Ahmed Zouari
email: ahmedzouarri@gmail.com

Valérie Briard-Bion

Frederic Gaucheron

Pierre Schuck

Claire Gaiani

Mehdi Triki

Hamadi Attia

Mohamed Ali Ayadi

Frédéric Gaucheron

Effect of pH on the physicochemical characteristics and the surface chemical composition of camel and bovine whey protein's powders

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Whey proteins are important hydrocolloids that offer interesting techno-functional properties in food formulations such as emulsifying, foaming, and gelling properties. Whey proteins are often obtained by membrane filtration, chemical separation or enzymatic process. Most of them are spray-dried to produce stable whey protein powders. The spray drying is a rapid dehydration technique that allows an effective preservation of the biological products, such as proteins (Roos, 2002a;[START_REF] Schuck | Relative humidity of outlet air: The key parameter to optimize\nmoisture content and water activity of dairy powders[END_REF].

The physical stability of whey powders is one of the most challenging step during storage. Indeed, these powders contain residual lactose which is in a thermodynamically instable amorphous state (i.e. nonequilibrium glassy state) (Roos, 2002a;[START_REF] Schuck | Water activity and glass transition in dairy ingredients[END_REF]. The exposure of whey powders to high relative humidity enhanced water adsorption from the ambient storage environment. This can induce glass transition mechanisms and amorphous lactose crystallization [START_REF] Jouppila | Glass transitions and crystallization in milk powders[END_REF][START_REF] Kelly | Influence of protein concentration on surface composition and physico-chemical properties of spray-dried milk protein concentrate powders[END_REF]Roos, 2002b). Since lactose crystallization is followed by the release of the adsorbed water, the sorption isotherms are important tools for characterizing changes in whey powders during and after the glass transition [START_REF] Foster | The prediction of moisture sorption isotherms for dairy powders[END_REF][START_REF] Jouppila | Glass transitions and crystallization in milk powders[END_REF][START_REF] Shrestha | Effect of protein concentration on the surface composition, water sorption and glass transition temperature of spray-dried skim milk powders[END_REF]. Several other studies have also considered the glass transition temperature (T g ) as a useful criterion to prevent physicochemical, nutritional and techno-functional deteriorations (e.g. stickiness, caking, non-enzymatic browning) of whey powders during storage [START_REF] Foster | The prediction of moisture sorption isotherms for dairy powders[END_REF][START_REF] Jouppila | Glass transitions and crystallization in milk powders[END_REF]. It was extensively demonstrated that the glass transition temperature and the sorption adsorption isotherm depend on the surface composition of whey powders, which was extensively studied using the X-ray photoelectron spectroscopy (XPS) [START_REF] Gaiani | Milk proteins differentiation and competitive adsorption during spray-drying[END_REF]. The XPS allows the measurement of the distribution of relative atomic elemental composition (carbon, oxygen, and nitrogen) at the studied surface (a layer of 5 to 10 nm). In case of whey powders, these elements are used to quantify the surface composition in terms of proteins, fats, and lactose [START_REF] Fäldt | The surface composition of spray-dried protein-lactose powders[END_REF]. Actually, two mechanisms are known to describe the transport of these previous compounds to the surface of whey powder. These mechanisms include the air/liquid interface interaction (through active substances) [START_REF] Fäldt | The surface composition of spray-dried protein-lactose powders[END_REF] and the solid/ solute segregation system [START_REF] Kim | On the mechanisms of surface formation and the surface compositions of industrial milk powders[END_REF]. In both systems, proteins start to accumulate preferentially at the surface (in the first 5 nm) of the whey powder particles [START_REF] Kim | On the mechanisms of surface formation and the surface compositions of industrial milk powders[END_REF][START_REF] Shrestha | Effect of protein concentration on the surface composition, water sorption and glass transition temperature of spray-dried skim milk powders[END_REF].

On the other hand, it was acknowledged that during drying the https://doi.org/10.1016/j.foodchem.2020.127514 Received 17 February 2020; Received in revised form 4 July 2020; Accepted 5 July 2020 temperature of the sprayed droplets increased to reach asymptotically the air outlet drying temperature [START_REF] Woo | Spray drying for food powder production[END_REF]. Several researchers demonstrated that the thermal denaturation of whey proteins depends on time, temperature, free thiol group (SH), and heat treatment intensity [START_REF] Manzo | Thermal markers arising from changes in the protein component of milk[END_REF]. The most denatured protein in bovine milk whey is the β-lactoglobulin [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF].

The particularity of camel milk whey is the lack of β-Lactoglobulin and the overexpression of α-lactalbumin [START_REF] Lajnaf | The effect of pH and heat treatments on the foaming properties of purified α-lactalbumin from camel milk[END_REF]. Recently, [START_REF] Felfoul | Proteomic profiling of camel and cow milk proteins under heat treatment[END_REF] indicated that the thermal degradation of camel milk proteins results in the denaturation of the camel serum albumin (CSA) followed by the peptidoglycan recognition protein (PGRP) and the α-lactalbumin. [START_REF] Lajnaf | The foaming properties of camel and bovine whey: The impact of pH and heat treatment[END_REF] demonstrated that camel whey proteins exhibited higher foaming properties as compared to bovine whey proteins', which can be an important foaming agent in food formulations. In addition, as far as the authors are aware of, the influence of pH on the denaturation extent, the surface chemical composition, the water sorption isotherm, and the glass transition temperature of camel whey protein's powders, is not yet studied. Thus, this study aimed at producing and characterizing camel whey's powders at neutral (=6.7) and acidic (=4.3) pH. As a first approach, the denaturation extent of spray drying on camel whey protein was analyzed using the LC-MS. Then, the influence of the pH on the sorption isotherms and the glass transition temperature was evaluated. The surface composition of the produced powders was investigated using the X-ray Photoelectron Spectroscopy (XPS). To create a well-based comparative study, we also assessed the bovine whey's powders (pH = 6.7 and 4.6 for neutral and acidic whey protein powders, respectively) produced under the same drying conditions and analyzed following the same evaluation techniques.

Materials and methods

Whey proteins preparation

Fresh skimmed camel (Camelus dromedarius) and bovine (Bos Taurus) milks were acidified with HCL 1 M to produce acidic whey proteins (isoelectric pH = 4.6 and 4.3 for skim bovine milk and skim camel milk, respectively), followed by centrifugation at 4000g for 15 min at 20 °C. Acidic camel and bovine milk's whey proteins were then collected and stored in sterile plastic bottles at -20 °C.

Sweet camel and bovine whey proteins (pH = 6.7) were obtained after a rennet enzyme coagulation (M. miehei, strength = 1:10,000, Laboratories Arrazi, Parachimic, Sfax, Tunisia) at 40 °C as described by [START_REF] Felfoul | A laboratory investigation of cow and camel whey proteins deposition under different heat treatments[END_REF]

Powder production

Acidic and sweet whey samples were spray-dried using a Bücchi mini spray dryer B-290 (Büchi Labortechnik AG, Flawil, Switzerland). The inlet and outlet drying temperatures were set up to 200 ± 2 °C and 80 ± 2 °C, respectively. During all experiments, the absolute humidity of air was equal to 5 g of water per kg of dry air; and the average droplets residence time in the drying chamber was equal to 1 s. The obtained powders are: ACWP: acidic camel whey powder, ABWP: acidic bovine whey powder, SCWP: sweet camel whey powder, and SBWP: sweet bovine whey powder. These powders were immediately stored at 4 °C in sterilized glass vials. The Size distribution of the produced powder was performed as detailed by [START_REF] Zouari | Changes in physical and biochemical properties of spray dried camel and bovine milk powders[END_REF]. The d 50 (diameter of 50% of the particles) was chosen as a size distribution indicator as recommended by [START_REF] Nikolova | Is it possible to modulate the structure of skim milk particle through drying process and parameters[END_REF].

Bulk and surface composition

The bulk composition of acidic and sweet camel and bovine whey powders was performed according to the AOAC [START_REF] Horwitz | Official Methods of Analysis of AOAC International[END_REF]. The surface composition of acidic and sweet camel and bovine whey powders were estimated using X-ray photoelectron spectroscopy. The XPS analyses were conducted using a Kratos Axis Ultra spectrometer (Kratos Analytical, Manchester, UK) working with a monochromatic AlKα X-ray source (E = 1486.6 eV; P = 90 W) as described by [START_REF] Gaiani | Milk proteins differentiation and competitive adsorption during spray-drying[END_REF]. The analyzed powder area was about 300 × 600 µm with a depth of 6 to 10 nm. The relative atomic concentration of carbon, oxygen and nitrogen in the powder surface was quantified and integrated in an equation system proposed by [START_REF] Fäldt | The surface composition of spray-dried protein-lactose powders[END_REF]. This equation system allows the quantification of the powder surface content in relation to lactose, proteins and fat.

LC-MS analysis

Before proceeding to the LC-MS analysis, the acidic and sweet whey powders were reconstituted in MiliQ water to reach the same solid content of unprocessed whey (6.5 w/w for acidic whey and 7.2 w/w for sweet whey). The mixtures were stirred (580 rpm) for 30 min at 25 °C. The reconstituted milks were stored at 4 °C overnight and then warmed up to the room temperature (25 °C). The total solids and the protein content of the reconstituted powders were then determined according to the AOAC [START_REF] Horwitz | Official Methods of Analysis of AOAC International[END_REF]. Afterwards, 100 µL of unprocessed whey and their corresponding reconstituted powders were diluted in 450 µL of buffer solution (Urea 4 M/Tris 25 mM pH8). Diluted samples were filtered through Millipore® Millex® filters HV PVDF membrane (0.45 µm Pore Size) and were subjected to a second dilution (½) in 5% of TFA solution. Then, 40 µL of each sample were injected in the LC-MS system.

