Holographic transform for tensor product of holomorphic discrete series

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Holographic transform for tensor product of holomorphic discrete series

Quentin Labriet

To cite this version:

Quentin Labriet. Holographic transform for tensor product of holomorphic discrete series. Interna- tional Journal of Mathematics, World Scientific Publishing, In press, �10.1142/S0129167X20500901�.

�hal-02555731v2�

Holographic transform for tensor product of holomorphic discrete series

Quentin Labriet

Abstract

We study holographic operators associated with Rankin-Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of the universal covering ofSL₂(R). Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.

1 Introduction

LetGbe a real reductive Lie group,G^′ a Lie subgroup ofGandπan irreducible unitary represen- tation ofGon a vector spaceV. The decomposition into irreducible representations (ρ, W) ofG^′of the restrictionπ|G^′ ofπtoG^′ is called the branching rule. Symmetry breaking operatorsare defined as elements of the space of linear continuous maps V →W that intertwineπ|G^′ andρ, denoted by HomG^′(π|G^′, ρ), for any given irreducible representation (ρ, W) ofG^′. Analogously, elements of the space HomG^′(ρ, π|G^′) are calledholographic operators (see for example (2.9)).

In the article [KP20], the authors investigate holographic operators in two different geometric settings : (G, G^′) = (SL·2(R)×SL·2(R),SL·2(R)) referred to as the diagonal case, and (G, G^′) = (SO0(2, n), SO0(2, n−1)) referred to as the conformal case. In both situations, they consider the restriction of holomorphic discrete series representations of G to G^′ and develop two explicit approaches to study corresponding holographic operators:

• The first method is based on the Laplace transform for tube domains, and leads to a new transform involving integration along a line segment in the complex upper half-plane Π (see 2.6) for the diagonal case.

• The second one uses the reproducing kernel technics for weighted Bergman spaces, and leads to the construction of a relative reproducing kernel in the conformal case (see Thm 3.10 in [KP20]).

It is known (see [DP07] for instance) that the symmetry breaking operators for the decomposition of the tensor products of holomorphic discrete series ofSL2(R) (what we call the diagonal case) are proportional to Rankin-Cohen bracketsRC_λ^λ′^′′′,λ^′′ (see (2.4)).

In this work, we extend the method of the relative reproducing kernel to the diagonal case and describe the corresponding holographic (RC_λ^λ′^′′′,λ^′′)^∗ operator as follows.

∗Laboratoire de Mathmatiques de Reims (LMR-UMR 9008 du CNRS), Universit de Reims Champagne-Ardenne Moulin de la Housse - BP 1039 51687 REIMS cedex 2, France. Email address: [email protected]

Theorem 1.1. Suppose λ^′, λ^′′, λ^′′′ >1 such that l = ¹₂(λ^′′′−λ^′−λ^′′) ∈N. Let w1, w2 ∈ Π, and g∈H_λ²′′′(Π). Then we have:

(RC_λ^λ′^′′′,λ^′′)^∗g(w1, w2) =C(λ^′, λ^′′) Z

Π

g(z)K_λ^λ′^′′′,λ^′′(z, w1, w2)dµ(z), (1.1) where K_λ^λ′^′′′,λ^′′(z, w1, w2) = (w2−w1)^{l w1}_2i^−¯z−(λ^′+l) w2−¯z

2i

−(λ^′′+l)

, dµ(x+iy) =y^{λ′′′−2} dxdy, and C(λ^′, λ^′′) =^(λ′−1)₂^l+12l+4^(λπ^′′2−1)l! ^l+1.

Notice that this kind of operators has already been considered by H. Rosengren in [Ros99]. We give two different proofs of this theorem, the first one is inspired by the proof of Theorem 3.10 in [KP20], and the second one, based on the Laplace transform, is new and can also be used in the conformal case.

Our second point (see Fact.4.3) explores a conceptual interpretation of the link between orthogo- nal polynomials and branching rules for restriction of discrete series representations. More precisely, Rankin-Cohen brackets can be expressed in terms of Jacobi polynomials as showed in [KP16b] (see Thm.8.1). This is due to the fact that the equivariance condition for symmetry breaking operators can be reduced, using the F-method (see [KP16a]), to the Jacobi ordinary differential equation for the symbols of such operators. In [KP20] p.15, the authors give yet another interpretation of the link between Rankin-Cohen brackets and Jacobi polynomials based on the fact that holographic transform can be expressed using the inversion of the usual Jacobi transform. In this work, we interpret this statement emphasising the fact that Jacobi polynomials form an orthogonal basis for the Hilbert spaceL²((−1,1),(1−v)^α(1 +v)^βdv) (see [KP20], section 5.1). For this, we use another realization of the holomorphic discrete series in which symmetry breaking operators are expressed in terms of the classical Jacobi transform.

