Hussein Hijazi
email: hussein.hijazi@gipsa-lab.inpg.fr

Laurent Ros
email: laurent.ros@gipsa-lab.inpg.fr

Bayesian Cramer-Rao Bound for OFDM Rapidly Time-varying Channel Complex Gains Estimation

Keywords: Bayesian Cramer-Rao Bound, OFDM, Rayleigh complex gains

In this paper, we consider the Bayesian Cramer-Rao bound (BCRB) for the dynamical estimation of multi-path Rayleigh channel complex gains in data-aided (DA) and non-dataaided (NDA) OFDM systems. This bound is derived in an on-line and off-line high Doppler scenarios for time-varying complex gains within one OFDM symbol, assuming the availability of prior information. In NDA context, whereas this true BCRB is hard to evaluate, we present a closed-form expression of a BCRB, i.e., the Modified BCRB (MBCRB). We discuss, based on the theoretical and simulation results, the benefit of using the a priori information and, the past and the future observations for the complex gains estimation.

I. INTRODUCTION

In the case of wideband Orthogonal Frequency Division Multiplexing (OFDM) mobile communication systems, high speeds of terminals cause Doppler effects that result intersub-carrier interference (ICI) and could seriously affect the performance. In such case, dynamic channel estimation [START_REF] Tang | Pilot-assisted timevarying channel estimation for OFDM systems[END_REF] [10] is a fundamental function, because the radio channel is frequency selective and time-varying. Channel estimation can be summarized to estimate certain physical propagation parameters, such as multi-path delays and multi-path complex gains. In Radio-Frequencies transmission, the delays are quasi invariant over several OFDM symbols but the complex gains may change significantly, even within one OFDM symbol. Exploiting the channel nature and assuming the availability of delay information, a lot of methods estimate the timevariations of the multi-path complex gains in OFDM [START_REF] Hijazi | Joint Data QR-Detection and Kalman Estimation for OFDM Time-varying Rayleigh Channel Complex Gains[END_REF] [4] [START_REF] Hijazi | Polynomial estimation of time-varying multipath gains with intercarrier interference mitigation in OFDM systems[END_REF] [6] [START_REF] Hijazi | Rayleigh Time-varying Channel Complex Gains Estimation and ICI Cancellation in OFDM Systems[END_REF] [START_REF] Hijazi | Time-varying Channel Complex Gains Estimation and ICI Suppression in OFDM Systems[END_REF] and CDMA [START_REF] Simon | Synchronization over rapidly timevarying multipath channel for CDMA downlink RAKE receivers in Time-Division mode[END_REF] systems.

In this context the question arises of the ultimate accuracy that can be achieved in channel estimation operations. Establishing bounds to such an accuracy is an important goal since it provides benchmarks for evaluating the performance of channel estimators. Tools to approach this problem are available from the parameters estimation theory [START_REF] Van Trees | Detection, estimation, and modulation theory: Part I[END_REF] [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF] in the form of Cramer-Rao Bounds (CRBs), which give fundamental lower limits to the variance of any parameter estimator. A Modified CRB (MCRB), easier to evaluate than the Standard CRB (SCRB), has been introduced in [START_REF] Gini | The modified Cramer-Rao bound in vector parameter estimation[END_REF] [START_REF] D'andrea | The modified Cramer-Rao bound and its application to synchronization problems[END_REF]. The MCRB proves useful when, in addition to the parameter to be estimated, the observed data also depend on other unwanted parameters. More recently, the problem of deriving CRBs suited to time-varying parameters has been addressed throughout the Bayesian context. In [START_REF] Tichavsky | Posterior Cramer-Rao bound for discret-time nonlinear filtering[END_REF], the authors propose a general framework for deriving analytical expression of online CRBs.

In [START_REF] Hijazi | Bayesian Cramer-Rao Bounds for complex gain parameters estimation of slowly varying rayleigh channel in OFDM systems[END_REF] [2], we have derived the expression of the online BCRB, in data-aided (DA) and non-data-aided (NDA) contexts, for the dynamic estimation of time-varying multipath Rayleigh channel complex gains with slowly variations (i.e., time-invariant complex gains within one OFDM symbol). In [3] [4], in order to evalute the quality of our complex gains estimator, we have just given the expression of the on-line BCRB in DA context for the case of rapidly time-varying channels.

In this contribution we investigate the BCRB related to the estimation of rapidly time-varying Rayleigh channel complex gains with Jakes spectrum for OFDM systems (i.e., timevarying complex gains within one OFDM symbol). Explicit expressions of the BCRB and its variant, MBCRB, are provided in NDA and DA contexts and, in on-line and off-line scenarios.

This paper is organized as follows: Section II sets the system model, whereas Section III recalls the general BCRB and the modified MBCRB. Section IV derives the BCRB and the MBCRB for "time-varying" multi-path complex gains estimation. Section V illustrates and interpretes different results. Finally, our conclusions are presented in Section VI.

The notations adopted are as follows: Upper (lower) bold face letters denote matrices (column vectors). [x] k denotes the kth element of the vector x, and [X] k,m denotes the [k, m]th element of the matrix X. We will use the matlab notation X[k1:k2,m1:m2] to extract a submatrix within X from row k 1 to row k 2 and from column m 1 to column m 2 . I N is a N × N identity matrix and 0 N is a N × N matrix of zeros. diag{x} is a diagonal matrix with x on its main diagonal, diag{X} is a vector whose elements are the elements of the main diagonal of X and blkdiag{X, Y} is a block diagonal matrix with the matrices X and Y on its main diagonal. The superscripts

II. OFDM SYSTEM AND CHANNEL MODELS

A. OFDM System Model

Consider an OFDM system with N sub-carriers, and a cyclic prefix length N g . The duration of an OFDM symbol is T = vT s , where T s is the sampling time and v = N +N g . Let

x (n) = £ x (n) [-N 2 ], x (n) [-N 2 +1], ..., x (n) [ N 2 -1] ¤ T be the nth transmitted OFDM symbol, where {x (n) [b]} are normalized symbols (i.e., E £ x (n) [b]x (n) [b] * ¤ = 1
). After transmission over a multi-path Rayleigh channel, the nth received OFDM symbol y

(n) = £ y (n) [-N 2 ], y (n) [-N 2 + 1], ..., y (n) [ N 2 -1] ¤ T is
given by [START_REF] Hijazi | Polynomial estimation of time-varying multipath gains with intercarrier interference mitigation in OFDM systems[END_REF] [7] [START_REF] Hijazi | Time-varying Channel Complex Gains Estimation and ICI Suppression in OFDM Systems[END_REF]:

y (n) = H (n) x (n) + w (n) (1)
where

w (n) = £ w (n) [-N 2 ], w (n) [-N 2 +1], ..., w (n) [ N 2 -1]
¤ T is a complex Gaussian noise vector with covariance matrix σ 2 I N and H (n) is a N × N channel matrix with elements given by:

[H (n) ] k,m = 1 N L X l=1 h e -j2π( m-1 N -1 2 )τ l N-1 X q=0 α (n) l (qT s )e j2π m-k N q i (2)
where L is the number of paths, α l is the lth complex gain of variance σ 2 α l and τ l × T s is the lth delay (τ L < N g ). The L individual elements of {α (n) l (qT s ) = α l (qT s + nT )} are uncorrellated. They are wide-sense stationary (WSS), narrowband complex Gaussian processes, with the so-called Jakes' power spectrum of maximum Doppler frequency f d , i.e.,

E [α l (q 1 T s )α * l (q 2 T s )] = σ 2 α l J 0 ¡ 2πf d T s (q 1 -q 2 ) ¢ [13]
. The average energy of the channel is normalized to one, i.e., P L l=1 σ 2 α l = 1.

B. Complex Gain Polynomial Modeling

In [START_REF] Mostofi | ICI mitigation for pilot-aided OFDM mobile systems[END_REF], a piece-wise linear method is used to approximate the equivalent discrete-time channel taps. In [START_REF] Hijazi | Polynomial estimation of time-varying multipath gains with intercarrier interference mitigation in OFDM systems[END_REF] [6], the authors show that the time-variation of Rayleigh channel complex gain, within N c OFDM symbols, can be approximated by a polynomial model of N c coefficients, choosen according to the Doppler spread f d T .

In this section, we show that, whatever f d T ≤ 0.5, each Rayleigh channel complex gain α

(n) l = £ α (n) l (-N g T s ), ..., α (n) l ¡ (N -1)T s
¢¤ T can be modeled as a polynomial time-variation of N c ≤ 5 coefficients (i.e., a (N c -1) degree polynomial), within one OFDM symbol. The optimal polynomial α (n) pol l , which is least-squares fitted (linear and polynomial regression) [START_REF]Wikipedia contributors[END_REF] to α

(n) l , and its N c coefficients c (n) l = £ c (n) 1,l , ..., c (n) N c ,l
¤ T are given by:

α (n) pol l = Q T c (n) l = Sα (n) l and c (n) l = ³ QQ T ´-1 Qα (n) l (3)
where

Q is a N c × v matrix of elements [Q] k,m = (m -N g - 1) (k-1) and S = Q T ³ QQ T ´-1 Q is a v×v matrix.
It provides the MMSE approximation for all polynomials containing N c coefficients, given by:

MMSE (0) l = 1 v Tr ³ MMSE (0)
l ´( 4)

MMSE (p) l = E h ξ (n) l ξ (n-p) l H i = ¡ I v -S ¢ R (p) α l ¡ I v -S T ¢ with ξ (n) l = α (n) l -α (n)
pol l is the model error and

R (p) αl = E • α (n) l α (n-p) l H ¸is the v × v correlation matrix of α (n) l
with elements given by: (n) l are correlated complex Gaussian variables with zeromeans and correlation matrix given by:

[R (p) α l ] k,m = σ 2 α l J 0 µ 2πf d T s (k -m + pv) ¶ (5)
R (p) cl = E[c (n) l c (n-p) l H ] = ³ QQ T ´-1 QR (p) αl Q T ³ QQ T ´-1 (6
) It should be noted that the last coefficients are very small. So, it is very difficult to find an estimator that can give a good estimation of the small coefficients in presence of noise. In the sequel, we will study the performance of the coefficients estimator in terms of N c and f d T .

Under this polynomial approximation, the observation model in [START_REF] Hijazi | Bayesian Cramer-Rao Bounds for complex gain parameters estimation of slowly varying rayleigh channel in OFDM systems[END_REF] for the nth OFDM symbol can be rewritten as:

y (n) = K (n) c (n) + w (n) (7)
where

c (n) = [c (n) 1 T , ..., c (n) L T ] T is a LN c × 1 vector, K (n) = 1 N [Z (n) 1 , ..., Z (n) L ] is a N × LN c matrix and Z (n) l = [M 1 diag{x (n) }f l , ..., M Nc diag{x (n) }f l ] is a N × N c matrix,
where f l is the lth column of the N × L Fourier matrix F and M d is a N × N matrix given by:

[F] k,l = e -j2π( k-1 N -1 2 )τ l and [M d ] k,m = N -1 X q=0 q d-1 e j2π m-k N q (8)

III. CRAMER-RAO BOUNDS (CRBS)

In this section, we present the family of Cramer-Rao Bounds (CRBs). The CRBs provide a lower bound on the Mean Square Error (MSE) achievable by any unbiased estimator. We give the general expression of the Bayesian CRB (BCRB) and its Modified Version (MBCRB). The BCRB is particularly suited for problems where a priori information is available.

