad(c⁻¹)E_αe = 1

2E_αe−1

2ερE_−α+1

2i(σE_−θ+ ρE_−β),

therefore Ad(c⁻¹)(E_α− ερEαe) and Ad(c⁻¹) · i(Eα+ ερE_αe) are elements of g, hence A_α = Eα− ερEαe, Bα = i(Eα + ερEα_e) belong to n^ij. Calculating dimensions we see that the vectors having the above form are a basis of n^ij. To write a basis of nij we shall use the formula (3.5) and compute Cα = 2Aα2ci and Dα = 2Bα2ci (see (3.1)). Take

X = 1

2(−δ(E^θ+ E−θ) − ε(E^β+ E−β)), then [X, EΓ] = Aα, therefore L(Aα) = X, and

Cα = 2([L(Aα), L(ci)] + L(Aαci))

= [X, Hγi] + X

= 1

2(δE_θ− δE−θ− εEβ + εE_−β)) + 1

2(−δ(Eθ+ E_−θ) − ε(Eβ+ E_−β))

= −δE^−θ− εE^β. Similarly we can compute:

D_α = i(δE_−θ− εEβ).

Our last step it to write a basis of n⁺₂. By [KW], Lemma 6.5, the map ψ = I + Ad(c²)τ is a real linear isomorphism of p⁺₂ onto n⁺₂. The dimension argument proves that vectors of the form

ψ(Eα), ψ(iEα) for α ∈ Qⁱ, are a basis of n⁺₂.

The vectors

{E^γi}, { 1

√2(Eα− ερEαe)}, { i

√2(Eα+ ερEα_e)}, { 1

√2ψ(Eα)}

{1

2H_γj}, { 1

√2(−δE−θ− εEβ)}, { i

√2(δE_−θ− εEβ)}, { 1

√2ψ(iE_α)}

form an orthonormal basis of s with respect to the Hermitian product Bτ. We denote the corresponding left-invariant vector fields on S respectively by:

Xi, X_α¹, X_α², X^α, Hj, Y_α¹, Y_α², Y^α,

and we introduce in p⁺ coordinates corresponding to the basis Xi, X_α¹, X_α², X^α: (3.26) w =X

j

wjXj(e) +X

i,j

X

α∈Q_ij

(w¹_αX_α¹(e) + w_α²X_α²(e)) +X

i

X

α∈Qi

wαψ⁻¹(X^α(e)),

Given a function f on ^cD let

f (s) = f (s · (c · o)),e then for a left-invariant vector field W on S we have

W ef (s) = ∂

∂tf (s exp tW · (c · o))|^t=0 and so

Xjf (e) = ∂ue jf (c · o), X_α^kf (e) = ∂ue ^k_αf (c · o), X^αf (e) = ∂ue αf (c · o), H_jf (e) = ∂ve _jf (c · o), Y_α¹f (e) = ∂ve ^k_αf (c · o), Y^αf (e) = ∂ve αf (c · o), (3.27)

where w = u + iv. Therefore the complex structure J on s, transported from^cD is:

J (X^j) = Hj, J (Hj) = −Xj, J (Xα^k) = Y_α^k,

J (Yα^k) = −Xα^k, J (X^α) = Y^α, J (Y^α) = −X^α. (3.28)

Finally,

Zj = Xj− iH^j, Z_α^k = X_α^k− iYα^k, Z^α = X^α− iY^α (3.29)

are holomorphic vector fields.

Let Z be one of the vectors fields Zj, Z_α^k, Z^α, w the corresponding coordinate wj, w^k_α or w_α and let ∆_Z be the unique left-invariant differential operator with the property

(3.30) ∆Zf (e) = ∂e w∂wf (c · o).

∆Z is real, second order, elliptic degenerate and annihilates holomorphic (consequently pluriharmonic) functions and any left-invariant operator with the above properties is a linear combination of such. Therefore ∆Z’s are building blocks for admissible operators.

∆_Z can be explicitly computed on the whole group S:

(3.31) ∆Z = ZZ − ∇^ZZ = X²+ (J X)² − ∇^XX − ∇^{J X}J X.

where ∇ denotes the Riemannian connection on S (see [DHP], [DHMP]).