The LC-MS analysis was performed using an Agilent-1100 Rp-HPLC system (Agilent Technologies, Waldbronn, Germany) coupled with a Q-Exactive™ Hybrid Quadrupole-Orbitrap™ (Thermo-Fisher scientific, Waltham, MA, USA) mass spectrometer [START_REF] Felfoul | Proteomic profiling of camel and cow milk proteins under heat treatment[END_REF]). The HPLC system was equipped with a column C4 (VYDAC, reference 214TP5215, length 150 mm, inner diameter 2.1 mm, pore size 300 Å, Grace™, Fisher-scientific, USA). A gradient from 37% to 90% of a solvent (acetonitrile: 80% (v/v) and TFA: 0.1% (v/v) in deionized water) was applied during 50 min for the elution of the proteins at a flow rate of 0.25 mL min -1 . Eluted proteins were then, electro-sprayed in a mass spectrometer Q-Exactive. The mass spectra acquisition speed was set up to a resolving power of 140,000 and an m/z ranging from 800 to 3000. Mass spectra were then analyzed using BioPharma Finder™ software (version 2.0.66.12, Thermo-Fisher scientific, Waltham, MA, USA). The Uniprot database (http://www.uniprot.org/) was used to identify the camel and bovine whey proteins (Camelus dromdarus, Taxon identifier: 9838 and Bos taurus, Taxon identifier 9913). The quantification of camel and bovine whey protein fractions was estimated based on the integrated peak areas of HPLC chromatographs (see Supplementary data, Fig. S1) and protein content of unprocessed whey and their corresponding reconstituted powders. The denaturation extent was evaluated as follow: = × Denaturation extent 100

Concentration before drying -Concentration after drying Concentration before drying

Dynamic vapor sorption

Dynamic vapor sorption was used to assess the water sorption isotherms at 25 °C. About 20 mg of each produced powder were loaded into a clean and a dry aluminum pan. The filled pans were subjected to water adsorption cycle using surface measurement DVS advantage (Surface Measurement Systems Ltd., London, UK) equipped with a Cahn microbalance. The DVS program was setup as described by [START_REF] Carpin | How does particle size influence caking in lactose powder[END_REF]. The lactose crystallization characteristics [water activity and necessary water for lactose crystallization (X c )] were identified.

The experimental data (up to 0.40 of a w ) were fitted to the GAB (Guggenheim, Andersen and de Boer) model using the OriginPro 8 software (OriginLab Corporation, Northampton, USA). The GAB model constants X m , K, and C were, then, calculated and used to interpret the obtained experimental data following the equation below:

= - - + X X C k a k a k a C k a m. . . w (1 . w). (1 . w . . w)
where, X m is the monolayer moisture capacity (g 100 g -1 ); the C constant is the water binding energy by the first layer; K constant describes the multilayer sorption ability; and a w is the studied water activity. The K constant was not considered in our study, since it described the multilayer sorption binding energy, up to 0.95 of a w .

Modulated dynamic scanning calorimetry (MDSC)

The MDSC was used to evaluate the glass transition temperature (T g ) of the produced whey powders at three a w (0.13, 0.23, and 0.33). Six milligrams of each samples were sealed in a hermitic aluminum pan and placed in a differential scanning calorimetry (DSC) Q1000 (TA Instruments, Eschborn, Germany). The 'heat mode only' (-30 to +200 °C) was used to assess the MDSC analysis against an empty aluminum pan as presented by [START_REF] Syll | Rehydration behaviour and ageing of dairy powders assessed by calorimetric measurements[END_REF]. The T g was identified as the midpoint temperature when change in heat capacity in the reversed MDSC profile was observed.

Statistical analysis

The whole experiment was carried out three times for statistical accuracy. All analyses and measurements in this work were conducted in triplicate. The statistical differences (p < 0.05) were examined using SPSS 19 software (IBM SPSS statistics, Version 19, USA) following descriptive tests including the student's t-test, the one-way ANOVA, and the Tukey post-hoc test with a confidence level of 95%.

Results and discussion

Bulk composition and size distribution

Table 1 shows the bulk composition of ACWP, ABWP, SCWP, and SBWP. All produced powders presented statistically the same water content, fat and lactose contents (p > 0.05, Table 1). The protein content of SCWP was slightly but significantly higher than SBWP's (14.5 ± 0.1 g 100 g -1 and 13.6 ± 0.1 g 100 g -1 , respectively, p < 0.05, Table 1). Lower protein content was observed in both ACWP and ABWP (p < 0.05, Table1) which was linked to the absence of the glucomacropetide. The latter existed in sweet whey as results of rennet coagulation (El-Salam, El-Shibiny, & Salem, 2009). Nevertheless, the acidic whey powders exhibited higher ash content than SCWP and SBWP (p < 0.05, Table 1). This observation was related to the additional quantity of colloidal minerals of casein micelles released into whey while acidifying.

The composition of acidic and sweet camel whey powders (ACWP and SCWP) has not been reported elsewhere. However, the bulk composition of acidic and sweet bovine whey powders (ABWP and SBWP) has been extensively studied [START_REF] Deeth | Chemistry of milk-Role of constituents in evaporation and drying[END_REF][START_REF] El-Salam | Factors affecting the functional properties of whey protein products: A review[END_REF]. According to these previous studies, the composition of bovine whey protein powders ranged from 9% to 93% of protein, 0.1% to 85% of lactose, 0.1% to 7% of fat, and 1.5% to 27% of ash. In this current study, the composition of ABWP, ACWP, SBWP, and SCWP was in line with the reported values in the literature [START_REF] Deeth | Chemistry of milk-Role of constituents in evaporation and drying[END_REF][START_REF] El-Salam | Factors affecting the functional properties of whey protein products: A review[END_REF].

Besides, the size distribution of the produced powder was analyzed (Table 1). The results indicated that ACWP and ABWP presented similar d 50 diameter (6.3 ± 0.4 and 6.0 ± 0.8, p > 0.05, Table 1). However, the sweet camel whey powder (SCWP) showed significantly higher d 50 than that of sweet bovine whey powder (SBWP) (25.4 ± 1.7 and 13.1 ± 1.0, p < 0.05, Table 1). Interestingly, sweet whey powders had higher d 50 than those of acidic ones. Several studies have reported that the d 50 mainly depends on the feed characteristics (e.g. composition and concentration) and the spray-dryer configuration [START_REF] Fitzpatrick | Glass transition and the flowability and caking of powders containing amorphous lactose[END_REF][START_REF] Nijdam | The effect of surface composition on the functional properties of milk powders[END_REF]. Indeed, in the current study the same drying conditions and the same dryer configuration were used to produce acidic and sweet camel or bovine whey powders. This fact indicated that the variation of whey composition (depending on the pH and the species) before drying was the main determining factor for the size distribution. Actually, it was demonstrated that a significant decrease in the particle size distribution of milk powder could be observed as a result of the decrease of the protein content and the increase of the ash quantity [START_REF] Mcsweeney | The influence of composition and manufacturing approach on the physical and rehydration properties of milk protein concentrate powders[END_REF]. This fact may explain the variation in the size distribution between acidic or sweet camel and bovine whey powders. Indeed, our findings indicated that decreasing the pH resulted in lower protein content and higher ash quantity for both camel and bovine whey powders (Table 1). These observations may explain the lower size distribution of acidic camel and bovine whey powders as compared to those of sweet ones (Table 1).

LC/MS of camel and bovine whey proteins before and after drying

The protein denaturation extent of acidic and sweet camel or bovine whey during drying was studied using LC-MS. The chromatograms profiles of unprocessed whey and reconstituted whey powders are shown in Supplementary data. By comparing these profiles, a slight and significant thermal denaturation was observed during the drying step. The proteins concentrations before and after reconstitution as well as their denaturation extent are presented in Table 2.

Analysis of the LC-MS profiles of SBWP and RSBWP (Supplementary data, Fig. S1) indicated that the bovine milk serum albumin protein (BSA) was co-eluted with α-lactalbumin (Peak 1, Table 2). These proteins were identified by comparing their molecular weight (results of mass spectroscopy) with the existing data in Uniprot database (http:// www.uniprot.org/). The serum albumin protein (BSA) and α-lactalbumin denaturation extent was equal to 13% (Peak 1, Table 2) in sweet bovine whey, meanwhile, these proteins were denatured to an extent of

Table 1

Bulk composition and size distrubition of sweet and acidic camel and bovine milk whey powders.

ACWP ABWP SCWP SBWP

Bulk composition

Water content (g 100 g -1 ) 4.8 ± 0.1 a 5.0 ± 0.1 a 4.9 ± 0.1 a 4.8 ± 0.1 a Proteins (g 100 g -1 )

12.1 ± 0.1 a 11.6 ± 0.1 b 14.5 ± 0.1 c 13.6 ± 0.1 d Fats (g 100 g -1 ) 0.1 ± 0.02 a 0.1 ± 0.02 a 0.1 ± 0.02 a 0.1 ± 0.02 a Lactose (g 100 g -1 )

72.1 ± 0.6 a 72.6 ± 0.5 a 71.7 ± 0.5 a 71.9 ± 0.5 a Ash (g 100 g -1 )

11.4 ± 0.1 a 10.5 ± 0.1 b 9.1 ± 0.1 b 7.6 ± 0.1 c Calcium (g L -1 ) 1.2 ± 0.01 a 1.2 ± 0.01 a 0.58 ± 0.04 b 0.57 ± 0.01 b Lactate (g L -1 ) 6.3 ± 0.8 a 5.7 ± 0.5 a 0.97 ± 0.02 b 0.99 ± 0.02 b Size distrubition: d 50 (µm) 6.3 ± 0.4 a 6.0 ± 0.8 a 25.4 ± 1.7 b 13.1 ± 1.0 c

Same letters in the same row represent the statistical data significance (p > 0.05).