Let ĉ(y) denotes an unbiased estimator of c using the set of measurements y. The estimation of c can be considered following two main scenarios off-line and on-line. In the offline scenario, the receiver waits until the whole observation frame, i.e., y = [y (1) T , ..., y (K) T ] T , has been received in order to estimate c = [c (1) T , ..., c (K) T ] T . In the on-line scenario, the receiver estimates c (n) based on the current and previous observations only, i.e., y = [y (1) T , ..., y (n) T ] T . In the sequel, the BCRB will be considered within the context of both the off-line and the on-line scenarios. The BCRB has been proposed in [START_REF] Van Trees | Detection, estimation, and modulation theory: Part I[END_REF] such that:

E y,c • ¡ ĉ(y) -c ¢¡ ĉ(y) -c ¢ H ¸≥ BCRB(c) (9)
The BCRB 1 is the inverse of the Bayesian Information Matrix (BIM), which can be written as:

B = E c £ F(c) ¤ + E c £ -∆ c c ln ¡ p(c) ¢¤ ( 10
)
where p(c) is the prior distribution and F(c) is the Fisher Information Matrix (FIM) defined as:

F(c) = E y|c £ -∆ c c ln ¡ p(y|c) ¢¤ (11)
where p(y|c) is the conditional probability density function of y given c. Unfortunately, in most cases of NDA context, the computation of F(c) is generally quite tedious because the p(y|c) cannot be carried out analytically due to the nuisance parameters x = [x (1) T , ..., x (K) T ] T , which are OFDM symbols in our case. In order to circumvent this problem, a Modified BCRB (MBCRB) has been proposed in [START_REF] Bobrovosky | Some classes of global Cramer-Rao bounds[END_REF]. This MBCRB is the inverse of the following information matrix:

C = E c £ G(c) ¤ + E c £ -∆ c c ln ¡ p(c) ¢¤ (12)
where G(c) is the modified FIM defined as:

G(c) = E x E y|x,c £ -∆ c c ln ¡ p(y|x, c) ¢¤ (13)
It should be noted that the MBCRB in NDA context is equal to the BCRB in DA context (i.e. the nuisance parameters x are a priori known). In our objective, we are interested in the estimation of the complex gains α = [α (1)

T , ..., α (K) T ] T , where

α (n) = h α (n) 1 T , ..., α (n) L T i T
. Actualy, α is related to c as:

α = Qc + ξ (14)
where

Q = blkdiag n Q T , ..., Q T o is a KLv × KLN c matrix and ξ = [ξ T (1) , ..., ξ T (K) ] T with ξ (n) = h ξ (n) 1 T , ..., ξ (n) L T i T .
Hence the estimation of α can be stated by: α = Qĉ. By neglecting the cross-covariance terms between the errors α polα and ξ, we can write:

E • ¡ α -α ¢¡ α -α ¢ H ¸= E • ¡ α -α pol ¢¡ α -α pol ¢ H ¸+ E £ ξξ H ¤ (15)
1 We recall that Standard Cramer-Rao Bound (SCRB) is the inverse of the Fisher Information Matrix (FIM) (the a priori information is not used).

where α pol = Qc. So, using the transformation of parameters property defined in [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF], we obtain the BCRB for the estimation of α from the BCRB for c as:

BCRB(α) = ¡ ∇ c α pol ¢ BCRB(c) ¡ ∇ c α T pol ¢ + E £ ξξ H ¤ = Q BCRB(c) Q T + MMSE (16)
where the KLv × KLv matrix MMSE is given by:

MMSE [i(l,p),i(l,p 0 )] = MMSE (p-p 0 ) l for l∈[1,L] p,p 0 ∈[0,K-1] (17)
with i(l, p) = 1 + (l -1)v + pLv : lv + pLv and MMSE (p) l is the correlation matrix of the model error ξ l (n) defined in (4). Notice that there are zero matrices between the block matrices MMSE (p) l since the L complex gains are uncorrellated. For K = L = 2, MMSE is given by:

MMSE = ⎡ ⎢ ⎢ ⎢ ⎣ MMSE (0) 1 0 v MMSE (-1) 1 0 v 0 v MMSE (0) 2 0 v MMSE (-1) 2 MMSE (1) 1 0 v MMSE (0) 1 0 v 0 v MMSE (1) 2 0 v MMSE (0) 2 ⎤ ⎥ ⎥ ⎥ ⎦
The computation of the off-line BCRB associated to the estimation of α (n) is given by:

BCRB(α (n) ) offline = Tr ³ BCRB(α) [i(n),i(n)]
´ [START_REF] Tichavsky | Posterior Cramer-Rao bound for discret-time nonlinear filtering[END_REF] where the sequence of indices i(n) = 1 + (n -1)Lv : nLv, with n ∈ [1, K]. The on-line BCRB associated to the observation vector y = [y (1) T , ..., y (K) T ] T is given by:

BCRB(α (K) ) online = Tr ³ BCRB(α) [i(K),i(K)] ´(19)
The definitions in ( 16), ( 18) and ( 19) will stand for the closed form of BCRB, i.e., MBCRB and ABCRB.

IV. BCRB FOR TIME-VARYING COMPLEX GAINS ESTIMATION

In this section, we present a closed-form expression for a BCRB related to the estimation of the polynomial coefficients c (n) of the multi-path complex gains in NDA OFDM systems. This bound is derived for time-varying complex gains within one OFDM symbol. In DA context, we deduce the computation of the true BCRB from the computation of the MBCRB in NDA.

Computation of E c

£ F(c) ¤ : The observation model is presented in [START_REF] Hijazi | Rayleigh Time-varying Channel Complex Gains Estimation and ICI Cancellation in OFDM Systems[END_REF]. Using the whiteness of the noise w = [w (1) T , ..., w (K) T ] T and the independence of the transmitted OFDM symbols x, we then obtain that:

∆ c c ln ¡ p(y|c) ¢ = K X n=1 ∆ c c ln ¡ p(y (n) |c (n) ) ¢ (20)
It is important to note that each term of the summation ( 20) is a KLN c × KLN c block diagonal matrix with only one nonzero LN c × LN c block matrix, namely:

∆ c c ln ¡ p(y (n) |c (n) ) ¢ [i 0 (n),i 0 (n)] = ∆ c (n) c (n) ln ¡ p(y (n) |c (n) ) ¢ (21)
where

i 0 (n) = 1 + (n -1)LN c : nLN c with n ∈ [1, K]. As a direct consequence, ∆ c c ln ¡ p(y|c)
¢ is a block diagonal matrix with the nth diagonal block given by [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation Theory[END_REF]. Moreover, because of the circularity of the observation noise, the expectation of ( 21) with respect to y (n) and c (n) does not depend on c (n) . We then obtain:

E c £ F(c) ¤ = blkdiag {J, J, ..., J} (22)
where J is a LN c × LN c matrix defined as:

J = E y,c £ -∆ c (n) c (n) ln ¡ p(y (n) |c (n) ) ¢¤ ( 23
)
The log-likelihood function in (23) can be expanded as:

ln ¡ p(y (n) |c (n) ) ¢ = ln ³ X x (n) p(y (n) |x (n) , c (n) )p(x (n) )
´( 24)

The vector y (n) for given x (n) and c (n) is a complex Gaussian vector with mean vector

m (n) = K (n) c (n) and covariance matrix σ 2 I N . Thus, p(y (n) |x (n) , c (n)
) is defined as:

p(y (n) |x (n) , c (n) ) = 1 |πσ 2 I N | e -1 σ 2 (y (n) -m (n) ) H (y (n) -m (n)) (25)
Since each element of the vector m (n) depends on all components of x (n) then, the computation of J is a demanding task. Hence, we resort to compute the MBCRB. Following the same reasoning as before, we have:

E c £ G(c) ¤ = blkdiag {J m , J m , ..., J m } (26)
where J m is a LN c × LN c matrix defined as:

J m = E y,x,c £ -∆ c (n) c (n) ln ¡ p(y (n) |x (n) , c (n) ) ¢¤ (27)
By taking the second derivative of the natural logarithm (ln) of ( 25) with respect to c (n) , we simply obtain that:

∆ c (n) c (n) ln ¡ p(y (n) |x (n) , c (n) ) ¢ = - 1 σ 2 K H (n) K (n) (28)
Consequently, we obtain that (see Appendix A):

J m = 1 σ 2 E x h K H (n) K (n) i = 1 N 2 σ 2 F H MF ( 29
)
where M and F are a NN c ×NN c and a NN c ×LN c matrices, respectively, defined as:

M = ⎡ ⎢ ⎣ M 1,1 • • • M 1,Nc . . . . . . . . . M Nc,1 • • • M Nc,Nc ⎤ ⎥ ⎦ (30) F = £ F 1 • • • F L ¤ ( 31
)
where M d,d 0 and F l are a N × N and a NN c × N c matrices, respectively, defined as:

M d,d 0 = diag © diag © M H d M d 0 ª ª (32) F l = blkdiag {f l , f l , ..., f l } (33)
Computation of E c £ -∆ c c ln ¡ p(c) ¢¤ : c is a complex Gaussian vector with zero mean and covariance matrix R c of size KLN c × KLN c defined as:

R c[i 0 (l,p),i 0 (l,p 0 )] = R (p-p 0 ) c l for l∈[1,L] p,p 0 ∈[0,K-1] ( 34
)
where i 0 (l, p) = 1 + (l -1)N c + pLN c : lN c + pLN c and R (p) c l is the correlation matrix of c l (n) defined in [START_REF] Hijazi | Polynomial Estimation of Time-varying Multipath Gains with ICI Mitigation in OFDM Systems[END_REF]. Thus, the probability density function p(c) is defined as:

p(c) = 1 |πR c | e -c H R -1 c c (35)
Taking the second derivative of the natural logarithm (ln) of ( 35) with respect to c and making the expectation over c, we simply obtain that:

E c £ -∆ c c ln ¡ p(c) ¢¤ = R -1 c (36)
The MBCRB for the estimation of c is given by:

MBCRB(c) = ³ blkdiag {J m , J m , ..., J m } + R -1 c
´-1

Notice that the MBCRB is usually looser than the BCRB. As in ( 16), the MBCRB for the estimation of α is given by:

MBCRB(α) = Q MBCRB(c) Q T + MMSE (38)
In data-aided (DA) context, the transmitted data symbols x (n) are known at the receiver. Hence, the matrix J is computed like J m , but without averaging over the data symbols x (n) , and consequently it depends on the nth transmitted OFDM symbol. Thus, J (n) is given by:

J (n) = 1 σ 2 K H (n) K (n) = 1 N 2 σ 2 F H (n) MF (n) (39)
where the matrix F (n) is computed like F but by replacing f l in equation (33) by diag{x (n) }f l . The BCRB for the estimation of c in DA context is given by:

BCRB(c) = ³ blkdiag © J (1) , J (2) , ..., J (K) ª + R -1 c ´-1 (40)
and consequently the BCRB for the estimation of α as [START_REF] Gini | The modified Cramer-Rao bound in vector parameter estimation[END_REF].

It should be noted that BCRB for the estimation of α in DA context depends on the transmitted data sequence x.

V. DISCUSSION In this section, we bring to the fore the behavior of the previous bounds, namely the off-line and the on-line MBCRBs (BCRB in DA) for the complex gains estimation. A normalized 4QAM OFDM system, N = 128 subcarriers, N g = N 8 subcarriers is used (note that SNR = 1 σ 2 ). The normalized channel model is Rayleigh with L = 6 paths of parameters given in [START_REF] Hijazi | Polynomial estimation of time-varying multipath gains with intercarrier interference mitigation in OFDM systems[END_REF] [6] [7] [START_REF] Hijazi | Time-varying Channel Complex Gains Estimation and ICI Suppression in OFDM Systems[END_REF]. We consider the case of high Doppler speed with 0.05 ≤ f d T ≤ 0.5, and 2 ≤ N c ≤ 5 for the polynomial modeling. In the following figures, the term MSE stands for Mean Square Error. Fig. 2 superimposes versus time index, the on-line MBCRB and the off-line MBCRB for f d T = 0.1, N c = 2 and different block-observation lengths K at SNR = 10dB . In the off-line context, we can see that the best complex gains estimation is achieved at the midblock, whereas the estimates are likely to be poorer at the block border. This stems from the fact that in the center position of the polynomial coefficients vector c we have more adjacent (past or future) and strongly correlated variables than at the border of the vector c. Concerning the online bound, at the beginning when the number of observations increases, the estimator takes into account more and more information and the estimation is improved; the bound thus decreases and converges to an asymptote. The estimation performance is then limited by the observation noise independently of the number of observations taken into account. However, in order to reach the asymptote, it is sufficient to use 3 past OFDM symbols for f d T = 0.1.