Notation: We use the Pochammer symbol, defined forλ∈Candn∈Nby : (λ)n=λ(λ+ 1)· · ·(λ+n−1) = Γ(λ+n)

Γ(λ) . and set the condition

λ^′, λ^′′, λ^′′′>1 such that l:=1

2(λ^′′′−λ^′−λ^′′)∈N. (C1)

2 Setting

In this section, we describe two different models for the holomorphic discrete series representations of the universal covering group of the Lie groupSL2(R), denotedSL·2(R), and their tensor products.

2.1 Holomorphic model

We denote by Π the Poincar upper half-plane endowed with the hyperbolic metric, and by H_λ²(Π) the weighted Bergman space defined by:

H_λ²(Π) =O(Π)∩L² Π, y^λ−2dxdy .

It is known that H_λ²(Π) ={0} for λ≤1, so we suppose that λ > 1. In this case H_λ²(Π) admits a reproducing kernel, also called the Bergman kernel, given by (see, for instance, [FK94], Prop.XIII.1.2, p.261):

Kλ(z, w) = λ−1 4π

z−w¯ 2i

^−λ

. (2.1)

We set, for λ^′, λ^′′ > 1, H_λ²′,λ^′′(Π×Π) ≃ H_λ²′(Π) ˆ⊗H_λ²′′(Π), where ˆ⊗ denotes the Hilbert space completion of the tensor product. This space also admits a reproducing kernel:

Kλ^′,λ^′′(z, w) =Kλ^′(z1, w1)·Kλ^′′(z2, w2) =(λ^′−1)(λ^′′−1) (4π)²

z1−w¯1

2i

^−λ′z2−w¯2

2i

^−λ′′

. (2.2) Forλ∈N\{0; 1}, the holomorphic discrete series representations πλ ofSL2(R) can be realized onH_λ²(Π) by the following formula:

(πλ(g)f) (z) = (cz+d)^−λf

Åaz+b cz+d ã

, (2.3)

where g⁻¹ = Åa b

c d ã

∈ SL2(R), and f ∈ H_λ²(Π). This representation lifts to a unitary and irreducible representation of the universal covering groupSL·2(R) forλ >1, through an appropriate choice of the determination of the power function.

The outer product representationπλ^′⊠πλ^′′ of the direct product groupSL·2(R)×SL·2(R) acts on the spaceH_λ²′,λ^′′(Π×Π). This representation is also irreducible and unitary for λ^′, λ^′′>1.

2.2 Rankin-Cohen operators

Letλ^′, λ^′′, λ^′′′ verify (C1), and define a differential operator R^λ_λ^′′′′,λ^′′ fromO(Π×Π) toO(Π×Π) by

R_λ^λ′′′′,λ^′′(f)(w1, w2) =

l

X

j=0

(−1)^j(λ^′+l−j)j(λ^′′+j)l−j

j!(l−j)!

∂^lf

∂z₁^l−j∂z₂^j(w1, w2). (2.4) TheRankin-Cohen operator is a map fromO(Π×Π) to O(Π) defined by :

RC_λ^λ′^′′′,λ^′′ =Rest◦R^λ_λ^′′′′,λ^′′, (2.5) where Rest is the restriction to the diagonal of Π×Π. The Rankin-Cohen operatorRC_λ^λ′^′′′,λ^′′ is a symmetry breaking operator for the outer product representationπλ^′ ⊠πλ^′′, more precisely it gen- erates the space Hom‡SL2(R)(πλ^′⊠πλ^′′|SL‡2(R), πλ^′′′), for λ^′, λ^′′, λ^′′′ satisfiying (C1) (see [KP16b] Cor.

9.3 for the precise statement).

The goal of this paper is to study the holographic operators associated to the Rankin-Cohen operators. In order to do so, we compute the adjoint operators (RC_λ^λ′^′′′,λ^′′)^∗:H_λ²′′′(Π)→H_λ²′,λ^′′(Π×Π) which give, in the unitary case, the holographic operators.

2.3 Holographic operators for the diagonal case

Letλ^′, λ^′′, λ^′′′ verify (C1), and define the operators Ψ^λ_λ^′′′′,λ^′′ : H_λ²′′′(Π)→H_λ²′,λ^′′(Π×Π) by Ψ^λ_λ^′′′′,λ^′′(g)(w1, w2) = (w1−w2)^l

2^{λ′+λ′′+2l−1}l!

Z 1

−1

g(w(v))(1−v)^λ′+l−1(1 +v)^{λ′′+l−1}dv, (2.6) where w(v) = ¹₂((w2−w1)v+ (w2+w1)).