12.5% and 7.4% during the drying of the acidic one (Peak 1, Peak 2, Table 2). The most denatured sweet bovine whey protein was β-Lactoglobulin (variant A or B) with a total denaturation extent of 74% (Peaks 2 and 3, Table 2). As such, this protein was denatured to an extent of 14.8% for variant A (Peak 5, Table 2) and 42.6% for variant B (Peak 4, Table 2) in acidic bovine whey, respectively. On the other hand, the analysis highlighted that the α-lactalbumin was the most denatured in acidic camel whey powders with a denaturation extent of 25.3%, respectively (Peak 1, Table 2). However, the peptidoglycan recognition protein (PGRP) and the camel milk serum albumin protein (CSA) presented the highest denaturation extent up to 54.2% and 56.4% for (Peak 2 and Peak 3, Table 2) in sweet camel whey, respectively. Lower denaturation extent was observed for these proteins in acidic camel whey (Peak 2, Peak 3, Table 2). These results are in agreement with [START_REF] Felfoul | Proteomic profiling of camel and cow milk proteins under heat treatment[END_REF], which showed that CSA (one free thiol group) is the most heat sensitive protein in camel whey followed by PGRP and α-Lactalbumin.

In a co-current drying configuration (system used in this study), several authors stated that the temperature of the particles increases until reaching asymptotically the air outlet drying temperature [START_REF] Schuck | Relative humidity of outlet air: The key parameter to optimize\nmoisture content and water activity of dairy powders[END_REF][START_REF] Wu | Towards spray drying of high solids dairy liquid: Effects of feed solid content on particle structure and functionality[END_REF]. At relatively low heat intensity, the denaturation of whey proteins was relatively low and mainly involved their initial free 'SH' groups. The free thiol function of β-Lactoglobulin (the main protein in bovine's whey) may explain its sensitivity to denaturation during drying [START_REF] Roefs | A model for the denaturation and aggregation of β-lactoglobulin[END_REF]. [START_REF] Felfoul | A laboratory investigation of cow and camel whey proteins deposition under different heat treatments[END_REF] indicated that the denaturation temperatures of β-Lactoglobulin was equal to 79.6 °C and 83.4 °C in sweet and acidic bovine whey, respectively, which were close to the outlet drying temperature in this study (80 °C). Recently, [START_REF] Lajnaf | The foaming properties of camel and bovine whey: The impact of pH and heat treatment[END_REF] demonstrated that acidic whey presented significantly lower free SH group after heating as compared to the sweet one. Therefore, both the denaturation temperature of β-Lactoglobulin (close to 80 °C) and the low free SH groups of bovine whey protein at acidic pH (=4.6) may explain its higher heat stability as compared to the neutral pH (=6.7) during drying.

Besides, results of this study highlighted the total absence of βlactoglobulin in both acidic and sweet camel whey, whereas the α-Lactalbumin was their main major protein. It is important to note that according to the literature, the camel and bovine α-Lactalbumin have no free thiol 'SH' function [START_REF] Lajnaf | The effect of pH and heat treatments on the foaming properties of purified α-lactalbumin from camel milk[END_REF][START_REF] Relkin | Thermal unfolding of β-lactoglobulin, α-lactalbumin, and bovine serum albumin[END_REF] and presented similar secondary structure [START_REF] Atri | Comparative study on heat stability of camel and bovine apo and holo alpha-lactalbumin[END_REF]. This explains the low denaturation extent of α-Lactalbumin during drying of both camel and bovine whey at neutral (=6.7) or acidic (=4.3 or 4.6) pH.

Surface elementary composition

The relative elemental composition of ACWP, ABWP, SCWP, and SBWP is presented in Table 3. Analysis of XPS survey of ACWP and ABWP indicated the existence of three peaks corresponding to Carbone 'C' (67.3 ± 3.4% and 65.5 ± 3.3%, p > 0.05), Azote 'N' (9.8 ± 0.5% and 9.1 ± 0.4%, p > 0.05), and Oxygen 'O' (21.3 ± 1.1% and 23.1 ± 1.1%, p > 0.05), respectively. The XPS analysis showed that SCWP and SBWP spectra are composed of C (68.9 ± 3.4% and 69.3 ± 3.5%, p > 0.05), N (8.5 ± 0.4% and 6.7 ± 0.5%, p < 0.05), and O (20.9 ± 1.1% and 22.6 ± 1.1, p > 0.05), respectively. These findings indicated that there were no statistical differences in Carbone and Oxygen amounts, while lower azote content (p < 0.05) was observed especially for camel whey powders at neutral pH (6.7). The calculated values for C/O were equal to 3.2 ± 0.2, 2.8 ± 0.2, 3.3 ± 0.2, and 3.1 ± 0.2 for ACWP, ABWP, SCWP, and SBWP, respectively (Table 3). The C/O ratio reflected the surface hydrophobicity of the studied powders. All whey powders displayed the same C/O ratio, indicating that their surfaces had the same hydrophobicity (Table 3), regardless the pH.

The deconvolution of C 1s spectra indicated the presence of four distinct sub-peaks (CeC, CeCOOH, CeO, C]O and OeC]O) in all samples. These sub-peaks corresponded to the different function of the macromolecules (proteins, fats, and lactose). Analysis of the surface composition showed that acidic whey protein powders of both camel and bovine milks statistically had the same composition regarding fats, proteins, and minerals (p > 0.05, Table 3). However, a significantly low surface lactose content was measured for ACWP as compared to ABWP (p < 0.05, Table 3). On the other hand, our findings highlighted that the surface of SCWP has a significantly higher protein and mineral content than SBWP's (p < 0.05, Table 3). The latter had higher lactose and fat content than SCWP's (p < 0.05, Table 3).

It was acknowledged that during drying, adsorption mechanisms at the water-air interface are undergone by proteins (interfacial properties) as the water is evaporated [START_REF] Fäldt | The surface composition of spray-dried protein-lactose powders[END_REF]. The difference in the interfacial properties of whey proteins of both camel and bovine milks can be related to the difference in protein composition as well as the modification of their flexibility as function of pH. [START_REF] Suttiprasit | The surface activity of α-lactalbumin, β-lactoglobulin, and bovine serum albumin: I. Surface tension measurements with single-component and mixed solutions[END_REF] proved that at neutral pH, the α-Lactalbumin is more efficient to reduce the surface tension than the β-Lactoglobulin, as it shows smaller size and more flexible molecular structure. [START_REF] Lajnaf | The foaming properties of camel and bovine whey: The impact of pH and heat treatment[END_REF] reported that the α-Lactalbumin dominates the air water interface of acidic bovine whey, whereas β-Lactoglobulin covers rapidly the interface at neutral pH. At acidic pH, camel 9.8 ± 0.5 a 9.1 ± 0.4 a,b 8.5 ± 0.4 b 6.7 ± 0.5 c C 1s (%) 67.3 ± 3.4 a 65.5 ± 3.3 a 68.9 ± 3.4 a 69.3 ± 3.5 a Ce(C,H) 47.1 ± 2.4 a,b 46.1 ± 2.3 a 48.7 ± 2.4 a,b 51.4 ± 2.6 b CeO 33.6 ± 1.7 a 35.3 ± 1.7 a 33.4 ± 1.7 a 31.8 ± 1.6 a C]O 13.9 ± 0.7 a 13.7 ± 0.7 a 13.2 ± 0.7 a 13.7 ± 0. and bovine whey proteins (especially α-lactalbumin) have higher interfacial properties than neutral pH due to the reduction of negative charges and electrostatic repulsions [START_REF] Marinova | Physico-chemical factors controlling the foamability and foam stability of milk proteins: Sodium caseinate and whey protein concentrates[END_REF]. In bovine's whey, β-Lactoglobulin and α-Lactalbumin act in synergy at the water/ air interface with a predominance of α-Lactalbumin due to its high flexibility at acid pH [START_REF] Marinova | Physico-chemical factors controlling the foamability and foam stability of milk proteins: Sodium caseinate and whey protein concentrates[END_REF]. This suggests that the surface of ABWP is covered by the α-Lactalbumin, whereas this protein coexisted with the residual β-Lactoglobulin (highly denatured, Table 2) in SBWP.

Either at neutral or acidic, the interface water/air in camel whey is suggested to be mainly covered by the α-Lactalbumin in the absence of β-lactoglobulin [START_REF] Lajnaf | The effect of pH and heat treatments on the foaming properties of purified α-lactalbumin from camel milk[END_REF]. The efficiency of the camel α-Lactalbumin to reduce the surface tension depends on both the pH value and the presence/absence of calcium fixation. Indeed, [START_REF] Lajnaf | The effect of pH and heat treatments on the foaming properties of purified α-lactalbumin from camel milk[END_REF] indicated that at acidic pH (=4.3) a partially denaturation of α-Lactalbumin occurred with the release of its chelated calcium, which enhanced its interfacial properties. The release of bounded calcium by α-Lactalbumin at acidic pH (=4.3) enhanced its presence at the surface of the ACWP (p < 0.05, Table 3). Furthermore, it was reported that the absence of β-Lactoglobulin in this serum reduces the rigidity properties of the protein film created at the air water interface [START_REF] Lajnaf | The foaming properties of camel and bovine whey: The impact of pH and heat treatment[END_REF]. This could promote the presence of the α-Lactalbumin at the surface of sweet and acidic camel whey powders.

Water sorption isotherms

The water adsorption isotherms of SCWP, SBWP, ACWP, and ABWP are shown in Fig. 1. These isotherms exhibited typically sigmoid curves showing a breaking point, which corresponded to the total crystallization of amorphous lactose. Similar observations were reported in several other studies [START_REF] Foster | The prediction of moisture sorption isotherms for dairy powders[END_REF][START_REF] Jouppila | Glass transitions and crystallization in milk powders[END_REF].

