Jean-Louis Clerc

Construction à la Ibukiyama of symmetry breaking differential operators, I

Keywords: 2020 MSC. Primary 32M15; Secondary 17C20, 22E46, 11F70 tube-type domain, Euclidean Jordan algebra, holomorphic representations, pluri-harmonic polynomial, symmetry breaking differential operator

The construction of symmetry breaking differential operators, using invariant pluri-harmonic polynomials, due to T. Ibukiyama in the context of the Siegel upper half space, is extended for scalar representations to general Hermitian symmetric spaces of tube-type. The new context is described in terms of Euclidean Jordan algebras and their representations. As an example, new and explicit differential operators are obtained for the restriction from the tube domain over the light cone to the product of two upper half-planes.

Introduction

In his seminal paper [START_REF] Ibukiyama | On differential operators on automorphic forms and invariant pluri-harmonic polynomials[END_REF] T. Ibukiyama introduced a construction of holomorphic differential operators, in the geometric context of the Siegel upper half space, its group of holomorphic diffeomorphisms (the symplectic group) and the holomorphic series of representations. The differential operators he constructs are examples of symmetry breaking differential operators in the sense of T. Kobayashi (see [START_REF] Kobayashi | A program for branching problems in the representation theory of real reductive groups[END_REF]). They can also be viewed as generalizations of the classical Rankin-Cohen brackets, and they play an important rôle in the theory of Siegel modular forms. For more on the subject, see [START_REF] Ibukiyama | Generic differential operators on Siegel modular forms and special polynomials[END_REF] and the bibliography therein.

1

In the present paper, a broader geometric context is considered, namely Hermitian symmetric spaces of tube-type. The theory of Euclidean Jordan algebras is very useful to handle these situation and the (lesser known) notion of representation of a Euclidean Jordan algebra, as introduced in [START_REF] Faraut | Analysis on Symmetric Cones[END_REF] and further studied in [START_REF] Clerc | Représentation d'une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel[END_REF][START_REF] Clerc | Kelvin transform and multi-harmonic polynomials[END_REF][START_REF] Clerc | Determinantally homogeneous polynomials on representations of Euclidean Jordan algebras[END_REF] is central to the present article.

The main theorem of [START_REF] Ibukiyama | On differential operators on automorphic forms and invariant pluri-harmonic polynomials[END_REF] is rephrased and proved in this broader context. However, we only explore the situation called (I) in [START_REF] Ibukiyama | On differential operators on automorphic forms and invariant pluri-harmonic polynomials[END_REF], and we only consider the case of scalar holomorphic representations.

As a test of the efficiency of the process to produce new and explicit examples, some symmetry breaking differential operators are obtained, in relation with the restriction from the tube-domain over the light cone to the product of two upper-half planes (or equivalently from the Lie ball to the bi-disc).

Examples in rank 2

To complete the introduction, here is a more precise description of this paper. Let D be a Hermitian symmetric space, and let Aut(D) be its group of holomorphic diffeomorphisms. Let D ′ be a Hermitian symmetric subspace of D. Let G be the subgroup of Aut(D) which preserves the smaller domain D ′ . Let π a representation of Aut(D) realized on space O(D) of holomorphic functions on D, and let π ′ be a representation of G which is realized on the space O(D ′ ). Finally let res : O(D) -→ O(D ′ ) be the restriction map. In this context, a symmetry breaking differential operator (SBDO for short) is a holomorphic differential operator D on D such that, for any g ∈ G (res •D) • π(g) = π ′ (g)(res •D) .

Among the Hermitian symmetric domains, there is the subclass of domains of tube-type, those which can be realized as Siegel domains of type I, i.e. complex tubes over convex symmetric cones in a Euclidean space. In turn, symmetric cones are related to Euclidean Jordan algebras. More precisely to any Euclidean Jordan algebra J is associated a convex symmetric cone Ω and a Hermitian symmetric space of tubetype T Ω , and vice versa, in a very functorial way (see [START_REF] Satake | Algebraic structures of symmetric domains[END_REF] Ch. I Section 9). The notion of complete system of orthogonal idempotents allows to construct specific Jordan subalgebras J ′ such that the associated tube-type domain T Ω ′ is a Hermitian subdomain of T Ω , both of the same rank. An example is the situation studied by Ibukiyama, where J = Symm(r) is the Jordan algebra of real symmetric matrices of size r, J ′ = Symm(r 1 ) ⊕ • • • ⊕ Symm(r k ) and r 1 + r 2 = • • • + r k = r. The corresponding tube-type domains are the Siegel upper half-space

T Ω = H r and T Ω ′ = H r 1 ⊕ • • • ⊕ H r k .
Section 1 is devoted to study this situation for a general Euclidean Jordan algebra and a general CSOI. The structure of the subgroup of Aut(T Ω ) which preserve the smaller tube-type domain T Ω ′ is precisely described.

The notion of representation of a Euclidean Jordan algebra, systematically introduced by J. Faraut and A. Korányi (see [START_REF] Faraut | Analysis on Symmetric Cones[END_REF]) offers a nice framework to reinterpret and generalize Ibukiyama's construction. The notion is recalled in Section 2, and further developped in the context of Section 1. The notion of pluri-harmonic polynomials is introduced in Section 3, and the classical Hecke formula (already extended in [START_REF] Kashiwara | On the Segal-Shale-Weil representation and harmonic polynomials[END_REF]) is further extended to the present situation.

Section 4 recalls the construction of the (scalar) holomorphic series of representations for the group Aut(T Ω ), in fact for a twofold covering of its neutral component.

Section 5 contains the main result, and reduces the analytic problem to an algebraic problem about a class of polynomials on J. The problem is hard to solve in general, but many cases can be investigated, using in particular the classical theory of compact simple Lie groups and theory of invariants (see [START_REF] Ibukiyama | Generic differential operators on Siegel modular forms and special polynomials[END_REF]).

Section 6 is devoted to an example, corresponding to the case where the Jordan algebra J is of rank 2. The domain T Ω is the tube-domain over the light cone, the subdomain T Ω ′ is a product of two upper halfplanes. The representations of J involved are interpreted as Clifford modules. In this case, we investigate the algebraic problem and give explicit solutions, producing new pluri-harmonic polynomials and new SBDO.

1 The algebraic/geometric setting Let J be a Euclidean Jordan algebra. The main reference for results and notation is [START_REF] Faraut | Analysis on Symmetric Cones[END_REF]. See also [START_REF] Satake | Algebraic structures of symmetric domains[END_REF].

1.1 Complete system of orthogonal idempotents and the associated subalgebra Let c = (c 1 , c 2 , . . . , c k ) be a CSOI of J. For each j, 1 ≤ j ≤ k, let J j be the Euclidean Jordan subalgebra defined by

J j = J(c j , 1) = {x ∈ J, c j x = x} .
Each J j is a subalgebra of J, and for any i, j such that 1

≤ i = j ≤ k, J i ∩ J j = {0} and J i J j = 0. Define J(c) = k j=1 J j .
Then J(c) is a Euclidean Jordan subalgebra of J, and we refer to it as the subalgebra associated to the CSOI c.

The group L(c)

Let Str(J) be the structure group of J. Its elements may be characterized as follows : an element ℓ ∈ GL(J) belongs to Str(J) if and only if ℓ preserves J × (the open set of invertible elements in J) and there exists h ∈ GL(J) such that for any x ∈ J × , (ℓx

) -1 = hx -1 .
Moreover, the element h is unique and equal to ℓ t -1 .

Consequently, the structure group Str(J) is a closed Lie subgroup of GL(J), stable by the Cartan involution ℓ -→ ℓ t -1 .

Let Ω be the symmetric cone in J which can be defined as the set of squares of invertible elements. The group G(Ω) of all linear transformations of J which preserve Ω is closely connected to Str(J). In fact, both groups have the same neutral component, henceforth denoted by L. The next result is introduced (with proof) because of lack of reference.

Proposition 1.1. Let J be a Euclidean Jordan algebra. The closed subgroup of GL(J) generated by {P (x), x ∈ Ω} is equal to L.

Proof. Let L 1 be the closed subgroup generated by {P (x), x ∈ Ω}. For x ∈ Ω, P (x) belongs to L, so that L 1 ⊂ L.