In [KP20], the authors prove that this is a holographic operator fromH_λ²′′′(Π) toH_λ²′,λ^′′(Π×Π). They also establish a Parseval-Plancherel type theorem for the corresponding Rankin-Cohen transorm and its associated holographic transform (see [KP20] Thm. 2.7). Finally, they show (Prop 2.23) that :

(RC_λ^λ′^′′′,λ^′′)^∗=CΨ^λ_λ^′′′′,λ^′′. (2.7) where C=₂2l+2^Γ(λπΓ(λ^{′+λ′′′}−1)Γ(λ^{+2l−1)′′}−1).

However, they also used another method to get a different expression for this map (see Lemma 3.1 below). This approach leads to a different type of integral transformations based on the relative reproducing kernelK_λ^λ′^′′′,λ^′′, defined forw2, w1, z∈Π by:

K_λ^λ′^′′′,λ^′′(z, w1, w2) = (w2−w1)^lw1−z¯ 2i

^−(λ′+l)w2−z¯ 2i

^{−(λ′′+l)}

. (2.8)

This relative reproducing kernel gives an explicit expression for the adjoint of the Rankin-Cohen operator as follows

Theorem 2.1. Setλ^′, λ^′′, λ^′′′ verify (C1). Letw1, w2∈Π, andg∈H_λ²′′′(Π). Then we have:

(RC_λ^λ′^′′′,λ^′′)^∗g(w1, w2) =C(λ^′, λ^′′) Z

Π

g(z)K_λ^λ′^′′′,λ^′′(z, w1, w2)dµ(z), (2.9) where C(λ^′, λ^′′) = ^(λ′−1)₂^l+12l+4^(λπ^′′2l!^−1)l+1.

We prove this theorem in the following sections (3.1 and 3.2) and make explicit the link with the operator Ψ^λ_λ^′′′′,λ^′′ introduced in (2.6). For this we need another model for the holomorphic discrete series representations ofSL·2(R).

2.4 L

-model of holomorphic discrete series

Forλ >1, the Laplace transform defined by:

Fg(z) = Z ∞

0

g(t)e^iztdt, (2.10)

is a one-to-one isometry (up to a constant) from L²_λ(R⁺) := L²(R⁺, t^1−λdt) to H_λ²(Π) (see for instance [FK94], Thm. XII.1.1). More precisely:

kFgk²_H²

λ(Π)=b(λ)kgk²_L²

λ(R⁺), for everyg∈L²_λ(R⁺), whereb(λ) = 2^2−λπΓ(λ−1).

Using this transform, we consider another realization for the holomorphic discrete series repre- sentations πλ ofSL·2(R) onL²_λ(R⁺), and call it theL²-model forπλ.

Forλ^′, λ^′′>1, we define

L²_λ′,λ^′′(R⁺×R⁺) :=L²_λ′(R⁺) ˆ⊗L²_λ′′(R⁺)≃L²(R⁺×R⁺, x^1−λ′y^1−λ′′dxdy).

We denoteF2=F ⊗ F the Laplace transform fromL²_λ′,λ^′′(R⁺×R⁺) toH_λ²′,λ^′′(Π×Π).

2.5 Holographic operators in the L

-model

Set λ^′, λ^′′, λ^′′′ satisfiying (C1). Using the Laplace transform (2.10), we define the analogue of the Rankin-Cohen operators in theL²-model by:

RCÿ_λ^λ′^′′′,λ^′′ :=F⁻¹◦RC_λ^λ′^′′′,λ^′′◦ F2. (2.11) It is a symmetry breaking operator for the SL·2(R)-action in theL²-model.

In [KP20], the authors prove that RCÿ_λ^λ′^′′′,λ^′′ is given by the following integral formula for F ∈ L²_λ′,λ^′′(R⁺×R⁺) (see [KP20], Prop. 2.13):

ÿRC_λ^λ′^′′′,λ^′′F(t) = t^l+1 2i^l

Z 1

−1

P_l^{λ′−1,λ′′−1}(v)F Åt

2(1−v),t 2(1 +v)

ã

dv, (2.12)

where P_l^{λ′−1,λ′′−1} denotes the Jacobi polynomials (see [KP20], section 5.1).