Our findings showed that, in both ACWP (Fig. 1A) and ABWP (Fig. 1B), lactose crystallized at 0.60 of water activity. Nevertheless, ABWP needed significantly higher water for total lactose crystallization (X c ) than ACWP's (19.0 ± 0.1 g 100 g -1 against 17.4 ± 0.2 g 100 g -1 , p < 0.05, Table 4). The analysis of adsorption isotherms for SCWP (Fig. 1C) and SBWP (Fig. 1D) indicated that both powders presented the same water activity (0.50, p > 0.05, Table 4) and the same amount of water to induce lactose crystallization (11.4 ± 0.2 and 11.1 ± 0.7, p > 0.05, Table 4).

In this study, SCWP and SBWP have the same a w and X c for lactose crystallization (Table 4). The main difference between SCWP and SBWP is the presence of β-lactoglobulin. At neutral pH, it was shown that the presence of β-lactoglobulin caused a w increase of the total lactose crystallization [START_REF] Ibach | Crystallization kinetics of amorphous lactose, wheypermeate and whey powders[END_REF][START_REF] Thomas | Lactose/β-lactoglobulin interaction during storage of model whey powders[END_REF]. Indeed, non-covalent interactions between lactose and β-lactoglobulin will occur during drying and storage indicating the lactosylation phenomenon during the Maillard reaction. These interactions have stabilized whey protein powders against lactose crystallization [START_REF] Ibach | Crystallization kinetics of amorphous lactose, wheypermeate and whey powders[END_REF][START_REF] Thomas | Lactose/β-lactoglobulin interaction during storage of model whey powders[END_REF]. However, camel whey is free of βlactoglobulin and has a high content of α-lactalbumin. In the absence of β-lactoglobulin, lactose should be less stable in sweet whey [START_REF] Ibach | Crystallization kinetics of amorphous lactose, wheypermeate and whey powders[END_REF][START_REF] Thomas | Lactose/β-lactoglobulin interaction during storage of model whey powders[END_REF]. This was not the case for sweet camel whey powder, which has the same water activity for lactose crystallization as for sweet bovine whey's powder. The explanation could be that proteins (mainly α-lactalbumin) dominated the surface of SCWP and encapsulated most of the amorphous lactose (Table 3). This could limit the water availability to induce lactose crystallization. Furthermore, it was demonstrated that the α-lactalbumin from camel milk was more hydrophobic than its bovine's milk counterpart [START_REF] Atri | Comparative study on heat stability of camel and bovine apo and holo alpha-lactalbumin[END_REF][START_REF] Lajnaf | The effect of pH and heat treatments on the foaming properties of purified α-lactalbumin from camel milk[END_REF]. This protein could constitute a hydrophobic barrier to water uptake and limits its contact with lactose. Hence, it caused the increase in lactose stability in sweet camel whey, even in the absence of β-Lactoglobulin. This hypothesis was checked by the analysis of the surface composition of SCWP. Indeed, as discussed in the Section 3.1, SCWP presented a protein content of 4.5 times higher that of lactose. However, this was only equal to 2.2 in SBWP. Thus, the presence of high protein content at the surface of SCWP could stabilize lactose against crystallization.

At acidic pH, lactose in ACWP (pH = 4.3) and ABWP (pH = 4.6) crystallized at the same water activity (a w = 0.60) and with the same water quantity. Moreover, in acidic whey powders, lactose was more stable than in sweet ones (a w = 0.50, p < 0.05, Table 4). The acidification process releases hydro-colloidal calcium (from casein micelles and from α-lactalbumin) and increases lactate content. The calcium and lactate contents were statically the same for ACWP and ABWP (Table 1) and were significantly higher than those of sweet ones. This could explain the increase in crystallization water activity (from 0.50 to 0.60, Table 4). Furthermore, as in the case of neutral pH, we assume that αlactalbumin adsorbs at the water-air interface during drying and saturates the surface of acidic camel and bovine whey powder particles, enhancing the lactose stabilization against crystallization.

On the other hand, the obtained sorption isotherms were adjusted to the Guggenheim-Anderson-de Boer (GAB) model (up to 0.40 of a w , Table 4, Fig. 1). In this study, ABWP (Fig. 1a) and ACWP (Fig. 1b) had statistically the same water monolayer content (X m = 2.6 ± 0.1 g 100 g -1 , p > 0.05) and the same water binding energy (7.3 ± 1.1, 7.5 ± 0.5, p > 0.05, Table 4). This could be explained by their similar d 50 diameter (Table 1). Likewise, SBWP (Fig. 1c) and SCWP (Fig. 1d) also showed similar water monolayer content (X m = 2.3 ± 0.3 g 100 g -1 and 1.9 ± 0.1 g 100 g -1 g 100 g -1 , p > 0.05, Table 4). However, SCWP had a significantly lower water binding energy (6.0 ± 2.7) than SBWP (14.9 ± 1.1, p < 0.05, Table This could be explained by the highest lactose content at the surface SBWP as compared to SCWP (p < 0.05, Table 2).

Glass transition temperature

The glass transition temperature (T g ) of acidic and sweet whey powders of camel and bovine's milk is presented in Table 5. The results indicated that the T g of all studied whey powders decreased with increasing the a w and the water content (Table 5). At 0.13 water activity, camel and bovine whey powders had the same T g value (p > 0.05, Table 5), regardless the pH. At 0.23 and 0.33 of a w , both camel and bovine whey powders' T g was significantly lower at acidic pH (p < 0.05, Table 5), although they adsorbed the same water amounts as measured at neutral pH (p > 0.05, Table 5).

Actually, it was demonstrated that the glass transition temperature (T g ) of dairy powders was relatively dominated by the physicochemical properties of lactose [START_REF] Shrestha | Effect of protein concentration on the surface composition, water sorption and glass transition temperature of spray-dried skim milk powders[END_REF]. Fan and Roos (2017) indicated that the glass transition of pure lactose was equal to 65.0 ± 4.0 °C, 40.0 ± 2.0 °C, and 30.0 ± 3.0 °C at 0.13, 0.23, and 0.33 of water activity, respectively. Results of this current study highlighted that the measured T g for sweet or acidic camel and bovine whey powders were mostly different from those of pure lactose (Table 5). Indeed, it has been reported that, while increasing water activity, lactose mobility is enhanced which induce the glass transition (Haque,

Table 4

Water adsorption properties of sweet and acidic camel and bovine milk whey powders derived from the obtained sorption isotherms at 25 °C. 17.4 ± 0.2 c 2.6 ± 0.1 a 7.5 ± 0.5 a X c : Total water of lactose crystallization (g 100 g -1 ); X m : The monolayer water capacity (g 100 g -1 ); C: Constant related to the water binding energy. Same letter in the same column represent the statistical data significance (p > 0.05). [START_REF] Haque | Glass transition and enthalpy relaxation of amorphous lactose glass[END_REF]. This mobility depends on the presence of other components in the powdered dairy matrix such as whey proteins. Indeed, whey proteins interact with water thanks to their polar and ionizable residues, which are likely to produce hydrogen bonds [START_REF] Foster | The prediction of moisture sorption isotherms for dairy powders[END_REF][START_REF] Hardy | Water activity and hydration of dairy powders[END_REF][START_REF] Jouppila | Glass transitions and crystallization in milk powders[END_REF]. By competing for the available water, whey proteins are likely to change the equilibrium balance between the amorphous and crystalline state of lactose. Therefore, depending on their quantity, hydrophobicity, and pH, whey protein could affect the thermal properties of lactose, its structural strength (α-relaxations) [START_REF] Fan | Water sorption-induced crystallization, structural relaxations and strength analysis of relaxation times in amorphous lactose/whey protein systems[END_REF][START_REF] Silalai | Dielectric and mechanical properties around glass transition of milk powders[END_REF] and its mobility [START_REF] Fan | Crystallization and structural relaxation times in structural strength analysis of amorphous sugar/whey protein systems[END_REF][START_REF] Maidannyk | Water sorption, glass transition and "strength" of lactose -Whey protein systems[END_REF]. This may explain the observed differences in the glass transition temperatures of acidic or sweet camel and bovine whey powders as compared to pure lactose.

Conclusion

Whey proteins are important byproduct with highly valuable techno-functional properties. These proteins are often dried and used in food formulations in powders. The present study aimed at studying the effect of pH on the thermal denaturation extent, the surface composition, and the water sorption isotherms and the glass transition temperature of camel and bovine whey powders. Analysis of camel and bovine whey powders before and after drying indicated that decreasing the pH led to the stabilization of β-Lactoglobulin against thermal denaturation. The α-Lactalbumin in camel or bovine whey powder was relatively heat stable regardless the pH, thanks to the absence of initial SH group. Analysis of the surface chemical composition of camel and bovine whey powders showed that regardless the pH, the α-Lactalbumin predominated the surface. This was related to its flexible structure and its capacity to adsorb to the interface air/water during water evaporation. Besides, analysis of the sorption isotherms indicated that decreasing the pH induced the stabilization of lactose against crystallization. Finally, the pH did not influence the glass transition of both whey powders at 0.13 of water activity. However, at acidic pH, the glass transition temperature of both camel and bovine whey powders was significantly lower than that at neutral pH at 0.23 and 0.33 of water activity.

This study provides valuable knowledge for better understanding of the physical and biochemical properties of camel and bovine whey powders as affected by pH. It will also provide some assistance in the development of whey protein powders application in whey-based food products.

Table 5

Glass transition temperatures of sweet and acidic camel and bovine milk whey powders derived from the obtained MDSC curves. SCWP 3.7 ± 0.4 a 58.3 ± 2.5 a 5.4 ± 0.9 a 47.6 ± 0.9 a 7.6 ± 0.9 a,b 29.7 ± 1.1 a SBWP 3.6 ± 0.5 a 55.0 ± 2.4 a 5.0 ± 0.5 a 45.8 ± 2.1 a 6.5 ± 0.7 a 31.5 ± 0.8 a ACWP 3.2 ± 0.3 a 56.5 ± 2.0 a 5.1 ± 0.7 a 41.2 ± 0.6 b 7.9 ± 0.2 b 24.6 ± 1.5 b ABWP 3.9 ± 0.4 a 55.0 ± 1.0 a 5.1 ± 0.6 a 41.5 ± 0.7 b 6.6 ± 0.6 a 23.1 ± 0.9 b *: moisture content in g.100 g -1 ; T g : glass transition temperature (°C). Same letter in the same column represent the statistical data significance (p > 0.05).

  the same row represent the statistical data significance (p > 0.05).

