Salem Ben Said

Jean-Louis Clerc

Khalid Koufany

Conformally

Salem Ben

S A Ïd

Jacques Faraut

Conformally covariant bi-differential

Keywords: 2000 Mathematics Subject Classification. Primary 43A85. Secondary 58J70, 22446 Rankin-Cohen brackets, covariant bi-differential operators, degenerate principal series, local Zeta functions, simple real Jordan algebras

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Introduction

The Rankin-Cohen brackets are well known examples of covariant bi-differential operators (see e.g. [START_REF] Cohen | Sums involving the values at negative integers of L-functions of quadratic characters[END_REF][START_REF] El Gradechi | Rankin-Cohen brackets and allied bi-differential operators[END_REF][START_REF] Peng | Tensor products of holomorphic representations and bilinear differential operators[END_REF][START_REF] Pevzner | Rankin-Cohen brackets and representations of conformal Lie groups[END_REF][START_REF] Zagier | Modular forms and differential operators[END_REF]). They appeared in the theory of modular forms as constant coefficients holomorphic bi-differential operators on the upper half-plane in C. Such operators are covariant with respect to representations of the holomorphic discrete series of the Lie group SL(2, R). They even have predecessors, known as transvectants (see e.g. [START_REF] Olver | Classical Invariant Theory[END_REF]), much used in the classical theory of invariants. The transvectants can be interpreted as constant coefficients bi-differential operators on R covariant with respect to representations of SL(2, R) containing a finite dimensional subrepresentation. More generally, the principal series representations of SL(2, R) are indexed by (λ, ε) ∈ C × {±}. Given two such representations π λ,ε and π µ,η , and given a positive integer N, there exists a family of constant coefficients bidifferential operators B (N ) λ,µ on R which are covariant with respect to (π λ,ε ⊗ π µ,η , π λ+µ+2N,εη ). Moreover, the family depends rationally on the parameters λ and µ. For special values of λ and µ (corresponding to cases where the representations are reducible), they coincide with the Rankin-Cohen brackets or the transvectants.

There are many efforts to study and construct such operators in more general geometric situations (see e.g. [START_REF] Choie | Rankin-Cohen brackets and invariant theory[END_REF][START_REF] Kobayashi | Differential symmetry breaking operators, I General theory and F-method[END_REF][START_REF] Kobayashi | Differential symmetry breaking operators: II. Rankin-Cohen operators for symmetric pairs[END_REF][START_REF] Ovsienko | Generalized transvectants, Rankin-Cohen brackets[END_REF][START_REF] Peng | Tensor products of holomorphic representations and bilinear differential operators[END_REF][START_REF] Zhang | Rankin-Cohen brackets, transvectants and covariant differential operators[END_REF]). In this paper, we describe a method for building such operators in the context of Jordan algebras. To be more explicit, let V be a simple real Jordan algebra, and let Co(V ) be its conformal group. It turns out to be more convenient to work with a group G which is locally isomorphic to Co(V ); more precisely G is a twofold covering of the proper conformal group (for details see Section 6). This group is a simple Lie group which acts rationally on V. The subgroup P of affine conformal maps is a maximal parabolic subgroup of G. The pair (G, P ) has the following properties :

the unipotent radical of P is abelian, -P is conjugate to its opposite subgroup P .

Conversely, a pair (G, P ), where G is a simple Lie group and P is a parabolic subgroup of G, satisfying the two above properties is known to be associated to a simple real Jordan algebra. For earlier use of Jordan algebras from this point of view, see e.g. [START_REF] Barchini | Positivity of zeta distributions and small unitary representations[END_REF][START_REF] Hilgert | Minimal representations via Bessel operators[END_REF][START_REF] Kostant | Jordan algebras and Capelli identities[END_REF][START_REF] Möllers | Minimal representations of conformal groups and generalized Laguerre functions[END_REF][START_REF] Sahi | Unitary Representations on the Shilov Boundary of a Symmetric Tube domain[END_REF][START_REF] Sahi | Jordan algebras and degenerate principal series[END_REF]. The map

V ∋ v → n v P ,
where n v is the translation x → x + v, has a dense open image in X = G/P (the big Bruhat cell).

The characters of P are parametrized by (λ, ε) ∈ C × {±} and we form the corresponding line bundles E λ,ε over X . The natural action of G on Γ(E λ,ε ), the space of sections of E λ,ε , gives raise to a smooth representation π λ,ε . These representations constitute the degenerate principal series of G. This compact realization of π λ,ε has another useful realization on functions defined on V, called the non-compact realization.

Let us come to the crucial result of this paper. Given two representations π λ,ε and π µ,η , we construct a differential operator on X × X

F (λ,ε),(µ,η) : Γ(E λ,ε ⊠ E µ,η ) -→ Γ(E λ+1,-ε ⊠ E µ+1,-η )
which is covariant with respect to (π λ,ε ⊗ π µ,η , π λ+1,-ε ⊗ π µ+1,-η ). To construct covariant bi-differential operators is then easy. In fact, let res be the restriction map from X × X to the diagonal {(x, x), x ∈ X } ≃ X . For any N ∈ N * , the bi-differential operator

B (N ) (λ,ε),(µ,η) := res •F (λ+N -1,ε),(µ+N -1,η) • • • • • F (λ,ε),(µ,η)
is covariant with respect to (π λ,ε ⊗ π µ,η , π λ+µ+2N,εη ). This idea of obtaining covariant bidifferential operators on X from a covariant differential operator on X × X is reminiscent of the use of the Ω-process in the classical construction of the transvectants.

The construction of F (λ,ε),(µ,η) follows a process that was introduced in [START_REF] Beckmann | Singular invariant trilinear forms and covariant (bi-)differential operators under the conformal group[END_REF] for V = R n,0 and G = SO 0 (1, n + 1) and further used in [START_REF] Clerc | Covariant bi-differential operators on matrix space[END_REF] for V = Mat(n, R) and G = SL(2n, R). This approach uses two main ingredients : the first one is the normalized Knapp-Stein operators J λ,ε for the degenerate principal series which intertwines π λ,ε and π 2n r -λ,ε , where n and r denote the dimension and the rank of V , respectively. The second ingredient is an operator M which in the non-compact realization becomes the multiplication operator is given by M f (x, y) = det(xy)f (x, y), x, y ∈ V × V.

Here det is the determinant polynomial of the Jordan algebra V. The operator M has a "universal" intertwining property due to the covariance of det(xy) under the diagonal action of the group G.

The operator F (λ,ε),(µ,η) corresponds to the following commutative diagram

H (λ,ε),(µ,η)
The fact that F (λ,ε),(µ,η) is covariant (for the diagonal action of G) is an obvious consequence of its definition. The main difficulty is to show that F (λ,ε),(µ,η) is a differential operator. This is done by working in the non-compact realization. Indeed, there are three steps. First, we prove the following main identity1 (Theorem 3.6):

Theorem 1 (Main identity). For (s, ε) and (t, η) in C × {±}, there exists a differential operator D s,t on V × V such that for all (x, y) ∈ V × × V × , we have

det ∂ ∂x - ∂ ∂y
• det(x) s,ε det(y) t,η = det(x) s-1,-ε det(y) t-1,-η • D s,t .

The differential operator D s,t has polynomial coefficients in x, y and also in s, t.

The second step uses the Fourier transform F on V . The distributions det(x) s,ε , which are defined for ℜ(s) large enough, can be extended by analytic continuation in s, yielding a meromorphic family of tempered distributions on V. Their Fourier transforms can be computed using the local Zeta functional equations on V (Theorem 4.2, Theorem 4.3 and Theorem 9.1). Let I s,ε be the convolution operator with the distribution det(x) s,ε , and denote by E s,t the differential operator with polynomials coefficients on V defined by

F • E s,t = D s,t • F.
The Fourier transform counterpart of the main identity is the following main theorem.

Theorem 2 (Main theorem). For (s, ε) and (t, η) in C × {±}, we have

M • (I s,ε ⊗ I t,η ) = κ(s, t) (I s+1,-ε ⊗ I s+1,-η ) • E -s-n r ,-t-n
r , where κ(s, t) is rational function on C × C.

The third step is to prove that for λ and µ generic, the local expression of F (λ,ε),(µ,η) in the non compact setting is equal to F λ,µ := E n r -λ, n r -µ , thus establishing that F (λ,ε),(µ,η) is a differential operator.

Many of our proofs (in particular the Fourier transform computations) depend on the type of the real Jordan algebra. However, for two simple real Jordan algebras V 1 and V 2 which are real forms of the same simple complex Jordan algebra V, the differential operator F λ,µ constructed for V 1 × V 1 and the corresponding one for V 2 × V 2 are restrictions of a common holomorphic differential operator on V × V. This remark advocates for a more algebraic construction (valid over C) of the objects we have constructed by analytical means.

Let us describe the content of this paper. After a general formulation of Leibnitz's formula for applying any constant coefficients differential operator to a product of two functions (Section 1) we obtain (Section 2) the Bernstein identity for det(x) s,ε in each of the four types of simple real Jordan algebras. We then construct (Section 3) the family of differential operators D s,t by proving the main identity above. The delicate Fourier calculations using the local Zeta functional equations for a real Jordan algebra are presented in Section 4. Although these matters are largely known, some aspects are new : the functional equation for det(x) s,- in the non-euclidean split case, the functional equation for det(x) s,± in the euclidean case and the role played by two "new" distributions Z even s and Z odd s (see (4.14)). In Section 5 we define the family E s,t and we prove Theorem 2 above. Up to this point, we mostly use To avoid confusion in the notation, we will apply the following conventional notation for constant coefficients differential operators on a finite dimensional vector space V over F = R or C endowed with an F-bilinear non-degenerate form (• , •) : for p ∈ C[V ] let p( ∂ ∂x ) to be the (uniquely determined constant coefficients) differential operator such that p ∂ ∂x e (x,y) (x) = p(y)e (x,y) .

Throughout this section we will assume that V is a real finite dimensional vector space endowed with a euclidean inner product (• , •).

Let P(V ) be the space of polynomials on V with real coefficients. Define on P(V ) the Fischer inner product by (p, q) F = p ∂ ∂x q (0).

The Fischer inner product is an euclidean inner product on P(V ) and it satisfies

(p, qr) F = r ∂ ∂x p, q F .
Fix p in P(V ) and let W(p) be the subspace of P(V ) generated by the partial derivatives of p. Let ρ = dim W(p) and choose an orthonormal basis (p 1 , . . . , p ρ ) of W(p). For any p ∈ P(V ), we define p ♭ by

p ♭ = p ∂ ∂x p. Theorem 1.1 (Generalized Leibnitz's formula). Let f, g ∈ C ∞ (V ). Then p ∂ ∂x (f g) = ρ j=1 p ♭ j ∂ ∂x f p j ∂ ∂x g . (1.1)
This formula can be rewritten as

p ∂ ∂x (f g) = ρ i=1 ρ j=1 p , p i p j F p i ∂ ∂x f p j ∂ ∂x g . (1.2)
Proof. Without loss of generality, it is possible to assume that f and g belong to P(V ). Since the differential operators p ∂ ∂x , p j ∂ ∂x and p ♭ j ∂ ∂x are invariant by translations, it is enough to prove (1.1) at x = 0.

From the definition of the Fischer inner product, we have

p ∂ ∂x (f g)(0) = (p , f g) F = f ∂ ∂x p , g F . Now f ∂ ∂x p belongs to W(p)
, and therefore

f ∂ ∂x p = ρ j=1 a j (f )p j ,
where

a j (f ) = f ∂ ∂x p , p j F = (p , f p j ) F = p j ∂ ∂x p , f F = (p ♭ j , f ) F = p ♭ j ∂ ∂x f (0). Hence p ∂ ∂x (f g)(0) = ρ j=1 p ♭ j ∂ ∂x f (0) p j ∂ ∂x g (0). Now formula (1.
2) can be obtained from (1.1) by expressing the polynomials p ♭ j in the basis (p 1 , . . . , p ρ ). Indeed, as p ♭ j belongs to W(p), we have

p ♭ j = ρ i=1 p ♭ j , p i F p i , with (p ♭ j , p i ) F = p j ∂ ∂x p, p i F = p, p i p j F .
Therefore (1.2) follows immediately from (1.1).

In the sequel we will also need a formula for computing

p ∂ ∂x (f gh). Proposition 1.1. Let f, g, h ∈ C ∞ (V ). Then p ∂ ∂x (f gh) = ρ i=1 ρ j=1 ρ k=1 a ijk p i ∂ ∂x f p j ∂ ∂x g p k ∂ ∂x h , (1.3)
where

a ijk = ρ ℓ=1 p, p i p ℓ F p ℓ , p j p k F .
Proof. This is a direct consequence of Theorem 1.1 where formula (1.2) is used twice.

2.

The Bernstein identity for det(x) s,ε

2.1. Background on simple real Jordan algebras. We will recall first some preliminary results on simple real Jordan algebras and fix notation. For more details we refer the reader to [START_REF] Bertram | The geometry of Jordan and Lie structures[END_REF]5,[START_REF] Faraut | Analysis on symmetric cones[END_REF][START_REF] Helwig | Halbeinfache reelle Jordan-Algebren[END_REF]. Let V be a n-dimensional real Jordan algebra with unit element 1 and of rank r (see Section 11). Denote by L(x) ∈ End(V ) the multiplication by x ∈ V and let P (x) be the quadratic operator defined by

P (x) := 2L(x) 2 -L(x 2 ).
(2.1)

The Jordan trace tr is a linear form on V and the Jordan determinant det is a homogeneous polynomial on V of degree r. In particular, they satisfy tr(1) = r and det(1) = 1.

A real Jordan algebra V is called semisimple if the symmetric bilinear form (x, y) := tr(xy)

is non-degenerate on V × V . If in addition V has no non-trivial ideal, then V is called simple. If the bilinear form (• , •) is positive definite, then V is called euclidean Jordan algebra. An involutive automorphism α of V is called Cartan involution of V if the symmetric bilinear form x, y := (α(x), y) (2.
2) is positive definite. For every semisimple Jordan algebra such a Cartan involution always exists and two Cartan involutions are conjugate by an automorphism of V (see [START_REF] Helwig | Halbeinfache reelle Jordan-Algebren[END_REF]). With respect to the involution α, the following orthogonal decomposition holds

V = V + ⊕ V -, where V + := {x ∈ V ; α(x) = x} and V -:= {x ∈ V ; α(x) = -x}.
The eigenspace V + is always a euclidean Jordan subalgebra of V with the same unit element 1. Notice that when V is euclidean then V + = V and V -= {0}. Denote by n + and r + the dimension and the rank of V + , respectively.

An element c ∈ V is said to be idempotent if c 2 = c. Further, two idempotents c 1 and c 2 are called orthogonal if c 1 c 2 = 0. A non-zero idempotent is called primitive if it cannot be written as the sum of two non-zero orthogonal idempotents.