As we have seen in DA context, the BCRB depends on the transmitted data sequence. We now study the DA bound behavior for different data sequences. Fig. 3 gives the DA BCRB versus SNR for f d T = 0.3 and N c = 3. We can observe that there is bad sequences (period infinity, i.e., all the bits are equals) and good sequences (Maximum length sequence MLS [START_REF] Peterson | Introduction of Spread Spectrum Communications[END_REF] with 13 shift registers). We have observed via experimentations that sequences with good deterministic autocorrelation (i.e, near Dirac impulse, as for exemple sequences of type MLS) are good sequences. In such case, the bounds are very similar. That is why in the paper we have given only one example (13 registers with feedback polynomial [20033] MLS [START_REF] Peterson | Introduction of Spread Spectrum Communications[END_REF]) of these sequences. However, if we use sequences with bad deterministic auto-correlation, like the sequences of Walsh-Hadamard (period K/8: bit = 0 0 0 during K.T/16 and bit = 0 1 0 during K.T/16), then the performance will be degraded as shown in this figure.

We now study the bound behavior versus the polynomial coefficients N c and SNR over a block of K = 10. Fig. 4 gives the MBCRBs for f d T = 0.5 in terms of SNR in (a) and in [START_REF] Hijazi | Polynomial estimation of time-varying multipath gains with intercarrier interference mitigation in OFDM systems[END_REF] X X X X X X X X terms of N c in (b). We observe in (a) that, whatever the SNR, the bound is not always decreasing in terms of N c but at high SNR, the bound converges to the MMSE (the model error). As we see in (b), for SNR =15dB, 25dB and 35dB, the minimum of the bound is obtained at N c = 3, 4 and 5 polynomial coefficients, respectively. This is due to the last coefficients which will be poorly estimated in presence of noise. Indeed, they are negligible compared to the noise level. Hence, in order to have a good estimation of the complex gains timevariation, we have to choose N c according to SNR and f d T .

The Table I terms of f d T . This is because the correlation between variables becomes stronger when f d T decreases. So, the estimation gain for slow channel variations is more significant.

VI. CONCLUSION

In this contibution, we have derived an analytical expression of a BCRB for the estimation of Rayleigh channel complex gains with time variations within one OFDM symbol. We have introduced a New BCRB (NBCRB) and we have shown that a good estimation of the complex gains time variation can be obtained by choosing the number of polynomial coefficients according to the noise level and the Doppler spread. These bounds are useful when analyzing the performance of complex gains estimators in DA and NDA contexts and in on-line and off-line scenarios. Moreover, we have shown the benefit of using the past and the future OFDM symbols in channel estimation process, whereas most methods use only the current symbol.

APPENDIX A EVALUATION OF J m

In this Appendix, we detail the calculus to obtain the expression of J m defined in (29). Using the definition of K (n) in section II, we have:

A = K H (n) K (n) = 1 N 2 ⎡ ⎢ ⎣ A 1,1 • • • A 1,L . . . . . . . . . A L,1 • • • A L,L ⎤ ⎥ ⎦(42)
where A l,l 0 = Z

(n) l H Z (n)
l 0 is a N c × N c matrix with elements given by:

£ A l,l 0 ¤ d,d 0 = f H l diag{x H (n) }M H d M d 0 diag{x (n) }f l 0 (43)
Taking the expectation of (43) over x, we obtain:

E x h £ A l,l 0 ¤ d,d 0 i = f H l M d,d 0 f l 0 (44)
since the symbols are normalized an uncorrelated with respect to each other. Consequently, we obtain that:

E x h A l,l 0 i = F H l M d,d 0 F l 0 (45)
and finally we obtain the expression of J m defined in (29).

Fig. 1 .

 1 Fig. 1. MMSE for a normalized channel with L = 6 paths and v = 144

Fig. 1

 1 Fig. 1 gives the MMSE in terms of f d T for different value of N c . As can be seen, for f d T ≤ 0.5 and N c = 5, we have MMSE < 4 • 10 -7 . This proves that, for high values of f d T , α (n) l can be represented by a polynomial model of N c ≤ 5 coefficients. c

Fig. 2 .Fig. 3 .

 23 Fig. 2. BCRBs vs K, for SNR =10dB, f d T = 0.1 and N c = 2

Fig. 4 .

 4 Fig. 4. (a) BCRBs vs SNR for f d T = 0.5 and Nc = 2 to 5 ; (b) BCRBs vs Nc for f d T = 0.5 and SNR =15dB, 25dB and 35dB

= 2 1 Nc = 3 2 Nc = 3 3 Nc = 3 4 Nc = 3

 213233343 Nc = 2 Nc = 2 Nc = 2 0.Nc = 3 Nc = 3 Nc = 3 0.Nc = 3 Nc = 3 Nc = 4 0.Nc = 3 Nc = 4 Nc = 4 0.Nc = 4 Nc = 4 Nc = 4

Fig. 5 .

 5 Fig. 5. NBCRBs vs f d T for SNR = 20dB

TABLE I THE

 I MINIMUM OF THE BCRB(α) FOR GSM CHANNEL

  shows how to select N c , for realistic values of SNR and different values of f d T , such that the bound is minimal.

	For example if f d T = 0.3, we choose N c = 3 and 4 for
	SNR ∈ [0; 27] and SNR ∈ [27; 40], respectively. Therefore, we introduce a New BCRB (NBCRB) which is independant
	of N c , defined as:
	NBCRB(α) = min


Jean-Louis Clerc

Bent Ørsted

Conformally invariant trilinear forms on the sphere

Keywords:

come

Conformally invariant trilinear forms on the sphere

Introduction

The motivation for this article came from the paper [START_REF] Bernstein | Estimates of automorphic functions[END_REF] by J. Bernstein and A. Reznikov. In order to estimate automorphic coefficients, they use trilinear invariant forms for G = P GL 2 (R). Their paper shows the importance of three related questions of harmonic analysis on P GL 2 (R) :

given three representations (π 1 , H 1 ), (π 2 , H 2 ), (π 3 , H 3 ) in the principal spherical 1 series of the group G, i) construct a trilinear invariant functional on H 1 × H 2 × H 3 ii) prove uniqueness (up to a scalar) of such a trilinear invariant functional iii) compute the value of the trilinear functional on the K fixed vectors of H 1 , H 2 , H 3 respectively.

The representations are realized on the unit circle, on which the group G acts projectively, and indexed by a complex number. One possible generalization consists in replacing the unit circle by the n -1 dimensional sphere S, under the action of the conformal group G = SO 0 (1, n). For this case, 1 with respect to the maximal compact subgroup K = P O(2) of G 1 we present here a construction of an invariant trilinear form (item i)), which uses an analytic continuation over three complex parameters and discuss the uniqueness statement (item ii)) for generic values of the parameters. The computation of the normalization factor (item iii)), even for more geometric situations, will be published elsewhere, (cf [START_REF] Clerc | Generalized Bernstein-Reznikov integrals[END_REF]).

In section 1, we recall elementary facts about conformal geometry of the sphere (in particular we give a description of the orbits of G in S × S × S), and introduce the representations which the paper is concerned with. In section 2, we define formally the trilinear invariant form, study the domain of convergence of the corresponding integral and determine the analytic continuation in the three complex parameters corresponding to the three representations. In section 3, we prove the uniqueness statement. The proof relies on Bruhat's theory, which we recall in an appendix at the end of the paper. In section 4, we give an alternative approach to the construction of an invariant trilinear form, using a realization of the tensor product of two of the representations involved, thus making connection with [START_REF] Molcanov | Tensor products of unitary representions of the threedimensional Lorentz group[END_REF].

The present paper only deals with regular values of the parameters. The residues at poles will yield new conformally invariant trilinear forms, supported by the singular orbits of G in S × S × S and involving differential operators akin the Yamabe operator on the sphere, worth of a further study. Other geometric situations are potential domains for similar results. Let P 1 , P 2 , P 3 be three parabolic subgroups of a semi-simple Lie group, such that G acts on G/P 1 × G/P 2 × G/P 3 with a finite number of orbits (see [START_REF] Littelmann | On spherical double cones[END_REF], [START_REF] Magyar | Multiple flag varieties of finite type[END_REF]). Take three representations induced by characters of P 1 , P 2 , P 3 . Invariant trilinear forms for these three representations can plausibly be studied along the same lines as in the present paper. The case of three copies of the Shilov boundary S of a bounded symmetric domain of tube type is specially appealing (see [START_REF] Clerc | Orbits of triples in the Shilov boundary of a bounded symmetric domain[END_REF] for a description of the orbits of G in S × S × S).

Let us mention the paper by A. Deitmar [START_REF] Deitmar | Invariant triple products[END_REF], which has some overlap with our results. Previous work on the subject also includes [START_REF] Oksak | Trilinear Lorentz invariant forms[END_REF] and [START_REF] Loke | Trilinear forms of gl 2[END_REF].

The first author thanks D. Barlet and L. Bérard Bergery for conversations on various aspects of this paper and the Mathematics Department of Aarhus University for welcoming him during the preparation of the present work.

1 Conformal geometry of the sphere Let S = S n-1 = {x = (x 1 , x 2 , . . . , x n ),

|x| 2 = x 2 1 + x 2 2 + • • • + x 2 n = 1}
be the unit sphere in R n . We usually (and tacitly) assume n ≥ 3, as the case n = 2 needs a few minor changes, which are occasionally mentioned in the text. The group K = SO(n) operates on S. Let 1 + = (1, 0, . . . , 0), 1 -= (-1, 0, . . . , 0).

The stabilizer of 1 + in K is the subgroup

M ≃ SO(n -1) = 1 0 0 u , u ∈ SO(n -1) .
With this notation, S ≃ K/M is a compact Riemannian symmetric space. Another realization of the sphere is useful. Let R 1,n be the real vector space R n+1 with the quadratic form

q(x) = [x, x] = x 2 0 -(x 2 1 + x 2 2 + • • • + x 2 n ) . (1)
To x = (x 1 , x 2 , . . . , x n ) in S associate x = (1, x 1 , x 2 , . . . , x n ) in R 1,n . The correspondance x -→ R x associates to a point in S an isotropic line in R 1,n . The correspondance is easily seen to be bijective. The group G = SO 0 (1, n) acts naturally on the set of isotropic lines, and hence on S. Explicitly, for x in S and g in G, g(x) is the unique point in S such that

g(x) = (g x) 0 -1 g x .
For x in S and g in G, set

κ(g, x) = (g x) 0 -1 . (2)
Let x, y be in S. The following identity holds

[ x, y] = 1 -x, y = 1 2 |x -y| 2 (3)
so that for g in G,

|g(x) -g(y)| = κ(g, x) 1 2 |x -y| κ(g, y) 1 2 . ( 4
)
The infinitesimal version of (4) is

|Dg(x) ξ| = κ(g, x)|ξ| (5)
for ξ any tangent vector to S at x, where g is in G and Dg(x) stands for the differential at x of the map x → g(x). Hence the action of G on S is conformal, and κ(g, x) is interpreted as the conformal factor of g at x.

We look at K as a subgroup of G. It is a maximal compact subgroup of G. The stabilizer in G of the point 1 + is the parabolic subgroup P = M AN , where

A =              a t =        cosh t sinh t 0 . . . 0 sinh t cosh t 0 . . . 0 0 0 1 . . . . . . . . . 0 0 1        , t ∈ R              and N =                n ξ =         1 + |ξ| 2 2 -|ξ| 2 2 ξ t |ξ| 2 2 1 -|ξ| 2 2 ξ t 1 ξ -ξ . . . 1         , ξ ∈ R n-1                .
The element a t (t ∈ R) acts on S by

a t     x 1 x 2 . . . x n     =      sinh t+x 1 cosh t cosh t+x 1 sinh t x 2 cosh t+x 1 sinh t . . . xn cosh t+x 1 sinh t      .
Let N be the subgroup image of N by the standard Cartan involution of G (g -→ (g t ) -1 ) :

N =                n ξ =         1 + |ξ| 2 2 |ξ| 2 2 ξ t -|ξ| 2 2 1 -|ξ| 2 2 -ξ t 1 ξ ξ . . . 1         , ξ ∈ R n-1                .
The map

c : ξ -→ n ξ (1 + ) =      1-|ξ| 2 1+|ξ| 2 2 1+|ξ| 2 ξ      . is a diffeomorphism from R n-1 onto S \ {1 -}.
Its inverse is the classical stereographic projection from the source 1 -onto the tangent space T 1 + S to S at 1 + . When using this chart on S, we refer to the noncompact picture.