Fig. 1 .

 1 Fig. 1. Representative adsorption isotherms of ABWP (A), ACWP (B), SBWP (C) and SCWP (D) determined at 25 °C. The experimental data from 0 to 0.40 of water activity were adjusted to the GAB model (R 2 > 0.98): ABWP (a), ACWP (b), SBWP (c), and SCWP (d).

Table 2

 2 Protein characterization from LC-MS peak integration for sweet and acidic camel and bovine milk whey protein before and after powders reconstitution.

	Peaks			Before reconstitution		Reconstituted powders		DE (%)
	Number Identification	MW (Da)	Area	Proportion (%) Concentration	Area	Proportion (%) Concentration (µg
				(AU × s)		(µg µL -1 )	(AU × s)		µL -1 )
	Acidic bovine whey
	1	α-La, V-B	14 176	8.8	26.4	2.112 ± 0.02 a	6.9	28.5	1.955 ± 0.02 b	7.4
	2	BSA	66 000	1.6	4.9	0.392 ± 0.01 a	1.2	4.9	0.343 ± 0.01 b	12.5
	3	Mixture of β-Lg	n.id	7.1	21.3	1.704 ± 0.02 a	5.1	21.1	1.477 ± 0.01 b	13.3
		variants
	4	β-Lg, V-B + 1 lactose	18 60118	3.1	9.3	0.744 ± 0.02 a	1.5	6.1	0.427 ± 0.01 b	42.6
		β-Lg, V-B	277
	5	β-Lg, V-A + 1 lactose	18 36318	11.2	33.6	2.688 ± 0.02 a	8	32.7	2.289 ± 0.02 b	14.8
		β-Lg, variant A	687
	Acidic camel whey
	1	α-La	14 421	9.7	80.2	9.624 ± 0.1 a	9.1	69.9	7.191 ± 0.07 b	25.3
	2	PGRP	19 137	0.5	4.2	0.504 ± 0.02 a	0.6	4.8	0.432 ± 0.01 b	14.3
	3	CSA	66 600	1.1	9.2	1.104 ± 0.02 a	1.2	9.4	0.846 ± 0.01 b	23.3
	Sweet bovine whey
	1	α-La, V-B	14 176	9.3	39.2	4.3 ± 0.05 a	5.3	41.7	3.7 ± 0.02 b	13
		BSA	66 000
	2	Mixture of β-Lg	n.id	4.5	28.3	2.6 ± 0.02 a	1.6	12.6	1.1 ± 0.01 b	56
		variants
	3	β-Lg, V-B + 1 lactose 18 601	9.1	28.5	2.2 ± 0.02 a	3.3	25.7	1.8 ± 0.02 b	18
		β-Lg, V-B	18 277
		β-Lg, V-A + 1 lactose 18 363
		β-Lg, V-A	18 687
	Sweet camel whey
	1	α-La	14 421	13.9	60.2	7.224 ± 0.07 a	11.6	59.4	4.94 ± 0.05 b	31.6
	2	PGRP	19 137	1.4	6.2	0.744 ± 0.01 a	4.5	2.3	0.341 ± 0.01 b	54.2
	3	CSA	66 600	3.1	13.6	1.632 ± 0.02 a	1.5	7.9	0.711 ± 0.01 b	56.4

n.id: non-identified; La: lactalbumin; Lg: Lactoglobulin; V: variant; BSA and CSA: Bovine and Camel Serum Albumin, respectively; PGRP: peptidoglycan recognition protein; DE: denaturation extent; MW: Molecular weight; Same letters in the same row represent the statistical data significance (p > 0.05).

Table 3

 3 Complete elemental composition and surface compotion of camel and bovine milk whey powders.

		ACWP	ABWP	SCWP	SBWP
	Elemental composition
	O 1s (%)	21.3 ± 1.1 a	23.1 ± 1.1 a	20.9 ± 1.1 a	22.6 ± 1.1 a
	N 1s (%)

Acknowledgement

The authors would like to thank Mr. Julien JARDIN (STLO-INRA, France) for helping in performing the proteomic analyses.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.foodchem.2020.127514.


Quentin Labriet
email: quentin.labriet@univ-reims.fr

Holographic transform for tensor product of holomorphic discrete series

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 of SL2(R). Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.

Introduction

Let G be a real reductive Lie group, G ′ a Lie subgroup of G and π an irreducible unitary representation of G on a vector space V . The decomposition into irreducible representations (ρ, W ) of G ′ of the restriction π| G ′ of π to G ′ is called the branching rule. Symmetry breaking operators are defined as elements of the space of linear continuous maps V → W that intertwine π| G ′ and ρ, denoted by Hom G ′ (π| G ′ , ρ), for any given irreducible representation (ρ, W ) of G ′ . Analogously, elements of the space Hom G ′ (ρ, π| G ′ ) are called holographic operators (see for example (2.9)).

In the article [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], 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 ′ ) = (SO 0 (2, n), SO 0 (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 [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF]).

It is known (see [START_REF] Van Dijk | Ring structures for holomorphic discrete series and Rankin-Cohen brackets[END_REF] for instance) that the symmetry breaking operators for the decomposition of the tensor products of holomorphic discrete series of SL 2 (R) (what we call the diagonal case) are proportional to Rankin-Cohen brackets RC 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.

1

Theorem 1.1. Suppose λ ′ , λ ′′ , λ ′′′ > 1 such that l = 1 2 (λ ′′′ -λ ′ -λ ′′ ) ∈ N. Let w 1 , w 2 ∈ Π, and g ∈ H 2 λ ′′′ (Π). Then we have:

(RC λ ′′′ λ ′ ,λ ′′ ) * g(w 1 , w 2 ) = C(λ ′ , λ ′′ ) Π g(z)K λ ′′′ λ ′ ,λ ′′ (z, w 1 , w 2 ) dµ(z), (1.1)
where K λ ′′′ λ ′ ,λ ′′ (z, w 1 , w 2 ) = (w 2 -w 1 ) l w1-z 2i -(λ ′ +l) w2-z 2i

-(λ ′′ +l) , dµ(x + iy) = y λ ′′′ -2 dxdy, and

C(λ ′ , λ ′′ ) = (λ ′ -1) l+1 (λ ′′ -1) l+1 2 2l+4 π 2 l! .
Notice that this kind of operators has already been considered by H. Rosengren in [START_REF] Rosengren | Multivariable orthogonal polynomials and coupling coefficients for discrete series representations[END_REF]. We give two different proofs of this theorem, the first one is inspired by the proof of Theorem 3.10 in [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], 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 orthogonal 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 [START_REF] Kobayashi | Differential symmetry breaking operators: II. Rankin-Cohen operators for symmetric pairs[END_REF] (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 [START_REF] Kobayashi | Differential symmetry breaking operators: I. General theory and F-method[END_REF]), to the Jacobi ordinary differential equation for the symbols of such operators. In [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF] 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 space L 2 ((-1, 1), (1 -v) α (1 + v) β dv) (see [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], 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 for λ ∈ C and n ∈ N by :

(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)

Setting

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

Holomorphic model

We denote by Π the Poincar upper half-plane endowed with the hyperbolic metric, and by H 2 λ (Π) the weighted Bergman space defined by:

H 2 λ (Π) = O (Π) ∩ L 2 Π, y λ-2 dxdy .
It is known that H 2 λ (Π) = {0} for λ ≤ 1, so we suppose that λ > 1. In this case H 2 λ (Π) admits a reproducing kernel, also called the Bergman kernel, given by (see, for instance, [START_REF] Faraut | Analysis on symmetric cones. Oxford Mathematical Monographs[END_REF], Prop.XIII.1.2, p.261):

K λ (z, w) = λ -1 4π z - w 2i -λ .
(2.1)

We set, for λ ′ , λ ′′ > 1,

H 2 λ ′ ,λ ′′ (Π × Π) ≃ H 2 λ ′ (Π) ⊗H 2 λ ′′ (Π)
, where ⊗ denotes the Hilbert space completion of the tensor product. This space also admits a reproducing kernel:

K λ ′ ,λ ′′ (z, w) = K λ ′ (z 1 , w 1 ) • K λ ′′ (z 2 , w 2 ) = (λ ′ -1)(λ ′′ -1) (4π) 2 z 1 -w1 2i -λ ′ z 2 -w2 2i -λ ′′ . (2.2)
For λ ∈ N\{0; 1}, the holomorphic discrete series representations π λ of SL 2 (R) can be realized on H 2 λ (Π) by the following formula:

(π λ (g)f ) (z) = (cz + d) -λ f Å az + b cz + d ã , (2.3)
where

g -1 = Å a b c d ã ∈ SL 2 (R)
, and f ∈ H 2 λ (Π). This representation lifts to a unitary and irreducible representation of the universal covering group • SL 2 (R) for λ > 1, through an appropriate choice of the determination of the power function.

The outer product representation

π λ ′ ⊠ π λ ′′ of the direct product group • SL 2 (R) × • SL 2 (R) acts on the space H 2 λ ′ ,λ ′′ (Π × Π)
. This representation is also irreducible and unitary for λ ′ , λ ′′ > 1.