4. The Hua system of second order operators and the main theorem Let {Ej} be any orthonormal basis of p⁺ and {Ej^∗} be a dual basis of p⁻ with respect to the Killing form B of g^C (for example { eEα}α∈Q⁺ and { eE−α}α∈Q⁺ are such bases). Then the Hua system is

(4.1) H =X

j,k

EjE_k^∗⊗ [E^k, E_j^∗].

The above definition was given by K. Johnson and A. Koranyi in [JK]. It is clearly an element of U^C⊗ k^C, where U^C is the complexification of the enveloping algebra of g. One can easily check that H does not depend on the chosen basis. For this reason we shall write always the operator H in terms of the base vectors { eEα}α∈Q⁺. We say that a function f defined on D is Hua-harmonic, if the corresponding function ef on G ( ef (g) = f (g · o)) is annihilated by the Hua system. Analogously f is annihilated by a left-invariant operator U on G if U ef = 0.

Now we are ready to formulate the main result of this paper:

The Main Theorem. Let D = G/K be an irreducible Hermitian symmetric domain of nontube type and let f be a real function on D. Then f is Hua–harmonic if and only if f is pluriharmonic.

Let us recall that f defined on D ⊂ Cⁿ is pluriharmonic if it is the real part of a holomorphic function. Pluriharmonicity is equivalent to being annihilated by all operators

∂zj∂zk (1 ≤ j, k ≤ n). Since E^jE_k^∗f (e) = ∂ze j∂zkf (0) (see [JK], formula 3.18), the Hua system annihilates pluriharmonic functions. For this reason we have only to prove that any Hua-harmonic function is pluriharmonic.

In fact we shall not use the whole Hua system, but only a part of it. More precisely for any basis {v^k} of k^C define elements Uvk of U^C by

(4.2) Uvk = X

α∈Q⁺

[vk, eEα] eE_α^∗. Then we have a simple proposition

Proposition 4.3 ([JK], [L]). Let f be a function on G. Then Hf = 0 if and only if for every k: U_vkf = 0.

By [JK] and [L] the Laplace-Beltrami operator is a linear combination of operators Uvk. Next we define second order differential operators Uk, which will be called strongly diagonal Hua operators:

U_k = X

α∈Q⁺

[ eH_γk, eE_α] eE_α^∗. (4.4)

Observe, that in view of proposition (2.16):

Uk= X

α∈Q⁺

α( eHγk) eEαEe_α^∗

= X

α∈Q⁺

hα, αi

2 α( ˜H_γk)E_αE_−α = X

α∈Q⁺

hα, αi

hγ^k, γkiα(H_γk)E_αE_−α

= 2EγkE−γk + X

α∈Qk

EαE−α+X

j

X

α∈Qjk

EαE−α. (4.5)

To prove pluriharmonicity of any Hua-harmonic function, we shall use only the Laplace - Beltrami operator and strongly diagonal Hua operators.

Now we explain the strategy of the proof. Using the classical result of Harish-Chandra [HC] we may expand f in terms of its projections on the spaces of K-finite vectors of type π:

f = X

π∈ bK

χπ∗^Kf,

where bK is the set of equivalence classes of irreducible unitary representations of K and χπ is the character of π. Now fπ = χπ ∗^Kf are K-finite functions and are clearly Hua–

harmonic. In the next section we prove (Theorem (5.1)) that every Hua-harmonic function is G-harmonic (i.e. annihilated by all G invariant operators on G/K), therefore fπ are G-harmonic. Each K–finite, G–harmonic function is Poisson integral of a continuous bounded function defined on the maximal Furstenberg boundary ([H3], Theorem V.6.1), therefore each f_π is bounded, and it is enough to prove the main theorem for bounded functions.

Having the growth assumption (which is invariant on biholomorphic mappings), we transfer our problem to the group S acting simply transitively on the Siegel domain^cD, and using techniques of [BDH], we obtain the result. The details are contained in sections 6 and 7.

5. G-harmonic functions The aim of this section is to prove the following theorem

Theorem 5.1. If f is a Hua-harmonic function, then f is G-harmonic.

The above theorem was proved in the tube case by [JK] and [L]. Furthermore the authors of the first cited paper remarked that this result holds also for nontube domains, but they didn’t give a proof. Our proof follows closely the argument of [L] and finally we get a system of equations that differs only by constants from the one considered by Lassalle. The main step is to prove the following theorem

Theorem 5.2. If Φ is K-biinvariant, Hua-harmonic function on G, then Φ is constant.