Proposition 1.1. The conformal factor κ(g, x) is a smooth function of both g and x, which satisfies the following properties :

i) ∀g 1 , g 2 ∈ G, x ∈ S, κ(g 1 g 2 , x) = κ(g 1 , g 2 (x)) κ(g 2 , x) (6)
ii

) ∀g ∈ G, x ∈ S κ(g, g -1 (x)) = κ(g -1 , x) -1 iii) ∀x ∈ S, k ∈ K κ(k, x) = 1 iv) ∀x ∈ S, t ∈ R, κ(a t , x) = (cosh t + x 1 sinh t) -1 .
Let g in G. As the dimension of the tangent space T x S is n -1, the Jacobian of g at x is given by j(g, x) = κ(g, x) n-1 .

(

) 7
The map c : R n-1 -→ S \ {-1} is also conformal. In fact, one has the following relation, valid for any ξ, η ∈ R n-1 :

|c(ξ) -c(η)| = 2|ξ -η| (1 + |ξ| 2 ) 1 2 (1 + |η| 2 ) 1 2 , (8)
and its infinitesimal version

|dc(ξ) ζ| = 2 1 + |ξ| 2 |ζ| (9) (ζ ∈ R n-1
). The corresponding integration formula reads

S f (x)dσ(x) = R n-1 f (c(ξ)) 2 n-1 (1 + |ξ| 2 ) n-1 dξ . ( 10
)
Later, we will need a description of the orbits of G in S × S × S (for the diagonal action of G). Recall first that the group G in its diagonal action on S × S has two orbits :

S 2 ⊤ = {(x, y) ∈ S × S, x = y}, ∆ S = {(x, x), x ∈ S}.
As base-point in S 2 ⊤ , choose (1 + , 1 -). The stabilizer of (1

+ , 1 -) in G is the subgroup M A. Proposition 1.2. Let n ≥ 3. There are 5 orbits of G in S × S × S, namely O 0 = {(x 1 , x 2 , x 3 ), x i = x j for i = j} O 1 = {(x 1 , x, x), x = x 1 } O 2 = {(x, x 2 , x), x = x 2 } O 3 = {(x, x, x 3 ), x = x 3 } O 4 = {(x, x, x)} .
Proof. The five subsets of S × S × S are invariant under the diagonal action of G. So it suffices to show that G is transitive on each of these sets.

• G is transitive on S, hence on O 4 . Choose (1 + , 1 + , 1 + ) as base-point. The stabilizer of the base-point in G is the subgroup P = M AN .

• The stabilizer P of 1 + is transitive on S \ {1 + } (the action of N is already transitive on S \ {1 + }), so that G is transitive on O j , for j = 1, 2, 3. As base-point in O 3 (similar choices can be made for O 1 and O 2 ) choose (1 + , 1 + , 1 -). The stabilizer of (1 + , 1 + , 1 -) is the subgroup M A.

• Let x 1 , x 2 , x 3 be in O 0 . We may assume w.l.o.g. that x 1 = 1 + , x 2 = 1 - and x 3 = 1 ± . The stabilizer of (1 + , 1 -) in G is M A. The orthogonal projection of x 3 on the hyperplane orthogonal to (1 + , 1 -) is not 0 and can be mapped by M to ce 2 , with 0 < c < 1, so that there exists t ∈ R such that x 3 is conjugate under M to the point (tanh t, 1 cosh t , 0, . . . , 0) = a t (e 2 ).

Hence any triplet in O 0 is conjugate under G to the triplet (1 + , 1 -, e 2 ). Its stabilizer in G is the compact subgroup {k ∈ M, ke 2 = e 2 } ≃ SO(n -2).

When n = 2 (S is the unit circle), then there are two open orbits in S × S × S unde the action of SO 0 (1, 2), each characterized by the value of the orientation index of the three points in S. It is possible to remedy to this fact by using the slightly larger (no longer connected) group O(1, 2) instead of SO 0 (1, 2).

Let dσ be the Lebesgue measure on S and let ω n-1 = S dσ(x). Also set ρ = n-1 2 . Under the action of G, the measure is dσ is transformed according to

S f g -1 (x) dσ(x) = S f (y)κ(g, y) 2ρ dσ(y) (11)
Let λ be in C. For f in C ∞ (S), the formula

π λ (g)f (x) = κ(g -1 , x) ρ+λ f (g -1 (x)) (12)
defines a representation of the group G, which will be denoted by π λ . It is a continous representation when the space C ∞ (S) is equipped with its natural Fréchet topology (see [START_REF] Takahashi | Sur les représentations unitaires des groupes de Lorentz généralisés[END_REF] for a systematic study of these representations).

The representations π λ and π -λ are dual in the sense that, for all ϕ, ψ ∈ C ∞ (S)

S π -λ (g)ϕ(s)π λ (g)ψ(s)ds = S ϕ(s)ψ(s)ds (13)
as can be deduced from the change of variable formula [START_REF] Molcanov | Tensor products of unitary representions of the threedimensional Lorentz group[END_REF]. For λ pure imaginary, the representation π λ can be extended continuously to L 2 (S) to yield a unitary representation of G (this is the reason for using ρ + λ in the definition of π λ ). Observe that the action of K is independant of λ and the constant fonction I S (equal to 1 evereywhere) is fixed by the action of K.

For α in C, let k α be the kernel on S × S defined by

k α (x, y) = |x -y| -ρ+α .
It satisfies the following transformation property under the action of an element g of G:

k α (g(x), g(y)) = κ(g, x) -ρ 2 + α 2 k α (x, y) κ(g, y) -ρ 2 + α 2 ( 14
)
for all x, y in S.

2 Construction of an invariant trilinear form

Formal construction

Let λ 1 , λ 2 , λ 3 be three complex numbers. Let T be a continuous trilinear form from

C ∞ (S) × C ∞ (S) × C ∞ (S) into C. The functional T is said to be invariant w.r.t. π λ 1 , π λ 2 , π λ 3 if, for every f 1 , f 2 , f 3 in C ∞ (S)
, and every g in G,

T (π λ 1 (g)f 1 , π λ 2 (g)f 2 , π λ 3 (g)f 3 ) = T (f 1 , f 2 , f 3 ) . ( 15
)
By Schwartz's kernel theorem, there exists a unique distribution T on S × S × S, such that

T (f 1 , f 2 , f 3 ) = T (f 1 ⊗ f 2 ⊗ f 3 ) (16)
where, as usual,

f 1 ⊗ f 2 ⊗ f 3 is the function on S × S × S defined by (f 1 ⊗ f 2 ⊗ f 3 )(x 1 , x 2 , x 3 ) = f 1 (x 1 )f 2 (x 2 )f 3 (x 3 ).
Let α 1 , α 2 , α 3 be complex numbers, and set α = (α 1 , α 2 , α 3 ). Let K α be the kernel on S × S × S defined by

K α (x 1 , x 2 , x 3 ) = k α 1 (x 2 , x 3 ) k α 2 (x 3 , x 1 ) k α 3 (x 1 , x 2 ) ( 17
)
For f 1 , f 2 , f 3 three functions in C ∞ (S), define the trilinear functional K α by

K α (f 1 , f 2 , f 3 ) = S×S×S K α (x 1 , x 2 , x 3 )f 1 (x 1 )f 2 (x 2 )f 3 (x 3 )dσ(x 1 ) dσ(x 2 ) dσ(x 3 ) (18) whenever it makes sense. Theorem 2.1. Let λ = (λ 1 , λ 2 , λ 3 ) ∈ C 3 . Define α = (α 1 , α 2 , α 3 ) by the relations α 1 = -λ 1 + λ 2 + λ 3 α 2 = -λ 2 + λ 3 + λ 1 α 3 = -λ 3 + λ 1 + λ 2 . (19) Then K α (π λ 1 (g)f 1 , π λ 2 (g)f 2 , π λ 3 (g)f 3 ) = K α (f 1 , f 2 , f 3 ) , (20)
whenever the integral on the right handside is defined.

The proof is obtained by the change of variables y j = g -1 (x j ) (j = 1, 2, 3) in the integral defining the left hand-side, using [START_REF] Takahashi | Sur les représentations unitaires des groupes de Lorentz généralisés[END_REF] and [START_REF] Molcanov | Tensor products of unitary representions of the threedimensional Lorentz group[END_REF]. Observe that the right-handside integral converges if, for i = j, the supports of f i and f j are disjoint, or if S×S×S |K α (x 1 , x 2 , x 3 )| dσ(x 1 )dσ(x 2 )dσ(x 3 ) < +∞.

The version of the trilinear functional in the noncompact picture will be useful. Let α = (α 1 , α 2 , α 3 ) ∈ C 3 and set for y 1 , y 2 , y 3 ∈ R n-1

J α (y 1 , y 2 , y 3 ) = |y 1 -y 2 | -ρ+α 3 |y 2 -y 3 | -ρ+α 1 |y 3 -y 1 | -ρ+α 2 (21)
and, whenever it makes sense, let J α be the associated distribution on

R n-1 × R n-1 × R n-1 given by J α (ϕ) = J α (y 1 , y 2 , y 3 ) ϕ(y 1 , y 2 , y 3 ) dy 1 dy 2 dy 3 , (22)
(ϕ ∈ C ∞ (R n-1 × R n-1 × R n-1 )). Moreover, let Ψ α be the function defined on R n-1 × R n-1 × R n-1 by Ψ α (y 1 , y 2 , y 3 ) = (1+|y 1 | 2 ) -ρ-α 2 2 - α 3 2 (1+|y 2 | 2 ) -ρ-α 3 2 - α 1 2 (1+|y 3 | 2 ) -ρ-α 1 2 - α 2 2 . Proposition 2.1. Let f ∈ C ∞ (S × S × S). Then K α (f ) = 2 3(n-1) J α (f • c)Ψ α ( 23
)
whenever the left hand side is defined.

This is merely the change of variable [START_REF] Magyar | Multiple flag varieties of finite type[END_REF] in the integral (18).

Integrability of the kernel K α

Theorem 2.2. The kernel K α is integrable on S × S × S if and only if

ℜα j > -ρ, j = 1, 2, 3 (24)
ℜα 1 + ℜα 2 + ℜα 3 > -ρ (25)
Proof. It is enough to study the integral when the α's are real, in which case the kernel K α is positive. Let U be a (small) neighborhood of (1

+ , 1 + , 1 + ) in S × S × S.
Let g be in G. As x varies in S, j(g, x) remains bounded from below and from above. Thanks to the transformation property of the kernel k α (14), the integrals of K α over U and over g(U ) are of the same nature. As U meets all G-orbits, a partition of unity argument shows that the integrability over S of the kernel K α is equivalent to its integrability over U . Now use the stereographic projection to see that the integrability of K α over U is equivalent to the integrability of J α over c -1 (U ), which is a (small) neighborhood of (0, 0, 0

) in R n-1 × R n-1 × R n-1 .
Hence, we are reduced to study the convergence of the integral

I 1 = |ξ 1 |<δ |ξ 2 |<δ |ξ 3 |<δ |ξ 1 -ξ 2 | α 3 -ρ |ξ 2 -ξ 3 | α 1 -ρ |ξ 3 -ξ 1 | α 2 -ρ dξ 1 dξ 2 dξ 3 ,
where δ is a small positive number. Set

y 1 = ξ 1 , y 2 = ξ 1 -ξ 2 , y 3 = -ξ 2 + ξ 3 .
Then, after integrating with respect to y 1 , the integral I 1 is seen to be of the same nature as the integral

I 2 = |y 2 |<δ |y 3 |<δ |y 2 | α 3 -ρ |y 3 | α 2 -ρ |y 2 -y 3 | α 1 -ρ dy 2 dy 3 .
Let Σ be the unit ball in R (n-1) × R (n-1) , and let dσ be the Lebesgue measure on Σ. Use polar coordinates (r, (σ 2 , σ 3 )) defined by

y 2 = rσ 2 , y 3 = rσ 3 , r 2 = |y 2 | 2 + |y 3 | 2 , |σ 1 | 2 + |σ 3 | 2 = 1
to obtain that I 2 is of the same nature as

I 3 = δ 0 r α 1 +α 2 +α 3 -3ρ+2(n-1)-1 dr × Σ |σ 2 | α 3 -ρ |σ 3 | α 2 -ρ |σ 2 -σ 3 | α 1 -ρ dσ .
The first factor converges if and only if condition (25) is satisfied. It remains to discuss the convergence of

I 4 = Σ |σ 2 | α 3 -ρ |σ 3 | α 2 -ρ |σ 2 -σ 3 | α 1 -ρ dσ .
Let δ > 0 and consider the following open subsets of Σ :

Σ 2 = {(σ 2 , σ 3 ), |σ 2 | < δ}, Σ 3 = {(σ 2 , σ 3 ), |σ 3 | < δ}, Σ 1 = {(σ 2 , σ 3 ), |σ 2 -σ 3 | < δ} .
For δ small enough, these sets are disjoint (recall that

|σ 2 | 2 + |σ 3 | 2 = 1). On Σ \ (Σ 1 ∪ Σ 2 ∪ Σ 3
) the function to be integrated is bounded from below. Hence the integral I 4 is convergent if and only the integrals

K j = Σ j |σ 2 | α 3 -ρ |σ 3 | α 2 -ρ |σ 2 -σ 3 | α 1 -ρ dσ are convergent for j = 1, 2, 3. Let j = 2. On Σ 2 , both |σ 3 | and |σ 2 -σ 3 | are bounded from below, so that it is equivalent to study the convergence of the integral Σ 2 |σ 2 | α 3 -ρ dσ .
We are reduced to a classical situation and K 2 converges if and only if α 3ρ > -(n -1). A similar study applies to K 1 and K 3 . This completes the proof of Theorem 2.2.