Rankin-Cohen operators

Let λ ′ , λ ′′ , λ ′′′ verify (C1), and define a differential operator

R λ ′′′ λ ′ ,λ ′′ from O(Π × Π) to O(Π × Π) by R λ ′′′ λ ′ ,λ ′′ (f )(w 1 , w 2 ) = l j=0 (-1) j (λ ′ + l -j) j (λ ′′ + j) l-j j!(l -j)! ∂ l f ∂z l-j 1 ∂z j 2 (w 1 , w 2 ).
(2.4)

The Rankin-Cohen operator is a map from O(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 operator RC λ ′′′ λ ′ ,λ ′′ is a symmetry breaking operator for the outer product representation π λ ′ ⊠ π λ ′′ , more precisely it generates the space Hom ‡

SL2(R) (π λ ′ ⊠ π λ ′′ | ‡ SL2(R)
, π λ ′′′ ), for λ ′ , λ ′′ , λ ′′′ satisfiying (C1) (see [START_REF] Kobayashi | Differential symmetry breaking operators: II. Rankin-Cohen operators for symmetric pairs[END_REF] Cor.

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 2 λ ′′′ (Π) → H 2 λ ′ ,λ ′′ (
Π×Π) which give, in the unitary case, the holographic operators.

Holographic operators for the diagonal case

Let λ ′ , λ ′′ , λ ′′′ verify (C1), and define the operators

Ψ λ ′′′ λ ′ ,λ ′′ : H 2 λ ′′′ (Π) → H 2 λ ′ ,λ ′′ (Π × Π) by Ψ λ ′′′ λ ′ ,λ ′′ (g)(w 1 , w 2 ) = (w 1 -w 2 ) l 2 λ ′ +λ ′′ +2l-1 l! 1 -1 g(w(v))(1 -v) λ ′ +l-1 (1 + v) λ ′′ +l-1 dv, (2.6)
where w(v) = 1 2 ((w 2 -w 1 )v + (w 2 + w 1 )). In [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], the authors prove that this is a holographic operator from H 2 λ ′′′ (Π) to H 2 λ ′ ,λ ′′ (Π×Π). They also establish a Parseval-Plancherel type theorem for the corresponding Rankin-Cohen transorm and its associated holographic transform (see [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF] Thm. 2.7). Finally, they show (Prop 2.23) that :

(RC λ ′′′ λ ′ ,λ ′′ ) * = CΨ λ ′′′ λ ′ ,λ ′′ . (2.7) where C = Γ(λ ′ +λ ′′ +2l-1) 2 2l+2 πΓ(λ ′ -1)Γ(λ ′′ -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 kernel K λ ′′′ λ ′ ,λ ′′ , defined for w 2 , w 1 , z ∈ Π by:

K λ ′′′ λ ′ ,λ ′′ (z, w 1 , w 2 ) = (w 2 -w 1 ) l w 1 - z 2i -(λ ′ +l) w 2 - 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. Set λ ′ , λ ′′ , λ ′′′ verify (C1). Let w 1 , w 2 ∈ Π, and g ∈ H 2 λ ′′′ (Π). Then we have:

(RC λ ′′′ λ ′ ,λ ′′ ) * g(w 1 , w 2 ) = C(λ ′ , λ ′′ ) Π g(z)K λ ′′′ λ ′ ,λ ′′ (z, w 1 , w 2 )dµ(z), (2.9)
where

C(λ ′ , λ ′′ ) = (λ ′ -1) l+1 (λ ′′ -1) l+1 2 2l+4 π 2 l! .
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 of • SL 2 (R).

L 2 -model of holomorphic discrete series

For λ > 1, the Laplace transform defined by: [START_REF] Faraut | Analysis on symmetric cones. Oxford Mathematical Monographs[END_REF], Thm. XII.1.1). More precisely:

F g(z) = ∞ 0 g(t)e izt dt, (2.10) is a one-to-one isometry (up to a constant) from L 2 λ (R + ) := L 2 (R + , t 1-λ dt) to H 2 λ (Π) (see for instance
F g 2 H 2 λ (Π) = b(λ) g 2 L 2 λ (R + ) , for every g ∈ L 2 λ (R + ), where b(λ) = 2 2-λ πΓ(λ -1).
Using this transform, we consider another realization for the holomorphic discrete series representations π λ of • SL 2 (R) on L 2 λ (R + ), and call it the L 2 -model for π λ . For λ ′ , λ ′′ > 1, we define

L 2 λ ′ ,λ ′′ (R + × R + ) := L 2 λ ′ (R + ) ⊗L 2 λ ′′ (R + ) ≃ L 2 (R + × R + , x 1-λ ′ y 1-λ ′′ dxdy). We denote F 2 = F ⊗ F the Laplace transform from L 2 λ ′ ,λ ′′ (R + × R + ) to H 2 λ ′ ,λ ′′ (Π × Π).

Holographic operators in the L 2 -model

Set λ ′ , λ ′′ , λ ′′′ satisfiying (C1). Using the Laplace transform (2.10), we define the analogue of the Rankin-Cohen operators in the L 2 -model by:

ÿ RC λ ′′′ λ ′ ,λ ′′ := F -1 • RC λ ′′′ λ ′ ,λ ′′ • F 2 . (2.11)
It is a symmetry breaking operator for the • SL 2 (R)-action in the L 2 -model.

In [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], the authors prove that ÿ RC λ ′′′ λ ′ ,λ ′′ is given by the following integral formula for

F ∈ L 2 λ ′ ,λ ′′ (R + × R + ) (see [KP20], Prop. 2.13): ÿ RC λ ′′′ λ ′ ,λ ′′ F (t) = t l+1 2i l 1 -1 P λ ′ -1,λ ′′ -1 l (v)F Å t 2 (1 -v), t 2 (1 + v) ã dv, (2.12)
where P λ ′ -1,λ ′′ -1 l denotes the Jacobi polynomials (see [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], section 5.1). We define the following operator which associates to a function g(t)

defined on R + a function of two variables Φ λ ′′′ λ ′ ,λ ′′ g on R + × R + : Φ λ ′′′ λ ′ ,λ ′′ g(x, y) := x λ ′ -1 y λ ′′ -1 (x + y) λ ′ +λ ′′ +l-1 P λ ′ -1,λ ′′ -1 l Å y -x x + y ã • g(x + y).
(2.13)

Once again, it is shown in [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], that Φ λ ′′′ λ ′ ,λ ′′ is a holographic operator between L 2 λ ′′′ (R + ) and L 2 λ ′ ,λ ′′ (R + × R + ). We summarize the framework in the two following commutative diagrams

L 2 λ ′ ,λ ′′ (R + × R + ) ◊ RC λ ′′′ λ ′ ,λ ′′ / / F2 L 2 λ ′′′ (R + ) F H 2 λ ′ ,λ ′′ (Π × Π) RC λ ′′′ λ ′ ,λ ′′ / / H 2 λ ′′′ (Π)
Symmetry breaking operators in the holomorphic and L 2 -models

L 2 λ ′ ,λ ′′ (R + × R + ) F2 L 2 λ ′′′ (R + ) F Φ λ ′′′ λ ′ ,λ ′′ o o H 2 λ ′ ,λ ′′ (Π × Π) H 2 λ ′′′ (Π) Ψ λ ′′′ λ ′ ,λ ′′ o o
Holographic operators in the holomorphic and L 2 -models

Inverse Laplace transform of the reproducing kernel

The following lemma gives the inverse Laplace transform for the Bergman kernels K λ (•, w).

Lemma 2.2. Suppose λ, λ ′ , λ ′′ > 1. The inverse Laplace transform of the reproducing kernel K λ is given by the following fomula:

F -1 K λ (•, z)(t) = 2 λ-1 2πΓ(λ -1) t λ-1 e -itz .
(2.14)

Consequently:

F -1 2 K λ ′ ,λ ′′ (•, (w 1 , w 2 ))(x, y) = 2 λ ′ +λ ′′ -2 4π 2 Γ(λ ′ -1)Γ(λ ′′ -1)
x λ ′ -1 y λ ′′ -1 e -i(x,y).(w1,w2) .

(2.15)

Proof. Consider the function g(t) = 2 λ-1 2πΓ(λ-1) t λ-1 e -itz . Then, the usual Laplace transform gives:

F g(z) = 2 λ-1 2πΓ(λ -1) ∞ 0 t λ-1 e it(w-z) dt = λ -1 4π Å 2i w - z ã λ = K λ (w, z),
which proves the first point. The second is obvious from the definitions.

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. Let D j (j = 1, 2) be some complex manifolds, and H j some Hilbert spaces of holomorphic functions on D j with reproducing kernels K (j) (•, •). If R : H 1 → H 2 is a continous linear map, then :

1. RK (1) (•, ζ)(τ ′ ) = (R * K (2) (•, τ ′ ))(ζ) pour ζ ∈ D 1 , τ ′ ∈ D 2 . 2. (R * g)(ζ) = (g, RK (1) (•, ζ)) H2 for g ∈ H 2 , ζ ∈ D 1 .
Thanks to this lemma, we only need to compute RC λ ′′′ λ ′ ,λ ′′ (K λ ′ ,λ ′′ (•, (w 1 , w 2 ))) 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.

First proof of Theorem 1.1

Our first proof of Theorem 1.1 reduces to a direct computation. Lemma 3.2. For l ∈ N and λ > 1, we have:

∂ l ∂z l K λ (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 on l. Then we are ready to prove Theorem 1.1.

Proof of Theorem 1.1.