Let l = Lie(L) = str(J) and l 1 = Lie(L 1 ). Let p = {L(x), x ∈ J}. Recall that for any x ∈ J P (exp x) = exp 2L(x). Moreover, Ω = exp J, so that l 1 ⊃ p, and hence

l 1 ⊃ [p, p] ⊕ p .
On the other hand, l = Der(J) ⊕ p where Der(J) is the space of derivations of J, which is known to be equal to [p, p], so that l = [p, p] ⊕ p .

and hence l 1 = l. As both L and L 1 are connected and L 1 ⊂ L, the conclusion follows.

Let J be a Euclidean Jordan algebra, let c = (c 1 , c 2 , . . . , c k ) be a CSOI and let J(c) be the associated subalgebra. Define

Str(c) = {ℓ ∈ Str(J), ℓ J(c) = J(c)} .
The group Str(c) is a Lie subgroup and its Lie algebra str(c) is given by str(c) = {T ∈ str(J), T J(c) ⊂ J(c) } .

Proposition 1.2. The group Str(c) is stable by the Cartan involution.

Proof. Let J × be the open subset of invertible elements in J, and let J(c) × the open subset of invertible elements of J(c). Then

J(c) × = J(c) ∩ J × . (1)
In fact, if x ∈ J(c) is invertible in J, its inverse belongs to R[x]. As e and x belong to J(c), R[x] ⊂ J(c), so that x is invertible in J(c).

Hence J(c) ∩ J × ⊂ J(c) × . The opposite inclusion J(c) × ⊂ J × is trivial. Now assume that ℓ belongs to Str(c). Let x ∈ J(c) ∩ J × . As ℓ t -1 belongs to Str(J),

ℓ t -1 x = (ℓx) -1 ∈ J × ∩ J(c)
and n(1) implies that ℓ t -1 maps J(c) × into itself. As J(c) × is dense in J(c), ℓ t -1 maps J(c) into itself. Hence the group Str(c) is stable by the involution ℓ -→ ℓ t -1 .

Notice that, as a consequence of Proposition 1.2, the Lie algebra str(c) is stable by the Cartan involution T -→ -T t . Proposition 1.3. Let T ∈ str(c). Then T maps each J j into itself.

Proof. If T ∈ l(c), then both 1 2 (T -T t ) and 1 2 (T + T t ) belong to l(c), so that it suffices to prove Proposition 1.3 separately for T = D ∈ l(c) ∩ Der(J) and for those T = L(v), v ∈ J which belong to l(c).

So let D be a derivation of J which maps J(c) into itself. For j, 1 ≤ j ≤ k, Dc j = Dc 2 j = 2c j Dc j so that Dc j ∈ J(c j , 1 2 ). As J(c j , 1 2 ) ∩ J(c) = {0}, Dc j = 0. Now for x ∈ J j , Dx = D(c j x) = c j Dx and hence Dx ∈ J j . As this is true for any j, the conclusion follows in this case.

Next, let T = L(v) for some v ∈ J and assume that L(v) maps

J(c) into itself. Let j, 1 ≤ j ≤ k and let v = v 1 + v 1 2
+ v 0 be its decomposition relative to the idempotent c j . Then

L(v)c j = L(c j )v = v 1 + 1 2 v 1 2 . As J(c j , 1 2 ) ∩ J(c) = {0}, this forces L(v)c j = v 1 ∈ J j . So, for any j, 1 ≤ j ≤ k, L(v)c j ∈ J j . Hence v = L(v)e = L(v)(c 1 + • • • + c k ) = L(v)c 1 + • • • + L(v)c k
belongs to J(c). But now for v ∈ J(c), L(v) maps each J j into itself and the conclusion follows.

Let L(c) be the neutral component of Str(c).

Proposition 1.4. Let ℓ ∈ L(c). Then ℓ maps J j into itself for any j, 1 ≤ j ≤ k. For each j, 1 ≤ j ≤ k, the induced map ℓ j : J j -→ J j belongs to L j , the connected component of Str(J j ).

Proof. Let ℓ t = exp tT be a one-parameter subgroup in L(c). Then by differentiation at t = 0, X belongs to l(c), which by Proposition 1.3 implies X(J j ) ⊂ J j for any j, 1 ≤ j ≤ k. Hence ℓ t (J j ) ⊂ J j for any t ∈ R. So the proposition is satisfied for all elements of L(c) sufficiently closed to the identity. As L(c) is connected, the property ℓ(J j ) ⊂ J j is valid for all ℓ ∈ L(c).

For the second part of the proposition, recall that for ℓ ∈ Str(J) and x ∈ J,

P (ℓx) = ℓP (x)ℓ t (2)
Let ℓ ∈ L(c) and let x ∈ J j , so that ℓx j ∈ J j . The operators ℓ, ℓ t , P (x) map J j into itself, so that by (2), P (ℓx) also maps J j into itself. By restriction to J j , (2) implies P j (ℓ j x) = ℓ j P j (x)(ℓ t ) j

By [START_REF] Faraut | Analysis on Symmetric Cones[END_REF] Lemma VIII.2.3 applied to J j , this implies ℓ j ∈ Str(J j ) and (ℓ t ) j = ℓ t j . As the restriction map ℓ -→ ℓ j is continuous, it follows that ℓ j belongs to L j .

The last proposition map allows to define the restriction map

L(c) ∋ ℓ -→ (ℓ 1 , ℓ 2 , . . . , ℓ k ) ∈ L 1 × L 2 × • • • × L k .
For each j, 1 ≤ j ≤ k let Ω j be the symmetric cone of J j . Proposition 1.5. Let x 1 ∈ J 1 , . . . , x j ∈ J j , . . . , x k ∈ J k and let x =

x 1 + x 2 + • • • + x k ∈ J(c). Then x belongs to Ω iff x j belongs to Ω j for 1 ≤ j ≤ k.
Proof. For any j, 1 ≤ j ≤ k let r j be the rank of J j . There exists a Jordan frame (e (1) j , . . . , e (r j ) j ) of J j such that

x j = a (1) j e (1) j + • • • + a (r j ) j e (r j ) j .
for some a

(i) j ∈ R, 1 ≤ i ≤ r j . The collection e (1) 1 , . . . , e (r 1 ) 1 , . . . , e (1)
k , . . . e (r k ) k is a Jordan frame of J and

x = x 1 + x 2 + • • • + x k = k j=1 r j i=1 a (i) j e (i) j .
Now x j belongs to Ω j if and only if a (i) j > 0 for 1 ≤ i ≤ r j , and x belongs to Ω if and only if a

(i) j > 0 for 1 ≤ j ≤ k and 1 ≤ i ≤ r j .
The proof of the equivalence of the two properties follows easily.

Consequently, let

Ω(c) = J(c) ∩ Ω = Ω 1 + . . . Ω j + • • • + Ω k . Proposition 1.6. Let c an idempotent of J and let x ∈ J(c, 1), y ∈ J(c, 0). Then P (x + y) maps J(c, λ) into itself (λ = 1, 1 2 , 0) and is equal to i) P 1 (x) on J(c, 1)) ii) 2 L(x)L(y) + L(y)L(x) on J c, 1 2
iii) P 0 (y) on J 0 (c)

.

Proof. As P (y) = 2L(y) 2 -L(y 2 ) we get

P (x + y) = 2L(x + y) 2 -L (x + y) 2 = 2L(x) 2 + 2L(y) 2 + 2 L(x)L(y) + L(y)L(x) -L(x 2 ) -L(y 2 ) -2L(xy) = P (x) + 2 L(x)L(y) + L(y)L(x) + P (y) .
Now, as P (c)x = x, P (x) = P P (c)x = P (c)P (x)P (c), P (x) maps J(c, 1) on itself and is 0 on J(c, 1 2 ) ⊕ J 0 (c). Permuting the role of c (resp. x) and (e -c) (resp. y), P (y) maps J(c, 0) on itself and is 0 on J(c, 1) ⊕ J(c, 1 2 ). Next, L(x) is 0 on J(c, 1) and L(y) is 0 on J(c, 1), so that 2(L(x)L(y) + L(y)L(x)) is 0 on J(c, 1) ⊕ J(c, 0), and finally both L(x) and L(y) maps J(c, 1 2 ) into itself, so that 2(L(x)L(y)+L(y)L(x)) maps J(c, 1 2 ) into itself. This completes the proof of Proposition 1.6.

Proposition 1.7. Let x 1 ∈ Ω 1 , . . . , x k ∈ Ω k and set x = x 1 + x 2 + • • • + x k .
Then P (x) belongs to L(c) and its image by the restriction map is equal to P 1 (x 1 ), P 2 (x 2 ), . . . , P (x k ) .