We define the following operator which associates to a functiong(t) defined onR⁺ a function of two variables Φ^λ_λ^′′′′,λ^′′g onR⁺×R⁺ :

Φ^λ_λ^′′′′,λ^′′g(x, y) := x^λ′−1y^λ′′−1

(x+y)^{λ′+λ′′+l−1}P_l^{λ′−1,λ′′−1} Åy−x

x+y ã

·g(x+y). (2.13) Once again, it is shown in [KP20], that Φ^λ_λ^′′′′,λ^′′ is a holographic operator between L²_λ′′′(R⁺) and L²_λ′,λ^′′(R⁺×R⁺). We summarize the framework in the two following commutative diagrams

L²_λ′,λ^′′(R⁺×R⁺)

◊

RC^λ′′′_{λ′,λ′′}

//

F2

L²_λ′′′(R⁺)

F

H_λ²′,λ^′′(Π×Π)

RC^λ_{λ′,λ′′}^′′′

//H²_λ′′′(Π)

Symmetry breaking operators in the holomorphic andL²-models L²_λ′,λ^′′(R⁺×R⁺)

F2

L²_λ′′′(R⁺)

F

Φ^λ_λ′^′′′_,λ′′

oo

H_λ²′,λ^′′(Π×Π) H_λ²′′′(Π)

Ψ^λ_{λ′,λ′′}^′′′

oo

Holographic operators in the holomorphic and L²-models

2.6 Inverse Laplace transform of the reproducing kernel

The following lemma gives the inverse Laplace transform for the Bergman kernelsKλ(·, w).

Lemma 2.2. Supposeλ, λ^′, λ^′′>1. The inverse Laplace transform of the reproducing kernel Kλ

is given by the following fomula:

F⁻¹Kλ(·, z)(t) = 2^λ−1

2πΓ(λ−1)t^λ−1e^−it¯z. (2.14) Consequently:

F₂⁻¹Kλ^′,λ^′′(·,(w1, w2))(x, y) = 2^{λ′+λ′′−2}

4π²Γ(λ^′−1)Γ(λ^′′−1)x^λ′−1y^λ′′−1e^{−i(x,y).(w1,w2)}. (2.15) Proof. Consider the functiong(t) =_{2πΓ(λ−1)}^2λ−1 t^λ−1e^−it¯z. Then, the usual Laplace transform gives:

Fg(z) = 2^λ−1 2πΓ(λ−1)

Z ∞

0

t^λ−1e^it(w−¯z)dt=λ−1 4π

Å 2i w−z¯

ãλ

=Kλ(w, z), which proves the first point. The second is obvious from the definitions.

3 Computation of the relative reproducing kernel

The goal of this section is to give two different proofs of Theorem 1.1. For this, we use Lemma 3.13 from [KP20]:

Lemma 3.1. LetDj(j= 1,2)be some complex manifolds, and Hj some Hilbert spaces of holomor- phic functions on Dj with reproducing kernels K^(j)(·,·). If R : H1 → H2 is a continous linear map, then :

1. RK⁽¹⁾(·, ζ)(τ^′) = (R^∗K⁽²⁾(·, τ^′))(ζ)pourζ∈D1, τ^′ ∈D2. 2. (R^∗g)(ζ) = (g, RK⁽¹⁾(·, ζ))H2 forg∈H2, ζ∈D1.

Thanks to this lemma, we only need to compute RC_λ^λ′^′′′,λ^′′(Kλ^′,λ^′′(·,(w1, w2))) in order to find an explicit formula for the holographic operator. We give two different ways for this computation.

First, we compute it directly and in a second time we use the Laplace transform to get our result.

3.1 First proof of Theorem 1.1

Our first proof of Theorem 1.1 reduces to a direct computation.

Lemma 3.2. For l∈Nandλ >1, we have:

∂^l

∂z^lKλ(z, w) =(−1)^l(λ−1)l+1

4π(2i)^l

z−w¯ 2i

^−(λ+l)

. (3.1)

This statement can be proved by induction onl. Then we are ready to prove Theorem 1.1.

Proof of Theorem 1.1.

R^λ_λ^′′′′,λ^′′Kλ^′,λ^′′(·,(w1, w2))(z1, z2)

=

l

X

j=0

(−1)^j(λ^′+l−j)j(λ^′′+j)l−j

j!(l−j)!

∂^l

∂z₁^l−j∂z₂^jKλ^′,λ^′′(·,(w1, w2))(z1, z2)

= (−1)^l (4π)²l!(2i)^l

Åz1−w1

2i

ã−(λ^′+l)Å z2−w2

2i

ã−(λ^′′+l)

×

l

X

j=0

(−1)^j(λ^′+l−j)j(λ^′′+j)l−j(λ^′−1)l−j+1(λ^′′−1)j+1

Çl j

å Åz2−w2

2i

ãl−jÅz1−w1

2i ãj

= (−1)^l(λ^′−1)l+1(λ^′′−1)l+1

(4π)²l!(2i)^2l

Åz1−w1

2i

ã−(λ^′+l)Åz2−w2

2i

ã−(λ^′′+l)

×

l

X

j=0

Çl j

å

(w1−z1)^j(z2−w2)^l−j

= (λ^′−1)l+1(λ^′′−1)l+1

(4π)²l!2^2l

Åz1−w1

2i

ã−(λ^′+l)Åz2−w2

2i

ã−(λ^′′+l)