R λ ′′′ λ ′ ,λ ′′ K λ ′ ,λ ′′ (•, (w 1 , w 2 ))(z 1 , z 2 ) = l j=0 (-1) j (λ ′ + l -j) j (λ ′′ + j) l-j j!(l -j)! ∂ l ∂z l-j 1 ∂z j 2 K λ ′ ,λ ′′ (•, (w 1 , w 2 ))(z 1 , z 2 ) = (-1) l (4π) 2 l!(2i) l Å z 1 -w 1 2i ã -(λ ′ +l) Å z 2 -w 2 2i ã -(λ ′′ +l) × l j=0 (-1) j (λ ′ + l -j) j (λ ′′ + j) l-j (λ ′ -1) l-j+1 (λ ′′ -1) j+1 Ç l j å Å z 2 -w 2 2i ã l-j Å z 1 -w 1 2i ã j = (-1) l (λ ′ -1) l+1 (λ ′′ -1) l+1 (4π) 2 l!(2i) 2l Å z 1 -w 1 2i ã -(λ ′ +l) Å z 2 -w 2 2i ã -(λ ′′ +l) × l j=0 Ç l å (w 1 -z 1 ) j (z 2 -w 2 ) l-j = (λ ′ -1) l+1 (λ ′′ -1) l+1 (4π) 2 l!2 2l Å z 1 -w 1 2i ã -(λ ′ +l) Å z 2 -w 2 2i ã -(λ ′′ +l) (z 1 -z 2 + w 2 -w 1 ) l .
Finally, we get by restriction to the diagonal z 1 = z 2 = z:

RC λ ′′′ λ ′ ,λ ′′ K λ ′ ,λ ′′ (•, (w 1 , w 2 ))(z) = Rest| z1=z2=z • R λ ′′′ λ ′ ,λ ′′ (K λ ′ ,λ ′′ (•, (w 1 , w 2 ))) (z) = (λ ′ -1) l+1 (λ ′′ -1) l+1 (4π) 2 l!2 2l Å w 1 -z 2i ã -(λ ′ +l) Å w 2 -z 2i ã -(λ ′′ +l) (w 2 -w 1 ) l .
Thus Lemma 3.1 implies the statement.

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. Set λ ′ , λ ′′ , λ ′′′ satisfying (C1).

RC λ ′′′ λ ′ ,λ ′′ K λ ′ ,λ ′′ (•, (w 1 , w 2 ))(z) = (3.2) C ′ (w 2 -w 1 ) l ∞ 0 F -1 K λ ′′′ (•, z)(t) 1 F 1 (λ ′ + l, λ ′ + λ ′′ + 2l; -i(w 2 -w 1 )t)e itw2 dt,
where

C ′ = Γ(λ ′ +λ ′′ +2l-1)B(λ ′ +l,λ ′′ +l) 2 2l+2 πl!Γ(λ ′ -1)Γ(λ ′′ -1)
and 1 F 1 is the Kummer function (see Appendix 5).

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

ÿ RC λ ′′′ λ ′ ,λ ′′ • F -1 2 (K λ ′ ,λ ′′ (•, (w 1 , w 2 ))) (t) = t λ ′ +λ ′′ +l-1 2 3 i l π 2 Γ(λ ′ -1)Γ(λ ′′ -1) 1 -1 P λ ′ -1,λ ′′ -1 l (v)e -it w 1 +w 2 2 +v w 2 -w 1 2 (1 -v) λ ′ -1 (1 + v) λ ′′ -1 dv = 2 λ ′ +λ ′′ -4 B(λ ′ + l, λ ′′ + l) π 2 Γ(λ ′ -1)Γ(λ ′′ -1)l! (w 2 -w 1 ) l t λ ′′′ -1 1 F 1 (λ ′ + l, λ ′ + λ ′′ + 2l; i(w 2 -w 1 )t) e -itw2 . As RC λ ′′′ λ ′ ,λ ′′ = F • ÿ RC λ ′′′ λ ′ ,λ ′′ • F -1 2
(see (2.10)), it gives:

RC λ ′′′ λ ′ ,λ ′′ K λ ′ ,λ ′′ (•, (w 1 , w 2 ))(z) = C ′ (w 2 -w 1 ) l R + 2 λ ′′′ -1 2πΓ(λ ′′′ -1) t λ ′′′ -1 e itz 1 F 1 (λ ′ + l, λ ′ + λ ′′ + 2l; i(w 1 -w 2 )t) e -itw2 dt.
Lemma 2.2 gives 2 λ ′′′ -1 2πΓ(λ ′′′ -1) t λ ′′′ -1 e -itz = F -1 K λ ′′′ (•, z)(t), which ends our proof.

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

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

Å ∂ ∂w ã n K λ ′′′ (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:

∞ 0 F -1 (K λ ′′′ (•, z)) (t)e itw2 1 F 1 (λ ′ + l, λ ′ + λ ′′ + 2l; -i(w 2 -w 1 )t) dt = ∞ n=0 (λ ′ + l) n (λ ′ + λ ′′ + 2l) n ∞ 0 F -1 (K λ ′′′ (•, z)(t)) e itw2 (-i(w 2 -w 1 )t) n e itw2 dt = ∞ n=0 (λ ′ + l) n (λ ′ + λ ′′ + 2l) n (w 2 -w 1 ) n Å ∂ ∂w ã n K λ ′′′ (w, z) w=w2 = λ ′′′ -1 4π ∞ n=0 (λ ′ + l) n (-1) n (2i) λ ′′′ (w 2 -z) -λ ′′′ -n (w 2 -w 1 ) n = λ ′′′ -1 4π w 2 - z 2i -λ ′′′ ∞ n=0 (λ ′ + l) n (-1) n Å w 2 -w 1 w 2 - z ã n = λ ′′′ -1 4π w 2 - z 2i -λ ′′′ Å 1 - w 2 -w 1 w 2 - z ã -λ ′ -l = λ ′′′ -1 4π w 2 - z 2i -λ ′′ -l w 1 - 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 arbitrary w 1 , w 2 , z ∈ Π. Proposition 3.3, implies the result. Finally, for the constant C(λ ′ , λ ′′ ), we have:

C(λ ′ , λ ′′ ) = C ′ (λ ′′′ -1) 4πl! = (λ ′′′ -1)Γ(λ ′′′ -1)Γ(λ ′ + l)Γ(λ ′′ + l) 2 2l+4 π 2 l!Γ(λ ′ -1)Γ(λ ′′ -1)Γ(λ ′′′ ) .

Link with the operator

Ψ λ ′′′ λ ′ ,λ ′′
We give another characterization of the operators Ψ λ ′′′ λ ′ ,λ ′′ (see (2.6)) as follows Proposition 3.4. Suppose λ ′ , λ ′′ , λ ′′′ verify (C1). Then we have the following identity:

Ψ λ ′′′ λ ′ ,λ ′′ (K λ ′′′ (•, z))(w 1 , w 2 ) = (-1) l (λ ′′′ -1)B(λ ′ + l, λ ′′ + l) 4πl! K λ ′′′ λ ′ ,λ ′′ (z, w 1 , w 2 ). (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, w 1 , w 2 ) = (w 2 -w 1 l w 1 - z 2i -(λ ′ +l) w 2 - z 2i -(λ ′′ +l) .
Proof. Similar argument as in the proof of Proposition 3.3 gives:

Ψ λ ′′′ λ ′ ,λ ′′ (K λ ′′′ (•, z))(w 1 , w 2 ) = (w 1 -w 2 ) l 2 λ ′′′ -1 l! 1 -1 Å ∞ 0 F -1 K λ ′′′ (•, z)(t)e itw(v) dt ã (1 -v) λ ′ +l-1 (1 + v) λ ′′ +l-1 dv = 2 2l+2 π(-1) l Γ(λ ′ -1)Γ(λ ′′ -1) Γ(λ ′′′ -1) RC λ ′′′ λ ′ ,λ ′′ K λ ′ ,λ ′′ (•, (w 1 , w 2 ))(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 every g ∈ H λ ′′′ (Π):

Ψ λ ′′′ λ ′ ,λ ′′ (g)(w 1 , w 2 ) = (w 1 -w 2 ) l 2 λ ′ +λ ′′ +2l-1 l! 1 -1 g(w(v))(1 -v) λ ′ +l-1 (1 + v) λ ′′ +l-1 dv = (w 1 -w 2 ) l 2 λ ′ +λ ′′ +2l-1 l! 1 -1 Π g(z)K λ ′′′ (z, w(v))y λ ′′′ -2 (1 -v) λ ′ +l-1 (1 + v) λ ′′ +l-1 dxdydv = Π g(z) Ç 1 -1 (w 1 -w 2 ) l 2 λ ′ +λ ′′ +2l-1 l! K λ ′′′ (w(v), z)(1 -v) λ ′ +l-1 (1 + v) λ ′′ +l-1 dv å y λ ′′′ -2 dxdy = Π g(z)Ψ λ ′′′ λ ′ ,λ ′′ (K λ ′′′ (•, z))(w 1 , w 2 ) y λ ′′′ -2 dxdy.

Geometrical interpretation

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

Cone stratification

Let σ be an involutive automorphism of a connected semi-simple Lie group G, which commutes with the Cartan involution θ of G. We use the same letters σ and θ to denote their differentials. Define a θ-stable subgroup of G by

G σ = {g ∈ G | σ(g) = g}. (4.1)
Consider K a maximal compact subgroup of G and g = k + p a Cartan decomposition of g = Lie(G). The space p is σ-stable, so we have the following diffeomorphism for the symmetric space G/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 [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF], Thm.1.1, p.252):

X + Y ∈ p σ ⊕ p -σ → exp(X + Y )K ∈ G/K. (4.3)
The automorphism σθ is also involutive, so one checks that:

p σ ≃ G σ /K σ and p -σ ≃ G σθ /K σθ . (4.4)
Finally, (4.2) and (4.4) gives the following diffeomorphism for G/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 -σ → (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 = ßÅ a 0 0 a -1 ã ; a ∈ R * ™ × ßÅ b 0 0 b -1 ã ; b ∈ R * ™ (which is the structure group of the Jordan algebra R × R ), and σ(g 1 , g 2 ) = (g 2 , g 1 ) with g = (g 1 , g 2 ) ∈ G. Then, G σ /K σ ≃ R + is diagonaly embeded in G/K ≃ R + × R + , and G σθ /K σθ ≃ (-1; 1). The following diffeomorphism ι from R + × (-1; 1) to R + × R + : ι : (t, v) → 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 of R + into R + × R + , and describes the stratification of the cone R + × R + by the segments orthogonal to this diagonal. In his recent preprint [START_REF] Clerc | Rankin-Cohen brackets on tube-type domains[END_REF], 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 [START_REF] Clerc | Rankin-Cohen brackets on tube-type domains[END_REF]) for an arbitrary Euclidean Jordan algebra.