Using this result we can easily prove theorem (5.1):

Proof of theorem (5.1). As we noticed in the previous section, among the Hua operators there is the Laplace-Beltrami operator. Hence f is an analytic function. Take dk to be unimodular normalized Haar measure on K. For fixed g ∈ G define a function on G

Φ(h) = Z

K

f (gkhK)dk.

Then Φ is K-biinvariant and Hua-harmonic, therefore by theorem (5.2) Φ is constant.

Hence

f (gK) = Z

f (gK)dk = Z

f (gkK)dk = Φ(e) = Φ(h) = Z

f (gkhK)dk

and using the Godement theorem ([H2], page 403) we deduce that f is G-harmonic. ¤ We are interested in studying how the Laplace - Beltrami operator and strongly diagonal Hua operators act on a K-biinvariant function f , defined on G. From KAK decomposition of G follows that f depends only on A. Hence its enough to compute radial parts of Uk, which we shall denote by ∆(Uk) ∈ U(a^C). For this we determine ∆(EαE−α) for all positive noncompact roots α, considering three cases when α belongs to Γ, Q_j or Q_ij. The main tool to compute radial parts of these operators will be the following proposition:

Proposition 5.3 ([L]). Let f ∈ C²(G) be a right K-invariant function and let X ∈ k. If f (a · exp t(X + Y )) = f(a)

for all t and some Y ∈ g then

Y²f (a) = ([Y, X]f )(a). 2 Using (2.8) we have

EαE−α= 1

4(Xα− iY^α)(Xα+ iYα) = 1

4(X_α² + Y_α²+ i[Xα, Yα]) = 1

4(X_α²+ Y_α²) mod ik Therefore it is enough in each case to compute ∆(X_α²) and ∆(Y_α²). Define

a^′ = {X =X

tkXγk : tk 6= 0; tⁱ ± t^j 6= 0 for i 6= j}

and put A^′ = exp a^′.

Proposition 5.4 ([L]). The radial part of EγkE−γk on A^′ is equal 1

4(X_γ²_k+ 2 coth 2t_kX_γk).

Proof. Take a ∈ A^′, then

f (a) = f (exp itH_γk · a) = f(a · Ada⁻¹(exp itH_γk)) = f (a exp Ada⁻¹(itH_γk)) Applying commutation relations:

[Xγk, iHγk] = −2Y^γk, [Xγk, Yγk] = −2iH^γk, we obtain

Ada⁻¹· iH^γk = Ad(exp −t^kXγk) · iH^γk

= exp ad(−tkX_γk) · iHγk = cosh(2t_k)iH_γk+ sinh(2t_k)Y_γk. Hence by proposition (5.3)

(sinh(2tk)Yγk)²f (a) = [sinh(2tk)Yγk, cosh(2tk)iHγk]f (a).

And finally by [Yγk, iHγk] = 2Xγk, we conclude

Y_γ²_kf (a) = 2 coth(2tk)Xγkf (a).

¤ Next we compute ∆(EαE−α), for α ∈ Qⁱ

Lemma 5.5. Let α ∈ Qⁱ. Then there exists U ∈ k such that [U, Xα] = εXγi,

Ada⁻¹· U = cosh tⁱU − ε sinh tⁱXα, where ε = ±1, and a ∈ A^′.

Proof. Take β = α − γⁱ and ε as in (3.24). Put U = Xβ (as defined in (2.9)). Then the following relations hold:

[Xγj, U ] = εδijXα, [Xγi, Xα] = εU,

[U, Xα] = εXγi. (5.6)

To compute them we use the definition of numbers Mα,β, and the observation (see the proof of proposition (2.16)) that the sum of some roots can not be a root. For instance α + γi, −α − γⁱ 6∈ ∆, for this reason

[Xγi, Xα] = [Eγi + E−γi, Eα+ E−α] = [Eγi, Eα] + [Eγi, E−α] + [E−γi, Eα] + [E−γi, E−α]

= −εE−β+ εE_β = εU.

Let a = exp(Pl

j=1tjXγj) ∈ A^′ and define an endomorphism M of g by M = ad(−

Xl j=1

tjXγj).

Then by (5.6),

Ada⁻¹· U = e^{ad(−PtjXγj)}U = e^MU, M²ⁿU = t²ⁿ_i U,

M²ⁿ⁺¹U = −εt²ⁿ⁺¹i Xα, and we get

Ada⁻¹· U = e^MU = Ã _∞

X

k=0

t^2k_i (2k)!