Proof. As x belongs to Ω, P (x) belongs to L. Let j, 1 ≤ j ≤ k and set y j = x -x j . Then x j ∈ J(c j , 1) and y j ∈ J(c j , 0). Hence, by Proposition 1.6, P (x) = P (x j + y j ) maps J(c j , 1) = J j into itself and the induced restriction on J(c j , 1) = J j is equal to P j (x j ).

Let K be the neutral component of the group of automorphisms (for the Jordan structure) of J. Recall that K is a maximal compact subgroup of L and its Lie algebra k is equal to Der(J). It is also the stabilizer of the neutral element e in L. For each j, 1 ≤ j ≤ k, the definition of K j is the same, adapted to the Jordan algebra J j .

Proposition 1.8. The image of K ∩ L(c) by the restriction map is equal to

K 1 × K 2 × • • • × K k . Proof. Let k ∈ K ∩ L(c) and let (k 1 , k 2 , . . . , k k )
be its image by the restriction map. Then ke = e, so that, for any j, k j c j = c j and hence k j ∈ K j . Next, let u j , v j be in J j . Then [L(u j ), L(v j )] is a derivation of J which preserves J(c) and induces (0, . . . , 0, [L j (u j ), L j (v j )], 0, . . . , 0) on J(c). As elements of the form [L(u j ), L(v j )] generate Der(J j ), it follows that Der(J 1 ) × Der(J 2 ) × • • • × Der(J k ) is contained in the image by the restriction map of Der(J) ∩ L(c). Hence the image by the restriction map of K ∩ L(c) contains a neigbourhood of the neutral element in

K 1 × K 2 × • • • × K k . As the image of K ∩ L(c) is compact and as K 1 × K 2 × • • • × K k is connected, the image is equal to K 1 × K 2 × • • • × K k . Let M (c) = {ℓ ∈ L(c), ℓx = x for all x ∈ J(c)} . Observe that M (c) is a closed normal subgroup of L(c), which is contained in Aut(J). Let Lie M (c) = m(c).
Proposition 1.9.

i) The restriction map

L(c) ∋ ℓ -→ (ℓ 1 , . . . , ℓ k ) ∈ L 1 × • • • × L k is a surjective homomorphism.
ii) The kernel of the restriction map is equal to M (c).

iii) L(c) is the closed subgroup of L generated by {P (x), x ∈ Ω(c)} and M (c) .

Proof. Any element of L j can be written as k j P (x j ) for some k ∈ K j and some x j ∈ Ω j . So i) follows easily from Proposition 1.7 and Proposition 1.8. Further, ii) is a merely a rephrasing of the definition of M (c).

Finally, let x j ∈ Ω j and set

x j = c 1 + • • • + c j-1 + x j + • • • + c k .
The image of P ( x j ) by the restriction map is equal to (Id 1 , . . . , Id j-1 , P j (x j ), Id j+1 , . . . , Id k ) .

As L j is the closed subgroup generated by {P j (x j ), x j ∈ Ω j }, the image by the restriction map of the closed subgroup generated by {P ( x j ),

x j ∈ Ω j } is equal to {Id 1 } × • • • × {Id j-1 } × L j × • • • × {Id k }
and iii) follows by repeating the argument for all j, 1 ≤ j ≤ k.

The group G(c)

Let J be the complexification of J, and let

T Ω = J + iΩ ⊂ J
be the tube over the cone Ω, which is a Hermitian symmetric domain. Let G be the neutral component of the group Aut(T Ω ) of all bi-holomorphic automorphisms of T Ω .

Let J j (resp. J(c)) be the complexification of J j (resp. J(c)). For j, 1 ≤ j ≤ k consider the tube-type domain T Ω j ⊂ J j . As a consequence of Proposition 1.5,

T Ω ∩ J(c) = T Ω 1 ⊕ • • • ⊕ T Ω j + • • • ⊕ T Ω k ,
and this domain will be denoted by T Ω(c) .

Let

G(c) be the neutral component of the subgroup of G defined by {g ∈ G, g(T Ω(c) ) ⊂ T Ω(c) } .
The subgroup G(c) is a closed Lie subgroup of G. We first determine the Lie algebra g(c) of G(c). Recall that the Lie algebra of g is given by

g = J ⊕ l ⊕ J = {X = (u, T, v), u ∈ J, T ∈ l, v ∈ J} with Lie bracket [(u 1 , T 1 , v 1 ), (u 2 , T 2 , v 2 )] = (u, T, v) u =T 1 u 2 -T 2 u 1 , T =[T 1 , T 2 ] + 2u 1 v 2 -2u 2 v 1 , v = -T t 1 v 2 + T t 2 v 1 . (3)
For X = (u, T, v) ∈ g, the vector field induced on T Ω by the oneparameter subgroup exp tX is equal to

X(z) = u + T z -P (z)v , z ∈ T Ω .
See [START_REF] Faraut | Analysis on Symmetric Cones[END_REF] X.5 page 211.

Proposition 1.10. The Lie algebra of G(c) is given by

g(c) = {X = (u, T, v), u ∈ J(c), T ∈ l(c), v ∈ J(c)}. Proof. Assume that X = (u, T, v) with u ∈ J(c), T ∈ l(c), v ∈ J(c). Let z ∈ T Ω(c)
. Then, by an elementary calculation u + T z + P (z)v belongs to J(c) and hence, by exponentiation, exp tX preserves T Ω(c) so that X ∈ g(c).

Conversely, suppose that the vector field X(z), z ∈ T Ω is associated to a one-parameter group of G(c). Then X(z) = u + T z -P (z)v must belong to J(c) whenever z belongs to T Ω(c) . As X(z) is polynomial in z, this property has to be valid for any z ∈ J(c). Set z = 0 to get u ∈ J(c). Next, for any t ∈ C × and z ∈ J(c)

1 t X(tz) -u = T z -tP (z)v ∈ J(c) .
As this has to be valid for any t ∈ C × , necessarily, T z ∈ J(c) and P (z)v ∈ J(c) for any z ∈ J(c). This implies that T maps J(c) into itself, hence belongs to l(c). Finally, set z = e ∈ J(c)

to get P (e)v = v ∈ J(c). Proposition 1.11. Let g ∈ G(c). Then there exists g 1 ∈ G 1 , . . . , g k ∈ G k such that for any z = z 1 + z 2 + • • • + z k ∈ T Ω(c) , g(z) = g 1 (z 1 ) + g 2 (z 2 ) + • • • + g k (z k ) . (4)
Proof. Let first assume that g = exp X, where X ∈ g(c). Then the vector field generated by the one parameter group g t = exp tX is given by

X(z) = u + T z -P (z)v , u, v ∈ J(c), T ∈ l(c) .
More explicitly, let

u = u 1 + u 2 + . . . u k , v = v 1 + v 2 + • • • + v k ,
where for any j, 1 ≤ j ≤ k, u j , v j ∈ J j . By Proposition 1.3, T maps each J j into itself and induces for each j, 1 ≤ j ≤ k an endomorphism T j , which belongs to l j . As a consequence, at any point

z = z 1 + z 2 + • • • + z k ∈ T Ω(c) , X(z) = X 1 (z 1 ) + X 2 (z 2 ) + • • • + X k (z k )
where X j (z j ) = u j + T j z j -P (z j )v j . Now let X j = X(u j , T j , v j ) ∈ g j and, for t ∈ R, let g j,t = exp tX j ∈ G j . Then by integration

g t (z 1 + z 2 + • • • + z k ) = g 1,t (z 1 ) + g 2,t (z 2 ) • • • + g k,t (z k ) for any z = z 1 + z 2 + . . . z k ∈ T Ω(c)
. So, the existence of a family (g 1 , . . . , g k ) which satisfies ( 4) is proven for any g in the image of the exponential map, in particular in some neighborhood of the identity in G(c). Now property (4) is stable by composition, i.e. if satisfied by two elements g and h of G(c), it is satisfied for gh. As G(c) is connected, the property (4) is satisfied for all elements of G(c). This finishes the proof of the proposition.

The last proposition allows to define the restriction map

G(c) ∋ g -→ (g 1 , g 2 , . . . , g k ) ∈ G 1 × G 2 × • • • × G k .
Theorem 1.1. The restriction map is surjective. Its kernel is equal to M (c).