(z1−z2+w2−w1)^l. Finally, we get by restriction to the diagonalz1=z2=z:

RC_λ^λ′^′′′,λ^′′Kλ^′,λ^′′(·,(w1, w2))(z) =Rest|z1=z2=z◦R^λ_λ^′′′′,λ^′′(Kλ^′,λ^′′(·,(w1, w2))) (z)

=(λ^′−1)l+1(λ^′′−1)l+1

(4π)²l!2^2l

Åw1−z 2i

ã−(λ^′+l)Å w2−z

2i

ã−(λ^′′+l)

(w2−w1)^l. Thus Lemma 3.1 implies the statement.

3.2 Second proof of Theorem 1.1

Here we give a second proof of Theorem 1.1 using the Laplace transform (2.10).

Proposition 3.3. Setλ^′, λ^′′, λ^′′′ satisfying (C1).

RC_λ^λ′^′′′,λ^′′Kλ^′,λ^′′(·,(w1, w2))(z) = (3.2) C^′(w2−w1)^l

Z ∞

0

F⁻¹Kλ^′′′(·, z)(t)1F1(λ^′+l, λ^′+λ^′′+ 2l;−i(w2−w1)t)e^itw2 dt, where C^′ =^Γ(λ₂^′2l+2^+λ′′πl!Γ(λ^{+2l−1)B(λ′}−1)Γ(λ^{′+l,λ′′}−1)^′′+l) and1F1 is the Kummer function (see Appendix 5).

Proof. Lemmas 2.2 and 5.1, and formula (2.12) imply:

ÿRC_λ^λ′^′′′,λ^′′◦ F₂⁻¹(Kλ^′,λ^′′(·,(w1, w2))) (t)

= t^{λ′+λ′′+l−1} 2³i^lπ²Γ(λ^′−1)Γ(λ^′′−1)

Z 1

−1

P_l^{λ′−1,λ′′−1}(v)e^{−itw1 +2w2+vw2−w2 1}(1−v)^λ′−1(1 +v)^λ′′−1dv

= 2^{λ′+λ′′−4}B(λ^′+l, λ^′′+l)

π²Γ(λ^′−1)Γ(λ^′′−1)l! (w2−w1)^lt^{λ′′′−1} 1F1(λ^′+l, λ^′+λ^′′+ 2l;i(w2−w1)t)e^−itw2.

As RC_λ^λ′^′′′,λ^′′ =F ◦ÿRC_λ^λ′^′′′,λ^′′◦ F₂⁻¹ (see (2.10)), it gives:

RC_λ^λ′^′′′,λ^′′Kλ^′,λ^′′(·,(w1, w2))(z)

=C^′(w2−w1)^l Z

R⁺

2^{λ′′′−1}

2πΓ(λ^′′′−1)t^{λ′′′−1}e^itz 1F1(λ^′+l, λ^′+λ^′′+ 2l;i(w1−w2)t)e^−itw2 dt.

Lemma 2.2 gives _2πΓ(λ^{2λ′′′ −}′′′−1)¹ t^{λ′′′−1}e^−it¯z=F⁻¹Kλ^′′′(·, z)(t), which ends our proof.

An alternative proof of Theorem 1.1 is based on the properties of Kummer special functions as follows.

Second proof of Theorem 1.1. First, remark that, for everyn∈N, we have:

Å ∂

∂w ãn

Kλ^′′′(w, z) = (−1)ⁿ(λ^′′′−1)(λ^′′′)n(2i)^λ′′′

4π (w−z)¯^{−λ′′′−n}. Then, we use the power series expansion for Kummer functions (5.1) to get:

Z ∞

0

F⁻¹(Kλ^′′′(·, z)) (t)e^itw2 1F1(λ^′+l, λ^′+λ^′′+ 2l;−i(w2−w1)t)dt

=

∞

X

n=0

(λ^′+l)n

(λ^′+λ^′′+ 2l)n

Z ∞

0

F⁻¹(Kλ^′′′(·, z)(t))e^itw2(−i(w2−w1)t)ⁿe^itw2 dt

=

∞

X

n=0

(λ^′+l)n

(λ^′+λ^′′+ 2l)n

(w2−w1)ⁿ Å ∂

∂w ãn

Kλ^′′′(w, z) _w=w

2

= λ^′′′−1 4π

∞

X

n=0

(λ^′+l)n(−1)ⁿ(2i)^λ′′′(w2−z)¯^{−λ′′′−n}(w2−w1)ⁿ

= λ^′′′−1 4π

w2−z¯ 2i

^{−λ′′′}X^∞

n=0

(λ^′+l)n(−1)ⁿ

Åw2−w1

w2−¯z ãn

= λ^′′′−1 4π

w2−z¯ 2i

^{−λ′′′}Å

1−w2−w1

w2−z¯ ã−λ^′−l

= λ^′′′−1 4π

w2−z¯ 2i

^{−λ′′−l}w1−z¯ 2i

^−λ′−l . Notice that the result is valid for

w2−w1

w2−¯z

<1, and use the analytic continuation to extend it for arbitraryw1, w2, z∈Π. Proposition 3.3, implies the result. Finally, for the constantC(λ^′, λ^′′), we have:

C(λ^′, λ^′′) =C^′(λ^′′′−1)

4πl! =(λ^′′′−1)Γ(λ^′′′−1)Γ(λ^′+l)Γ(λ^′′+l) 2^2l+4π²l!Γ(λ^′−1)Γ(λ^′′−1)Γ(λ^′′′) .

3.3 Link with the operator Ψ

We give another characterization of the operators Ψ^λ_λ^′′′′,λ^′′ (see (2.6)) as follows

Proposition 3.4. Supposeλ^′, λ^′′, λ^′′′ verify (C1). Then we have the following identity:

Ψ^λ_λ^′′′′,λ^′′(Kλ^′′′(·, z))(w1, w2) =(−1)^l(λ^′′′−1)B(λ^′+l, λ^′′+l)

4πl! K_λ^λ′^′′′,λ^′′(z, w1, w2). (3.3) where B(x, y) denotes the usual Euler beta function, and K_λ^λ′^′′′,λ^′′ is the relative reproducing kernel (2.8) given by

K_λ^λ′^′′′,λ^′′(z, w1, w2) = (w2−w1)^lw1−z¯ 2i

^−(λ′+l)w2−z¯ 2i

^{−(λ′′+l)} . Proof. Similar argument as in the proof of Proposition 3.3 gives:

Ψ^λ_λ^′′′′,λ^′′(Kλ^′′′(·, z))(w1, w2)

= (w1−w2)^l 2^{λ′′′−1}l!

Z 1

−1

ÅZ ∞ 0

F⁻¹Kλ^′′′(·, z)(t)e^itw(v) dt ã

(1−v)^λ′+l−1(1 +v)^{λ′′+l−1}dv

= 2^2l+2π(−1)^lΓ(λ^′−1)Γ(λ^′′−1)

Γ(λ^′′′−1) RC_λ^λ′^′′′,λ^′′Kλ^′,λ^′′(·,(w1, w2))(z).

Remark: This proposition gives another proof of the link between Ψ^λ_λ^′′′′,λ^′′ and (RC_λ^λ′^′′′,λ^′′)^∗ (see formula (2.7)). Indeed, for everyg∈Hλ^′′′(Π):

Ψ^λ_λ^′′′′,λ^′′(g)(w1, w2)

= (w1−w2)^l 2^{λ′+λ′′+2l−1}l!

Z 1

−1

g(w(v))(1−v)^λ′+l−1(1 +v)^{λ′′+l−1} dv

= (w1−w2)^l 2^{λ′+λ′′+2l−1}l!

Z 1

−1

Z

Π

g(z)Kλ^′′′(z, w(v))y^{λ′′′−2}(1−v)^λ′+l−1(1 +v)^{λ′′+l−1} dxdydv

= Z

Π

g(z) ÇZ 1

−1

(w1−w2)^l

2^{λ′+λ′′+2l−1}l!Kλ^′′′(w(v), z)(1−v)^λ′+l−1(1 +v)^{λ′′+l−1}dv å

y^{λ′′′−2} dxdy

= Z

Π

g(z)Ψ^λ_λ^′′′′,λ^′′(Kλ^′′′(·, z))(w1, w2)y^{λ′′′−2}dxdy.

4 Geometrical interpretation

In this section, we give a geometrical interpretation of the diffeomorphism ι betweenR⁺×(−1; 1) and R⁺×R⁺ introduced in [KP20] (equation (2.21), p.17), and describe yet another model for the tensor product πλ^′⊗πλ^′′.

4.1 Cone stratification

Letσbe an involutive automorphism of a connected semi-simple Lie groupG, which commutes with the Cartan involutionθ ofG. We use the same letters σandθ to denote their differentials. Define a θ-stable subgroup of Gby

G^σ={g∈G|σ(g) =g}. (4.1)

ConsiderKa maximal compact subgroup ofGandg=k+pa Cartan decomposition ofg= Lie(G).