Every set {c 1 , . . . , c k } of orthogonal primitive idempotents in V + with the additional condition [START_REF] Faraut | Analysis on symmetric cones[END_REF]Theorem III.1.2] the cardinal of a Jordan frame is always equal to the rank r + of V + , and two Jordan frames are conjugate by an automorphism of V + (see [START_REF] Faraut | Analysis on symmetric cones[END_REF]Corollary IV.2.7]).

c 1 + • • • + c k = 1 is called a Jordan frame in V + . By
Fix a Jordan frame {c 1 , . . . , c r + } in V + . By [START_REF] Faraut | Analysis on symmetric cones[END_REF]Proposition III.1.3] the spectrum of the multiplication operator L(c k ) by c k is {0, 1 2 , 1}. Further, the operators L(c 1 ), . . . , L(c r + ) commute and therefore are simultaneously diagonalizable. This yield the following Peirce decomposition

V = 1≤i≤j≤r + V ij , (2.3)
where

V ii := V (c i , 1), 1 ≤ i ≤ r + , V ij := V (c i , 1 2 ) ∩ V (c j , 1 2 ), 1 ≤ i < j ≤ r + .
Here V (c, λ) denotes the eigenspace of L(c) corresponding to the eigenvalue λ. Since the operators L(c k ), for 0 ≤ k ≤ r + , are symmetric with respect to the inner product (2.2), the direct sum (2.3) is orthogonal.

Denote by d the common dimension of the subspaces V ij (i < j) and by e + 1 the common dimension of the subalgebras

V ii . Then dim V = n = r + (e + 1) + r + (r + -1) d 2 .
The Jordan algebra V is called split if V ii = Rc i for every 1 ≤ i ≤ r + (equivalently e = 0), otherwise V is called non-split. By [START_REF] Helwig | Halbeinfache reelle Jordan-Algebren[END_REF], if V is split then r = r + , otherwise r = 2r + . We pin down that every euclidean Jordan algebra is split.

There is a classification of simple real Jordan algebra given in [START_REF] Helwig | Halbeinfache reelle Jordan-Algebren[END_REF] (see also [START_REF] Hilgert | Minimal representations via Bessel operators[END_REF][START_REF] Loos | Bounded symmetric domains and Jordan pairs[END_REF]). We refer the reader to Section 10 for the complete list. More precisely, we have four types of algebras :

Type I : V is euclidean. Type II : V is split non-euclidean. Type III : V is non-split with no complex structure. Type IV : V is non-split with a complex structure. Notice that every simple real Jordan algebra is either :

(a) a real form of a simple complex Jordan algebra (it is the case of type I, II, III); or (b) a simple complex Jordan algebra viewed as a real one (it is the case of type IV). When V is a simple real Jordan algebra, the integers n, r, d and e are called the structure constants of V .

When dealing with a simple Jordan algebra over C we use the symbol V and we denote by n, r and d its structure constants (over C).

Below we will establish the Bernstein identity for det(x) s,ε , which takes slightly different forms depending on the type of the Jordan algebra.

We first consider the case where V is a euclidean Jordan algebra where we recall from [START_REF] Faraut | Analysis on symmetric cones[END_REF] the Bernstein identity on the (open) cone of squares Ω. We then use it to give the corresponding identity for a simple complex Jordan algebra which is essentially obtained by a holomorphic extension. This formula is the key ingredient to get the Bernstein identity on the set of invertible elements in each of the four types of simple real Jordan algebras.

2.2. The Bernstein identity for the cone Ω. Let V be a simple euclidean Jordan algebra and let Ω be the associated symmetric cone (see [START_REF] Faraut | Analysis on symmetric cones[END_REF] for more details on euclidean Jordan algebras). Throughout this paper, we will denote the Jordan determinant det of a euclidean Jordan algebra by ∆. We recall that ∆(x) > 0 for all x ∈ Ω.

For k, l ∈ N, let b k,l be the polynomial defined on C by

b k,l (s) = s s + l 2 . . . s + (k -1) l 2 . ( 2
∆ ∂ ∂x ∆(x) s = b r,d (s)∆(x) s-1 .
(2.5) 2.3. The Bernstein identity for a complex Jordan algebra. Let V be a simple complex Jordan algebra. Then V has a simple euclidean real form V . The structure constants n, r and d of V (viewed as a complex Jordan algebra) coincide with the structure constants n, r and d of V (see Section 11). As above, we will denote the determinant of V by ∆. Then the determinant of V is the holomorphic extension of ∆, still denoted by ∆ (see Section 11).

Let V × := {z ∈ V, ∆(z) = 0}
be the open set of invertible elements in V.

Proposition 2.2. For s ∈ C and for z ∈ V × , we have

∆ ∂ ∂z ∆(z) s = b r,d (s)∆(z) s-1 , (2.6)
where the powers of ∆ are computed using a local branch of log ∆ near z.

Proof. Let z 0 ∈ V × . The subset V × being pathwise-connected, it is possible to choose a simply connected neighborhood

O 0 ⊂ V × of z 0 such that O 0 ∩ Ω = ∅.
Here Ω denotes the symmetric cone in V . By analytic continuation, there is a branch of log ∆(z) on O 0 which coincides with ln ∆(x) on O 0 ∩ Ω. For s ∈ C, use this branch to define ∆(z) s for z ∈ O 0 by ∆(z) s = e s log ∆(z) and similarly for ∆(z) s-1 . On O 0 ∩ Ω, (2.6) reduces to (2.5). Now, both hand sides of (2.6) are holomorphic functions on O 0 . As they coincide on O 0 ∩ Ω, they have to coincide on O 0 , and in particular at z 0 . Since z 0 was arbitrary chosen in V × , the conclusion follows. Needless to say, if the identity is valid for one local branch of log ∆(z), then it is true for any local branch.

2.4. The Bernstein identity for a real Jordan algebra of type IV. Let V be a simple complex Jordan algebra viewed as a simple real Jordan algebra. Recall that its (real) determinant det satisfies det(z) = ∆(z)∆(z), where ∆ is the (complex) determinant of V . Denote by n, r and d the structure constants of V viewed as a complex Jordan algebra, and let e, n, r and d be the structure constants of V viewed as a real Jordan algebra. Then, e = 1, n = 2n, r = 2r and d = 2d. Proposition 2.3. For s ∈ C and for z ∈ V × , we have

∆ ∂ ∂z ∆ ∂ ∂z det(z) s = b r,d (s) 2 det(z) s-1 .
(2.7)

Proof. As det(z) > 0 on V × , then det(z) s is well defined. For any local branch of log ∆(z), we have ∆(z) s ∆(z) s = det(z) s .

First apply the holomorphic differential operator ∆ ∂ ∂z to this equality using (2.6). Then apply the conjugate holomorphic operator ∆ ∂ ∂z to get (2.7).

2.5. The Bernstein identity for a real Jordan algebra of type I and II. Let V be a split simple real Jordan algebra (either euclidean or non-euclidean, see Section 10). Then its complexification V is a simple complex Jordan algebra. Let ∆ be the determinant of V. In this case, the determinant det of V (which is the restriction of ∆ to V ) takes both positive and negative values. Hence we introduce on V × the two expressions det(x) s,+ and det(x) s,- using the convention that for ξ ∈ R * and s ∈ C,

ξ s,ε = |ξ| s for ε = + sign(ξ)|ξ| s for ε = - (2.8)
In this case the relation between the structure constants of V and V is given by r = r, n = n and d = d.

Proposition 2.4. For (s, ε) ∈ C × {±} and for x ∈ V × , we have

det ∂ ∂x det(x) s,ε = b r,d (s)det(x) s-1,-ε .
(2.9)

Proof. Let x 0 ∈ V × and assume first that det(x 0 ) = ∆(x 0 ) > 0. Let Log z be the principal branch of the logarithm function over C (-∞, 0]. Then, for z in a small neighborhood of x 0 in V × , we may choose Log∆(z) as a branch of log ∆(z), and therefore,

det(x) s,-ε = ∆(x) s , det(x) s-1,ε = ∆(x) s-1 ,
for all x ∈ V × close to x 0 . Hence, on the small neighborhood of x 0 , (2.9) follows from (2.6). We now assume that det(x 0 ) < 0. For z in a neighborhood of x 0 in V, we choose Log(-∆(z)) + π √ -1 as a local branch of log ∆(z). Then, for

x ∈ V × close to x 0 , ∆(x) s = |det(x)| s e sπ √ -1 , ∆(x) s-1 = |det(x)| s-1 e (s-1)π √ -1 . For ε = +, we have det(x) s,ε = |det(x)| s = e -sπ √ -1 ∆(x) s , and det(x) s-1,-ε = -|det(x)| s-1 = e -sπ √ -1 ∆(x) s-1 . For ε = -, we have det(x) s,ε = -|det(x)| s = e (-s+1)π √ -1 ∆(x) s , and det(x) s-1,-ε = |det(x)| s-1 = e -(s-1)π √ -1 ∆(x) s-1 .
Hence, on the small neighborhood of x 0 , the identity (2.9) follows from (2.6). As x 0 was chosen arbitrary in V × , the conclusion holds true for every x ∈ V × .

2.6. The Bernstein identity for a real Jordan algebra of type III. Let V be a nonsplit simple real Jordan algebra without complex structure and let det be its determinant.

Then its complexification V is a simple complex Jordan algebra and we denote by ∆ its determinant. In this case, det(x) = ∆(x) ≥ 0 for all x ∈ V (see Section 11), n = n, r = r and d = 1, 2, k -2 (k ≥ 2) in accordance with d = 4, 8, 0 (see Section 10).

Proposition 2.5. Let s ∈ C and x ∈ V × , we have

det ∂ ∂x det(x) s = b r,d (s) det(x) s-1 .
(2.10)

Proof. Let x 0 ∈ V × . Then ∆(x 0 ) = det(x 0 ) > 0, so that ℜ ∆(z) > 0 for all z in a small neighborhood of x 0 in V × . Hence we may use Log∆(z) as a local branch of log det(z) near x 0 . For this choice of branch and for x near x 0 , we have

∆(x) s = det(x) s ,
and therefore (2.10) follows from (2.6).

3. Construction of the family D s,t 3.1. Construction for the cone Ω. Let V be a simple euclidean Jordan algebra with structure constants n, r, and d. As above, let ∆ be the determinant polynomial of V, and let P = P(V ) be the space of real-valued polynomials on V. Let W(∆) be the subspace of P generated by the partial derivatives of ∆ (cf. Section 1). Let Str(V ) be the structure group of V defined as the set of g ∈ GL(V ) such that (gx) -1 = (g -1 ) t (x -1 ), for all x ∈ V × .

The group Str(V ) acts on the space P by

π(g)p = p • g -1 , g ∈ Str(V ).
The space W(∆) is invariant under this action of Str(V ). The representation π (more exactly its complex extension) was studied in [START_REF] Faraut | Pseudo-Hermitian symmetric spaces of tube-type, Topics in Geometry[END_REF], showing that it is multiplicity free and producing its explicit decomposition in irreducible components. Fix a Jordan frame {c 1 , c 2 , . . . , c r } of V . Let ∆ 1 , . . . , ∆ r = ∆ be the associated principal minors on V (see [START_REF] Faraut | Analysis on symmetric cones[END_REF]Page 114]). For our convenience we write ∆ 0 = 1. For 0 ≤ k ≤ r, we denote by W(∆ k ) the subspace of P generated by π(g)∆ k , for g ∈ Str(V ). By [11, Proposition 6.1], the subspaces W(∆ k ) are absolutely irreducible under the action of Str(V ).

For p ∈ W(∆), denote by p ♯ the function defined on V × by

p ♯ (x) = p(x -1 )∆(x).
The function p ♯ extends as a polynomial function on V .

Proposition 3.1 (see [12, Proposition XI.5.1]). Let p ∈ W(∆ k ). Then, for x ∈ Ω, we have p ∂ ∂x ∆ s (x) = b k,d (s) p ♯ (x) ∆(x) s-1 , (3.1)
where b k,d (s) is given by (2.4).

Let

d k = dim W(∆ k ) and choose a basis {p j,k , 1 ≤ j ≤ d k } of W(∆ k )
which is orthonormal for the Fischer inner product. In the present context, the Leibnitz formula (1.3) can be written for p = ∆ as following : Proposition 3.2. Let f, g, h be three smooth functions on V . Then there exist real numbers a

(lmn) ijk such that ∆(f gh) = l,m,n≥0 l+m+n=r d l i=1 dm j=1 dn k=1 a (lmn) ijk p i,l ∂ ∂x f p j,m ∂ ∂x g p k,n ∂ ∂x h .
The vector space V × V is naturally endowed with an inner product, and so is the vector space P(V × V ) of polynomial functions on V × V. To each p ∈ P(V × V ), we associate a differential operator p ∂ ∂x , ∂ ∂y . When p is given by p(x, y) = q(xy), with q ∈ P(V ), we use the notation p ∂ ∂x , ∂ ∂y = q ∂ ∂x -∂ ∂y .

Theorem 3.1. For any s, t ∈ C, there exists a differential operator D s,t on V × V such that for any smooth function f on

Ω × Ω, ∆ ∂ ∂x - ∂ ∂y ∆(x) s ∆(y) t f (x, y) = ∆(x) s-1 ∆(y) t-1 (D s,t f ) (x, y). (3.2)
The operator D s,t has polynomial coefficients in x, y and in s, t.

Proof. Let ϕ and ψ be two functions so that

ϕ(u, v) = ψ(u, v -u) or equivalently ϕ(x, x + y) = ψ(x, y), where (u, v) ∈ {(u, v) ∈ V × V : u ∈ Ω, v -u ∈ Ω} or equivalently x, y ∈ Ω. Then ∆ ∂ ∂x - ∂ ∂y ψ(x, y) = ∆ ∂ ∂u ϕ(u, v) u=x, v=x+y . Assume that ψ(x, y) = ∆(x) s ∆(y) t f (x, y). Then ϕ(u, v) = ∆(u) s ∆(v -u) t f (u, v -u). Fix v and apply now Proposition 3.2 on the open set Ω ∩ (-v + Ω) of V to get ∆ ∂ ∂u ∆(u) s ∆(v -u) t f (u, v -u) = r l=0 d l i=1 q i,l (u, v; s, t) p i,l ∂ ∂u f (u, v -u),
where

q i,l (u, v; s, t) = m,n≥0 m+n=r-l dm j=1 dn k=1 a (lmn) ijk p j,m ∂ ∂u ∆(u) s p k,n ∂ ∂u ∆(v -u) t . Since p k,n ∂ ∂u ∆(v -u) t = (-1) n p k,n ∂ ∂u ∆( • ) t (v -u), it follows from (3.1) that p j,m ∂ ∂u ∆(u) s p k,n ∂ ∂u ∆(v -u) t = (-1) n b m,d (s)b n,d (t)∆(u) s-1 ∆(v -u) t-1 p ♯ j,m (u) p ♯ k,n (v -u). (3.3)
The final formula is now a matter of putting the pieces together.