! U − ε

à _∞ X

k=0

t^2k+1_i (2k + 1)!

! Xα

= cosh tiU − ε sinh tⁱXα.

¤ Lemma 5.7. Suppose α ∈ Qi, then there exists V ∈ k such that:

[V, Yα] = εXγi,

Ada⁻¹· V = cosh tⁱV − ε sinh tⁱYα, where ε has the same value like in the previous lemma.

Proof. Taking V = Yβ (for β as in the previous lemma) and repeating the proof we

conclude the lemma. ¤

Proposition 5.8. For α ∈ Qⁱ the radial part of EαE−α on A^′ is:

1

2coth tiXγi.

Proof. Using K invariance of the function f and lemma (5.5) we obtain f (a) = f (a exp(cosh tiU − ε sinh tⁱXα)).

Hence by proposition (5.3):

(−ε sinh tⁱXα)² = [−ε sinh tⁱXα, cosh tiU ] = −ε sinh tⁱcosh ti[Xα, U ] = sinh ticosh tiXγi. Therefore

X_α² = coth tiXγi. Analogously, applying lemma (5.7):

Y_α² = coth tiXγi. So we have

∆(EαE−α) = 1

4(∆(X_α²) + ∆(Y_α²)) = 1

4(coth tiXγi+ coth tiXγi) = 1

2coth tiXγi.

¤

The last case is α ∈ Q^ij. Then for our purpose it is enough to determinate ∆(EαE−α+ E_eαE_−eα), for eα defined by (3.22).

Lemma 5.9. If α ∈ Qij then there exist U, ˜U , V, ˜V ∈ k such that

Ada⁻¹U = cosh(ti+ ρσtj)U − sinh(tⁱ+ ρσtj)(εXα+ σXα_e), Ada⁻¹U = cosh(t˜ _i− ρσtj) ˜U − sinh(ti− ρσtj)(εX_α− σXαe), Ada⁻¹V = cosh(ti− ρσt^j)V − sinh(tⁱ− ρσt^j)(εYα+ σYα_e), Ada⁻¹V˜ = cosh(t_i+ ρσt_j)V − sinh(ti+ ρσt_j)(εY_α− σYαe), [εXα+ σXα_e, U ] = −2(X^γi + ρσXγj),

[εXα− σXαe, ˜U ] = −2(X^γi − ρσX^γj), [εY_α+ σY_αe, V ] = −2(Xγi − ρσXγj), [εYα− σYαe, ˜V ] = −2(X^γi + ρσXγj), where ε, σ, ρ, δ ∈ {−1, 1}.

Proof. Take β, θ, ε, δ, ρ, σ, as in (3.23) and (3.24). Next set U = Xβ+ Xθ, ˜Uβ = Xβ−X^θ, V = Yβ + Yθ, ˜Vβ = Yβ − Y^θ. Then applying the methods of the proof of lemma (5.5), and using (3.25) we can easily compute all needed brackets relations, and conclude the

lemma. ¤

Proposition 5.10. Let α ∈ Qij, then on A’

∆(EαE−α+ Eα_eE_−eα) = 1

2(coth(ti+ tj)(Xγi + Xγj) + coth(ti− t^j)(Xγi − X^γj)).

Proof. By lemma (5.9) we get

f (a) = f (a exp(cosh(ti + ρσtj)U − sinh(tⁱ+ ρσtj)(εXα+ σXα_e))) In view of proposition (5.3)

(− sinh(tⁱ+ ρσtj)(εXα+ σXα_e))² = −[sinh(tⁱ+ ρσtj)(εXα+ σXα_e), cosh(ti+ ρσtj)U ].

Accordingly

(εXα+ σXα_e)² = 2 coth(ti+ ρσtj)(Xγi+ ρσXγj), and similarly

(εXα− σXαe)² = 2 coth(ti− ρσt^j)(Xγi− ρσX^γj), which gives us

2(X_α²+ X_eα²) = (εXα+ σXα_e)²+ (εXα− σXαe)²

= 2(coth(ti+ tj)(Xγi + Xγj) + coth(ti− t^j)(Xγi − X^γj)).

Analogous computations entail

Y_α²+ Y_αe² = coth(ti+ tj)(Xγi + Xγj) + coth(ti− t^j)(Xγi− X^γj).