Proof. The restriction map induces an homomorphism from g(c) into

g 1 × g 2 × • • • × g k .
From the proof of Proposotion 1.11, this homomorphism is given by

(X = (u, T, v) -→ (u 1 , T 1 , v 1 ), . . . , (u j , T j , v j ), . . . , (u k , T k , v k ) where u = u 1 + u 2 + • • • + u k , u j ∈ J j , v = v 1 + • • • + v k , v j ∈ J j and T j = T |J j .
By (the Lie algebra version of) Proposition 1.9, this homomorphism is surjective. Hence the image of restriction map contains a neighborhood of the neutral element in

G 1 × • • • × G k .
As for any j, 1 ≤ j ≤ k, G j is connected, this implies the surjectivity. Now let g be an element of G(c) which is in the kernel of the restriction map. Then, as c 1 + c 2 + . . . c k = e, the point ie belongs to T Ω(c) and hence by assumption, g(ie) = ie. The differential Dg(ie) of g at ie belongs to L. The tangent space to T (Ω(c)) at ie is equal to J(c), and Dg(ie) induces the identity on it, hence belongs to M (c), say Dg(ie) = m with m ∈ M (c). Now consider g ′ = g • m -1 . Then g ′ is an isometry of the tube T Ω for the Bergman metric which fixes the point ie and has differential Dg ′ (ie) = id J . Hence g ′ = id and so g ∈ M (c).

c-homogeneous polynomials and L(c)-covariant differential operators

In this section we assume that J is a simple Euclidean Jordan algebra. Proposition 1.12. Let J be a simple Euclidean Jordan algebra and let c be an idempotent of J. Then J(c, 1) is a simple Jordan algebra.

Proof. Let c = d 1 + d 2 + • • • + d m
be a decomposition of c as a sum of primitive idempotents of J. For any 1 ≤ j ≤ m, as cd j = d j , d j belongs to J(c, 1). This family of primitive orthogonal idempotents can be completed to obtain a Jordan frame of J, namely {d 1 , d 2 , . . . , d m , d m+1 , . . . , d r }. The corresponding Peirce decomposition yields

J = 1≤i≤j≤r J ij .
Now by an elementary check,

J ij ∩ J(c, 1) = {0} if and only if 1 ≤ i ≤ j ≤ m .
Hence J(c) = 1≤i≤j≤k J ij . Now assume that J(c, 1) could be decomposed as a sum of two non trivial Jordan subalgebras J(c, 1) = J 1 ⊕J 2 . Let e 1 (resp. e 2 ) be the neutral element of J 1 (resp. J 2 ). Then c = e 1 + e 2 is a decomposition of c as a sum of two orthogonal idempotents, and there is a Jordan frame of J(c, 1) say {d 1 , d 2 , . . . , d m } such that

e 1 = d 1 + • • • + d p , e 2 = d p+1 + • • • + d m .
Notice that d 1 , . . . , d p ∈ J 1 , d p+1 , . . . , d m ∈ J 2 . The space V 1,p+1 is not reduced to {0} (a consequence of the fact that J is assumed to be simple). Let

w ∈ V 1,p+1 , w = 0. Then w = w 1 + w 2 , w 1 ∈ J 1 , w 2 ∈ J 2 . Now d 1 w 1 = d 1 w = 1 2 w, hence w ∈ J 1 . But similarly d p+1 w 2 = 1 2 w and hence w ∈ J 2 . As w = 0 whereas J 1 ∩ J 2 = {0}, this yield a contradiction. Q.E.D.
Let c = (c 1 , c 2 , . . . , c k ) be a complete system of idempotents. and let J(c) be the associated algebra. As a consequence of Proposition 1.12, J j = J(c j , 1) is simple for any j, 1 ≤ j ≤ k.

There exists a character χ on L, such that

for ℓ ∈ L, x ∈ J, det(ℓx) = χ(ℓ) det x , (5)
and similarly for any j,

1 ≤ j ≤ k for ℓ j ∈ L j , x ∈ J j , det j (ℓ j x j ) = χ j (ℓ j )det j x j .
Proposition 1.13. Let ℓ ∈ L(c) and let (ℓ 1 , ℓ 2 , . . . , ℓ k ) be its restriction to J(c). Then

χ(ℓ) = χ 1 (ℓ 1 )χ 2 (ℓ 2 ) . . . χ k (ℓ k ) . Proof. Let x = x 1 + x 2 + • • • + x k be in ∈ J(c). Then det x = det 1 x 1 det 2 x 2 . . . det k x k . Similarly, ℓx = ℓ 1 x 1 + ℓ 2 x 2 + • • • + ℓ k x k and det ℓx = det 1 (ℓ 1 x 1 )det 2 (ℓ 2 x 2 ) . . . det k (ℓ k x k ) .
and the statement follows easily by choosing x such that det x = 0.

Definition 1.2. A polynomial q on J is said to be c-homogenous of degree (p 1 , p 2 , . . . , p k ), if for any ℓ ∈ L(c), for any

x ∈ J q(ℓx) = χ 1 (ℓ 1 ) p 1 . . . χ k (ℓ k ) p k q(x) , (6)
where (ℓ 1 , ℓ 2 , . . . , ℓ k ) is the restriction of ℓ to J(c).

Let q be a polynomial on J, which we extend as a holomorphic polynomial on J. Let D q be the unique constant coefficients holomorphic differential operator on J such that for any v ∈ J D q e (z,v) = q(v) e (z,v) .

(

) 7
The polynomial q is said to be the algebraic symbol of D q .

Proposition 1.14. Let q be a polynomial on J and assume that q is c-homogeneous of multidegree (p 1 , p 2 , . . . , p k ). Then the differential operator D q satisfies for all f ∈ C ∞ (J)

D q (f • ℓ -1 ) = k j=1 χ j (ℓ j ) -p j (D q f ) • ℓ -1 . ( 8
)
where ℓ ∈ L(c) and (ℓ 1 , ℓ 2 , . . . , ℓ k ) is its restriction to J(c).

Proof. For v ∈ J let f v be the function defined on J by

f v (z) = e (z,v)
It is enough to prove (8) for the family of functions

(f v ), v ∈ J. Now f v • ℓ -1 = f ℓ -1 t v , so that D q (f v • ℓ -1 ) = D q f ℓ -1 t v = q(ℓ -1 t v)f ℓ -1 t v whereas D q f v = q(v)f v , D q f v • ℓ -1 = q(v)f ℓ -1 t v .
Use the covariance property (6) of q to conclude.

2 Jordan algebra representations

Generalities

Recall that a representation of J is a triple (E, . , . , Φ) where (E, . , . ) is a finite dimensional Euclidean vector space and Φ : J → Sym(E) a unital Jordan algebra morphism, i.e. Φ satisfies

i) Φ is linear ii) Φ(xy) = 1 2 Φ(x)Φ(y) + Φ(y)Φ(x) , for all x, y ∈ J iii) Φ(e) = id E iv) Φ(x)ξ, η = ξ, Φ(x)η , for all x ∈ J, ξ, η ∈ E .
For general results on the subject, see [START_REF] Faraut | Analysis on Symmetric Cones[END_REF], IV.4 and ch. XVI, and [START_REF] Clerc | Représentation d'une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel[END_REF].

The assumptions imply that Φ(x) is invertible if x is invertible and then Φ(x) -1 = Φ(x -1 ) , (

Φ(P (x)y) = Φ(x)Φ(y)Φ(x), for all x, y ∈ J .

To a representation E of J is associated a symmetric bilinear map H : E × E → J define by for all x ∈ J, H(ξ, η), x = Φ(x)ξ, η .

Denote by Q : E → J the associated quadratic map, defined by

Q(ξ) = H(ξ, ξ) .
Let us recall some results which will be necessary later. For proofs, see [START_REF] Clerc | Représentation d'une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel[END_REF].