Corollary 2.1. Let λ 1 , λ 2 , λ 3 be three complex numbers satisfying the condition 0 ≤ ℜ(λ j ) < ρ .

Define α by the relations (19). Then the kernel K α is integrable, and the corresponding trilinear form is invariant for

π λ 1 , π λ 2 , π λ 3 .
Proof. The conditions on λ = (λ 1 , λ 2 , λ 3 ) imply that ℜ(α j ) > -ρ for j = 1, 2, 3 and ℜ(α

1 + α 2 + α 3 ) ≥ 0. Hence α is in the domain of integrability of K α .
The corollary covers all interesting cases for spherical unitary series, provided one excludes the trivial representation. In fact, the parameter λ for such a representation can be chosen either as pure imaginary (principal series) or satisfying 0 < λ < ρ (complementary series, excluding the trivial representation).

Analytic continuation of K α

The main result of this section concerns the analytic continuation of K α beyond its domain of integrability.

Theorem 2.3. The map α -→ K α , originally defined for α in in the domain described by the conditions (24) and (25), valued in D ′ (S × S × S) can be extended meromorphically with at most simple poles along the family of hyperplanes in C × C × C defined by the equations

α j = -ρ -2k, j = 1, 2, 3, k ∈ N (26)
α 1 + α 2 + α 3 = -ρ -2k, k ∈ N (27)
Proof. Let f be in C ∞ (S × S × S) and consider the integral

S×S×S K α (x 1 , x 2 , x 3 )f (x 1 , x 2 , x 3 )dσ(x 1 )dσ(x 2 )dσ(x 3 ) .
to be meromorphically continued. Repeating the argument given supra during the discussion of the integrability of the kernel K α , we may assume that f has its support contained in a small neigborhood of the point (1

+ , 1 + , 1 + ).
Further, transfer the integral to the noncompact picture (cf ( 23)), and study the analytic continuation of

α -→ J α (f • c)Ψ α Now ϕ = f • c is in C ∞ c (R n-1 × R n-1 × R n-1
), and both α -→ Ψ α and α -→ Ψ α -1 are entire fonctions on C 3 , so that it is equivalent to study the meromorphic continuation of J α as a distribution on

R n-1 × R n-1 × R n-1 .
The kernel J α is invariant by translations by "diagonal vectors". To take advantage of this remark, make the change of variables

z 1 = y 1 , z 2 = y 1 -y 3 , z 3 = y 1 -y 2 in the integral J α (ϕ) = |y 1 -y 2 | -ρ+α 3 |y 2 -y 3 | -ρ+α 1 |y 3 -y 1 | -ρ+α 2 ϕ(y 1 , y 2 , y 3 ) dy 1 dy 2 dy 3 to obtain J α (ϕ) = R n-1 ×R n-1 |z 2 | -ρ+α 2 |z 3 | -ρ+α 3 |z 2 -z 3 | -ρ+α 1 ψ(z 2 , z 3 ) dz 2 dz 3
where we have set

ψ(z 2 , z 3 ) = R n-1 ϕ(z 1 , z 1 -z 3 , z 1 -z 2 )dz 1 . Now observe that ψ is in C ∞ c (R n-1 × R n-1
). Hence we are reduced to studying the analytic continuation of the distribution I α on R n-1 × R n-1 defined by

I α (ψ) = R n-1 ×R n-1 I α (z 2 , z 3 )ψ(z 2 , z 3 ) dz 2 dz 3 (28) for ψ in C ∞ c (R n-1 × R n-1
), where we set

I α (z 2 , z 3 ) = |z 2 | -ρ+α 2 |z 3 | -ρ+α 3 |z 2 -z 3 | -ρ+α 1 . ( 29
) For δ > 0, consider the following open subsets of R n-1 × R n-1 U 0 = {(z 2 , z 3 ), |z 2 | < δ, |z 3 | < δ, |z 2 -z 3 | < δ} U 1 = {(z 2 , z 3 ), |z 2 | > δ 2 , |z 3 | > δ 2 , |z 2 -z 3 | < δ 2 } U 2 = {(z 2 , z 3 ), |z 2 | < δ 2 , |z 3 | > δ 2 , |z 3 -z 2 | > δ 2 } U 3 = {(z 2 , z 3 ), |z 2 | > δ 2 , |z 3 | < δ 2 , |z 2 -z 3 | > δ 2 } U ∞ = {(z 2 , z 3 ), |z 2 | > δ 2 , |z 3 | > δ 2 , |z 2 -z 3 | > δ 2 } .
The family of these five open sets form a covering of R n-1 × R n-1 . Let study the restriction of the distribution I α to each of these five open subsets. If Supp(ψ) ⊂ U ∞ , there the I α (ψ) extends as an entire function, because I α has no singularity on U ∞ . Next assume that Supp(ψ) ⊂ U 2 . Set

φ α (z 2 ) = R n-1 |z 3 | -ρ+α 3 |z 2 -z 3 | -ρ+α 1 ψ(z 2 , z 3 )dz 3 , so that I α (ψ) = R n-1 |z 2 | -ρ+α 2 φ α (z 2 )dz 2 . ( 30
)
As

|z 3 | and |z 2 -z 3 | are bounded from below on U 2 , the function φ α is in C ∞ c (R n-1
) and α -→ φ α is an entire function on C 3 . On R n-1 , the distribution-valued function s -→ |z 2 | s extends meromorphically on C, with simple poles at s = -(n -1) -2k, k ∈ N (see [START_REF] Gelfand | Generalized functions[END_REF]), so that the integral (30) extends meromorphically to C 3 with simple poles along the hyperplanes α 2 = -ρ -2k. A similar analysis can be done over U 1 and U 3 .

To sum up what we have already proved, introduce the family M of meromorphic functions on C 3 having at most simple poles along the hyperplanes {α j = -ρ -2k}, j = 1, 2, 3, k ∈ N. Notice that they are the hyperplanes corresponding to the conditions (26).

Proposition 2.2. Let ψ be in C ∞ c (R n-1 ×R n-1
) and assume that Supp(ψ) ∋ (0, 0) . Then the function α → I α (ψ) belongs to the class M.

Proof. In fact, choose δ small enough so that Supp(ψ) ∩ U 0 = ∅. Use a partition of unity to write ψ as

ψ = ψ 1 + ψ 2 + ψ 3 + ψ ∞ where ψ j ∈ C ∞ c (R n-1 × R n-1
) and Supp(ψ j ) ⊂ U j , j = 1, 2, 3, ∞. The previous analysis shows that α -→ I α (ψ j ) extends meromorphically on C 3 with at most simple poles along the hyperplanes

α j = -ρ -2k, k ∈ N for j = 1, 2, 3, whereas I α (ψ ∞ ) is an entire function of α.
Now we use a priori the existence of the meromorphic continuation of such integrals (see [START_REF] Sabbah | Polynômes de Bernstein-Sato à plusieurs variables[END_REF]). Moreover, the poles are located on a locally finite family of affine hyperplanes (of a rather specific type, but we won't need this result). Let H be such an hyperplane (to be determined), but not included in the family of hyperplanes given by conditions (26). Let α 0 = (α 0 1 , α 0 2 , α 0 3 ) be a regular point in H, (i.e. not contained in any other hyperplane of poles). The Laurent coefficients at α 0 are distributions on R n-1 × R n-1 , and, by Lemma 2.2, their supports have to be contained in {0, 0}, hence they are derivatives of the Dirac measure δ (0,0) . So, if I α does have a pole at α 0 , there exists a smooth function ρ on R n-1 × R n-1 with compact support and identically equal to 1 in a neigbourhood of (0, 0), and a polynomial P on R n-1 × R n-1 , homogeneous of degree k such that I α (ρP ) does not extend holomorphically at α 0 . For t ∈ R * , let ρ t (z 2 , z 3 ) = ρ(tz 2 , tz 3 ). Use the change of variables (z 2 → tz 2 , z 3 → tz 3 ) to get 2 , for α = α 0

I α (ρ t P ) = R n-1 ×R n-1 |z 2 | -ρ+α 3 |z 3 | -ρ+α 2 |z 2 -z 3 | -ρ+α 1 ρ(tz 2 , tz 3 )P (z 2 , z 3 ) dz 2 dz 3 = |t| -α 3 -α 2 -α 1 -ρ t -k I α (ρP ) .
2 Following a traditional way, we write Iα(ϕ) as an integral. What is really used here is merely the homogeneity of Iα.

Hence

(1 -|t| -α 3 -α 2 -α 1 -ρ t -k )I α (ρP ) = I α (ρ -ρ t )P . ( 31
)
Now observe that the support of (ρ-ρ t )P does not contain (0, 0), and hence, α -→ I α (ρρ t )P belongs to M. The assumption that I α (ρP ) does have a pole at α 0 forces the condition

∀t ∈ R * 1 -|t| -α 0 3 -α 0 2 -α 0 1 -ρ t -k = 0 .
In turn, this condition amounts to

k ∈ 2N, α 0 1 + α 0 2 + α 0 3 = -ρ -k .
Moreover, (31) shows that I α has at most a simple pole along the corresponding hyperplane. This achieves the proof of Theorem 2.3.

As the invariance condition remains true by analytic continuation, Theorem 2.3 can be reformulated for trilinear invariant functionals (cf Theorem 2.1).

Theorem 2.4. Let λ = (λ 1 , λ 2 , λ 3 ) in C 3 . Assume that -λ 1 + λ 2 + λ 3 / ∈ -ρ -2N λ 1 -λ 2 + λ 3 / ∈ -ρ -2N λ 1 + λ 2 -λ 3 / ∈ -ρ -2N λ 1 + λ 2 + λ 3 / ∈ -ρ -2N (32)
Set α = (α 1 , α 2 , α 3 ) where (cf (19))

α 1 = -λ 1 + λ 2 + λ 3 α 2 = -λ 2 + λ 3 + λ 1 α 3 = -λ 3 + λ 1 + λ 2 . Then (f 1 , f 2 , f 3 ) -→ T λ (f 1 , f 2 , f 3 ) = K α (f 1 ⊗ f 2 ⊗ f 3 ) is a well defined non trivial trilinear invariant functional w.r.t. the representations (π λ 1 , π λ 2 , π λ 3 ).
The next result was obtained some time ago by the present authors, and has been generalized to other geometric situations in a collaboration with T. Kobayashi and M. Pevzner (see [START_REF] Clerc | Generalized Bernstein-Reznikov integrals[END_REF]). To state the result, consider the evaluation of the functional K α against the function I S ⊗ I S ⊗ I S , where I S is the function which is identically 1 on S. Let

I(λ) = K(α) = S S S k α 1 (x 2 , x 3 )k α 2 (x 3 , x 1 )k α 3 (x 1 , x 2 )dσ(x 1 )dσ(x 2 )dσ(x 3 )
where α and λ are related by the relations (19).