3.2. Extension of D s,t to a complex Jordan algebra. Let V be a simple complex Jordan algebra, and let V be a euclidean real form of V. The differential operator D s,t constructed in Theorem 3.1 has a natural extension to V × V as a holomorphic differential operator D s,t by replacing ∂ ∂x and ∂ ∂y by ∂ ∂z and ∂ ∂w , respectively, and extending holomorphically the polynomial coefficients to V × V. Theorem 3.2. For any smooth function f on V × × V × , we have

∆ ∂ ∂z - ∂ ∂w ∆(z) s ∆(w) t f (z, w) = ∆(z) s-1 ∆(w) t-1 (D s,t f ) (z, w), (3.4)
where the powers of ∆ are computed with respect to a (any) local branch of log ∆( . ) near z and near w.

Proof. To prove this equality between two holomorphic differential operators, it is enough to prove the equality for a holomorphic function f on V × × V × . But then, arguing as in the proof of (2.6), the equality follows from Theorem 3.1 by analytic continuation, 3.3. The construction for a real Jordan algebra of type IV. Let V be a simple complex Jordan algebra, for which we keep notation as in the previous subsection. In particular ∆ denotes its determinant. When V is viewed as a (simple) real Jordan algebra V, its determinant polynomial is given by det(z) = ∆(z)∆(z). For a holomorphic differential operator D on V with polynomial coefficients, we let D be its associated conjugate-holomorphic differential operator.

Theorem 3.3. For any smooth function f defined on V × × V × , we have det ∂ ∂z - ∂ ∂w det(z) s det(w) t f (z, w) = det(z) s-1 det(w) t-1 D s,t D s,t f (z, w). (3.5)
Proof. We argue as in the proof of (2.7). Indeed, Theorem 3.2 establishes the equality of two holomorphic differential operators. Compose each side with its conjugate differential operator and use the fact that a holomorphic differential operator commutes with its conjugateholomorphic differential operator to obtain (3.5).

3.4.

The construction for a real Jordan algebra of type I and II. Let V be a euclidean or a split non-euclidean Jordan algebra (see Section 10). Then its complexification V is a simple complex Jordan algebra. Let D = p z, ∂ ∂z be a holomorphic differential operator on V. Then its restriction to V is the differential operator D = p(x, ∂ ∂x ). We extend this notation to differential operators on

V × V . Theorem 3.4. For any smooth function f on V × × V × , we have det ∂ ∂x - ∂ ∂y det(x) s,ε det(y) t,η f (x, y) = det(x) s-1,-ε det(y) t-1,-η D s,t f (x, y). (3.6)
Proof. To prove the equality of two differential operators, it is enough to prove that they coincide on polynomial functions. Hence we may assume that f is the restriction to V × × V × of a holomorphic polynomial function on V × V. We now argue as in the proof of (2.9) to deduce (3.6) from Theorem 3.2.

3.5.

The construction for a real Jordan algebra of type III. Let V be a non-split simple real Jordan algebra without complex structure. Then its complexification V is a simple complex Jordan algebra. For a holomorphic differential operator D on V × V, we denote by D its restriction to V × V.

Theorem 3.5. For any smooth function

f on V × × V × , we have det ∂ ∂x - ∂ ∂y det(x) s det(y) t f (x, y) = det(x) s-1 det(y) t-1 D s,t f (x, y). (3.7)
Proof. The proof goes along the same lines as that of the Bernstein identity (2.10).

3.6. General formulation of the main identity. In summary we have proved the following statement.

Theorem 3.6 (Main identity). Let V be a simple real Jordan algebra V . For (s, ε) and (t, η) in C × {±}, there exists a differential operator D s,t on V × V such that for any smooth function

f on V × × V × , det ∂ ∂x - ∂ ∂y det(x) s,ε det(y) t,η f (x, y) = det(x) s-1,-ε det(y) t-1,-η D s,t f (x, y).
The differential operator D s,t has polynomial coefficients in x, y and in s, t.

Local Zeta functional equations

Let V be a simple real Jordan algebra. Let S(V ) be the space of rapidly decreasing smooth functions on V and let S ′ (V ) be its dual, the space of tempered distributions on V .

For (s, ε) ∈ C × {±}, we consider the following local Zeta integrals

Z s,ε (f ) = V f (x)det(x) s,ε dx, f ∈ S(V ), (4.1)
where dx is the Lebesgue measure on V and det(x) s,ε is defined by (2.8). For s ∈ C with ℜ s > 0, the function |det(x) s,ε | is locally integrable on V . Therefore, (4.1) defines a tempered distribution on S ′ (V ).

When ε = +, it is known that the S ′ (V )-valued function s → Z s,+ extends to a meromorphic function on C (see for instance [START_REF] Faraut | Analysis on symmetric cones[END_REF] for the euclidean case, [START_REF] Barchini | Positivity of zeta distributions and small unitary representations[END_REF] for the non-euclidean case except R p,q and [13] for R p,q ). Suppose now ε = -. When V is non-split (that is of type III and IV), det(x) is nonnegative on V , therefore Z s,-= Z s,+ and we may drop the index ± in the notation. When V is split (that is of type I and II), the meromorphic extension of Z s,-can be deduced from the Bernstein identity (2.9) for the split case.

Below, we will compute the Fourier transform of the distributions Z s,ε . It is a classical subject in the literature under the name of local Zeta functional equation (see e.g. [START_REF] Barchini | Positivity of zeta distributions and small unitary representations[END_REF][START_REF] Bopp | Local zeta functions attached to the minimal spherical series for a class of symmetric spaces[END_REF][START_REF] Kayoya | Zeta functional equation on Jordan algebras of type II[END_REF][START_REF] Muller | Décomposition orbitale des espaces préhomognes réguliers de type parabolique commutatif et application[END_REF][START_REF] Satake | The functional equation of zeta distributions associated with formally real Jordan algebras[END_REF][START_REF] Sato | On zeta functions associated with prehomogeneous vector spaces[END_REF]).

The dual space V ′ of V will be identified with V via the non-degenerate bilinear form (x, y) = tr(xy).

The Fourier transform of f ∈ S(V ) is defined by

F(f )(x) = f (x) = V e 2 √ -1π(x,y) f (y)dy, (4.2)
and extend it by duality to the space S ′ (V ) of tempered distributions. For p ∈ P(V ), we recall the following classical formulas:

F p ∂ ∂x f (x) = p -2π √ -1x F(f )(x), F (pf ) (x) = p 1 2π √ -1 ∂ ∂x F(f )(x). (4.3)
4.1. Zeta functional equations for a non-euclidean Jordan algebra except R p,q . In this subsection V is of type II, III and IV except R p,q . The case V = R p,q will be treated in Section 9. Recall from above that the determinant polynomial det takes only positives values whenever V is of type III and IV (V is non-split), while det takes positives as well as negative values whenever V is of type II (V is split).

Recall that r + denotes the split rank of V , and therefore r = r + if V is split and r = 2r

+ if V is non-split; see Subsection 2.1. For s ∈ C, let Γ V (s) := r + k=1 Γ s 2 -(k -1) d 4 . (4.4)
The following result can be found in [START_REF] Barchini | Positivity of zeta distributions and small unitary representations[END_REF]Theorem 4.4]. Our det is related to ∇ in [START_REF] Barchini | Positivity of zeta distributions and small unitary representations[END_REF] by the relation |det| = ∇ whenever V is split and by |det| = ∇ 2 whenever V is non-split. Theorem 4.1. For every s ∈ C, the Fourier transform of the tempered distribution Z s,+ is given by

F(Z s,+ ) =        π -rs-n 2 Γ V s + n r Γ V (-s) Z -s-n r ,+
Type II (split case),

π -rs-n 2 Γ V 2s + 2n r Γ V (-2s) Z -s-n r ,+
Type III and IV (non-split case). Now let us consider the case ε = -. Recall that when V is non-split non-euclidean (that is of type III and IV), Z s,-= Z s,+ . Theorem 4.2. Assume that V is a split non-euclidean Jordan algebra ( ∼ = R p,q ) of dimension n and rank r. We have

F(Z s,-) = ( √ -1) r π -rs-n 2 Γ V (s + 1 + n r ) Γ V (-s + 1) Z -s-n r ,-.
Proof. Recall from the Bernstein identity (2.9) that

2 det(x) s,-= 1 b r,d (s + 1) det ∂ ∂x det(x) s+1,+ .
In view of (4.3) and the functional equation (4.5) for Z s,+ , we obtain

F(Z s,-) = 1 b r,d (s + 1) det(-2π √ -1x)F(Z s+1,+ ) = (-2π √ -1) r b r,d (s + 1) π -r(s+1)-n 2 Γ V (s + 1 + n r ) Γ V (-s -1) det(x)Z -s-1-n r ,+ = (-2 √ -1) r π -rs-n 2 b r,d (s + 1) Γ V (s + 1 + n r ) Γ V (-s -1) Z -s-n r ,- = ( √ -1) r π -rs-n 2 Γ V (s + 1 + n r ) Γ V (-s + 1) Z -s-n r ,-.
In summary we have proved that

F(Z s,ε ) = c(s, ε)Z -s-n r ,ε , (4.6)
where 4.2. Zeta functional equations for euclidean Jordan algebras. Let V be a n-dimensional simple euclidean Jordan algebra (that is of type I) of rank r, and denote as usual by ∆ its determinant. Let {c 1 , . . . , c r } be a Jordan frame of V. It is known (see for instance [START_REF] Faraut | Analysis on symmetric cones[END_REF]) that every x ∈ V can be written as

c(s, ε) =                π -rs-n 2 Γ V s + n r Γ V (-s) if ε = + and V of type II (split) ( √ -1) r π -rs-n 2 Γ V (s + 1 + n r ) Γ V (-s + 1) if ε = -and V of type II (split) π -rs-n 2 Γ V 2s + 2n r Γ V (-2s) if ε = ± and V of type III and IV (non-split). (4.7) Remark 4.1. One can prove that the function s → Z s,ε := c(s, ε) -1 Z s,ε where c(s, ε) =      Γ V (s + n r ) if ε = + and V of type II (split) Γ V (s + 1 + n r ) if ε = -and V of type II (split) Γ V (2s + 2n r ) if ε = ± and V of
x = k r i=1 λ j c j , (4.8)
where λ 1 ≥ • • • ≥ λ r and k is an element of the identity component of the group of automorphisms of V . The λ i 's in (4.8) are uniquely determined by x. Further, x is invertible if and only if λ i = 0 for all i. We say that x is of signature (r

-i, i) if λ 1 ≥ • • • ≥ λ r-i > 0 > λ r-i+1 ≥ • • • ≥ λ r .
Denote by Ω i the set of all elements of signature (ri, i). Then the set of invertible elements V × decomposes into the disjoint union as

V × = r i=0 Ω i .
In particular Ω 0 coincides with the symmetric cone Ω of V .

For s ∈ C, let Γ Ω (s) = (2π) n-r 2 r j=1 Γ s -(j -1) d 2
be the Gindikin gamma function. For s ∈ C, f ∈ S(V ) and 0 ≤ i ≤ r, the Zeta integrals Z i (f, s) defined by

Z i (f, s) = Ω i f (x)|∆(x)| s dx
converges for ℜ(s) > 0, and has a meromorphic continuation to C. Further, they satisfy the functional equation

Z i ( f , s - n r ) = (2π) -rs e rs 2 Γ Ω (s) r j=0 u ij (s)Z j (f, -s), (4.9)
where u ij (s) are polynomials in e (-s/2) with e(z) := e 2π √ -1z ; see [START_REF] Satake | The functional equation of zeta distributions associated with formally real Jordan algebras[END_REF][START_REF] Sato | On zeta functions associated with prehomogeneous vector spaces[END_REF]. Put x = e(-s/2) and write u ij (x) for u ij (s). Then, it is proved in [START_REF] Satake | The functional equation of zeta distributions associated with formally real Jordan algebras[END_REF] that the matrix coefficients u ij (x) satisfy r i=0

y i u ij (x) = ξ -(r-j) P j (ξx, y)P r-j (1, ξxy), ∀y ∈ R, (4.10)
where ξ := ( √ -1) d(r+1) and

P j (x, y) = (x + y) j if d is even, (x + y) ⌊ j 2 ⌋ (y -x) j-⌊ j 2 ⌋ if d is odd . (4.11)
Recall that if x ∈ Ω i , then x is of signature (ri, i), and therefore ∆(x) = (-1) i |∆(x)|. Thus we may rewrite the local Zeta integrals (4.1) in terms of the Z i 's as follows

Z s,+ (f ) = V f (x)|∆(x)| s dx = r i=0 Z i (f, s), Z s,-(f ) = V f (x)sgn(∆(x))|∆(x)| s dx = r i=0 (-1) i Z i (f, s). (4.12) That is Z s,+ (f ) = ⌊ r 2 ⌋ k=0 Z 2k (f, s) + ⌊ r-1 2 ⌋ k=0 Z 2k+1 (f, s), Z s,-(f ) = ⌊ r 2 ⌋ k=0 Z 2k (f, s) - ⌊ r-1 2 ⌋ k=0 Z 2k+1 (f, s). (4.13)
Let us introduce two more tempered distributions :

Z even s (f ) = Z e s (f ) = ⌊ r 2 ⌋ k=0 (-1) k Z 2k (f, s) and Z odd s (f ) = Z o s (f ) = ⌊ r-1 2 ⌋ k=0 (-1) k Z 2k+1 (f, s). (4.14)
We are now in a position to examine the functional equations for Z s,+ and Z s,-.