The spacep isσ-stable, so we have the following diffeomorphism for the symmetric spaceG/K:

G/K≃p=p^σ⊕p^−σ, (4.2)

where p^±σ = {X ∈ p | σ(X) =±X}. This diffeomorphism is explicit using the exponential map (see [Hel01], Thm.1.1, p.252):

X+Y ∈p^σ⊕p^−σ7→exp(X+Y)K∈G/K. (4.3) The automorphismσθ is also involutive, so one checks that:

p^σ≃G^σ/K^σ andp^−σ≃G^σθ/K^σθ. (4.4) Finally, (4.2) and (4.4) gives the following diffeomorphism forG/K:

G/K≃G^σ/K^σ×G^σθ/K^σθ, (4.5)

where the diffeomorphism for the right-hand side is explicitly given by

X+Y ∈p^σ⊕p^−σ7→(exp(X)K^σ,exp(Y)K^σθ)∈G^σ/K^σ×G^σθ/K^σθ. (4.6) Example 4.1. Consider the case where G=

ßÅa 0 0 a⁻¹

ã

;a∈R^∗

™

×

ßÅb 0 0 b⁻¹

ã

;b∈R^∗

™ (which is the structure group of the Jordan algebra R×R), andσ(g1, g2) = (g2, g1)withg= (g1, g2)∈G.

Then, G^σ/K^σ≃R⁺ is diagonaly embeded inG/K≃R⁺×R⁺, andG^σθ/K^σθ≃(−1; 1).

The following diffeomorphism ιfrom R⁺×(−1; 1)toR⁺×R⁺: ι: (t, v)7→ι(t, v)≡(t

2(1−v),t

2(1 +v)) (4.7)

is then an explicit realization of the decomposition (4.5).

This diffeomorphism corresponds to the diagonal embedding ofR⁺ intoR⁺×R⁺, and describes the stratification of the cone R⁺×R⁺ by the segments orthogonal to this diagonal. In his recent preprint [Cle20], J-L. Clerc gives a generalization of this construction for the diagonal embedding of a symmetric cone Ω into the direct product Ω×Ω (see [Cle20]) for an arbitrary Euclidean Jordan algebra.

Letα, β >1, anddµα,β(v) := (1−v)^α−1(1 +v)^β−1 dvbe a measure on the segment (−1,1). We define the Hilbert spaces

L²_α,β(R⁺×(−1; 1)) :=L²(R⁺×(−1; 1), t^α+β−1 dtdµα,β(v)).

For a functionf defined onR⁺×R⁺, we define a function Tι(f) onR⁺×(−1,1) by Tι(f)(t, v) =

Åt 2

ã2−λ^′−λ^′′

(1−v)^1−λ′(1 +v)^1−λ′′·f◦ι(t, v). (4.8) One checks that it is a one-to-one isometry from L²(R⁺×R⁺, x^1−λ′y^1−λ′′ dxdy) to L²_λ′,λ^′′(R⁺× (−1; 1)). The inverse map is given by the following formula:

T_ι⁻¹(h)(x, y) =x^λ′−1y^λ′′−1·h◦ι⁻¹(x, y). (4.9)

We use this operator in order to define the stratifiedL²-model by transfering the restriction of the outer product of holomorphic discrete series representations of SL·2(R) on the space L²_λ′,λ^′′(R⁺× (−1; 1)). The groupSL·2(R) acts onL²_λ′,λ^′′(R⁺×(−1; 1)) by the operators:

Tθ◦πÿλ^′,λ^′′(g)◦T_θ⁻¹, (4.10) where g∈SL·2(R) andπÿλ^′,λ^′′(g) =F₂⁻¹◦πλ^′⊗πˆ λ^′′(g, g)◦ F2.

Finaly, we introduce the map Θ for any function defined onR⁺: Θ(h)(t, v) =t^{−(λ′+λ′′+l−1)}P_l^{λ′−1,λ′′−1}(v)

kP_l^{λ′−1,λ′′−1}kh(t). (4.11) This is a one-to-one isometry from L²_λ′′′(R⁺) to the Hilbert space

V_l^{λ′,λ′′}:=L²Ä

R⁺, t^{λ′+λ′′−1} dtä

⊗ˆ C·P_l^{λ′−1,λ′′−1}(v), (4.12) whose inverse is given by:

Θ⁻¹(f×P_l^{λ′−1,λ′′−1})(t) =kP_l^{λ′−1,λ′′−1}kt^{λ′+λ′′+l−1}f(t), (4.13) where kP_l^{λ′−1,λ′′−1}k=₂α+β+1Γ(l+α+1)Γ(l+β+1)

l!(2l+α+β+1)Γ(l+α+β+1)

¹2

is the norm of Jacobi polynomials in the Hilbert spaceL²((−1; 1), dµλ^′,λ^′′(v)).

Analogously to (4.10), we use this map to transfer the holomorphic discrete series representation ofSL·2(R) and define, for everyg∈SL·2(R), the following operator onV_l^{λ′,λ′′}:

Θ◦π◊λ^′′′(g)◦Θ⁻¹. (4.14)

where ◊πλ^′′′(g) =F⁻¹◦πλ^′′′(g)◦ F.