Proposition 2.1. An element ξ ∈ E is said to be regular if one of the four equivalent propositions is satisfied :

i) x ∈ J, Φ(x)ξ = 0 =⇒ x = 0 ii) Q(ξ) ∈ Ω iii) det Q(ξ) = 0 iv) the differential d ξ Q of Q at ξ is surjective.
Proposition 2.2. A representation Φ on E is said to be regular if one of the three equivalent propositions is verified :

i) ∃ ξ ∈ E, Q(ξ) = e ii) ∃ ξ ∈ E, ξ regular iii) Q(E) ⊃ Ω .
Proposition 2.3. Let J be a simple Euclidean Jordan algebra of rank r, and let (E, Φ) be a representation of J of dimension N . i) for a primitive idempotent d of J, q := rank(Φ(d)) is independent of d and N = rq.

ii) for any idempotent c of rank r c , rank(Φ(c)) = r c q .

iii)

for x ∈ J, Det Φ(x) = (det x) N r .
Proof. For i) and iii), see [START_REF] Faraut | Analysis on Symmetric Cones[END_REF]. For ii), let c = d 1 + • • • + d rc be a Peirce decomposition of c as a sum of mutually orthogonal idempotents. Then Φ(c) = Φ(d 1 )+• • •+Φ(d rc ) is a sum of mutually orthogonal projectors, each of rank q. Hence rank Φ(c) = r c q.

Let us give two typical examples of representations. Let J = Symm(r) the space of real symmetric matrices of size r, with the Jordan product x.y =1 2 (xy + yx). Let q be an integer, q ≥ 1 and let E = Mat(r, q) be the space of real matrices of size (r, q), equipped with its standard inner product ξ, η = tr(ξη t ). For x ∈ J and ξ ∈ E, define Φ(x)ξ = xξ .

Then Φ is a representation of J on E. The representation is regular if and only if q ≥ r. This example plays a fundamental rôle in Ibukiyama's work.

The second example will be studied in Section 7. Let J be the quadratic algebra of dimension n, i.e. J = R ⊕ R n-1 , with the Jordan product

(s, x)(t, y) = (st + x.y, sy + tx), s, t ∈ R, x, y ∈ R n-1 ,
where x.y = x 1 y 1 + . . . x n-1 y n-1 .

Let Cl(R n-1 ) be the Clifford algebra of R n-1 , generated by R n-1 and the relations 1

x, y ∈ R n-1 , xy + yx = 2 x.y .
The representations of J coincide with the Clifford modules for the algebra Cl(R n-1 ). In fact, If E is a Clifford module, then for x ∈ J and ξ ∈ E, Φ(s, x) ξ = s ξ + x ξ defines a representation of J on E, and vice versa. The regularity of these representations is a rather subtle question, see [START_REF] Clerc | Représentation d'une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel[END_REF].

Restriction to J(c)

We now consider the situation studied in Section 1. So let c = (c 1 , c 2 , . . . , c k ) be a CSOI of J.

Proposition 2.4. The operators {Φ(c j ), 1 ≤ j ≤ k} form a complete family of orthogonal projectors on E.

The proof is similar to the proof given for the case of a Jordan frame of J in [START_REF] Faraut | Analysis on Symmetric Cones[END_REF] Section IV.4 or in [START_REF] Clerc | Représentation d'une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel[END_REF].

Let E = E 1 ⊕ E 2 ⊕ • • • ⊕ E k be
the corresponding orthogonal decomposition of the space E. Proposition 2.5. Let j, 1 ≤ j ≤ k, and let x ∈ J j . Then i)

for i = j, Φ(x)E i = 0 ii) Φ(x)E j ⊂ E j
iii) The map Φ j : J j -→ End(E j ) given by Φ j (x) = Φ(x) |E j yields a representation of the Jordan algebra J j .

Proof. i) Let i = j and let ξ i ∈ E i . Then Φ(x)ξ i = Φ(c j x)ξ i = 1 2 Φ(c j )Φ(x) + Φ(x)Φ(c j ) ξ i = 1 2 Φ(c j )Φ(x)ξ i .
But 2 is not an eigenvalue of Φ(c j ), so that Φ(x)ξ i = 0.

ii

) Let ξ ∈ E j . Then Φ(x)ξ = Φ(c j x)ξ = 1 2 Φ(c j )Φ(x)+Φ(x)Φ(c j ) ξ = 1 2 Φ(c j )Φ(x)ξ+ 1 2 Φ(x)ξ , so that Φ(x)ξ = Φ(c j )Φ(x)ξ
and hence Φ(x) ξ belongs to E j .

iii) The verification of this result is elementary.

For each j, 1 ≤ j ≤ k, let Q j : E j -→ J j be the quadratic map associated to the representation E j , Φ j on E j .

Let proj(c) be the orthogonal projection from J onto J(c).

Proposition 2.6. For any

ξ = ξ 1 + ξ 2 + • • • + ξ k ∈ E, proj(c)Q(ξ) = k j=1 Q j (ξ j ) . ( 11
) Proof. Let x = x 1 + • • • + x k ∈ J(c) and let ξ ∈ E. Then x, Q(ξ) = k j=1 x j , Q(ξ) = k j=1 Φ(x j )ξ, ξ = k j=1 Φ j (x j )ξ j , ξ j = k j=1 x j , Q j (ξ j ) .
As k j=1 Q j (ξ j ) belongs to J(c), the statement follows.

Proposition 2.7. If the representation (E, Φ) is regular, then for any j, 1 ≤ j ≤ k the representation (E j , Φ j ) of J j is regular.

Proof. By assumption, there exists ξ 0 ∈ E such that Q(ξ 0 ) = e. Now, use (11) to obtain

e = proj(c)e = proj(c)Q(ξ 0 ) = k j=1 Q j (ξ 0 j ),
and hence for each j,

1 ≤ j ≤ k Q j (ξ 0 j ) = c j ,
so that the representation (E j , Φ j ) is regular.

Proposition 2.8. Let J be a simple Euclidean Jordan algebra, and let c = (c 1 , c 2 , . . . , c k ) be a CSOI, and let r j be the rank of J j . Let (E, Φ) be a representation of J, and let N = rq be its dimension. Then, for any j,

1 ≤ j ≤ k, dim(E j ) = N j = r j q . ( 12
)
Proof. This is a direct application of Proposition 2.3.

Proposition 2.9. Let q be a polynomial on J which is c-homgeneous of multidegree (p 1 , . . . , p k ). Let p be the polynomial on E defined by p = q • Q. Then p satisfies for any x =

x 1 + x 2 + • • • + x k ∈ J(c), p(Φ(x)ξ) = k j=1 det j (x j ) 2p j p(ξ) . ( 13
)
Proof. Let x = x 1 + x 2 + • • • + x k ∈ Ω(c
). Then P (x) belongs to L(c) and its restriction to E j is equal to P j (x j ). As χ j ((P j (x j )) = det j (x j ) 2 follows

p(Φ(x)ξ) = q Q(Φ(x)ξ) = q P (x)Q(ξ) = k j=1 det j (x j ) 2p j q(Q(ξ)) .
As both sides of (13) are polynomial in x, and Ω(c) is an open set of of J(c), the identity is valid on J(c).

Pluriharmonic polynomials

Let J be a Euclidean Jordan algebra, (E, Φ) a representation of J, and let Q : E × E -→ J be the associated quadratic map.

Definition 3.1. A polynomial p on E is said to be pluri-harmonic if for all x ∈ J ∆ E (p • Φ(x)) = 0 , (14)
where ∆ E is the Laplacian on E.

The Hecke formula for pluri-harmonic polynomials

A key result to be used later is a generalization of the classical Hecke formula, which we recall now, with a proof. For earlier occurrences of such results, see [START_REF] Kashiwara | On the Segal-Shale-Weil representation and harmonic polynomials[END_REF][START_REF] Ibukiyama | On differential operators on automorphic forms and invariant pluri-harmonic polynomials[END_REF]. Let us also mention the paper [START_REF] Clerc | A generalized Hecke identity[END_REF], where a similar result in an even more general context is proved.

Let J be a simple Euclidean Jordan algebra, and assume that (E, Φ) is a regular representation of J of dimension N . Proposition 3.1. Let p be a pluri-harmonic polynomial on E. Then for x ∈ Ω,

E e i ξ,η e -1 2 (x,Q(ξ)) p(ξ) dξ = (2π) N 2 (det x) -N 2r e -1 2 (x -1 , Q(η)) p iφ(x -1 )η) (15)
Proof. First notice that for x ∈ Ω, the quadratic form (x,

Q(ξ)) = Φ(x)ξ, ξ = Φ(x 1 2 )ξ, Φ(x 1
2 )ξ is positive definite on E. Hence the integral on the left hand side converges. Next, recall that for any harmonic polynomial q on a Euclidean space F , F e i ξ,η e -1 2 ξ,ξ q(ξ)dξ = (2π)

N 2 e -1 2 η,η q(iη) . (16)
Let

ξ ′ = Φ(x 1
2 )ξ. Then

E e i ξ,η e -1 2 (x,Q(ξ)) p(ξ) dξ = E e i ξ ′ ,Φ(x -1 2 η e -1 2 ξ ′ ,ξ ′ p(Φ(x -1 2 )ξ ′ ) Det Φ(x -1 2 ) dξ ′ . 20 = (2π) N 2 (det x) -N 2r e -1 2 (x -1 , Q(η)) p iφ(x -1 )η ,
where we apply (16

) to q = p • Φ(x -1 2 ), η ′ = Φ(x - 1
2 )η and notice that by Proposition 2.3 iii), Det Φ(x -1

2 ) = (det x) -N 2r .