According to the classification of simple euclidean Jordan algebras (see [START_REF] Faraut | Analysis on symmetric cones[END_REF][START_REF] Satake | The functional equation of zeta distributions associated with formally real Jordan algebras[END_REF] or Section 10), we will consider the following (all) possibilities :

F Z s,+ Z s,- = 2 r γ(s + n r )A(s) Z -s-n r ,+ Z -s-n r ,- , where
A(s) =   cos r ( π 2 (s + n r )) 0 0 ( √ -1) r sin r ( π 2 (s + n r ))   .
Case (a') : If d ≡ 2 (mod 4) and r even, then

F Z s,+ Z s,- = 2 r γ(s + n r )A(s) Z -s-n r ,+ Z -s-n r ,- , where
A(s) =   ( √ -1) r sin r ( π 2 (s + n r )) 0 0 cos r ( π 2 (s + n r ))   .
Case (b -1) : If r = 2 and d ≡ 1 (mod 4), then

F Z s,+ Z s,- = 4 √ 2γ(s + n r )A(s) Z -s-n r ,+ Z -s-n r ,- , (4.15)
where

A(s) =    sin( π 2 (s + n + 1 2 )) cos( π 2 (s + n 2 )) -sin( π 2 (s + n + 1 2 )) sin( π 2 (s + n 2 )) cos( π 2 (s + n + 1 2 )) cos( π 2 (s + n 2 )) cos( π 2 (s + n + 1 2 )) sin( π 2 (s + n 2 ))    .
Case (b -2) : If r = 2 and d ≡ 3 (mod 4), then

F Z s,+ Z s,- = 4 √ 2γ(s + n r )A(s) Z -s-n r ,+ Z -s-n r ,- , where
A(s) =    cos( π 2 (s + n + 1 2 )) cos( π 2 (s + n 2 )) cos( π 2 (s + n + 1 2 )) sin( π 2 (s + n 2 )) sin( π 2 (s + n + 1 2 )) cos( π 2 (s + n 2 )) -sin( π 2 (s + n + 1 2 )) sin( π 2 (s + n 2 ))    .
Case (c -1) : If d = 1 and r ≡ 3 (mod 4), then

F Z s,+ Z s,- = -(-2 √ -1) ⌊ r 2 ⌋ γ(s + n r ) sin ⌊ r 2 ⌋ (π(s + n r ))B(s) Z e -s-n r Z o -s-n r , (4.16)
where

B(s) =   √ -1 sin( π 2 (s + n r )) - √ -1 sin( π 2 (s + n r )) cos( π 2 (s + n r )) cos( π 2 (s + n r ))   .
Case (c -2) : If d = 1 and r ≡ 1 (mod 4), then

F Z s,+ Z s,- = (-2 √ -1) ⌊ r 2 ⌋ γ(s + n r ) sin ⌊ r 2 ⌋ (π(s + n r ))B(s) Z e -s-n r Z o -s-n r , where
B(s) =   cos( π 2 (s + n r )) cos( π 2 (s + n r )) √ -1 sin( π 2 (s + n r )) - √ -1 sin( π 2 (s + n r ))   .
Case (c -3) : If d = 1 and r ≡ 0 (mod 4), then

F Z s,+ Z s,- = 2 r-1 2 e √ -1 π 4 γ(s + n r ) cos r 2 (π(s + n r ))A(s) Z -s-n r ,+ Z -s-n r ,- , where
A(s) = - √ -1 1 1 - √ -1 .
Case (c -4) : If d = 1 and r ≡ 2 (mod 4), then

F Z s,+ Z s,- = 2 r-1 2 e √ -1 π 4 γ(s + n r ) cos r 2 (π(s + n r ))A(s) Z -s-n r ,+ Z -s-n r ,-
, where

A(s) = 1 - √ -1 - √ -1 1 .
Proof. Recall from above that x = e (-s/2). In view of (4.12) and (4.9) we have

Z s-n r ,+ ( f ) = (2π) -rs e rs 2 Γ Ω (s) r j=0 r i=0 u ij (x) Z j (f, -s), (4.17)
and

Z s-n r ,-( f ) = (2π) -rs e rs 2 Γ Ω (s) r j=0 r i=0 (-1) i u ij (x) Z j (f, -s). (4.18)
We will use (4.10) and (4.11) to compute r i=0 u ij (x) and r i=0 (-1) i u ij (x) for any fixed j,

0 ≤ j ≤ r. Observe first that ξ = ( √ -1) d(r+1) reduces to ξ =          1 in case (a) -1 in case (a') ( √ -1) 3d in case (b) ( √ -1) r+1 in case (c).
-Case (a) : Here d ≡ 0 (mod 4) or d ≡ 2 (mod 4) and r is odd. In this case

r i=0 u ij (x) = (x + 1) j (1 + x) r-j = (1 + x) r = 2 r e - rs 4 cos r πs 2 ,
and

r i=0 (-1) i u ij (x) = (x -1) j (1 -x) r-j = (-1) j (1 -x) r = (-1) j (2 √ -1) r e - rs 4 sin r πs 2 .
Hence (4.17) reduces to

Z s-n r ,+ ( f ) = 2 r (2π) -rs e rs 4 cos r πs 2 Γ Ω (s)Z -s,+ (f ),
and (4.18) to

Z s-n r ,-( f ) = (2 √ -1) r (2π) -rs e rs 4 sin r πs 2 Γ Ω (s)Z -s,- (f ).
-Case (a') : Here d ≡ 2 (mod 4) and r is even. In this case,

r i=0 u ij (x) = (-1) -(r-j) (-x + 1) j (1 -x) r-j = (-1) j (1 -x) r = (-1) j (2 √ -1) r e - rs 4 sin r πs 2 ,
and

r i=0 (-1) i u ij (x) = (-1) -(r-j) (-x -1) j (1 + x) r-j = (1 + x) r = 2 r e - rs 4 cos r πs 2 .
Then, by (4.17) and (4.18), one has

Z s-n r ,+ ( f ) = (2 √ -1) r (2π) -rs e rs 4 sin r πs 2 Γ Ω (s)Z -s,-(f ),
and

Z s-n r ,-( f ) = 2 r (2π) -rs e rs 4 Γ Ω (s) cos r πs 2 Z -s,+ (f ).
-Case (b) : In this case r = 2 and ξ = ( √ -1) 3d with d odd. By [32, page 481] we have

2 i=0 y i u ij (x) =      1 + x 2 y 2 for j = 0 x + ξ(1 -x 2 )y + xy 2 , for j = 1 x 2 + y 2 , for j = 2
for any real number y. Then, for ε = ±, (4.17) and (4.18) become

Z s-n 2 ,ε ( f ) = γ(s) (1 + x 2 ){Z 0 (f, -s) + Z 2 (f, -s)} + (2x + εξ(1 -x 2 ))Z 1 (f, -s) . Since Z -s,+ (f ) = Z 0 (f, -s) + Z 1 (f, -s) + Z 2 (f, -s) and Z -s,-(f ) = Z 0 (f, -s) -Z 1 (f, -s) + Z 2 (f, -s), the above equality for ε = + becomes Z s-n 2 ,+ ( f ) = γ(s) 2 (1 + x) 2 + ξ(1 -x 2 ) Z -s,+ (f ) + (1 -x) 2 -ξ(1 -x 2 ) Z -s,-(f ) ,
and for ε =it becomes

Z s-n 2 ,-( f ) = γ(s) 2 (1 + x) 2 -ξ(1 -x 2 ) Z -s,+ (f ) + (1 -x) 2 + ξ(1 -x 2 ) Z -s,-(f ) .
Since d is odd, therefore d ≡ 1 (mod 4) or d ≡ 3 (mod 4).

(b -1) : If d ≡ 1 (mod 4), then ξ = -√ -1, and we obtain

Z s-n 2 ,+ ( f ) = c 1 (s) cos( πs 2 )Z + (f, -s) -sin( πs 2 )Z -s,-(f ) , Z s-n 2 ,-( f ) = c 2 (s) cos( πs 2 )Z + (f, -s) + sin( πs 2 )Z -s,- (f )
,

where c 1 (s) = 2 √ 2γ(s) sin( πs 2 + π 4 ) and c 2 (s) = 2 √ 2γ(s) cos( πs 2 + π 4 ). (b -2) : If d ≡ 3 (mod 4), then ξ = √ -1 and Z s-n 2 ,+ ( f ) = c 2 (s) cos( πs 2 )Z + (f, -s) + sin( πs 2 )Z -s,-(f ) , Z s-n 2 ,-( f ) = c 1 (s) cos( πs 2 )Z + (f, -s) -sin( πs 2 )Z -s,-(f ) ,
with the same c 1 (s) and c 2 (s) as in (b -1).

-Case (c) : In this case d = 1, r is arbitrary and ξ = ( √ -1) r+1 . Then we shall consider four cases.

(c -1) : If r ≡ 3 (mod 4), then r is odd, say r = 2ρ + 1, and ξ = 1. Thus

r i=0 u ij (x) = P j (x, 1)P r-j (1, x) = (x + 1) ⌊ j 2 ⌋ (1 -x) j-⌊ j 2 ⌋ (x + 1) ⌊ r-j 2 ⌋ (x -1) r-j-⌊ r-j 2 ⌋
, and

r i=0 (-1) i u ij (x) = P j (x, -1)P r-j (1, -x) = (x -1) ⌊ j 2 ⌋ (-1 -x) j-⌊ j 2 ⌋ (1 -x) ⌊ r-j 2 ⌋ (-x -1) r-j-⌊ r-j 2 ⌋ . If j is even, say j = 2k, then r i=0 u ij (x) = (-1) k (x 2 -1) ρ (x -1) and r i=0 (-1) i u ij (x) = (-1) k+1 (x 2 -1) ρ (x + 1). If j is odd, say j = 2k + 1, then r i=0 u ij (x) = (-1) k+1 (x 2 -1) ρ (x -1) and r i=0 (-1) i u ij (x) = (-1) k+1 (x 2 -1) ρ (x + 1).
Therefore, by (4.17) and by (4.18), we have

Z s-n r ,+ ( f ) = c 3 (s) ρ k=0 (-1) k Z 2k (f, -s) - ρ k=0 (-1) k Z 2k+1 (f, -s) = c 3 (s) Z e -s (f ) -Z o -s (f ) , where c 3 (s) = (-2 √ -1) ρ+1 γ(s) sin ρ (πs) sin( πs 2
), and

Z s-n r ,-( f ) = -c 4 (s) ρ k=0 (-1) k Z 2k (f, -s) + ρ k=0 (-1) k Z 2k+1 (f, -s) = -c 4 (s) Z e -s (f ) + Z o -s (f ) , where c 4 (s) = 2(-2 √ -1) ρ γ(s) sin ρ (πs) cos( πs 2
). (c -2) : If r ≡ 1 (mod 4), then r is odd, say r = 2ρ + 1, and ξ = -1. Then r i=0 u ij (x) = (-1) -(r-j) P j (-x, 1)P r-j (1, -x), and r i=0 (-1) i u ij (x) = (-1) -(r-j) P j (-x, -1)P r-j (1, x).

If j is even, say j = 2k, we have

r i=0 u ij (x) = (-1) k (x 2 -1) ρ (x + 1) and r i=0 (-1) i u ij (x) = (-1) k+1 (x 2 -1) ρ (x -1). If j is odd, say j = 2k + 1, then r i=0 u ij (x) = (-1) k (x 2 -1) ρ (x + 1) and r i=0 (-1) i u ij (x) = (-1) k (x 2 -1) ρ (x -1).
Therefore, by (4.17), (4.18) and (4.14), we have

Z s-n r ,+ ( f ) = c 4 (s) ρ k=0 (-1) k Z 2k (f, -s) + ρ k=0 (-1) k Z 2k+1 (f, -s) = c 4 (s) Z e -s (f ) + Z o -s (f ) , and
Z s-n r ,-( f ) = c 3 (s) - ρ k=0 (-1) k Z 2k (f, -s) + ρ k=0 (-1) k Z 2k+1 (f, -s) = c 3 (s) -Z e -s (f ) + Z o -s (f )
, where the c 3 (s) and c 4 (s) are the same as in (c -1).

(c -3) : If r ≡ 0 (mod 4), then r is even, say r = 2ρ, and

ξ = √ -1. Thus r i=0 u ij (x) = ( √ -1) -(r-j) P j ( √ -1x, 1)P r-j (1, √ -1x),
and

r i=0 (-1) i u ij (x) = ( √ -1) -(r-j) P j ( √ -1x, -1)P r-j (1, - √ -1x).
If j is even, then

r i=0 u ij (x) = (1 + x 2 ) ρ and r i=0 (-1) i u ij (x) = (1 + x 2 ) ρ .
If j is odd, then

r i=0 u ij (x) = - √ -1(1 + x 2 ) ρ and r i=0 (-1) i u ij (x) = √ -1(1 + x 2 ) ρ .
Therefore, by (4.17), (4.18) and (4.13), we have

Z s-n r ,+ ( f ) = c 5 (s) ρ k=0 Z 2k (f, -s) - √ -1 ρ k=0 Z 2k+1 (f, -s) = 1 √ 2 c 5 (s)e √ -1π 4
- √ -1Z -s,+ (f ) + Z -s,-(f ) , and
Z s-n r ,-( f ) = c 5 (s) ρ k=0 Z 2k (f, -s) + √ -1 ρ k=0 Z 2k+1 (f, -s) = 1 √ 2 c 5 (s)e √ -1π 4
Z -s,+ (f ) - √ -1Z -s,-(f ) ,
where c 5 (s) = 2 ρ γ(s) cos ρ (πs).

(c -4) : If r ≡ 2 (mod 4), then r is even, say r = 2ρ, and ξ = -√ -1. Thus

r i=0 u ij (x) = (- √ -1) -(r-j) P j (- √ -1x, 1)P r-j (1, - √ -1x),
and

r i=0 (-1) i u ij (x) = (- √ -1) -(r-j) P j (- √ -1x, -1)P r-j (1, √ -1x).
If j is even,

r i=0 u ij (x) = (1 + x 2 ) ρ and r i=0 (-1) i u ij (x) = (1 + x 2 ) ρ . If j is odd, r i=0 u ij (x) = √ -1(1 + x 2 ) ρ and r i=0 (-1) i u ij (x) = - √ -1(1 + x 2 ) ρ .
Therefore, similarly to the case (c -3), we can prove that

Z s-n r ,+ ( f ) = c 5 (s) ρ k=0 Z 2k (f, -s) + √ -1 ρ k=0 Z 2k+1 (f, -s) = 1 √ 2 c 5 (s)e √ -1π 4
Z -s,+ (f ) - √ -1Z -s,-(f ) , and
Z s-n r ,-( f ) = c 5 (s) ρ k=0 Z 2k (f, -s) - √ -1 ρ k=0 Z 2k+1 (f, -s) = 1 √ 2 c 5 (s)e √ -1π 4 - √ -1Z -s,+ (f ) + Z -s,-(f ) ,
where c 5 (s) is the same as in (c -3).

Construction of the family E s,t

Recall from Theorem 3.6 the definition of the differential operator D s,t with (s, t) ∈ C × C. Let E s,t be the differential operator with polynomial coefficients defined on the Schwartz space S(V × V ) by

F • E s,t = D s,t • F. (5.1) For (s, ε) ∈ C × {±} let J s,ε f (x) := V f (y)det(x -y) s,ε dy, f ∈ S(V ).
(5.2)

This integral, initially well defined for ℜ(s) ≫ 0, can be extended meromorphically to C. We may think of J s,ε as a convolution operator :

J s,ε f = Z s,ε * f,
where Z s,ε is the tempered distribution defining the local Zeta integral (4.1).

Let M be the multiplication operator defined on S(V × V ) by M f (x, y) = det(xy)f (x, y).

(5.3) Theorem 5.1. For generic (s, ε) and (t, η) in C × {±}, we have

M • [J s,ε ⊗ J t,η ] = κ(s, t)[J s+1,-ε ⊗ J t+1,-η ] • E -s-n r ,-t-n
r , where κ(s, t) is a meromorphic function on C × C.

5.1.