4.2 Symmetry breaking and holographic operator

According to the orthogonality of Jacobi polynomials, we have the following isomorphisms of Hilbert spaces:

L²_λ′,λ^′′(R⁺×(−1; 1))

≃ L²Ä

R⁺, t^{λ′+λ′′−1}dtä

⊗Lˆ ²((−1; 1), dµλ^′,λ^′′(v))

≃ X

l≥0

⊕L²Ä

R⁺, t^{λ′+λ′′−1} dtä

⊗ˆ C·P_l^{λ′−1,λ′′−1}(v) (4.15)

≃ X

l≥0

⊕L²Ä

R⁺, t^{λ′+λ′′−1} dtä

⊗ˆ V_l^{λ′,λ′′}.

where P⊕ stands for an orthogonal Hilbert sum. The last isomorphism is due to the fact that the Jacobi polynomials form a Hilbert basis forL²((−1; 1), dµλ^′,λ^′′(v)) if λ^′, λ^′′>1.

This allows us to consider the orthogonal projection J_l^{λ′,λ′′} from L²_λ′,λ^′′(R⁺×(−1; 1)) on the Hilbert subspaceV_l^{λ′,λ′′} which corresponds to the usual Jacobi transform:

J_l^{λ′,λ′′}(h)(t, v) = P_l^{λ′−1,λ′′−1}(v) kP_l^{λ′−1,λ′′−1}k²

Z 1

−1

h(t, u)P_l^{λ′−1,λ′′−1}(u)dµλ^′,λ^′′(u). (4.16) We summarize the situation in the following Proposition.

Proposition 4.2. Letλ^′, λ^′′, λ^′′′ verify (C1). The following diagram is commutative

L²_λ′,λ^′′(R⁺×R⁺)

c1RC◊_{λ′,λ′′}^λ′′′

//

Tι

L²_λ′′′(R⁺)

Θ

c2Φ^λ_{λ′,λ′′}^′′′

//L²_λ′,λ^′′(R⁺×R⁺)

L²_λ′,λ^′′(R⁺×(−1; 1))

J_l^{λ′,λ′′}

//V^{λ′,λ′′}_l ( ^T

−1 ι

55

❦

❦❦

❦

❦❦

❦❦

❦❦

❦

❦❦

❦❦

❦

where c1= ^{2λ′+λ′′−3il}

kP_l^{λ′−1,λ′′ −1}k andc2= ¹

kP_l^{λ′ −1,λ′′ −1}k.

Proof. Letf ∈L²_λ′,λ^′′(R⁺×R⁺). On one hand side we have:

Φ^λ_λ^′′′′,λ^′′◦RCÿ_λ^λ′^′′′,λ^′′f(x, y)

= x^λ′−1y^λ′′−1

2i^l(x+y)^{λ′+λ′′−2}P_l^{λ′−1,λ′′−1} Åy−x

x+y ãZ 1

−1

P_l^{λ′−1,λ′′−1}(u)F(ι(x+y), u)du.

On the other hand side we have:

T_ι⁻¹◦ J_l^{λ′,λ′′}◦Tι(f)(x, y)

=T_ι⁻¹ Ç

(t, v)7→

Åt 2

ã2−λ^′−λ^′′

P_l^{λ′−1,λ′′−1}(v) Z 1

−1

f(ι(t, u))P_l^{λ′−1,λ′′−1}(u)du å

= 2^{λ′+λ′′−2} kP_l^{λ′−1,λ′′−1}k²

x^λ′−1y^λ′′−1

(x+y)^{λ′+λ′′−2}P_l^{λ′−1,λ′′−1} Åy−x

x+y ãZ 1

−1

P_l^{λ′−1,λ′′−1}(u)F(ι(x+y), u)du.

A direct computation shows thatT_ι⁻¹◦Θ =c2Φ^λ_λ^′′′′,λ^′′.

The fact thatRCÿ_λ^λ′^′′′,λ^′′and Φ^λ_λ^′′′′,λ^′′are intertwinning operators (see [KP20], Thm.2.11), and Propo- sition 4.2 lead to the following:

Proposition 4.3. The orthogonal projectionJ_l^{λ′,λ′′} is a symmetry breaking operator for theSL·2(R) action (4.10) in the stratified L²-model and the canonical injection corresponds to the associated holographic operator.

In [KP20], the authors explore the link between the symmetry breaking operator and Jacobi transform in this geometric setting (see Rmk.2.15). Proposition 4.3 directly relates the L²-space associated with Jacobi polynomials to the construction of symmetry breaking operators for the