Let J be the complexified Jordan algebra, E the complexification of E, and extend Φ C-linearly to J. The previous formula has an extension to this complex setting. Proposition 3.2. Let p be a pluriharmonic polynomial on E and extend it holomorphically to E. Then for any z ∈ T Ω ,

E e i ξ,η e i 2 (z,Q(ξ)) p(ξ) dξ = (2π) N 2 det( z i ) -N 2r e i 2 (-z -1 , Q(η)) p φ(-z -1 )η) , (17)
where det( z i )

-N
2r is computed using the determination which is equal to (det y) -N 2r for z = iy, y ∈ Ω.

Proof. The left hand side of ( 17) is a convergent integral depending holomorphically on the parameter z ∈ T Ω . The right handside is also holomorphic and both sides coincide on iΩ ⊂ T Ω by ( 16). The conclusion follows.

c-pluri-harmonic polynomials

Let J be a simple Euclidean Jordan algebra, let be a CSOI of J, and let (E, Φ) a representation of J.

Definition 3.2. A polynomial p on E is said to be c-pluri-harmonic (with respect to Φ) if for any x ∈ J(c)

for any x ∈ J(c), ∆ E p • Φ(x) = 0 . (18)
Equivalently, for any

ξ 1 ∈ E 1 , . . . , ξ j-1 ∈ E j-1 , ξ j+1 ∈ E j+1 , • • • + ξ k ∈ E k , the polynomial ξ j -→ p(ξ 1 , ξ 2 , . . . , ξ j-1 , ξ j , ξ j+1 , . . . , ξ k ) is pluriharmonic on E j (w.r.t. Φ j ). (19)
In fact, assume that p is c-pluri-harmonic. Let x j ∈ J j . For

ξ = ξ 1 + . . . ξ k ∈ E, Φ(c 1 + • • • + c j-1 + x j + c j+1 + • • • + c k )(ξ) = ξ 1 + • • • + ξ j-1 + Φ j (x j )ξ j + ξ j+1 + • • • + ξ k ,
and hence the equivalent definition is satisfied.

Conversely, assume p is a polynomial on E which satisfies the conditions (19). Let

x = x 1 + • • • + x k ∈ J(c). For ξ = ξ 1 + • • • + ξ k ∈ E, p(Φ(x)ξ) = p Φ 1 (x 1 )ξ 1 , . . . , Φ(x k )ξ k . So, if p satisfies the condition (19), ∆ E j (p • Φ(x)) = 0 for any j, 1 ≤ j ≤ k and hence ∆ E p • Φ(x) = 0, as ∆ E = ∆ E 1 + • • • + ∆ E k .

Holomorphic representations

Let J be a simple Euclidean Jordan algebra, denote by J its complexification. Form the corresponding tube-type domain T Ω = J + iΩ ⊂ J. Let G(T Ω ) be the group of bi-holomorphic automorphisms of T Ω , which turns out to be a Lie group. Its neutral component G(T Ω ) 0 is generated by

• the translations

t u : z → z + u, u ∈ J • the group L = Str(J) 0 = G(Ω) 0 • the inversion ι : z → -z -1
(see [START_REF] Faraut | Analysis on Symmetric Cones[END_REF] Ch. X).

It turns out to be wise to work with a two-fold covering of the group G = G(T Ω ) 0 . Notice that already for the upper-half plane H = R+iR + in C, the relevant group is SL(2, R), whereas the group of holomorphic diffeomorphisms of H is P SL(2, R) = SL(2, R)/{± id}.

Viewing J as a complex Jordan algebra, there is a corresponding structure group Str(J) which is a complexification of Str(J). Let L be its neutral connected component, which can be called the complexification of L. The character χ, being defined by an algebraic condition [START_REF] Faraut | Analysis on Symmetric Cones[END_REF], has a natural complex extension to L.

We now recall the construction of a two-fold covering group of G(T Ω ) 0 . For g ∈ G(T Ω ) 0 and z ∈ T Ω denote by D(g, z) the differential of g at z. A remarkable fact is that Dg(z) belongs to L. This is obtained by verifying the property for the generators of G and extending to the full group by the chain rule.

Let χ(g, z) = χ Dg(z) . This is a smooth cocycle, and the covering is defined using a square root of this cocycle. Let

G = {(g, ψ g ), g ∈ G(T Ω ) 0 , ψ g : T Ω → C holomorphic, ψ g (z) 2 = χ(g, z)}, (20)
with the group law

(g, ψ g )(h, ψ h ) = (gh, (ψ g • h) ψ h ) .
Then G has a natural structure of Lie group, and the projection (g, ψ g ) -→ g is a twofold covering of G. Elements of G will denoted simply by g and we let ψ(g, z) = ψ g (z) be the corresponding choice of a square root of χ(g, z).

The group G has generators very similar to those for G.

• For u ∈, χ(t u , z) ≡ 1, so that the element (t u , 1) belongs to G and is (with some abuse of notation) still denoted by t u .

• For the inversion ι, χ(ι, z) = (det z) -2 , so (ι, (det z) -1 ) belongs to G and is (again with some abuse of notation) denoted by ι.

• Finally, let L = {(ℓ,

ψ ℓ = ±χ(ℓ) 1/2 ), ℓ ∈ L = Str(J) 0 }, a twofold covering of L.
The group G is generated by the translations {t u , u ∈ J}, the group L and the inversion ι.

It turns out to be necessary to consider also the twofold covering L of L = Str(J) 0 , constructed exactly as the twofold covering of L, using a square root of the complex-valued character χ of Str(J). Denote by ψ the corresponding (well-defined) character on L. It is worthwile to notice that for any g ∈ G, the function z -→ ψ(g, z) -1 is (the restriction to T Ω of) a polynomial function on J.

Let m be an integer. For any holomorphic function F on T Ω and any element g ∈ G, let

π m (g)F (z) = ψ(g -1 , z) m F g -1 (z) .
This defines a smooth representation of G on the space O(T Ω ) of holomorphic functions on T Ω , equipped with the Montel topology.

Let us write the expression of the representation π m for generators of the group G.

For translations

t v , v ∈ J, π m (t v )F (z) = F (z -v) , (21)
for an element ℓ ∈ L π m (ℓ)F (z) = ψ(ℓ) -m F (ℓ -1 z) , (22)
for the inversion ι,

π m (ι)F (z) = (det z) -m F (-z -1 ) . ( 23
)
This construction of a twofold covering can be done in the context of the algebra J(c) for c a complete system of mutually orthogonal idempotents of J. We skip details, but just mention that all the results proved in Section 1 (and particularly Theorem 1.1) have an almost trivial extension to the groups L, G, M c and L j , G j . Similarly, for each j, 1 ≤ j ≤ k, we may define representations of the group G j on the space of holomorphic O(T Ω j ) by π (j) m j (g)F (z j ) = ψ j (g -1 , z j ) m j F g -1 (z j ) . Given m = (m 1 , m 2 , . . . , m k ), consider the tensor product representation π

(1)

m 1 ⊗ π (2) m 2 ⊗ • • • ⊗ π (k) m k of G 1 × • • • × G k defined on O(T Ωc ) by π (1) m 1 (g 1 ) ⊗ π m 2 (g 2 ) ⊗ • • • ⊗ π m k (g k )F (z 1 , z 2 , . . . z k ) = k j=1 ψ j (g -1 j , z j ) m j F g -1 1 (z 1 ), g -1 2 (z 2 ), . . . , g -1 k (z k ) .
m ⊗π

⊗ . . . π (k) m of G 1 × • • • × G k . (2) m
Proof. First notice that for g ∈ G(c) and z ∈ T Ωc , π m (g)F (z) only depends on g mod M c . Next use Proposition 1.13 (more exactly its analog for the characters ψ and ψ j ) and the conclusion follows.