Proof of Theorem 5.1 in the non-euclidean case except R p,q (type II, III and IV). Recall from (4.6) that

F(Z s,ε ) = c(s, ε)Z -s-n r ,ε
where c(s, ε) is as in (4.7). Now, by the main identity in Theorem 3.6 for the non-euclidean case we have

F M • (J s,ε ⊗ J t,η )f (x, y) = 1 (2π √ -1) r det ∂ ∂x - ∂ ∂y F(Z s,ε )(x)F(Z t,η )(y)Ff (x, y) = c(s, ε)c(t, η) (2π √ -1) r det ∂ ∂x - ∂ ∂y Z -s-n r ,ε (x)Z -t-n r ,η (y)Ff (x, y) = c(s, ε)c(t, η) (2π √ -1) r Z -s-1-n r ,-ε (x)Z -t-1-n r ,-η (y)D -s-n r ,-t-n r Ff (x, y) = c(s, ε)c(t, η) (2π √ -1) r Z -s-1-n r ,-ε (x)Z -t-1-n r ,-η (y)F(E -s-n r ,-t-n r f )(x, y) = c(s, ε)c(t, η) (2π √ -1) r c(s + 1, -ε)c(t + 1, -η) F(Z s+1,-ε )(x)F(Z t+1,-η )(y)F(E -s-n r ,-t-n r f )(x, y) = c(s, ε)c(t, η) (2π √ -1) r c(s + 1, -ε)c(t + 1, -η) F (J s+1,-ε ⊗ J t+1,-η ) • E -s-n r ,-t-n r f (x, y).
By the injectivity of the Fourier transform we get

M • (J s,ε ⊗ J t,η ) = κ(s, t)(J s+1,-ε ⊗ J t+1,-η ) • E -s-n r ,-t-n r , where κ(s, t) = c(s, ε)c(t, η) (2π √ -1) r c(s + 1, -ε)c(t + 1, -η) =        (2π √ -1) r b r,d (s + 1)b r,d (s + 1) split case (-8π √ -1) r b 2r,d (-2s -2n r )b 2r,d (2s + 2)b 2r,d (-2t -2n r )b 2r,d (2t + 2)
non-split case.

5.2. Proof of Theorem 5.1 in the euclidean case (type I). The proof of Theorem 5.1 in the cases (a) and (a') is similar to the one in the non-euclidean case. We will give a proof only in the cases (b -1) and (c -1). We leave the remaining cases to the reader.

-Case (b -1) : Here r = 2 and d ≡ 1 (mod 4). Recall from (4.15) that

F Z s,+ Z s,- = 4 √ 2γ(s + n 2 )A(s) Z -s-n 2 ,+ Z -s-n 2 ,- , where
A(s) = a +,+ (s) a +,-(s) a -,+ (s) a -,- (s)
, and

a +,+ (s) = sin π 2 (s + n + 1 2 ) cos π 2 (s + n 2 ), a +,-(s) = -sin π 2 (s + n + 1 2 ) sin π 2 (s + n 2 ), a -,+ (s) = cos π 2 (s + n + 1 2 ) cos π 2 (s + n 2 ), a -,-(s) = cos π 2 (s + n + 1 2 ) sin π 2 (s + n 2 ).
Observe that for any ε, η = ±, we have

a ε,η (s) = -a -ε,-η (s + 1). (5.4)
Let f ∈ S(V × V ). Since J s,ε is a convolution operator with Z s,ε , we have

F(M • (J s,ε ⊗ J t,η )f )(x, y) = 1 (2π √ -1) r ∆ ∂ ∂x - ∂ ∂y {F(Z s,ε )(x)F(Z t,η )(y)Ff (x, y)} = c(s, t) (2π √ -1) r ∆ ∂ ∂x - ∂ ∂y a ε,+ (s)Z -s-n 2 ,+ (x) + a ε,-(s)Z -s-n 2 ,- (x)
a η,+ (t)Z -t-n 2 ,+ (y) + a η,-(t)Z -t-n 2 ,- (y
) Ff (x, y) , where c(s, t) = 32γ(s + n 2 )γ(t + n 2 ). In view of (5.4) and Theorem 3.4 we get

F(M • (J s,ε ⊗ J t,ε )f )(x, y) = c(s, t) (2π √ -1) r a -ε,+ (s + 1)Z -s-1-n 2 ,+ (x) + a -ε,-(s + 1)Z -s-1-n 2 ,- (x)
a -η,+ (t + 1)Z -t-1-n 2 ,+ (y) + a -η,-(t + 1)Z -t-1-n 2 ,-(y) D -s-n 2 ,-t-n 2 (Ff )(x, y), = c(s, t) (2π √ -1) r c(s + 1, t + 1) F(Z s+1,-ε )(x)F(Z t+1,-η )(y)F(E -s-n 2 ,-t-n 2 f )(x, y) = c(s, t) (2π √ -1) r c(s + 1, t + 1) F((J s+1,-ε ⊗ J t+1,-η ) • (E -s-n 2 ,-t-n 2 f
))(x, y). Now, using the inverse Fourier transform, we obtain

M • [J s,ε ⊗ J t,η ] = κ(s, t)[J s+1,-ε ⊗ J t+1,-η ] • E -s-n r ,-t-n r , where κ(s, t) = (2π √ -1) r b r,d (s + 1)b r,d (t + 1)
.

-Case (c -2) : Here d = 1 and r ≡ 1 (mod 4). Recall from (4.16) that

F Z s,+ Z s,- = (-2 √ -1) ⌊ r 2 ⌋ γ(s + n r ) sin ⌊ r 2 ⌋ (π(s + n r ))B(s) Z e -s-n 2 Z o -s-n 2 ,
where

B(s) = a e + (s) a o + (s) a e -(s) a o - (s)
,

and a e + (s) = cos( π 2 (s + n 2 )), a o + (s) = cos( π 2 (s + n 2 )). a e -(s) = √ -1 sin( π 2 (s + n 2 )), a o -(s) = - √ -1 sin( π 2 (s + n 2 )
), Observe that for any ε = ±, we have

a e ε (s) = - √ -1a e -ε (s + 1), a o ε (s) = √ -1a o -ε (s + 1). (5.5) Let f ∈ S(V × V ). We have, F[(M • (J s,ε ⊗ J t,η ))f ](x, y) = 1 (2π √ -1) r det ∂ ∂x - ∂ ∂y F(Z s,ε )(x)F(Z t,η )(y)F(f )(x, y), = c(s, t) (2π √ -1) r det ∂ ∂x - ∂ ∂y (a e ε (s)Z e -s-n 2 (x) + a o ε (s)Z o -s-n 2 (x)) × ×(a e η (t)Z e -t-n 2 (y) + a o η (t)Z o -t-n 2 (y))F(f )(x, y), where c(s, t) = (-2 √ -1) r-1 γ(s + n r )γ(t + n r ) sin ⌊ r 2 ⌋ (π(s + n r )) sin ⌊ r 2 ⌋ (π(t + n r )). Using the fact that det ∂ ∂x -∂ ∂y [(Z e -s-n r ⊗ Z e -t-n r )F(f )] = [(Z e -s-1-n r ⊗ Z e -t-1-n r )D -s-n r ,-t-n r F(f )], det ∂ ∂x -∂ ∂y [(Z o -s-n r ⊗ Z o -t-n r )F(f )] = [(Z o -s-1-n r ⊗ Z o -t-1-n r )D -s-n r ,-t-n r F(f )], det ∂ ∂x -∂ ∂y [(Z e -s-n r ⊗ Z o -t-n r )F(f )] = -[(Z e -s-1-n r ⊗ Z o -t-1-n r )D -s-n r ,-t-n r F(f )], det ∂ ∂x -∂ ∂y [(Z o -s-n r ⊗ Z e -t-n r )F(f )] = -[(Z o -s-1-n r ⊗ Z e -t-1-n r )D -s-n r ,-t-n r F(f )]
, and the identities (5.5) we obtain

F[(M • (J s,ε ⊗ J t,η ))f ](x, y) = - c(s, t) (2π √ -1) r c(s + 1, t + 1) F(Z s+1,-ε )F(Z t+1,-η )D -s-n r ,-t-n r F(f )(x, y) = - c(s, t) (2π √ -1) r c(s + 1, t + 1) F[(J s+1,-ε ⊗ J t+1,-ε ) • E -s-n r ,-t-n r (f )](x, y)
. By the injectivity of the Fourier transform we obtain the desired result.

6. The degenerate principal series and the Knapp-Stein operators 6.1. The conformal group of a real Jordan algebra. Let V be a simple real Jordan algebra. Recall that Str(V ) denotes the structure group of V (see Subsection 3.1). If ℓ ∈ Str(V ), then for any x ∈ V det(ℓ(x)) = χ(ℓ)det(x), (

for some χ(ℓ) ∈ R * , and the function ℓ → χ(ℓ) is a character of Str(V ). It is known that for x ∈ V × , the quadratic operator P (x) (see (2.1)) belongs to Str(V ), and for any x, y ∈ V det P (x)y) = det(x) 2 det(y),

which implies χ P (x) = det(x) 2 . (6.2)
The inversion ı is a rational transformation of V defined on V × by

ı(x) = -x -1 .
Its differential at x ∈ V × is given by

Dı(x) = P (x -1 ) = P (x) -1 . (6.3)
For a ∈ V , denote by n a the translation x → x + a. Let N := {n a , a ∈ V } be the (abelian Lie) group of all translations. The conformal group Co(V ) of V is by definition the group of rational transforms of V generated by Str(V ), N and the inversion ı. It can be shown that Co(V ) is a simple Lie group (see [START_REF] Bertram | The geometry of Jordan and Lie structures[END_REF][START_REF] Koecher | The Minnesota notes on Jordan algebras and their applications[END_REF]). We will denote by Str(V ) + the subgroup of Str(V ) defined by Str(V ) + = {ℓ ∈ Str(V ), χ(ℓ) > 0}.

Define the proper conformal group Co(V ) + to be the group generated by Str(V ) + , N and the inversion ı.

Let Aff(V ) = Str(V ) ⋉ N (resp. Aff(V ) + = Str(V ) + ⋉ N ) be the group generated by Str(V ) (resp. Str(V ) + ) and N . Then Aff(V ) is a parabolic subgroup of Co(V ) equal to the normalizer of N in Co(V ). As Aff(V ) + = Co(V ) + ∩ Aff(V ), Aff(V ) + is the normalizer of N in Co(V ) + , and hence a parabolic subgroup of Co(V ) + .

The center of Str(V ) is the group of dilations

{δ t : v → tv, t ∈ R * }. Let A := {δ t , t ∈ R >0 }. As χ(δ t ) = t r , A is contained in Str(V ) + . For g ∈ Co(V ) + , let Θ(g) = α • ı • g • ı • α, (6.4)
where α is a Cartan involution of V (see Subsection 2.1). As α is an automorphism of V , χ(α) = 1 and α ∈ Str(V ) + . So Θ is an automorphism of Co(V ) + , which can be shown to be a Cartan involution of Co(V ) + . Moreover,

Θ(ı) = ı and ı • δ t • ı = δ 1/t ,
so that ı belongs to the maximal compact subgroup (Co(V ) + ) Θ of Co(V ) + and is a representantative of the non trivial Weyl group element for (Co(V ) + , Aff(V ) + ). Since α commutes to ı, the involution Θ preserves Str(V )

+ . Let N := Θ(N ) = ı • N • ı, P := Θ(P ) = Str(V ) + ⋉ N .
For any g ∈ Co(V ), denote by V g the (open, dense) subset of V where g is defined. It is known that for x ∈ V g , the differential Dg(x) of g at x belongs to Str(V ). Moreover for any g ∈ Co(V ), the map x → Dg(x) -1 extends polynomially from V into End(V ). As a consequence of the chain rule and using (6.3) and (6.2), for any g ∈ Co(V ) + and any x ∈ V g , the differential Dg(x) belongs to Str(V ) + . For g ∈ Co(V ) + and x ∈ V g , let c(g, x) := χ Dg(x)

-1 .

In view of the chain rule, the map g → c(g, x) satisfies the following cocycle relation :

c(g 1 g 2 , x) = c g 1 , g 2 (x) c(g 2 , x) g 1 , g 2 ∈ Co(V ) + , x ∈ V (6.5)
whenever both sides are defined. Moreover, (i) For any ℓ ∈ Str(V )

+ and x ∈ V c(ℓ, x) = χ(ℓ) -1 . (ii) For any v ∈ V and x ∈ V c(n v , x) = 1. (iii) For any x ∈ V × , c(ı, x) = det(x) 2 .
The cocycle property (6.5) satisfied by c implies that, for any g ∈ Co(V ) + , the function x → c(g, x) is a rational function on V . On the other hand, for ℓ ∈ Str(V ) + , χ(ℓ)

2n r = Det(ℓ) 2 , so that for x ∈ V g c(g, x) 2n r = χ Dg(x) -1 2n r = Det Dg(x) -1 2 .
Hence, for any g ∈ Co(V ) + , the function x -→ c(g, x)

2n r extends as a polynomial on V. But if a power3 of a rational function coincides on a Zariski open subset with a polynomial, then the rational function has to be a polynomial, and therefore x -→ c(g, x) extends as a polynomial on V. Proposition 6.1. For any g ∈ Co(V ) + , there exists a polynomial p g ∈ P(V ) such that, for

x ∈ V g c(g, x) = p g (x) 2 . (6.6)
The polynomial p g is unique up to a sign ±.

Proof. In fact, -if g = n v is a translation, then p nv ≡ 1 satisfies (6.6), -if g = ℓ ∈ Str(V ) + , then p ℓ ≡ χ(ℓ) -1 2 satisfies (6.6), -if g = ı, then p ı (x) = det(x) satisfies (6.6).

Recall that Co(V ) + is generated by Str(V ) + , N and ı. Hence, for any g ∈ Co(V ) + , the existence of a rational function satisfying (6.6) follows from the cocycle relation (6.5). But if the square of a rational function is a polynomial, then the rational function has to be a polynomial. For the uniqueness, let g ∈ Co(V ) + . If p g and q g are two such that p 2 g (x) = q 2 g (x) = c(g, x) for x ∈ V g , then by continuity, p 2 g = q 2 g or equivalently (p g + q g )(p gq g ) = 0, and p g = ±q g follows.

Let G = (g, p g ); g ∈ Co(V ) + , p g ∈ P(V ), p g (x) 2 = c(g, x) ∀x ∈ V ,
and define the product of two elements g 1 = (g 1 , p g 1 ) and g 2 = (g 2 , p g 2 ) of G by

g 1 g 2 = g 1 g 2 , (p g 1 • g 2 )p g 2 .g
To see that this definition makes sense, observe that (p g 1 • g 2 ) p g 2 is a rational function on V . Its square (wherever defined) satisfies

(p g 1 • g 2 ) p g 2 2 (x) = c(g 1 , g 2 (x)) c(g 2 , x) = c(g 1 g 2 , x),
which extends as a polynomial on V. As a consequence, (p g 1 • g 2 ) p g 2 itself extends to V as a polynomial, and its square equals c(g 1 g 2 , x), this proves that g 1 g 2 , (p g 1 •g 2 )p g 2 is an element of G. Now G can be endowed with a Lie group structure yielding a twofold covering of Co(V ) + . The various notions defined (and results obtained) for Co(V ) + have their counterparts for G. The subgroups A and N of Co(V ) + are identified with subgroups of G via

δ t -→ (δ t , t -r 2 ), n -→ (n, 1).
The inversion ı is identified with (ı, det(x)) and the Cartan involution α is identified with (α, 1). Finally, let L and P be the subgroups of G defined by

L = { ℓ, ± χ(ℓ) -1 2 ; ℓ ∈ Str(V ) + }, P = { p, ± χ(p) -1 2 ; p ∈ Aff(V ) + },
which are twofold (trivial) coverings of respectively Str(V ) + and Aff(V ) + . Above we have extended the character χ of Str(V ) to a character of Aff(V ) + = Str(V ) + ⋉ N by letting it acting trivially on N . Then P is a maximal parabolic subgroup of G with Langlands decomposition P = LN . We keep the notation Θ for the Cartan involution of G given by Θ

(g) = α • ı • g • ı • α. Bruhat decomposition of G takes the form G = P ıP ∪ P. (6.7)
For a given g = (g, p g ) ∈ G and x ∈ V g , we set g(x) := g(x). Further, we wil use the following notation a( g, x) := p g (x), x ∈ V. (6.8) Then a( g, . ) is a polynomial on V satisfying the cocycle relation

a( g 1 g 2 , x) = a g 1 , g 2 (x) a( g 2 , x), g 1 , g 2 ∈ G, x ∈ V.
(6.9)

We will abuse notation and denote elements of G without tilde. Remark that for all g ∈ G, g defined at x ⇐⇒ a(g, x) = 0.