Let α, β > 1, and dµ α,β (v) := (1 -v) α-1 (1 + v) β-1 dv be a measure on the segment (-1, 1). We define the Hilbert spaces

L 2 α,β (R + × (-1; 1)) := L 2 (R + × (-1; 1), t α+β-1 dtdµ α,β (v)).
For a function f defined on R + × R + , we define a function T ι (f ) on R + × (-1, 1) by

T ι (f )(t, v) = Å t 2 ã 2-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 L 2 (R + × R + , x 1-λ ′ y 1-λ ′′ dxdy) to L 2 λ ′ ,λ ′′ (R + × (-1; 1)). The inverse map is given by the following formula:

T -1 ι (h)(x, y) = x λ ′ -1 y λ ′′ -1 • h • ι -1 (x, y). (4.9)
We use this operator in order to define the stratified L 2 -model by transfering the restriction of the outer product of holomorphic discrete series representations of • SL 2 (R) on the space L 2 λ ′ ,λ ′′ (R + × (-1; 1)). The group • SL 2 (R) acts on L 2 λ ′ ,λ ′′ (R + × (-1; 1)) by the operators:

T θ • ÿ π λ ′ ,λ ′′ (g) • T -1 θ , (4.10)
where g ∈ • SL 2 (R) and ÿ π λ ′ ,λ ′′ (g) = F -1 2 • π λ ′ ⊗π λ ′′ (g, g) • F 2 . Finaly, we introduce the map Θ for any function defined on R + :

Θ(h)(t, v) = t -(λ ′ +λ ′′ +l-1) P λ ′ -1,λ ′′ -1 l (v) P λ ′ -1,λ ′′ -1 l h(t). (4.11)
This is a one-to-one isometry from L 2 λ ′′′ (R + ) to the Hilbert space

V λ ′ ,λ ′′ l := L 2 Ä R + , t λ ′ +λ ′′ -1 dt ä ⊗ C • P λ ′ -1,λ ′′ -1 l (v), (4.12)
whose inverse is given by:

Θ -1 (f × P λ ′ -1,λ ′′ -1 l )(t) = P λ ′ -1,λ ′′ -1 l t λ ′ +λ ′′ +l-1 f (t), (4.13)
where

P λ ′ -1,λ ′′ -1 l = 2 α+β+1 Γ(l+α+1)Γ(l+β+1) l!(2l+α+β+1)Γ(l+α+β+1)
1 2 is the norm of Jacobi polynomials in the Hilbert space L 2 ((-1; 1), dµ λ ′ ,λ ′′ (v)).

Analogously to (4.10), we use this map to transfer the holomorphic discrete series representation of • SL 2 (R) and define, for every g ∈ • SL 2 (R), the following operator on

V λ ′ ,λ ′′ l : Θ • ◊ π λ ′′′ (g) • Θ -1 . (4.14)
where ◊ π λ ′′′ (g) = F -1 • π λ ′′′ (g) • F .

Symmetry breaking and holographic operator

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

L 2 λ ′ ,λ ′′ (R + × (-1; 1)) ≃ L 2 Ä R + , t λ ′ +λ ′′ -1 dt ä ⊗L 2 ((-1; 1), dµ λ ′ ,λ ′′ (v)) ≃ l≥0 ⊕ L 2 Ä R + , t λ ′ +λ ′′ -1 dt ä ⊗ C • P λ ′ -1,λ ′′ -1 l (v) (4.15) ≃ l≥0 ⊕ L 2 Ä R + , t λ ′ +λ ′′ -1 dt ä ⊗ V λ ′ ,λ ′′ l .
where ⊕ stands for an orthogonal Hilbert sum. The last isomorphism is due to the fact that the Jacobi polynomials form a Hilbert basis for L 2 ((-1; 1),

dµ λ ′ ,λ ′′ (v)) if λ ′ , λ ′′ > 1.
This allows us to consider the orthogonal projection J λ ′ ,λ ′′ l from L 2 λ ′ ,λ ′′ (R + × (-1; 1)) on the Hilbert subspace V λ ′ ,λ ′′ l which corresponds to the usual Jacobi transform:

J λ ′ ,λ ′′ l (h)(t, v) = P λ ′ -1,λ ′′ -1 l (v) P λ ′ -1,λ ′′ -1 l 2 1 -1 h(t, u)P λ ′ -1,λ ′′ -1 l (u) dµ λ ′ ,λ ′′ (u). (4.16)
We summarize the situation in the following Proposition.

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

L 2 λ ′ ,λ ′′ (R + × R + ) c1 ◊ RC λ ′′′ λ ′ ,λ ′′ / / Tι L 2 λ ′′′ (R + ) Θ c2Φ λ ′′′ λ ′ ,λ ′′ / / L 2 λ ′ ,λ ′′ (R + × R + ) L 2 λ ′ ,λ ′′ (R + × (-1; 1)) J λ ′ ,λ ′′ l / / V λ ′ ,λ ′′ l @ Ø T -1 ι 5 5 ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ where c 1 = 2 λ ′ +λ ′′ -3 i l P λ ′ -1,λ ′′ -1 l and c 2 = 1 P λ ′ -1,λ ′′ -1 l . Proof. Let f ∈ L 2 λ ′ ,λ ′′ (R + × R + ).
On one hand side we have:

Φ λ ′′′ λ ′ ,λ ′′ • ÿ RC λ ′′′ λ ′ ,λ ′′ f (x, y) = x λ ′ -1 y λ ′′ -1 2i l (x + y) λ ′ +λ ′′ -2 P λ ′ -1,λ ′′ -1 l Å y -x x + y ã 1 -1 P λ ′ -1,λ ′′ -1 l (u)F (ι(x + y), u) du.
On the other hand side we have:

T -1 ι • J λ ′ ,λ ′′ l • T ι (f )(x, y) = T -1 ι Ç (t, v) → Å t 2 ã 2-λ ′ -λ ′′ P λ ′ -1,λ ′′ -1 l (v) 1 -1 f (ι(t, u))P λ ′ -1,λ ′′ -1 l (u) du å = 2 λ ′ +λ ′′ -2 P λ ′ -1,λ ′′ -1 l 2 x λ ′ -1 y λ ′′ -1 (x + y) λ ′ +λ ′′ -2 P λ ′ -1,λ ′′ -1 l Å y -x x + y ã 1 -1 P λ ′ -1,λ ′′ -1 l (u)F (ι(x + y), u) du. A direct computation shows that T -1 ι • Θ = c 2 Φ λ ′′′ λ ′ ,λ ′′ .
The fact that ÿ RC λ ′′′ λ ′ ,λ ′′ and Φ λ ′′′ λ ′ ,λ ′′ are intertwinning operators (see [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], Thm.2.11), and Proposition 4.2 lead to the following: Proposition 4.3. The orthogonal projection J λ ′ ,λ ′′ l is a symmetry breaking operator for the • SL 2 (R) action (4.10) in the stratified L 2 -model and the canonical injection corresponds to the associated holographic operator.

In [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], 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 L 2 -space associated with Jacobi polynomials to the construction of symmetry breaking operators for the holomorphic discrete series representations of • SL 2 (R) and explains the structure of the corresponding branching rules.

Remark: We should mention that we have similar result in the conformal case (G, G ′ ) = (SO 0 (2, n), SO 0 (2, n -1)) (see [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF] for more detatils). For this we use the stratification of the time-like cone Ω(n) given by the diffeomorphism ι : Ω(n -1) × (-1, 1) → Ω(n) (y ′ , v) → Ä y ′ , -vQ 1,n-2 (y ′ ) 1 2 ä . (4.17)

5 Appendix: a lemma about Kummer function

The Kummer function, denoted 1 F 1 , is a special case of generalized hypergeometric functions defined for complex parameter a, b where b is not a negative integer by

1 F 1 (a, b; x) = ∞ n=0 (a) n (b) n
x n n! .

(5.1)

For more results about Kummer function, we refer to [START_REF] Beals | Special functions[END_REF], for example. The Kummer function admits the following integral representation in terms of Jacobi polynomials:

Lemma 5.1. Let α, β > 1, l ∈ N and x ∈ R + . Then:

C α,β x l e ix 1 F 1 (α + l, α + β + 2l; -2ix) = 1 -1 P α-1,β-1 l (v)e ivx (1 -v) α-1 (1 + v) β-1 dv, (5.2)
where C α,β = 2 α+β+l-1 i l l! B(α + l, β + l), and B(x, y) is the usual Euler beta function. where we put v = 1 -2s. Then, the Rodrigues formula for Jacobi polynomials (see [START_REF] Kobayashi | Inversion of Rankin-Cohen operators via holographic transform[END_REF], section 5.1) and integration by part give:

1 -1 P α-1,β-1 l (v)e ivx (1 -v) α-1 (1 + v) β-1 dv = (-1) l 2 l l! 1 -1 e ixv Å d dv ã l (1 -v) α+l-1 (1 + v) β+l-1 dv = 1 2 l l! 1 -1 Å d dv ã l e ixv (1 -v) α+l-1 (1 + v) β+l-1 dv = i l x l 2 l l! 1 -1
e ixv (1 -v) α+l-1 (1 + v) β+l-1 dv.

The result follows from the integral representation (5.3) for the Kummer function with c = α + β + 2l > α + 2l > 0.

Proof.

  The following integral representation for the Kummer function, for Re(c) > Re(a) > 0 is classical (see[START_REF] Beals | Special functions[END_REF] p.190):1 F 1 (a, c; x) = 1 B(a, c -a) 1 0 s a-1 (1 -s) c-a-1 e sx ds 2 (1 -v) a-1 (1 + v) c-a-1 dv,(5.3)