Proposition 2.3. Let λ = (λ 1 , λ 2 , λ 3 ) in C 3 and assume that the conditions (32) are satisfied. Then

I(λ) = √ π 2 3(n-1)
2 λ 1 +λ 2 +λ 3 . . . . . . Γ( λ 1 +λ 2 +λ 3 +ρ 2 )Γ( -λ 1 +λ 2 +λ 3 +ρ 2 )Γ( λ 1 -λ 2 +λ 3 +ρ 2 )Γ( λ 1 +λ 2 -λ 3 +ρ 2 ) Γ(ρ + λ 1 )Γ(ρ + λ 2 )Γ(ρ + λ 3 ) (33)
Remark Both sides of the formula are meromorphic functions on C 3 , and they are equal where defined. Notice that I(λ) = K(α) has simple poles exactly as prescribed by Theorem 2.3.

This result allows to strengthen the previous theorem. Define

K α = K α Γ( λ 1 +λ 2 +λ 3 +ρ 2 )Γ( -λ 1 +λ 2 +λ 3 +ρ 2 )Γ( λ 1 -λ 2 +λ 3 +ρ 2 )Γ( λ 1 +λ 2 -λ 3 +ρ 2
) .

Theorem 2.5. The distribution-valued function α → K α extends as an entire holomorphic function on C 3 . The trilinear functional Tλ defined by

T λ (f 1 , f 2 , f 3 ) = K α (f 1 ⊗ f 2 ⊗ f 3 ) on C ∞ (S)×C ∞ (S)×C ∞ (S)
) is invariant with respect to (π λ 1 , π λ 2 , π λ 3 ), where (λ 1 , λ 2 , λ 3 ) are related to (α 1 , α 2 , α 3 ) by the relations (19). The trilinear form T λ is not identically 0 provided the two following conditions are not simultaneously realized • ∃j, 1 ≤ j ≤ 3, λ j ∈ -ρ -N • (at least) one of the conditions (32) is satisfied .

Proof. The function α -→ K α extends holomorphically near any regular point of the hyperplanes defined by conditions ( 26) and ( 27), so is holmomorphic outside of the set where ar least two hyperplanes of poles meet. But this set is of codimension 2 and hence α -→ K α extends as a holomorphic function to all of C 3 . If none of the conditions (32) is satisfied, then K α is a multiple ( = 0) of K α which is different from 0 on Ω. If λ j / ∈ -ρ + N for j = 1, 2, 3, then I(λ) = 0 and hence K α is not identically 0.

3 Uniqueness of the invariant trilinear form

Induced representations and line bundles

For this part, it is useful to realize the representation π λ as acting on smooth sections of a line bundle over S.

Recall that the stabilizer of 1 + in G is the parabolic subgroup P = M AN . The left invariant Haar measure on P is

P f (p)dp = M A N f (man)dmdadn . ( 34
)
For λ ∈ C, denot by χ λ the character of P defined by

χ λ (ma t n) = e tλ .
The modular function of P is given by δ P (man) = e -2ρ(log a) = χ -2ρ (p) (see e.g. [START_REF] Warner | Harmonic analysis on semi-simple Lie groups I[END_REF] Lemma 5.5.1.1), so that, for any q in

P P f (pq) dp = χ 2ρ (q) P f (p) dp (35)
Let E λ be the space of functions f in C ∞ (G) which, for all g in G, p in P satisfy

f (gp) = χ -(ρ+λ) (p)f (g) (36)
Then G acts on

E λ by Π λ (g)f (γ) = f (g -1 γ) (37)
To any function f in E λ , associate the function f defined on S by the formula

f (s) = f (k) ( 38
)
where k is any element in K satisfying k1 + = s. As f (km) = f (m) for any m in M , the right handside of (38) does not depend on m, but merely on k1 + = s. Now, let g be in G, let s be in S, and let k be in K such that k1

+ = s ∈ S. Let g -1 k = k(g -1 k) a(g -1 k) n be the Iwasawa decomposition of g -1 k. Then k(g -1 k)1 + = (g -1 k)(1 + ) = g -1 (s) and κ(g -1 , s) = κ(g -1 k, 1 + ) = χ -1 (a(g -1 k)) .
Hence,

Π λ (g)f (s) = f (g -1 k) = f (k(g -1 k) a(g -1 k) n) = κ(g -1 , s) ρ+λ f (g -1 (s))
so that f → f is an intertwining operator for Π λ and π λ . Let χ be a character of A. Denote by C χ the representation of P on C given by man.z = χ(a)z .

Form the line bundle L = L χ = G × P C χ over S, and let L = L ∞ χ be the space of C ∞ sections of L. Then G acts naturally on L by left translations. As sections of L can be identified with functions on G transforming by χ -1 under the right action of P , the representation of G on L is equivalent to π λ if χ = χ ρ+λ . Now take three characters χ 1 = χ ρ+λ 1 , χ 2 = χ ρ+λ 2 , χ 3 = χ ρ+λ 3 of A, set λ = (λ 1 , λ 2 , λ 3 ) and form the "exterior" product bundle L λ = L χ 1 ⊠ L χ 2 ⊠ L χ 3 as a line bundle over S × S × S. Let L λ be the space of C ∞ sections of this bundle. Let the group G acts naturally on this space by diagonal action. Then a trilinear invariant functional on C ∞ (S) × C ∞ (S) × C ∞ (S) for π λ 1 , π λ 2 , π λ 3 corresponds to an invariant linear form on L λ .

In turn, such an invariant linear functional on L λ can be viewed as an invariant distribution density for the dual bundle L * λ (see Appendix). The main tool to study these invariant distributions is Bruhat's theory, which is presented in the Appendix. We use heavily the description of orbits of G in S × S × S (cf Proposition 1.2). Theorem 3.1. Let λ = (λ 1 , λ 2 , λ 3 ) be three complex numbers, let α = (α 1 , α 2 , α 3 ) be defined as in (19) and assume that they satisfy the following generic conditions i) α j / ∈ -ρ -2N for j = 1, 2, 3 ii) α 1 + α 2 + α 3 / ∈ -ρ -2N. Then any trilinear invariant form for π λ 1 , π λ 2 , π λ 3 is proportional to the form T λ .

Proof. Denote by T = T λ the distribution density on S × S × S for the bundle L * λ corresponding to an invariant trilinear form for π λ 1 , π λ 2 , π λ 3 .

Step 1 : contribution of O 0

We use freely of the notation presented in the appendix. Consider the restriction T ′ of T to the open orbit O 0 . Then O 0 is a homogeneous space under G, the stabilizer H of the base point (1 + , 1 -, e 2 ) is compact. There is an invariant measure on G/H, the group H acts trivially on the fiber of L * λ , so there exists exactly one (up to constant) invariant distribution given by a smooth density. But we already know that K α has the right transformation property. Hence on O 0 , after multiplication by a constant, we may assume that T ′ coincides with the restriction of K α to O 0 . But the assumptions on λ guarantee that K α can be extended (by analytic continuation) as an invariant distribution on S × S × S. Hence, in order to prove that T is a multiple of K α , (i.e. to prove the uniqueness statement) we need only to prove that an invariant distribution which vanishes on O 0 is identically 0. In other words, we may (and hence do) assume that

Supp(T ) ⊂ ∪ 1≤j≤4 O j .
and proceed further to prove that T = 0.

Let us mention that the argument given here should be modified for the case of the circle (i.e. as n = 2), because there are two orbits for SO 0 (1, 2) in S × S × S. To restore uniqueness, one can also consider the full group O(1, 2) instead. The rest of the proof is unchanged.

Step

2 : contribution of O 1 , O 2 , O 3 We now show that Supp(T ) ∩ O 3 = ∅ .
Observe that for 1 ≤ j ≤ 3

O j = O j ∪ O 4 so that (O 1 ∪ O 2 ∪ O 4 ) is a closed subset of S × S × S. Let X = S × S × S \ (O 1 ∪ O 2 ∪ O 4 ) .
Then X is an open submanifold of S × S × S, acted by G, and O 3 is a closed orbit of G in X. Let T ′′ be the restriction of T to X, so that Supp(T ′′ ) ⊂ O 3 . Now apply Bruhat's theory.

The normal space at the base point (1

+ , 1 + , 1 -) is identified (via the Riemannian metric on S × S × S) with N o = {(ξ, -ξ, 0), ξ ∈ T 1 S} ≃ R n-1
The stabilizer of of the base point is H = AM , and there is an invariant measure on O 3 , so that χ 0 ≡ 1. The group M acts on N 0 by its natural action. As M modules, N 0 and its dual are equivalent, so that the spaces S k (N 0 ) and P k (N 0 ) are equivalent M -modules. The space of M -invariants in P k (N 0 ) is 0 if k is odd (and one-dimensional, generated by |ξ| k if k is even, but we won't need this fact). On the other hand, an element a t of A acts on N 0 by dilation by e -t , so that it acts on S k (N 0 ) by multiplication by e -kt . The element a t acts on the fiber L 0 of L χ 1 ,χ 2 ,χ 3 at (1 + , 1 + , 1 -) by e (ρ+λ 1 +ρ+λ 2 -ρ-λ 3 )t . Hence the element a t acts on S k (N 0 ) ⊗ L * 0 by multiplication by e (-k-ρ-(λ 1 +λ 2 -λ 3 ))t .

The assumptions on λ (more precisely

α 3 = λ 1 + λ 2 -λ 3 / ∈ -ρ -2N) guarantee that S k (N 0 ) ⊗ L * 0 ⊗ C χ -1 0 H = 0 for any k ∈ N.

Third step : contribution of O 4

Here we take X = S ×S ×S, as O 4 is closed. The stabilizer of the base-point

(1 + , 1 + , 1 + ) is P = M AN .
The character χ 0 is given by

χ 0 (ma t n) = e -2ρt .
The normal space at (1 + , 1 + , 1 + ) can be identified with

N 0 = {(ξ 1 , ξ 2 , ξ 3 ), ξ j ∈ T 1 S, ξ 1 + ξ 2 + ξ 3 = 0} ≃ R n-1 ⊕ R n-1 .
The group M acts on N 0 by its natural action on each factor. Again, as M modules, the space S k (N 0 ) is isomorphic to P k (N 0 ). The algebra of SO(n-1) invariant polynomials on R n-1 ⊕R n-1 is generated (as an algebra) by |ξ| 2 , |η| 2 , ξ, η . Hence S k (N 0 ) M = {0} if k is odd. On the other hand, an element a t of A acts on N 0 by multiplication by e -t , hence on S k (N 0 ) by multiplication by e -kt . It acts on the fiber L 0 by e (ρ+λ 1 +ρ+λ 2 +ρ+λ 3 )t . Hence

a t acts on S k (N 0 ) ⊗ L * 0 ⊗ C χ -1 o by e (-k-ρ-(λ 1 λ 2 +λ 3 ))t .
The assumptions on λ (namely

λ 1 + λ 2 + λ 3 / ∈ -ρ -2N) guarantee that S k (N 0 ) ⊗ L * 0 ⊗ C χ -1 0 H = {0}
for any k ∈ N. Hence, by Corollary A1, there is no non-trivial invariant distribution supported in O 4 . The uniqueness statement follows.

Another construction of a trilinear invariant form

In this section we present a different construction of the invariant trilinear form T λ . Trilinear forms are connected with tensor products of representations. Roughly speaking, a trilinear form on H 1 × H 2 × H 3 can be realized as an invariant bilinear pairing between H 1 ⊗ H 2 and H 3 . Although this point of view breaks the (sort of) symmetry between the three factors, it produces interesting relations with questions about tensor products of representations (cf [START_REF] Molcanov | Tensor products of unitary representions of the threedimensional Lorentz group[END_REF], which was the main source of inspiration for this section). Our presentation of the construction is formal and we don't work out the estimates and analytical aspects of the construction, which would follow along similar lines as in previous sections.