The following equalities are immediate from above

a(n v , x) = 1 a v ∈ N a((ℓ, ± χ(ℓ) -1 2 ), x) = ±(χ(ℓ)) -1 2 ℓ ∈ Str(V ) + a(ı, x) = det(x).
Proposition 6.2. Let g ∈ G and x, y ∈ V such that g is defined at x and y. Then det g(x)g(y) = a(g, x) -1 det(xy) a(g, y) -1 .

(6.10)

Proof. For g = (n v , 1) the identity (6.10) is obvious. For g = (ℓ, ±χ(ℓ) -1 2 ) we have

det ℓ(x -y) = χ(ℓ)det(x -y) = ± χ(ℓ) -1 2 ) -1 det(x -y) ± χ(ℓ) -1 2 ) -1 .
Finally, for g = (ı, det(x)), we have

det ı(x) -ı(y) = det(-x -1 + y -1 ) = (det x) -1 det(x -y)(det y) -1 ,
which is essentially Hua's formula. The general formula follows by using the cocycle relation (6.9). 6.2. The degenerate principal series and Knapp-Stein intertwining operators. The degenerate principal series is the family of representations of G (smoothly) induced by the characters of the parabolic subgroup opposed to P . In this section, we will consider the non-compact realization of these representations.

For (λ, ε) ∈ C × {±} and for g ∈ G let

π λ,ε (g)f (x) = a(g -1 , x) -λ,ε f g -1 (x) . (6.11)
In this formula, f is a smooth function on V . As the action of G on V is not defined everywhere, we let H λ,ε to be the subspace of functions f ∈ C ∞ (V ) such that ∀g ∈ G : V g ∋ x -→ π λ,ε (g)f (x) extends as a smooth function on V.

By the Bruhat decomposition G = P ∪ P ıP , the space H λ,ε can be realized as the subspace of functions f ∈ C ∞ (V ) such that

V × ∋ x -→ π λ,ε (ı)f (x) extends as a smooth function on V .
The space H λ,ε is equipped with a topology using the semi-norms defining the usual topology of C ∞ (V ) for both f and π λ,ε (ı)f . Then (6.11) defines a continuous representation of G on H λ,ε .

The space H λ,ε is not well fitted in order to use Fourier analysis on V . So we will replace the representation of the group by its infinitesimal version.

Let f be in the space D(V ) of smooth functions on V with compact support. Then for g close to the neutral element of G, g -1 is defined on the support of f , so that π λ,ε (g)f is a well defined smooth function on V . Denote by g the Lie algebra of G.

For X ∈ g, let dπ λ,ε (X)f = d dt π λ,ε (exp(tX))f t=0 .
This defines a representation of g on D(V ). If g is close enough to the identity, then a(g -1 , x) > 0 for all x ∈ Supp(f ), and therefore the infinitesimal representation dπ λ,ε does not depend on ε. Hence we will drop the index ε from dπ λ,ε .

Finally, it is well-known (see e.g. [START_REF] Pevzner | Analyse conforme sur les algèbres de Jordan[END_REF]) that dπ λ (X) is a first order differential operator on V with polynomial coefficients of degree ≤ 2. As a consequence we let the infinitesimal representation act on the Schwartz space S(V ).

Let (λ, ε) ∈ C × {±}. For f ∈ S(V ), define the Knapp-Stein intertwining operator by

I λ,ε f (x) := V det(x -y) -2n r +λ,ε f (y)dy. (6.12)
For ℜλ ≫ 0, the function det(x) -2n r +λ,ε is locally integrable and has polynomial growth at infinity, so that it can be considered as a tempered distribution on V , and hence (6.12) defines an operator from S(V ) into S ′ (V ). Next, the definition is extended meromorphically to C by analytic continuation in the parameter λ, thus defining a family of operators from S(V ) into S ′ (V ) depending meromorphically on λ.

Let f ∈ D(V ). For any g ∈ G such that g is defined on the support of f ,

I λ,ε • π λ,ε (g) f = π 2n r -λ,ε (g) • I λ,ε
f, as can be proved first for ℜλ ≫ 0, using (6.10), followed by analytic continuation in the parameter λ. In particular it implies that

I λ,ε • dπ λ (X) = dπ 2n r -λ (X) • I λ,ε , ∀X ∈ g, (6.13)
as an equality of operators from S(V ) into S ′ (V ).

7. The covariance property of the operators F λ,µ 7.1. The infinitesimal covariance property. Recall that M denotes the multiplication operator on S(V × V ) defined by M f (x, y) = det(xy)f (x, y).

Lemma 7.1. Let f be a smooth function on V × V with compact support. Let g ∈ G close to the neutral element so that g (acting diagonally on V × V ) is defined on the support of f . Then, for any (λ, ε) and (µ, η) in C × {±}, we have

M • π λ,ε (g) ⊗ π µ,η (g) f = π λ-1,-ε (g) ⊗ π µ-1,-η (g) • M f.
Proof. This is an immediate consequence of the covariance property (6.10) of the function det(xy).

For X ∈ g, let d(π λ ⊗π µ )(X) = dπ λ (X)⊗Id+Id⊗dπ µ (X) be the infinitesimal representation of g induced by π λ,ε ⊗ π µ,η . The infinitesimal version of the above lemma reads : Proposition 7.1. For any X ∈ g, we have

M • d(π λ ⊗ π µ )(X) = d(π λ-1 ⊗ π µ-1 )(X) • M.
To make connection with the notation of Section 5 and Section 6, first observe that

I λ,ε = J -2n
r +λ,ε , where J s,ε was defined in (5.2). For notational convenience, we let

F λ,µ = E n r -λ, n r -µ ,
where E s,t is defined by (5.1). Then we may rephrase Theorem 5.1 as follows: Proposition 7.2. For all (λ, ε) and (µ, η) in C × {±}, we have

M • (I λ,ε ⊗ I µ,η ) = k(λ, µ)(I λ+1,-ε ⊗ I µ+1,-η ) • F λ,µ ,
where k(λ, µ) is a meromorphic function on C × C.

Further, we have the following covariance property of the operators F λ,µ .

Theorem 7.1. For all λ, µ ∈ C and for any X ∈ g, we have

F λ,µ • (d(π λ ⊗ π µ )(X) = (d(π λ+1 ⊗ π µ+1 )(X) • F λ,µ (7.1)
Proof. In view of Proposition 7.1, Proposition 7.2 and the identity (6.13), we have

I λ+1,-ε ⊗ I µ+1,-η • F λ,µ • d(π λ ⊗ π µ )(X) = 1 κ(λ, µ) M • (I λ,ε ⊗ I µ,η ) • d(π λ ⊗ dπ µ )(X) = 1 κ(λ, µ) M • (d(π 2n r -λ ⊗ π 2n r -µ )(X)) • I λ,ε ⊗ I µ,η = 1 κ(λ, µ) d(π 2n r -λ-1 ⊗ π 2n r -µ-1 )(X) • M • I λ,ε ⊗ I µ,η = d(π 2n r -λ-1 ⊗ π 2n r -µ-1 )(X) • I λ+1,-ε ⊗ I µ+1,-η • F λ,µ = I λ+1,-ε ⊗ I µ+1,-η • d(π λ+1 ⊗ π µ+1 )(X) • F λ,µ .
For generic λ, the operator I λ,ε is injective on S(V ), so that (7.1) follows. By continuity, (7.1) extends to all λ, µ ∈ C.

The global covariance property.

There is a global formulation for the covariance property of the operators F λ,µ , but it is nicer to work in the compact setting. Using the notation of Subsection 6.1, let P = Θ(P ) be the opposite parabolic subgroup of G. Let X = G/P . The map

V ∋ v -→ n v P
is a diffeomorphism onto an open dense subset of X (for that reason X is usually called the conformal compactification of V ). Let χ 1 2 be the character of P which is trivial on N = Θ(N ) and which on L is defined by

χ 1 2 ℓ, ±χ(ℓ) -1 2 = ±χ(ℓ) 1 2 ,
where χ is the character of Str(V ) defined by (6.1).

For (λ, ε) ∈ C × {±}, consider the character χ λ,ε

1 2
of L and form the line bundle

E λ,ε = G ⊗ P C χ λ,ε 1 2
.

Finally denote by Γ(E λ,ε ) the space of smooth sections of E λ,ε . The natural actions of G on Γ(E λ,ε ) defines a smooth representation of G which is the compact realization of the representation π λ,ε considered previously in (6.11). We keep same notation for this representation.

Similarly, consider the normalized Knapp-Stein intertwining operators J λ,ε : Γ(E λ,ε ) -→ Γ(E 2n r -λ,ε ) (see [START_REF] Knapp | Representation theory of semisimple groups. An overview based on examples[END_REF]). They form a holomorphic family of intertwining operators. It is known that for generic λ, the representation π λ,ε is irreducible, so that

J 2n r -λ,ε • J λ,ε = d ε (λ) id (7.2)
for some holomorphic function d ε . By (6.10), the kernel det(xy) correspond to a G-invariant section of E -1,-⊗ E -1,-and hence, for any (λ, ε) and (µ, η), the corresponding multiplication operator

M : Γ(E λ,ε ⊗ E µ,η ) -→ Γ(E λ-1,-ε ⊗ E µ-1,-η ) intertwines π λ,ε ⊗ π µ,η and π λ-1,-ε ⊗ π µ-1,-η .
Let (λ, ε), (µ, η) ∈ C × {±} and consider the operator

F (λ,ε),(µ,η) : Γ(E λ,ε ⊗ E µ,η ) -→ Γ(E λ+1,-ε ⊗ E µ+1,-η ) defined by F (λ,ε),(µ,η) = J 2n r -λ-1,-ε ⊗ J 2n r -µ-1,-η • M • J λ,ε ⊗ J µ,η .
Notice that the family F (λ,ε),(µ,η) depends holomorphically on λ and µ.

Theorem 7.2. The operator F (λ,ε),(µ,η) is a differential operator which intertwines the representations π λ,ε ⊗ π µ,η and π λ+1,-ε ⊗ π µ+1,-η .

Proof. The intertwining property of F (λ,ε),(µ,η) follow immediately from its construction. Now, by composing F (λ,ε),(µ,η) with J λ+1,-ε ⊗ J µ+1,-η from the left and using (7.2), we get

J λ+1,-ε ⊗ J µ+1,-η • F (λ,ε),(µ,η) = d -ε (λ + 1)d -η (µ + 1)M • J λ,ε ⊗ J µ,η .
When translating this relation in the non-compact setting, the local expression of J λ,ε is (up to a function of λ) equal to I λ,ε and M corresponds to the multiplication by det(xy).

Comparing with the result obtained in Section 7, this implies that the local expression of F (λ,ε),(µ,η) in the non-compact setting is equal to F λ,µ , up to a meromorphic function of λ and µ. Hence F (λ,ε),(µ,η) is a differential operator. This works for generic λ and µ and hence for every λ and µ by continuity.

Conformally covariant bi-differential operators

Let res : C ∞ (V × V ) → C ∞ (V ) be the restriction map defined by res(ϕ(x)) = ϕ(x, x), x ∈ V. It satisfies res • (π λ,ε (g) ⊗ π µ,η (g)) = π λ+µ,εη (g) • res, ∀g ∈ G.
For λ, µ ∈ C and N ∈ N * we define the bi-differential operator

B (N ) λ,µ : C ∞ (V × V ) -→ C ∞ (V ) by B (N ) λ,µ = res • F λ+N -1,µ+N -1 • • • • • F λ,µ .
The covariance property of the operators F λ,µ and of res imply the following statement.

H (λ,ε),(µ,η) F (N) (λ,ε),(µ,η) -------→ H (λ+N,ε N ),(µ+N,η N ) J λ,ε ⊗ Jµ,η     J 2n r -λ-N,ε N ⊗ J 2n r -µ-N,η N H ( 2n r -λ,ε),( 2n r -µ,η) M N ----→ H ( 2n r -λ-N,ε N ),( 2n r -µ-N,η N )
with the convention ε N = (-1) N ε. It is a generalization of the diagram presented in the introduction which corresponds to the case N = 1. The operator F (N ) (λ,ε),(µ,η) is by construction covariant with respect to π λ,ε ⊗ π µ,η and π λ+N,ε N ⊗ π µ+N,η N . By induction on N , it is possible to prove that F (N ) (λ,ε),(µ,η) is a differential operator on X × X . In fact F (N ) (λ,ε),(µ,η) coincides (up to a meromorphic function in λ and µ) with the operator

F (λ+N,ε N ),(µ+N,η N ) • • • • • F (λ,ε),(µ,η) .
The corresponding bi-differential operator is res •F (N ) (λ,ε),(µ,η) , and its expression in the non-compact realization coincides (up to a meromorphic function in λ and µ) with B (N ) λ,µ . 9. The case of R p,q with p ≥ 2 and q ≥ 1 Let E be a real vector space of dimension n -1, endowed with a non degenerate symmetric bilinear form β : E × E → R of signature (p -1, q) where p + q = n. Then V := R × E is a simple real Jordan algebra with multiplication given by (λ, v) • (µ, w) = (λµβ(v, w), λw + µv).