The main theorem

In this section, we want to construct a differential operator D on T Ω such that res(c) • D intertwines the restriction of π m to G(c) and k j=1 π (j) m j for an appropriate choice of m = (m 1 , m 2 , . . . , m k ).

The data and the statement of the main theorem

Theorem 5.1. Let (E, Φ) be a regular representation of J and assume that N = dim E = 2rm for some m ∈ N.

Let q be a polynomial on J which satisfies

i) q is c-homogeneous of multidegree p = (p 1 , p 2 , . . . , p k ) ii) p = q • Q is c-pluriharmonic on E . (24)
Let

m 1 = m + 2p 1 , m 2 = m + 2p 2 , . . . , m k = m + 2p k .
Let D q be the holomorphic constant coefficients differential operator on J with algebraic symbol q. Then D q satisfies, for any g ∈ G(c) whose restriction to J(c) is equal to (g 1 , g 2 , . . . , g k )

res •D q • π m (g) = (π (1) m 1 (g 1 ) ⊗ • • • ⊗ π (k) m k (g k )) • res •D q . ( 25
)

The proof of the main theorem

It suffices to verify (25) for generators of G(c). First, the operator D has constant coefficients, hence commutes to translations and (25) follows for g = t v , v ∈ J(c). When g is in L(c), the proof to get (25) is longer and we state it as a lemma. Notice however that the intertwining property (25) is trivially satsified for m ∈ M (c).

Lemma 5.1. For ℓ ∈ L(c) and for z ∈ T Ω(c) ,

D • π m (ℓ)F (z) = (π (1) m 1 (ℓ 1 ) ⊗ • • • ⊗ π (k) m k (ℓ k ))(DF |T Ω(c) )(z) . (26)
Proof. As D is a constant coefficients differential operator, it suffices to prove (26) for the family of functions f v , v ∈ J, where f v (z) = e (z,v) . Let ℓ ∈ L(c). For any z ∈ T Ω ,

π m (ℓ)f v (z) = ψ(ℓ) -m f ℓ -1 t v (z) . As Df v = q(v)f v , D • π m (ℓ)f v (z) = ψ(ℓ) -m q(ℓ -1 t v)f ℓ -1 t u (z) , so that, using (6)
D • π m (ℓ)f v (z) = k j=1 ψ j (ℓ j ) -m-2p j q(v)f ℓ -1 t v (z) . ( 27
) Let further proj(c)v = v 1 + v 2 + . . . v k , where v j ∈ J j . For z = z 1 + z 2 + • • • + z k ∈ T Ω(c) , f v (z) = k j=1 e i(z j ,v j ) ,
and hence

DF |T Ω(c) (z) = q(v) k j=1 e i(z j ,v j ) , so that (π (1) m 1 (ℓ 1 )⊗• • •⊗π (k) m k (ℓ k ))(DF |T Ω(c) )(z) = q(v) k j=1 ψ j (ℓ j ) -m j e (z j ,ℓ -1 j t u j ) .
(28) Now compare ( 27) and (28) to finish the proof of Lemma 5.1.

It remains to prove (25) for g = ι, which we also state as a lemma.

Lemma 5.2. Let z ∈ T Ω(c) . Then D • π m (ι)F (z) = (π (1) m 1 (ι 1 ) ⊗ • • • ⊗ π (k) m k (ι k ))(DF |T Ω(c) )(z) . (29)
Proof. For ξ ∈ E, let

F ξ (z) = e i 2 z,Q(ξ) .
As the representation (E, Φ) is assumed to be regular, it is enough to prove (29) for the family of functions F ξ , ξ ∈ E.

For

z ∈ T Ω π m (ι)F ξ (z) = (det z) -m e i 2 -z -1 ,Q(ξ)
Notice that for z = x + iy ∈ T Ω the quadratic form

y, Q(ξ) = Φ(y 1 2 )ξ, Φ(y 1
2 )ξ is positive-definite, and hence, ξ -→ F ξ (z) is in the Schwartz class S(E). Use (3.2) for p ≡ 1 on E and notice that m = N 2r to get

π m (ι)F ξ (z) = (2π) -N 2 (-i) rm E e i η,ξ e i 2 (z,Q(η)) dη . Now use (7) to get Dπ m (ι)F ξ (z) = (2π) -N 2 (-i) rm 2 -r.p E q Q(η) e i ξ,η e i 2 z,Q(η) dη ,
where r.p = k j=1 r j p j .

Next, assume that z

= z 1 + z 2 + • • • + z k ∈ T Ω(c) . Then, as proj c (Q(η)) = j=1 Q j (η j ), Dπ m (ι)F ξ (z) = (2π) -N 2 (-i) rm 2 -r.p E 1 • • • E k p(η 1 , . . . , η k ) k j=1
e i ξ j ,η j e i 2 z j ,Q j (η j ) dη 1 . . . dη k .

Use Fubini theorem and (3. This achieves the proof of the main theorem.

Let E = E 1 ⊕ E 2 the decomposition of E with respect to the CSOI c = (c 1 , c 2 ). For v ∈ R n-2 , c 1 (0, 0, v) = 1 2 (0, 0, v) and hence Φ (0, 0, v) = 2Φ c 1 (0, 0, v) = Φ(c 1 )Φ (0, 0, v) + Φ (0, 0, v) Φ(c 1 ), so that Φ (0, 0, v) Φ(c 2 ) = Φ(c 1 )Φ (0, 0, v) .

Hence Φ (0, 0, v) permutes E 1 and E 2 .

The quadratic map Q is given in the original basis of J by For the next statements, let ∂ j = ∂ ∂y j be the partial derivative (on J) with respect to the coordinate y j .

Q(ξ 1 + ξ 2 ) = 1 2 (|ξ 1 | 2 ) + |ξ 2 | 2 ), 1
Proposition 6.1. Let q be a polynomial on J and let p be the polynomial on E defined by p = q • Q. Then for ξ = (ξ 1 , ξ 2 )

∆ 1 p(ξ 1 , ξ 2 ) = (δ 1 q) Q(ξ 1 , ξ 2 )
where

δ 1 = 2N 1 ∂ 1 + 4 y 1 ∂ 2 1 + 4 n j=3 y j ∂ 1 ∂ j + 2y 2 n j=3 ∂ 2 j . (34)
Proof. Consider p as a polynomial on E 1 ⊕ E 2 , so that p(ξ 1 , ξ 2 ) = q |ξ 1 | 2 , |ξ 2 | 2 , . . . , 2 ξ 1 , Φ(f j )ξ 2 , . . .

Then, for any k, 1 ≤ k ≤ N 1 ∂ ∂ξ 1,k p (ξ 1 , ξ 2 ) = 2 ξ 1,k ∂ 1 q + 2 n j=3 ǫ k , Φ(f j )ξ 2 ∂ j q
Proof. The polynomial p = q • Q is c-pluriharmonic if and only if

δ 1 q = 0, δ 2 q = 0 , (38)
where

δ 2 = 2N 1 ∂ 2 + 4y 2 ∂ 2 2 + 4 j=3 y j ∂ 2 ∂ j + 2y 1 n j=3 ∂ 2 j .
Use (35) to compute δ 1 q and simlarly for δ 2 q. The conditions (38) are satisfied if and only if, for any k, 0 ≤ k ≤ inf(p 1 , p 2 ) -1

(k + 1)(k + n -1 2 )a k+1 + (p 1 -k) k + N 1 2 + p 1 -1 a k = 0 (k + 1)(k + n -1 2 )a k+1 + (p 2 -1) k + N 1 2 + p 2 -1 a k = 0 . (39)
Hence p 1 = p 2 and (37) follows.

From Proposition 6.2 follows

a j = - (p + 1 -j) j + N 1 2 + p -2 j j + n-3 2 a j-1 (40)
and hence a j = (-1) j p . . . (p -(j -1)) ( N 1 2 + p -1) . . . ( N 1 2 + p -1 + (j -1)) 1 2 . . . j ( n-1 2 ) . . . n-1 2 + (j -1) a 0

(41) For n, m, p ∈ N, let for j ∈ N, 0 ≤ j ≤ p a n,m,p j = (-1) j p! j!(p -j)! (m + p -1) . . . m + p -1 + (j -1) ( n-1 2 ) . . . ( n-1 2 + j -1)

. Theorem 6.1. Let E be a Clifford module for Cl(n -1) and assume that the corresponding representation of J is regular. Assume moreover that dim E = 4m for some m ∈ N.