4.1

The space S 2 ⊤ as a symmetric space

Recall that S 2 ⊤ = {(x, y) ∈ S × S, x = y} is the orbit of (1 + , 1 -) under G. The stabilizer in G of (1 + , 1 -) is the subgroup H = M A. Let J =        -1 0 0 . . . 0 0 -1 0 . . . 0 0 0 1 0 . . . . . . . . . . . . 0 0 0 . . . 1        . Then J t = J -1 = J. The map g -→ JgJ
is an involutive automorphism of G. The set of fixed points of this involution is the subgroup H = H ⊔ H -,

H -= h =   cosh t -sinh t 0 sinh t -cosh t 0 0 0 k   , t ∈ R, k ∈ O(n -1), det k = -1 .
So H is the neutral component of the fixed points of an involutive automorphism of G. In other words, S 2 ⊤ can be realized as the symmetric space

G/H via G/H ∋ g -→ g(1 + ), g(1 -) .
There exists a unique (up to a positive real number) G-invariant measure on S 2 ⊤ , namely

G/H f (x)dµ(x) = S S f (s, t) ds dt |s -t| 2(n-1) . ( 39
)
Another description of G/H is as follows. Let X be the set of all 2dimensional oriented subspaces Π in R 1,n such that the restriction of q to Π is of signature (1, 1). The space X is an open set in the Grassmannian G(2, n + 1) of oriented 2-dimensional subspaces in R 1,n . The group G operates transitively on X . To the couple (s, t) in S 2 ⊤ , associate the 2-dimensional space Π(s, t) = R s⊕R t, with the orientation for which ( s, t) is a direct basis.

Proposition 4.1. The mapping Π is a diffeomorphism from S 2 ⊤ onto X . Proof. Let s, t be in S 2
⊤ . Then Π(s, t) cannot be totally isotropic for q, because the maximally isotropic subspaces are of dimension 1. As Π(s, t) contains two independant isotropic vectors ( s and t), the signature of q on Π(s, t) has to be (1, 1). Hence Π(s, t) belongs to X . On the other hand, let Π be in X . Then the isotropic cone in Π decomposes as the union of two distinct isotropic lines, say D ∪ ∆. The line D (resp. ∆) intersects the hyperplane {x 0 = 1} at a point s (resp. t) with s (resp. t) in S. Permuting D and ∆ if needed, we may assume that the basis ( s, t) corresponds to the orientation of Π. Then clearly Π(s, t) = Π. Thus Π is a 1-1 correspondance. The fact that it is a diffeomorphism is standard and left to the reader.

An equivariant realization of π σ ⊗ π τ

The group H is connected, isomorphic to R × SO(n -1). For ζ any complex number, the function ν ζ on H defined by

ν ζ   cosh t sinh t 0 sinh t cosh t 0 0 0 k   = e tζ (t ∈ R, k ∈ M ) is a character of H. Form the line bundle L ζ = G × H C ζ over G/H. A section of L ζ can be viewed as a C ∞ function F : G -→ C which satisfies F (gh) = ν ζ (h) -1 F (g) . (40)
I ζ (γ)F (g) = F (γ -1 g) . When ζ is pure imaginary, the character ν ζ is unitary. If F is in L ζ , then |F (gh)| = |F (g)| for any h in H. Hence the expression F 2 = G/H |F (g)| 2 dµ(gH) , (41)
is well-defined (maybe +∞) and is finite if (say) F has compact support modulo H. The representation I ζ extends as a unitary representation for this inner product. Let σ, τ be two complex numbers, and consider the representations π σ and π τ . As C ∞ (S) is a Fréchet space, the projective and inductive topologies on the tensor product C(S) ⊗ C(S) coincide, and the (completed) tensor product is realized as C ∞ (S × S). Hence the tensor product π σ ⊗ π τ is naturally realized on C ∞ (S × S). Explicitly,

π σ ⊗ π τ (g)f (x 1 , x 2 ) = κ(g -1 , x 1 ) ρ+σ κ(g -1 , x 2 ) ρ+τ f (g -1 (x 1 ), g -1 (x 2 )) (42) for g ∈ G, f ∈ C ∞ (S × S), x 1 , x 2 ∈ S.
For f in C ∞ (S × S), let P σ,τ f be the function on G defined by

P σ,τ f (g) = κ(g, 1 + ) ρ+σ κ(g, 1 -) ρ+τ f g(1 + ), g(1 -) (43)
Proposition 4.2. P σ,τ f satisfies the relation

P σ,τ f (gh) = ν σ-τ (h) -1 P σ,τ f (g), (44)
for g in G and h in H.

Proof. Recall that the elements of H fix both 1 + and 1 -. Moreover,

κ(a t , 1 ± ) = e ∓t , t ∈ R . (45)
Now (44) follows [START_REF] Gelfand | Generalized functions[END_REF].

The map P σ,τ will be regarded as a map from C ∞ (S × S) into L σ-τ . Let σ and τ be pure imaginary. Observe that

|g(1 + ) -g(1 -)| = 2 κ(g, 1 + ) 1 2 κ(g, 1 -) 1 2
, so that

P σ,τ f 2 = 2 -2(n-1) S×S |s -t| 2(n-1) |f (s, t)| 2 ds dt |s -t| 2(n-1) = 2 -2(n-1) f 2
and P σ,τ extends as an isometry (up to a constant) from L 2 (S × S) onto the space of square-integrable elements of L σ-τ for the inner product associated to (41).

G P σ,τ • (π σ ⊗ π τ )(g) = I σ-τ (g) • P σ,τ (46)
Proof. Let f be in C ∞ (S × S). For γ in G,

P σ,τ (π σ ⊗ π τ )(γ)f )(g) = κ(g, 1 + ) ρ+σ κ(g, 1 -) ρ+τ κ(γ -1 , g(1 + )) ρ+σ κ(γ -1 , g(1 -)) ρ+τ . . . . . . f γ -1 (g(1 + ), g(1 -)) .
On the other hand,

I σ-τ (γ)P σ,τ f (g) = κ(γ -1 g, 1 + ) ρ+σ κ(γ -1 g, 1 -) ρ+τ f γ -1 (g(1 + ), g -1 (1 -)) .
The two expressions are easily seen to be equal by using (6).

Construction of a (H, ν ζ ) covariant function for π λ

The next step consists in finding in the representation space for π λ an element Θ λ,ζ which transforms under the action of H by the character ν ζ . In general, it will be a distribution on S (a commun fact in harmonic analysis on semisimple symmetric space (cf [START_REF] Van Den Ban | The principal series for a reductive symmetric space. I. H-fixed distribution vectors[END_REF]). We use a geometric approach through the realization of G/H as X (see Proposition 4.1).

Let Π be an element of X and s be in S. Observe that Π ⊥ is a an (n -1)-dimensional space, complementary to Π, and the restriction of q to Π ⊥ is negative-definite. Define Ψ(Π, s) = 2q(proj Π ⊥ s) (47) Proposition 4.4. Let (s 1 , s 2 ) be in S 2 ⊤ . Then, for any s 3 in S,

Ψ(Π(s 1 , s 2 ), s 3 ) = 2 |s 1 -s 3 ||s 2 -s 3 | |s 1 -s 2 | . ( 48
)
Proof. Let σ 3 = proj Π ⊥ s 3 . Then there exist real numbers α 1 and α 2 such that σ 3 = s 3α 1 s 1α 2 s 2 . The numbers α 1 and α 2 are determined by the conditions

[σ 3 , s 1 ] = [σ 3 , s 2 ] = 0 .
Hence,

1 -s 3 .s 2 -α 1 (1 -s 1 .s 2 ) = [ s 3 , s 2 ] -α 1 [ s 1 , s 2 ] = [σ 3 + α 2 s 2 , s 2 ] = 0 so that α 1 = 1 -s 2 .s 3 1 -s 1 .s 2 = |s 2 -s 3 | |s 1 -s 2 | 2
and similarly

α 2 = 1 -s 3 .s 1 1 -s 1 .s 2 = |s 1 -s 3 | |s 1 -s 2 | 2 . Now -q(σ 3 ) = -[ s 3 -α 1 s 1 -α 2 s 2 , s 3 -α 1 s 1 -α 2 s 2 ] = 2α 1 [ s 1 , s 3 ] + 2α 2 [ s 2 , s 3 ] -2α 1 α 2 [ s 1 , s 2 ]) = |s 1 -s 3 ||s 2 -s 3 | |s 1 -s 2 | 2 .
Proposition 4.5. Let g be in G. Then, for any Π in X and s in S,

Ψ(g(Π), g(s)) = κ(g, s)Ψ(π, s) (49)
Proof. As g preserves the form q and hence the orthogonality relative to q, for any Π in X g

• proj Π ⊥ = proj g(Π) ⊥ • g . Hence q(proj g(Π) ⊥ g(s)) = (g s) -2 0 q(proj g(Π) ⊥ g s) = (g s) 0 ) -2 q(proj Π ⊥ s)
so that Ψ(g(Π), g(s)) = (g s) -1 0 Ψ(Π, s) = κ(g, s)Ψ(π, s)

Let Π 0 = Π(1 + , 1 -) = x = (x 0 , x 1 , 0, . . . , 0), x 0 , x 1 ∈ R}, and let

Ψ 0 (s) = Ψ(Π 0 , s) = |1 + -s||1 --s| = 2(x 2 2 + • • • + x 2 n ) 1 2 . ( 50
)
As a consequence of the previous proposition, the function Ψ 0 has a nice transformation law under the action of H.

Let λ and ζ be two complex numbers. For s in S define

Θ λ,ζ (s) = Ψ 0 (s) -ρ-λ 1 + -s 1 --s ζ = |1 + -s| -ρ-λ+ζ |1 --s| -ρ-λ-ζ . ( 52
)
Proposition 4.6. The function Θ λ,ζ satisfies

π λ (h)Θ λ,ζ = ν ζ (h)Θ λ,ζ (53)
for all h in H.

Proof. Let h be in H. Then

π λ (h)Θ λ,ζ (s) = κ(h -1 , s) ρ+λ Θ λ,ζ (h -1 (s)) = κ(h -1 , s) ρ+λ Ψ 0 (h -1 (s)) -ρ-λ 1 + -h -1 (s) 1 --h -1 (s) ζ = κ(h -1 , s) ρ+λ κ(h -1 , s) -ρ-λ Ψ 0 (s) -ρ-λ κ(h -1 , 1 + ) κ(h -1 , 1 -) ζ 2 1 + -s 1 --s ζ = Ψ 0 (s) -ρ-λ (e 2t ) ζ 2 1 + -s 1 --s ζ = ν ζ (h) Θ λ,ζ (s)
.

by using (45).

4.4

The duality between I ζ and π λ Now define the corresponding Fourier transform F λ,ζ : for F a smooth section of L ζ with compact support modulo H, define F ζ,λ F by

F λ,ζ F (s) = G/H F (g) π λ (g)Θ λ,ζ (s) dµ(gH) . (54)
Observe that, thanks to (53) the integrand is a function on G/H, so that the integral makes sense, the result being in general a distribution on S.

Proposition 4.7. For any g in G

F λ,ζ I ζ (g) = π λ (g)F λ,ζ, . (55)
Proof. Set Θ = Θ λ,ζ , F = F λ,ζ , let F be in L ζ with compact support modulo H and let γ be in G. Then

F • I ζ (γ)F (s) == G/H F (γ -1 g) κ(g -1 , s) ρ+λ Θ(g -1 (s))dµ(gH) .
Set γ -1 g = l and use the invariance of the measure dµ to obtain

F • I ζ (γ)F (s) = G/H F (l)κ(l -1 γ -1 , s) ρ+λ Θ(l -1 (γ -1 (s))dµ(lH) = κ(γ -1 , s) ρ+λ G/H F (l)κ l -1 , γ -1 (s) ρ+λ Θ l -1 (γ -1 (s) dµ(lH) = κ(γ -1 , s) ρ+λ G/H F (l)π λ (l)Θ(γ -1 (s))dµ(lH) = κ(γ -1 , s) ρ+λ FF (γ -1 (s)) = π λ (γ)FF (s) .

Application to trilinear forms

Theorem 4.1. Let λ 1 , λ 2 , λ 3 be three complex numbers. For f 1 , f 2 , f 3 three functions in C ∞ (S), let T (f 1 , f 2 , f 3 ) be defined by

T (f 1 , f 2 , f 3 ) = F -λ 3 , λ 1 -λ 2 P λ 1 ,λ 2 (f 1 ⊗ f 2 ), f 3 . ( 56
)
Then, for any g ∈ G,

T (π λ 1 (g)f 1 , π λ 2 (g)f 2 , π λ 3 (g)f 3 ) = T (f 1 , f 2 , f 3 ) ,
whenever the right hand side is well defined.