The dimension of V equals n = p + q and its rank is 2. The neutral element of V is e = (1, 0 E ) and its determinant is given by

det(λ, v) = λ 2 + β(v, v),
which is a quadratic form of signature (p, q) and the trace is

tr(λ, v) = 2λ.
Therefore, (λ, v) ∈ V is invertible if and only if λ 2 + β(v, v) = 0 and its inverse is given by

(λ, v) -1 = (λ 2 + β(v, v)) -1 (λ, -v).
Fix a basis (e 2 , . . . , e n ) of E with coordinates (x 2 , . . . , x n ) chosen so that, for v = (x 2 , . . . , x n ),

β(v, v) = x 2 2 + • • • + x 2 p -x 2 p+1 -• • • -x 2
n . Below we will denote V by R p,q and its determinant by

P (x) = x 2 1 + x 2 2 + • • • + x 2 p -x 2 p+1 -• • • -x 2 n
, where x = (x 1 , x 2 , . . . , x n ) ∈ V. In this notation the neutral element is e = e 1 . 9.1. The Zeta functional equation. Let V ′ be the dual space of V . The symmetric bilinear form on V associated to P induces an isomorphism of V ′ with V , and so we can transfer the quadratic form on V to a quadratic form on V ′ , which we denote also by P . More explicitly, let (e ′ 1 , . . . , e ′ n ) be the basis of V ′ which is the dual to the canonical basis of V , and denote by (ξ 1 , . . . , ξ n ) the coordinates of an arbitrary element ξ ∈ V ′ . Let

P (ξ) = ξ 2 1 + • • • + ξ 2 p -ξ 2 p+1 -• • • -ξ 2 n .
The corresponding symmetric bilinear form on V ′ × V ′ is given by

P (ξ, ζ) = ξ 1 ζ 1 + • • • + ξ p ζ p -ξ p+1 ζ p+1 -• • • -ξ n ζ n .
In a departure from our general convention (4.2), we define in this section the Fourier transform F : S(V ) -→ S(V ′ ) by

Ff (ξ) = V e i(ξ,x) f (x)dx,
and extend it by duality to S ′ (V ).

Below, for (s, ε) ∈ C × {±}, P s,ε (x) stands for det(x) s,ε .

Theorem 9.1. For every s ∈ C, we have F P s,+ P s,-= γ(s)A(s)

P -s-n 2 ,+ P -s-n 2 ,- (9.1)
where Proof. Introduce

A(s) =    cos (p -q)π 4
P + (x) =    P (x) on {P (x) > 0} 0 on {P (x) < 0} P -(x) =    0 on {P (x) > 0} -P (x) on {P (x) < 0}
.

For s ∈ C with ℜ(s) > -1, the functions P s + and P s -are locally integrable with moderate growth at infinity. Further, they can be extended, as tempered distributions, meromorphically for s ∈ C. Their Fourier transforms are given by

F(P s + ) = γ(s) -sin q 2 + s π P -s-n 2 + + sin pπ 2 P -s-n 2 - , and
F(P s -) = γ(s) sin qπ 2 P -s-n 2 + -sin s + p 2 π P -s-n 2 -
, where γ(s) = 2 2s+n π n 2 -1 Γ(s + 1)Γ(s + n 2 ). See [13, (2.8) and (2.9)] (or [START_REF] Strichartz | Fourier transforms and non-compact rotation groups[END_REF]). Now P s,+ = P s + + P s -and P s,-= P s + -P s -, and (9.1) follows by routine computation.

We will use the following notation for the coefficients of the matrix A(s) :

A(s) = a +,+ (s) a +,-(s) a -,+ (s) a -,- (s)
.

Observe that the matrix-valued function A(s) is periodic of period 2 and the coefficients a ε,η satisfy

a ε,η (s + 1) = -a -ε,-η (s), (9.3)
for every ε, η = ±. 9.2. The main identity for R p,q . Recall from Theorem 3.6 that there exists a differential operator D s,t on V ′ × V ′ such that

P ∂ ∂ξ - ∂ ∂ζ P (ξ) s,ε P (ζ) t,η f (ξ, ζ) = P (ξ) s-1,-ε P (ζ) t-1,-η D s,t f (ξ, ζ), (9.4)
for every f in S(V ′ × V ′ ).

Theorem 9.2. For V = R p,q , the differential operator D s,t is given explicitly by Elementary calculations show that for f a smooth function on {P (ξ) > 0}, we have Putting all pieces together yields Proposition 9.1. 9.4. The main theorem for R p,q . For (s, ε) ∈ C×{±}, recall the Knapp-Stein intertwining operator

D s,t = P (ξ)P (ζ) P ∂ ∂ξ - ∂ ∂ζ +4s P (ζ) n i=1 ξ j ∂ ∂ξ j - ∂ ∂ζ j + 4t P (ξ) n j=1 ζ j ∂ ∂ζ j - ∂ ∂ξ j +2t(2t -2 + n)P (ξ) -8stP (ξ, ζ) + 2s(2s -2 + n)P (ζ).
P ∂ ∂ξ P (ξ) s f (ξ) = 2s(2s -2 + n)P (ξ) s-1 f (ξ) + 4s n i=1 ξ i P (ξ)
J s,ε f (x) = V P (x -y) s,ε f (y)dy.
The integral is well defined for f ∈ S(V ) when ℜs > -1. It defines a convolution operator from S(V ) into S ′ (V ) and the function s -→ J s,ε can be extended meromorphically on C.

Let M be the multiplication operator given for

f ∈ S(V × V ) (or S ′ (V × V )) by M f (x, y) = P (x -y)f (x, y).
In view of the functional equation (9.1) of P s,ε and the relation (9.3), the proof of the statement below goes along the same lines as that of Theorem 5.1 Theorem 9.3. For (s, ε) and (t, η) in C × {±}, we have

M • (J s,ε ⊗ J t,η ) = κ(s, t) (J s+1,-ε ⊗ J t+1,-η ) • E -s-n 2 ,-t-n 2 , where κ(s, t) = 1 16(s + 1)(s + n 2 )(t + 1)(t + n 2 )
. 9.5. The conformal group of V = R p,q . For t ∈ R * , let δ t be the dilation given by V ∋ v -→ tv. The structure group of V is equal to

Str(V ) = {h • δ t , h ∈ O(p, q), t ∈ R >0 } = O(p, q) × R * /∼
where ∼ is the equivalence relation (h, δ t ) ∼ (-h, δ -t ). As

P (h • δ t )(x) = t 2 P (x),
the character χ of Str(V ) is given by χ(h • δ t ) = t 2 and therefore Str(V )

+ = Str(V ). Let W = R × V × R. Set e 0 = ( 1
, 0, 0) and e n+1 = (0, 0, 1). We will use the following convention : for w

= (α, v, β) ∈ W, α(w) = α, (w) V = v, β(w) = β. On W define the quadratic form Q(α, v, β) = P (v) -αβ.
Notice that Q is of signature (p + 1, q + 1). Let us consider the proper isotropic cone Ξ = w ∈ W, w = 0, Q(w) = 0 , and let X = Ξ/R * , a real projective quadric. For w ∈ W {0}, let w be its image in the projective space P(W ).

Lemma 9.1. The map κ : V -→ X defined by

V ∋ v -→ κ(v) = (1, v, P (v)) (9.6)
is a diffeomorphism of V onto a dense open subset of X.

Proof. Since for v ∈ V , (1, v, P (v)) belongs to Ξ and is = 0, the map is well defined. It is smooth and injective. Let (α, v, β) be in Ξ. If α = 0, then (α, v, β) = (1, v ′ , P (v ′ ) where v ′ = α -1 v. Hence the image of the map given by (9.6) is equal to X ∩ {α(w) = 0} and the lemma follows.

In the sequel we let ∞ = e n+1 and o = e 0 , both points of X.

Let G = O(Q) ≃ O(p + 1, q + 1) be the orthogonal group of the form Q. Then G preserves Ξ and commutes to the dilations, so acts on the space4 X. This allows to define a (rational) action of G on V by setting

g(x) = α gκ(x) -1 gκ(x) V .
Let L be the subgroup of G given by

L =      t -1 0 0 0 h 0 0 0 t   , h ∈ O(p, q), t ∈ R *    .
The elements ±   t -1 0 0 0 h 0 0 0 t   both act on V by v -→ t hv, which realizes L as a twofold covering of the structure group Str(V ). For a ∈ V, let n a be the linear transform of W defined by n a (α, v, β) = (α, αa + v, αP (a) + 2P (a, v) + β).

It is easily verified that n a belongs to G. The action of n a on V is given by n

a (v) = v + a. Further, for a, b ∈ V , we have n a • n b = n a+b so that N = {n a , a ∈ V } is an abelian subgroup of G. Lemma 9.2. Let P = LN . The stabilizer G ∞ of ∞ in G is equal to P. Proof. First, clearly P stabilizes ∞. Next let g ∈ G ∞ . Then g preserves the subspace (Re n+1 ⊥ = V ⊕ Re n+1 . Let ge 0 = (α, v, β). As Q(e 0 , e n+1 ) = 1, ge 0 / ∈ (Re n+1 ⊥ and α is different from 0. Moreover, 0 = Q(e 0 ) = Q(ge 0 ) = P (v) -αβ, so that (δ 1 α • t v )(1, 0, 0) = (α, v, β). Let g 1 = δ 1 α • t v -1
• g, so that g 1 e 0 = e 0 . Then g 1 stabilizes both ∞ and o. Hence g 1 stabilizes (Re 0 ⊕ Re n+1 ) ⊥ = V and the restriction of g 1 to V preserves the quadratic form

Q |V = P . So the matrix of g 1 is of the form   s 0 0 0 h 0 0 0 t   ,
where h ∈ O(p, q) and st = 1. In other words, g 1 belongs to L, and hence g ∈ P . The conclusion follows.

Let ı be the element of G defined by

  0 0 1 0 I 1,n-1 0 -1 0 1   where I 1,n-1 = diag(-1, 1, . . . , 1). Then for x = (x 1 , x 2 , . . . , x n ) ∈ V × , ı(x) = -x P (x) = -x -1
, where x = (x 1 , -x 2 , . . . , -x n ). The group G is generated by P and ı.

The character χ of Str(V ) is given on L by

χ     t -1 0 0 0 h 0 0 0 t     = t 2 ,
so that we choose χ 1 2 to be defined by

χ 1 2     t -1 0 0 0 h 0 0 0 t     = t.
It is then easily verified that the cocycle a(g, x), defined in the general situation by (6.8), is given by a(g, x) = α gκ(x) .

For (λ, ε) ∈ C × {±}, the principal series representations π λ,ε is given by

π λ,ε (g)f (x) = a(g, x) -λ,ε f (g -1 (x)).
The Knapp-Stein intertwining operator is given by

I λ,ε f (x) = V f (
y)P (xy) -n+λ,ε dy and satisfies

I λ,ε • π λ,ε (g) = π n-λ,ε (g) • I λ,ε .
9.6. The operator F λ,µ . As in the general case, the main Theorem 9.3 can be reinterpreted to give a covariance property for the differential operator F λ,µ (or its global version as a differential operator on X × X), just by the change of parameters s = n 2λ and t = n 2µ. The differential operator F λ,µ is given by

F λ,µ = -P (x -y)P ∂ ∂x P ∂ ∂y +4(-λ + n 2 -1) n j=1 (x j -y j ) ∂ ∂x j P ∂ ∂y + 4(-µ + n 2 -1) n j=1 (y j -x j ) ∂ ∂x j P ∂ ∂x +4λ(-λ + n 2 -1)P ∂ ∂y + 4µ(-µ + n 2 -1)P ∂ ∂x +8(-λ + n 2 -1)(-µ + n 2 -1)P ∂ ∂x , ∂ ∂y .
Theorem 9.4. For (λ, ε) and (µ, η) in C × {±}, the operator F λ,µ is covariant with respect to (π λ,ε ⊗ π µ,η ) and (π λ+1,-ε ⊗ π µ+1,-η ).

The construction of covariant bi-differential operators for O(p + 1, q + 1) is then obtained as in the general case. Let us state the formula for B

(1) λ,µ , which is covariant with respect to (π λ,ε ⊗ π µ,η , π λ+µ+2,εη ) :

B (1) λ,µ = 4 res µ(-µ + n 2 -1)P ∂ ∂x + λ(-λ + n 2 -1)P ∂ ∂y + 2(-λ + n 2 -1)(-µ + n 2 -1)P ∂ ∂x , ∂ ∂y .
We point out that these covariant bi-differential operators were already introduced in [ Because of lack of a convenient reference, the purpose of this appendix is to clarify the relations between a real simple Jordan algebra and its complexification. 11.1. Rank, generic minimal polynomial and determinant. Let V be a unital Jordan algebra over F = R or C and denote by 1 its unit element. Recall that the rank of an element x ∈ V is defined by rk F (x) = min{k > 0, (e, x, x 2 , . . . , x k ) are F-linearly dependant} The rank of V is defined as rk F (V ) = max{rk F (x), x ∈ V }. Proposition 11.1. Let V be a unital real Jordan algebra and V its complexification. Then rk R (V ) = rk C (V).

(m = 2ℓ) Type IV V Sym(m, R) × Sym(2ℓ, R) ∩ Mat(ℓ, H) Sym(m, C) co(V ) sp(m, R) × sp(ℓ, ℓ) sp(m, C str(V ) sl(m, R) ⊕ R × sp(2ℓ) ⊕ R sl(m, C) ⊕ C n m(m + 1)/2 × ℓ(2ℓ + 1) m(m + 1) r m × 2ℓ 2m d 1 × 4 2 e 0 × 2 1 V + Sym(m, R) × Herm(ℓ, C) Sym(m, R) r + m × ℓ m d + 1 × 2 1 V Herm(m, C) Mat(m, R) Mat(ℓ, H) Mat(m, C) co(V ) su(m, m) sl(2m, R) su * (4ℓ) sl(2m, C) str(V ) sl(m, R) ⊕ R sl(m, R) ⊕ sl(m, R) ⊕ R su * (2ℓ) ⊕ su * (2ℓ) ⊕ R sl(m, C) ⊕ sl(m, C) ⊕ C n m 2 m 2 4ℓ 2 2m 2 r m m 2ℓ 2m d 2 2 8 4 e 0 0 3 1 V + Herm(m, C) Sym(m, R) Herm(ℓ, H) Herm(m, C) r + m m ℓ m d + 2 1 4 2 V Herm(m, H) Skew(2m, R) × Skw(2m, C) co(V ) so * (4m) so(2m, 2m) × so(4m, C) str(V ) su * (2m) ⊕ R sl(2m, R) ⊕ R × sl(2m, C) ⊕ C n m(2m -1) m(2m -1) × 2m(2m -1) r m m × 2m d 4 4 × 8 e 0 0 × 1 V + Herm(m, H) Herm(m, C) × Herm(m, H) r + m m × m d + 4 2 × 4 V R 1,k-1 R p,q R k,0 C k co(V ) so(2, k) so(p + 1, q + 1) so(k + 1, 1) so(k + 2, C) str(V ) so(1, k -1) ⊕ R so(p, q) ⊕ R so(k) ⊕ R so(n, C) ⊕ C n m p + q = k k 2k r 2 2 2 4 d k -2 p + q -2 0 2(k -2) e 0 0 k -1 1 V + R 1,k-1 R 1,q R 1,0 R 1,k-1 r + 2 2 1 2 d + k -2 q -1 0 k -2 V Herm(3, O) Herm(3, O s ) × Herm(3, O) C co(V ) e 7
Proof. Let rk R (V ) = r. By assumption, there exists an element x ∈ V such that

1 ∧ x ∧ • • • ∧ x r-1 = 0, (11.1)
whereas for all elements y ∈ V ,

1 ∧ y ∧ • • • ∧ y r = 0. ( 11.2)
Let ρ = rk C (V). Now (11.1) implies that ρ ≥ r. Next the map

V ∋ z -→ 1 ∧ z ∧ • • • ∧ z r ∈ Λ r+1 (V)
is holomorphic and vanishes on V by (11.2), hence everywhere. This shows that ρ ≤ r. We can conclude that r = ρ.