Let D be the holomorphic diffferential operator defined on T Ω by

D = p j=0 a n,m,p j ∂ 2 ∂z 2 0 - ∂ 2 ∂z 2 1 p-j ∆ j n-2 . ( 42
)
For any g ∈ G(c), whose restriction to T Ω(c) is equal to (g 1 , g 2 ), the operator D satisfies

(res •D) • π m (g) = π m+2p (g 1 ) ⊗ π m+2p (g 2 ) • (res •D) . (43)

CONTENTS 1 . 1 . 2 1 . 3 1 2 . 2 1 2

 1121312212 The algebraic/geometric setting 1.1 Compete system of orthogonal idempotents and the associated Jordan subalgebra The group L(c) The group G(c) 1.4 c-plurihomogeneous polynomials and L(c)-covariant differential operators 2. Jordan algebra representations 2.Generalities Restriction to J(c) 3. Pluri-harmonic polynomials 3.1 The Hecke formula for puri-harmonic polynomials 3.2 c-pluri-harmonic polynomials 4. Holomorphic representations 5. The main theorem 5.The data and the statement of the main theorem 5.The proof of the main theorem

Definition 1 . 1 .

 11 A complete system of orthogonal idempotents (CSOI for short) of J is a family c = (c 1 , c 2 , . . . , c k ) of mutually orthogonal idempotents of J such that e = c 1 + c 2 + • • • + c k .

Proposition 4 . 1 .

 41 The restriction map res : F -→ F |T Ωc intertwines the restriction of π m to G(c) and the tensor representation π

2 )- 1 jj e i 2 -z - 1 jm 1 (ι 1 )-m j e i 2 z

 2121112 repeatedly for j = 1, . . . , k, take into account that N j = 2r j m (see (12)) to getDπ m (ι)F ξ (z) = 2 -r.p ,Q j (ξ j ) p(. . . , Φ(-z -1 j )ξ j , . . . )Now use (12) and (13) to obtainDπ m (ι)F ξ (z) = 2 -r.p ,Q j (ξ j ) p(ξ 1 , . . . , ξ k ) .(30) On the other hand, for z ∈ T Ω , DF ξ (z) = q( Q(ξ) = 2 -r.p p(ξ)e i 2 z,Q(ξ) .Hence for z= z 1 + z 2 • • • + z k ∈ T Ω(c) DF ξ (z) = 2 -r.p p(ξ) ⊗ • • • ⊗ π (k) m k (ι k ) DF ξ |T Ω(c) (z 1 , z 2 , . . . , z k ) = 2 -r.p p(ξ) j ,Q(ξ j ) .(31)Compare (30) and (31) to conclude.

  2 (|ξ 1 | 2 -|ξ 2 | 2 ), . . . , Φ(e j )ξ 1 , ξ 2 , . . . . and hence in the basis {f 1 , f 2 , . . . , f n } byQ(ξ 1 + ξ 2 ) = |ξ 1 | 2 , |ξ 2 | 2 , . . . , 2 Φ(f j )ξ 1 , ξ 2 , . . . . Let {ǫ k , 1 ≤ k ≤ N 1 }be an orthogonal basis of E 1 and denote by (ξ 1,k ) 1≤k≤N 1 the corresponding coordinates of a generic element ξ 1 ∈ E 1 . Finally, denote by ∆ 1 the partial Laplacian on E related to E 1 , i.e.

	∆ 1 =	N 1 k=1	∂ 2 1,k ∂ξ 2	.

Notice that it differs from the usual convention by the absence of a sign -

Here is the reason to exclude the case n =

3, as SO(1) = {id} and the invariance by M (c) imposes no condition in this case.

Examples in rank 2

In this section, let J = J n be the simple Euclidean Jordan algebra of rank 2 and dimension n, i.e. J n = R ⊕ R n-1 , with the Jordan product (s, x 1 , . . . , x n-1 )(t, y 1 , . . . , y n-1 ) = (st+x.y, sy 1 +tx 1 , . . . , sy n-1 +tx n-1 ) , where x.y = x 1 y 1 + • • • + x n-1 y n-1 .

We will assume that n ≥ 4. In fact, for n = 2, the algebra J 2 is not simple. For n = 3, J is isomorphic to Symm(2, R) and so this case has been studied in [START_REF] Ibukiyama | On differential operators on automorphic forms and invariant pluri-harmonic polynomials[END_REF] and differs slightly from the general case (see next footnote).

The trace and the determinant are given by tr(s,

The structure group of J is the product R * × O(1, n -1) and the group denoted by L in the general case is equal to R + ×Spin 0 (1, n-1).

The cone Ω = Ω n is the forward cone

The tube-type domain is

the group of biholomorphic automorphisms is isomorphic to O(2, n)/{± id}, and the group G is isomorphic to Spin 0 (2, n), see [START_REF] Satake | Algebraic structures of symmetric domains[END_REF]. The domain has a bounded realization known as the Lie ball.

It will be convenient to use for J the standard inner product on a Euclidean Jordan algebra, which in this case is given by (s, x), (t, y) = 2(st + x.y) .

An idempotent of rank 1 is of the form ( 1 2 , x), with |x| = 1 2 . Up to an isomorphism of J, there is only one (non-trivial) CSOI, namely

, 0, . . . , 0 .

Consequently,

The corresponding Peirce decomposition is given by

where

A well-adapted orthonormal basis of J is given by

where

2 is in the j-th place. A generic element of J will be denoted by y = (y 1 , y 2 , . . . , y j , . . . ,

where y j is the j-th coordinate of the element in the new basis {f 1 , . . . , f n }.

The formula for the base change is given by

The group L(c) preserves J(c) and hence also its orthogonal

L 1 and L 2 are both isomorphic to R + , and the restriction map is given by

The group M (c) is isomorphic to Spin(n -2). Lemma 6.1. A polynomial q on J is c-homogeneous of multi-degree (p 1 , p 2 ) if and only if q is of the form

for some a j ∈ C.

Proof. The group M (c) is isomorphic to Spin(n -2), fixes the subalgerba J(c) and acts as SO(n -2) on J 1

2

. Hence 2 a c-homogeneous polynomial can be written as a polynomial in y 1 , y 2 and |y ′ | 2 . Now consider the elementary polynomial y m 1 1 y m 2 2 |y ′ | 2m 3 . It satisfies the required conditions for c-homogeneity if and only if m 1 , m 2 , m 3 satisfy

Hence m 1 = p 1 -j, m 2 = p 2 -j, m 3 = j for some j, 0 ≤ j ≤ p 1 , p 2 and the conclusion follows.

The cone Ω(c) is equal to {a 1 c 1 + a 2 c 2 , a 1 , a 2 ∈ R} is a product of two positive half-lines, the tube-domain T Ω(c) is equal to

and is a product of two complex upper half-planes, the groups G 1 and G 2 are isomorphic to Sl(2, R).

Let (E, Φ) be a representation of

Hence E is a Clifford module for the Clifford algebra Cl(n -1) generated by R n-1 with the relation (beware of the absence of sign -)

Conversely, if E is a Clifford module for Cl(n -1), then set

to obtain a representation of J. For a deeper study of these representations, see [START_REF] Clerc | Représentation d'une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel[END_REF].

Sum over k, 1 ≤ k ≤ N 1 and use the formula

The formula follows from these observations. Lemma 6.2. Let p 1 , p 2 ∈ N and let k ∈ N, 0 ≤ k ≤ inf(p 1 , p 2 ). Then

The verification is elementary and left to the reader. Proposition 6.2. Let q ≡ 0 be the polynomial defined by

Then p(ξ) = q(Q(ξ)) is c-pluri-harmonic if and only if p 1 = p 2 = p and the coefficients a j satisfy the relation

Proof. Recall that N 1 = dim E 1 = 1 2 dim E = 2m, and hence N 1 2 = m. Theorem 6.1 is merely a transcription of Theorem 5.1 in the present context.

The Clifford modules for Cl(n-1) viewed as representations of the algebra J are studied in [START_REF] Clerc | Représentation d'une algèbre de Jordan, polynômes invariants et harmoniques de Stiefel[END_REF], and in particular the regular representations are classified (Théorème 3), thus giving for each n the values of m for which Theorem (6.1) is valid. Address Jean-Louis Clerc, Université de Lorraine, CNRS, IECL, F-54000 Nancy, France jean-louis.clerc@univ-lorraine.fr