Proof. Let g be in G and let f 1 , f 2 , f 3 be three functions in C ∞ (S). Then

T (π λ 1 (g)f 1 , π λ 2 (g)f 2 , π λ 3 (g)f 3 ) = F -λ 3 ,λ 2 -λ 1 , P λ 1 ,λ 2 (π λ 1 (g)f 1 ⊗π λ 2 (g)f 2 ), π λ 3 (g)f 3 = F -λ 3 ,λ 1 -λ 2 I λ 1 -λ 2 (g)P λ 1 ,λ 2 (f 1 ⊗ f 2 ), π λ 3 (g)f 3 (use (46)) = π -λ 3 (g) F -λ 3 ,λ 1 -λ 2 P λ 1 ,λ 2 (f 1 ⊗ f 2 ) , π λ 3 (g)f 3 (use (55)) = T (f 1 , f 2 , f 3 ) .
(use ( 13)).

Making explicit the right handside of (56) shows that it coincides (up to a constant) with the former expression of T λ where λ = (λ 1 , λ 2 , λ 3 ) (see Theorem 2.1).

The space of distribution densities on X is denoted by D ′ (X). A smooth density is a density the local expressions of which are C ∞ functions. The smooth densities are C ∞ sections of a line bundle called the density bundle Ω(X). It is very similar to the bundle of differential forms of maximal degree on X, in the sense that their transition functions just differ by an absolute value.

This definition can be extended to the case of vector bundles. Let π : L → X be a C ∞ vector bundle over M , with model fiber E 0 . Let (X i ) be a family of open subsets of X such that over each X i the bundle can be trivialized. Let Ψ i : π -1 (X i ) → X i × E 0 and Ψ j : π -1 (X j ) → X j × E 0 be two trivializations of the bundle over two overlapping subsets X i and X j . Then the map Ψ

i • Ψ -1 j over (X i ∩ X j ) × L 0 → (X i ∩ X j ) × L 0 is of the form (x, v) → (x, g ij (x) v)
where g ij (x) is in GL(L 0 ) and the map (transition functions of the bundle)

g ij : X i ∩ X j ∈ x → g ij (x) ∈ GL(E 0 ) is C ∞ . A distribution density for the bundle L is a system (u i ) of distribution densities on X i with values in L 0 such that u i = g ij u j in X i ∩ X j .
Denote by D ′ (X, L) the space of distribution densities for the bundle L.

Let L ∞ c be the space of C ∞ sections with compact support of the bundle L. Then the dual of L ∞ c is identified with the space D ′ (X, L * ). The smooth elements in the dual (those given locally by integration against a smooth function) are the C ∞ sections of the bundle Ω(X) ⊗ L * .

A.2 Invariant distribution : the case of a homogeneous vector bundle

Let G be a Lie group acting transitively on a manifold X. Let o be a base-point in X, and H = G o be its stabilizer in G, so that X ≃ G/H. An element h of H acts on X and fixes o, so that by differentiation, it acts on its tangent plane by (say) τ 0 (h). The tangent space T 0 X can be identified with g/h. The element h acts on g by the adjoint action Ad g (h), and preserves the subalgebra h on which it acts by Ad h h. Hence its acts on g/h, and this action coincides with τ 0 (h). This action satisfies

det τ 0 (h) = det(Ad g h) det(Ad h h) (59)
The modular function δ G of a Lie group G is defined by

δ G (g) = | det Ad(g -1 )| . so that | det τ 0 (h)| := χ 0 (h) = δ H (h) δ G (h) . (60)
A homogeneous vector bundle L over X is a vector bundle L together with an action of the group G on L by bundle isomorphisms. If g is in G and x in X, then g maps the fiber L x into L g(x) by a linear isomorphism.

In particular, H acts on L o by a representation (say) τ . Conversely, given a representation τ of H in a vector space E, then one constructs the bundle G × τ E as G × E/ ∼, where ∼ is the equivalence relation defined by the right action of H on G × E (g, v) ∼ (gh -1 , τ (h)v) for some h ∈ H , g in G and v ∈ E. Any homogeneous vector bundle over X is of that sort, in the sense that the bundle L is isomorphic to G × τ L 0 (see [START_REF] Wallach | Harmonic analysis on homogeneous spaces[END_REF]).

A section s : X -→ L can be realized as a L 0 -valued function f s on G which satisfies

f s (gh) = τ -1 (h)f s (g) (61)
and, conversely, such a function f gives raise to a section of L. Let L ∞ c be the space of smooth sections of L with compact support. The space L ∞ c is G-equivariantly isomorphic to C ∞ c (G, H, τ ), the space of C ∞ functions on G which satisfy (61) and have compact support modulo H. The group G acts by left translations on L ∞ c , and this action is equivariant with the left action of G on C ∞ c (G, H, τ ). The tangent bundle T X of X is an example of such a homogeneous bundle. The action of H on the fiber T 0 X is τ 0 . Another important homogenous bundle is the bundle Ω(X) of densities over X. It is a line bundle, corresponding to the character of H given by | det (τ (h) -1 ) t | = χ o (h) -1 (cf (60)), so that, in this context, we denote the fiber at o of the bundle Ω(X) by C χ -1 0 . Theorem A1. There exists a non trivial invariant continous linear forms on L ∞ c if and only if there exists a non trivial linear form ξ on E such that, for all h ∈ H τ (h -1 ) t ξ = χ 0 (h)ξ .

(62)

More precisely,

L ∞ c * G = D ′ (X, L * ) G ≃ (L * 0 ⊗ C χ -1 0 ) H . ( 63
)
Proof. (Sketch of) An element of the dual of L ∞ c is a distribution density for the bundle L * . If it is invariant by G, then the corresponding distribution density turns out to be smooth. Hence we are looking for a G-invariant smooth section of the bundle L * ⊗ Ω(X). But this is equivalent to an H invariant element in the fiber at o.

A.3 Invariant distribution supported in a submanifold

Let X be a manifold and G a Lie group acting on X. Let Q be an orbit of G in X and assume that Q is closed. Let L be a homogeneous vector bundle over X. Let N be the normal bundle of Q (i.e. the quotient bundle T X |Q /T Q). Fix a base-point o in Q, let G o = H be the stabilizer of o. Then H acts on the tangent space T o X, preserving the subspace T o Q, and hence acts on the normal space N o = T o X/T o Q at o.

Let T be a distribution on Q. Then the map

C ∞ c (X) ∋ ϕ -→ (T, ϕ |Q )
defines a distribution on X, which we still denote by T . Let L ∞ c be the space of smooth sections of L with compact support. Let T be a continuous linear form on L ∞ c supported in Q. Choose a local coordinate system on X (u 1 , . . . , u s , v 1 , . . . , v r ) such that v j = 0, 1 ≤ j ≤ r are local equations for Q. The (u i ) 1≤i≤s form a coordinate system of Q near o. On the other hand let N be the normal bundle of Q. The family ( ∂ ∂v j ) 1≤j≤r gives a local trivialization of the normal bundle N . Choose a local trivialization of the bundle L, and denote by e * 1 , . . . , e * l the corresponding coordinates on the fiber. By Schwarz's local structure theorem for distributions supported in a vector subspace, there exists an integer k ∈ N (the local transversal order of the distribution), and for each j, 1 ≤ j ≤ l and each multidiindex α = (α 1 , . . . Assume now that T is invariant by G. Obersve that invariance insures that the local transversal degree of T is the same for all charts of Q.

Denote by D ′ k (Q, L) G the space of G-invariant elements of D ′ k (Q, L). The conjonction of Theorem A1 and Theorem A2 gives some estimate of its dimension.

Theorem A3. For any k ∈ N,

dim D ′ k (Q, L) G /D ′ k-1 (Q, L) G ≤ dim S k (N 0 ) ⊗ L * 0 ⊗ C χ -1 0 H .
The theorem is mostly used in through the following corollary.

Corollary A1. Assume that for any k ∈ N

S k (N 0 ) ⊗ L * 0 ⊗ C χ -1 0 H = {0} .
Then there exists no non trivial continuous G-invariant linear form on L ∞ c such that Supp(T ) ⊂ Q.

  Denote by L ζ the space of smooth sections of L ζ over G/H. The natural action of G acts on L ζ by left translation is denoted by I ζ :

Proposition 4 . 3 .

 43 P σ,τ intertwines the representation π σ ⊗ π τ and the representation I σ-τ , i.e. for any g in

Lemma 4 . 1 . cosh t sinh t 0 sinh t cosh t 0 0 0 k 

 41k Let h =  , where t is in R, and k is in SO(n -1). For any s in S Ψ 0 (h(s)) = κ(h, s)Ψ 0 (s) .

  , α r ) with |α| = α 1 + • • • + α r ≤ k uniquely determined distributions T j α on (some open subset of) R s ⊂ R s × R r such that T = 1≤j≤l |α|≤k (-1) |α| D α T j α e * jLet us consider the "top terms" subcollection (T j α , |α| = k, 1 ≤ j ≤ l). This collection can be interpreted as the local expression of a distribution density for the bundle S k (N ) ⊗ L * |Q on Q. This is obtained by checking the way the collection transforms under• change of local trivialization of the bundle L (change the e * j 's) • change of local trivialization of the normal bundle (change the v j 's)• change of the local coordinate system on Q (change the u j 's). Denote by σ (k) (T ) the section of the bundle S k (N ) ⊗ L * |Q associated to T .Denote by D ′ k (Q, L) the space of continuous linear forms on L ∞ c which are supported on Q and of transversal order ≤ k on any local chart of X .Theorem A2. The map T -→ σ (k) (T ) is a linear map from D ′ k (Q, L) in D ′ Q, S k (N ) ⊗ L * |Q , with kernel D ′ k-1 (Q, L). Moreover, if Φ is a bundle diffeomorphism of the bundle L, which maps Q into Q, then σ (k) (Φ * T ) = Φ * σ (k) (T ) ,where Φ * denotes the action naturally induced by Φ on D ′ (Q, L) or of the restriction of Φ |L |Q on D ′ (Q, S k (N ) ⊗ L * |Q ).

  Hence, by Corollary A1, there is no non-trivial invariant distribution supported in O 3 . Repeating the argument for O 2 and O 1 , we may (and hence do) assume now that Supp(T ) ⊂ O 4 .

Appendix : Invariant distributions supported by a submanifold

Here is a presentation of the main results in Bruhat's theory [START_REF] Bruhat | Sur les représentations induites des groupes de Lie[END_REF], but written in terms of vector bundles and distribution densities (in the sense of [START_REF] Hörmander | The analysis of linear partial differential operators I[END_REF] ch. VI). We sketch the main steps of the proof, following [START_REF] Warner | Harmonic analysis on semi-simple Lie groups I[END_REF].

A.1 Distribution densities for a vector bundle over a manifold

First recall the composition of a distribution with a diffeomorphism. Let X 1 and X 2 be two open sets of R N and let Φ :

Then there is a unique continous linear map Φ * : D ′ (X 2 ) → D ′ (X 1 ) which extends the composition of functions, i.e. such that

A distribution density on a manifold X is by definition a continuous linear form on C ∞ c (X). Let u be a distribution density on X. Let (X κ , κ) be a local chart, i.e. κ is a diffeomorphism of an open set X κ of X onto an open set X κ of R N . Then the formula

for ϕ ∈ C ∞ c ( X κ ) defines a distribution u κ on X κ , called the local expression of u in the chart (X κ , κ) . Further, let (X κ , κ) and (X κ ′ , κ ′ ) be two overlapping charts, and let u κ and u κ ′ the corresponding local expressions of u.

be the change of coordinates (equal to κ • κ ′-1 ). Then

Conversely, suppose we have an atlas F of charts (X κ , κ) covering the manifold M and suppose that for each κ we are given a distribution u κ ∈ D ′ ( X κ ). Assume further that for any two overlapping charts (X κ , κ) and (X κ ′ , κ ′ ), the condition (58) is satisfied. Then the system (u κ ), κ ∈ F defines a unique distribution density u on X such that, for κ in F and ϕ ∈ C ∞ c ( X κ ) condition (57) is satisfied.