An element x ∈ V is said to be regular if rk F (x) = rk F (V ). In the case where V is a real Jordan algebra, an element of V is regular if and only if it is regular as an element of V.

Let x ∈ V . Then the subalgebra F[x] generated by 1 and x is commutative and power associative. Let I(x) be the ideal of F[x] defined by I(x) = {p ∈ F[T ], p(x) = 0}.

Since F[T ] is a principal ring, I(x) is generated by a monic polynomial, called the minimal polynomial of x and denoted by p x . Proposition 11.2. There exists polynomials a 1 , a 2 , . . . , a r ∈ F[T ] such that the minimal polynomial p x of every regular element of V is given by p x (T ) = T ra 1 (x)T r-1 + • • • + (-1) r a r (x).

The polynomial m(T, x) = m x (T ) = T ra 1 (x)T r-1 + • • • + (-1) r a r (x) is called the generic minimal polynomial of V at x. The linear form a 1 is the trace of V and the polynomial a r is the determinant of V , tr(x) = a 1 (x), det(x) = a r (x).

Proposition 11.3. Let V be a real Jordan algebra and V its complexification. The generic minimal polynomial of V restricts to the generic minimal polynomial of V .

Proof. This is a consequence of the local expression of these coefficients near a regular element, namely a j (x) = (-1) j-1 Det(1, x, . . . , x j-1 , x r , x j+1 , . . . , x r-1 , e r+1 , . . . , e n ) Det(1, x, x 2 , . . . , x r-1 , e r+1 , . . . , e n ) ,

where e r+1 , . . . , e n are elements completing e, x, x 2 , . . . , x r-1 to a basis of V (see [12, proof of Proposition II.2.1]).

Corollary 11.1. Let V be a real Jordan algebra, and let V be its complexification. Then the restriction to V of the determinant of V coincides with the determinant of V .

11.2. Primitive idempotents. Let x ∈ V. A complex number λ is called an eigenvalue of x if λ is a root of the minimal polynomial p x . An element is said to be semi-simple if its minimal polynomial has only simple roots.

Proposition 11.4. Let (d 1 , d 2 , . . . , d l ) be a complete system of orthogonal idempotents, and let (µ 1 , . . . , µ l ) be l distinct complex numbers. Let x = l j=1 µ j d j . Then x is semisimple, d j ∈ C[x] for every 1 ≤ j ≤ l and the eigenvalues of x are µ 1 , . . . , µ l . For 1 ≤ j ≤ l, let p j be the unique polynomial of degree l -1 such that p j (µ i ) = δ ij . Then This shows that p belongs to the ideal I(x) if and only if p is a multiple of Π(T ) = l j=1 (Tµ j ). In other words Π is the minimal polynomial of x and the conclusion follows.

Proposition 11.5. Let x ∈ V be a semi-simple element, with distinct eigenvalues λ 1 , . . . , λ k . There exists a unique (up to permutation of the indices) system of orthogonal idempotents (c 1 , c 2 , . . . , c k ) in C[x] such that

c 1 + c 2 + • • • + c k = 1, x = λ 1 c 1 + • • • + λ k c k .
This is (part of) [START_REF] Faraut | Analysis on symmetric cones[END_REF]Proposition VIII.3.2]. The uniqueness statement comes from the fact that necessarily c j = p j (x) where p j is the polynomial of degree k -1 which satisfies p j (λ i ) = δ ij . Proposition 11.6. Let V be a complex simple Jordan algebra.The set of semi-simple regular elements of V is open and dense in V.

Proof. V has a real form V which is euclidean and simple. Let r be the rank of V and let (c 1 , c 2 , . . . , c r ) be a Jordan frame of V . Choose r distinct complex numbers λ 1 , λ 2 , . . . , λ r , and let x = λ 1 c 1 + • • • + λ r c r . Then x is semi-simple, and x is regular as the minimal polynomial of x is of degree r. The set of regular elements is an open Zariski subset of V (see [START_REF] Faraut | Analysis and geometry on complex homogeneous domains[END_REF]Proposition IV.1.1]). For x regular, the minimal polynomial is equal to the generic minimal polynomial. The set where the generic minimal polynomial m x has only simple roots is a Zariski open subset, as this is the set where the discriminant of m x (which is polynomial in x) vanishes. Hence the set of semi-simple regular elements is a Zariski open subset, which is non empty by the first part and hence dense.

A real Jordan algebra V is said to be complex, if there exists a linear isomorphism J of V such that J 2 = -Id, J(xy) = (Jx)y, ∀x, y ∈ V.

Proposition 11.7. Let V be a simple real Jordan algebra. Let V be its complexification. Then V is simple, unless V has a complex structure in which case V is isomorphic to V ⊕ V opp .

Proof. Assume that V is not simple. There exists a non trivial ideal J ⊂ V. Let σ be the conjugation with respect to V . Then σ(J) is also an ideal of V. Let H = J ∩ σ(J). Then H is an ideal of V, which moreover is σ-stable. So H = H ⊕ iH for some subspace H of V . Now H is an ideal of V . Hence H = 0 or H = V . But the second assumption leads to H = V and hence J = V, a contradiction. Hence J ∩ σ(J) = {0}. Along the same lines, one can prove that J + σ(J) = V. As a result, the Jordan algebra V splits as V = J ⊕ σ(J). Now consider the (real linear) map J ∋ x -→ x + σ(x) ∈ V, it is both injective and surjective, and hence an isomorphism. But as J and σ(J) are ideals of V, (x + σ(x))(y + σ(y)) = (xy + σ(xy)) and hence V is isomorphic to J. Moreover, as σ is C-conjugate linear, the complex structure of σ(J) is the opposite complex of that of J. Proposition 11.8. Let V be a real simple Jordan algebra. Let c be a primitive idempotent element of V . Then c, viewed as an element of V, is either a primitive idempotent or it can be decomposed as c = d + σ(d), where σ is the conjugation w.r.t. V , and d and σ(d) are orthogonal idempotents of V.

Proof. Assume c is not primitive in V.

Let V 1 = V 1 (c) and V 1 = V 1 (c). Now V 1
is a real semisimple Jordan algebra. As V has c as its unique idempotent, V 1 is simple. By [14, Section 6], V 1 is a Jordan field, and there exists a positive-definite bilinear β form on V 1 such that xy = β(x, c)y + β(c, y)xβ(x, y)c.

Then β(c, c) = 1. Let W = (Rc) ⊥ = {0}. Rewrite elements of V 1 as sc + v where s ∈ R and v ∈ W , so that the Jordan product of V 1 can be written as (sc + v)(tc + w) = (stβ(v, w))c + (sw + tv). Now let V 1 the complexification of V 1 which can be seen as

  type III and IV (non-split) , admits an analytic continuation as entire function of s in C. The case ε = + goes back to[START_REF] Barchini | Positivity of zeta distributions and small unitary representations[END_REF].

  Case (a) : d ≡ 0 (mod 4) or d ≡ 2 (mod 4) and r odd, Case (a') : d ≡ 2 (mod 4) and r even, Case (b) : r = 2 and d odd, Case (c) : r arbitrary and d = 1. Theorem 4.3. For s ∈ C, let γ(s) := (2π) -rs e( rs 4 )Γ Ω (s). Then the following functional equations hold. Case (a) : If d ≡ 0 (mod 4) or d ≡ 2 (mod 4) and r odd, then

Theorem 8 . 1 .

 81 For all (λ, ε) and (µ, η) in C × {±} and for all N in N * , the operatorB (N ) λ,µ is covariant with respect to (π λ,ε ⊗ π µ,η , π λ+µ+2N,εη ), i.e. B (N ) λ,µ • (π λ,ε (g) ⊗ π µ,η (g)) = π λ+µ+2N,εη (g) • B (N ) λ,µ , ∀g ∈ G.It is possible to give a slightly different presentation of the bi-differential operators B (N ) λ,µ . Below we will use the notation of Subsection 7.2. For N ≥ 1, define the operator F (N ) (λ,ε),(µ,η) by the following diagram

(9. 5 )

 5 Proof. It is enough to prove the identity on the open subset {(ξ, ζ), P (ξ), P (ζ) > 0}.

  Proof. Let p ∈ C[T ]. Then p(x) = l j=1 p(µ j )d j .

  p j (x) = d j , so that d j ∈ C[x]. Now let p ∈ C[T ]. Then p(x) = 0 ⇐⇒ p(µ j ) = 0.

V 1 =

 1 {(λc + ξ), λ ∈ C, ξ ∈ W} with Jordan product given by(λc + ξ)(µc + η) = λµβ(ξ, η) c + µξ + λη, where β is the C-bilinear extension of β to V 1 × V 1 . As (λc + ξ) 2 = λ 2β(ξ, ξ) c + 2λξ ,

  Explicit form of the operator E s,t . Recall from (5.1) the definition of the differential operator E s,t defined on V × V by E s,t = F -1 • D s,t • F. Proposition 9.1. In the V = R p,q case, the differential operator E s,t is given explicitly by

	Next, and for f a smooth function on {P (ξ) > 0, P (ζ) > 0}, we have P ∂ ∂ξ , ∂ ∂ζ P (ξ) s P (ζ) t f (ξ, ζ) +2s P (ξ) s-1 P (ζ) t n i=1 ξ i ∂f ∂ζ i + 2tP (ξ) s P (ζ) t-1 +P (ξ) s P (ζ) t P ∂ ∂ξ , ∂ζ f (ξ, ζ). ∂ = 4st P (ξ, ζ)P (ξ) s-1 P (ζ) t-1 f (ξ, ζ) P ∂ ∂y n i=1 ∂ ∂x i (x i -y i ) = nP ∂ ∂y -2P ∂ ∂x , ∂ ∂y + n j=1 (x j -y j ) ∂x j i=1 n ∂	ζ i P	∂ξ i ∂f ∂ ∂y	.
	Now the identity (9.5) is a matter of putting pieces together.
	9.3. E s,t = -P (x -y)P		∂ ∂x	P		∂ ∂y		+
		+4(s -1)	n j=1	(x j -y j )	∂ ∂x j	P		∂ ∂y	+ 4(t -1)	n j=1	(y j -x j )	∂ ∂y j	P	∂ ∂x
		-2(s -1)(2s -n)P	∂ ∂y	+ 8(s -1)(t -1)P	∂ ∂x	,	∂ ∂y	-2(t -1)(2t -n)P	∂ ∂x	.
	Proof. From (4.3) it follows that
		E s,t = -P		∂ ∂x	P	∂ ∂y	P (x -y)
			+4sP	∂ ∂y	n i=1	∂ ∂x i	(x i -y i ) + 4tP	∂ ∂x	n i=1	∂ ∂y i	(y i -x i )
			-2t(2t -2 + n)P	∂ ∂x	+ 8stP	∂ ∂x	,	∂ ∂y	-2s(2s -2 + n)P	∂ ∂y	.
	It remains to put the differential operator in normal form, multiplication preceding differen-
	tiation. First
	P	∂ ∂x	P	∂ ∂y	P (x -y)
		= P (x -y)P +2nP ∂ ∂x	∂ ∂x -8P P		∂ ∂y ∂ ∂x ,	+ 4 ∂y ∂ + 2nP n (x j -y j ) j=1 ∂ ∂y	s-1 ∂f ∂ξ i ∂ ∂x j P .	(ξ) + P (ξ) s P ∂ ∂y -∂ ∂y j	P	∂ ∂x ∂ξ ∂ f (ξ),

  26]. 10. Appendix A : Classification of simple real Jordan algebras

	Euclidean	Non-Euclidean	Non-Complex type	Complex type
	(split)	(split)	(non split)	(non-split)
	Type I	Type II	Type III

We use the notation, for any a ∈ R * , a s,+ = |a| s and a s,-= sign(a)|a| s .

In this case r = r and d = d.

Recall that 2n r is an integer.

Observe that this is a twofold covering of O(Q)/{± Id}, which already acts on X.

idempotents of V 1 are of the form ( 1 2 c + ξ) where β(ξ, ξ) = -1 4 . Now choose w ∈ W such that β(w, w) = 1 4 . Then let d = 1 2 c + iw and f = d = 1 2 ciw. Both d and f are idempotents, df = 0 and c = d + f , thus justifying the lemma. 11.3. Complex simple Jordan algebra viewed as a real Jordan algebra. Let V be a simple complex Jordan algebra, and let ∆ be its determinant. When viewed as a real Jordan algebra, it is still simple. Let det be its determinant.

Lemma 11.1. For any z ∈ V det(z) = ∆(z)∆(z).

(11.3)

Proof. Let x ∈ V and let p be its minimal polynomial over C. Then pp(x) = 0, and pp is a polynomial with real coefficients. As deg(p) ≤ r, where r is the rank of V, the degree of the minimal polynomial over R of an element is ≤ 2r. Conversely, consider a complete system of primitive idempotents (c 1 , c 2 , . . . , c r ) of V. Let x = i λ j c j with λ i ∈ C. Suppose that p is a polynomial with real coefficients such that p(x) = 0. Then, necessarily p(λ j ) = 0 for 1 ≤ j ≤ r. But as p is real valued, p(λ j ) = 0 for 1 ≤ j ≤ r. Assume that λ j , λ j are all distinct. Then p is a multiple of r j=1 (T -

. Hence the identity (11.3) is valid for the elements of the form considered. But clearly these elements form a dense set in V, hence the identity holds in general. 11.4. Non complex non-split simple real Jordan algebra. The algebra V is said to be split if a (hence any) primitive idempotent is primitive in the complexification V. Otherwise it is said to be non-split.

Let V be a non-split simple real Jordan algebra. Let (c 1 , c 2 , . . . , c ρ ) be a maximal set of orthogonal primitive idempotents. Then there exists d j , f j ∈ V(c j ) , such that

By Proposition, 11.8, c j decomposes as a sum c j = d j + f j , and so (d j , f j ) 1≤j≤ρ is a maximal family of orthogonal idempotents of V. Hence r = 2ρ.

Proposition 11.9. Let V be a real simple Jordan algebra which has no complex structure and which is non-split. Then det(x) ≥ 0 for every x ∈ V .

Proof. Let x ∈ V be a regular semi-simple element. There exists a maximal set (c 1 , c 2 , . . . , c ρ ) of orthogonal primitive idempotents such that x = ρ j=1 t j c j . For each j, 1 ≤ j ≤ ρ, let d j , f j ∈ V(c j ) , such that f j = σ(d j ), c j = d j + f j , 1 ≤ j ≤ ρ, so that x = ρ j=1 t j d j + ρ j=1 t j f j and hence det V (x) = det V (x) = ρ j=1 t 2 j ≥ 0. The conclusion follows as regular semi-simple elements are dense in V .