Conformally invariant trilinear forms on the sphere (Q416006)

We set \(G:=SO_0(1,n), \) \(S:=G/MAN=\) unit sphere in \(\mathbb R^n\) centered at the origin. For each \(\lambda \in \mathbb C\) let \(\pi_\lambda\) denote the spherical principal series representation of \(G\) realized in \(C^\infty (S).\) For three values \(\lambda_1, \lambda_2, \lambda_3\) the authors construct a trilinear form \(T_{\lambda_1, \lambda_2,\lambda_3}\) in \(C^\infty (S) \times C^\infty (S) \times C^\infty (S)\) which is invariant by \(\pi_{\lambda_1} \otimes \pi_{\lambda_2} \otimes\pi_{\lambda_3}. \) The authors show: For a parameter \(\vec{\lambda}\) outside a specific countable union of hyperplanes the trilinear form is unique up to a constant. The function \(T_{\lambda_1, \lambda_2,\lambda_3}\) is meromorphic with simple poles located in an explicit countable union of hyperplanes. The construction of the function \(T_{\lambda_1, \lambda_2,\lambda_3}\) is achieved in two ways. The first one defines \(T_{\lambda_1, \lambda_2,\lambda_3}\) to be the integration along the Lebesgue measure on \(S\times S\times S\) with respect to an explicit kernel (product of three functions \(| x - y|^r\)) on which \(r\) depends linearly on \(\vec{\lambda}\) and so that the integral converges for parameters in an explicit open subset of \(\mathbb C^3 . \) To obtain the analytic continuation, the authors consider the five orbits of \(G\) in \(S\times S \times S\) and show that it is enough to consider the continuation of the integral for functions with support contained in the open orbit. Now, by techniques of classical analysis, the authors obtain the meromorphic continuation as well as the structure of the singularities of the continuation. The second construction relies on an explicit equivalence \(P_{\lambda_1, \lambda_2 }\) as smooth representations NEWLINE\[NEWLINE\pi_{\lambda_1} \otimes \pi_{\lambda_2} \vec{\cong} Ind_{MA}^G (1\otimes e^{\lambda_1 - \lambda_2}), NEWLINE\]NEWLINE a distribution \(\Theta_{\lambda, \zeta} \) on \(S\) which satisfies \(\pi_\lambda (h) \Theta_{\lambda, \zeta} = e^{\zeta t} \Theta_{\lambda, \zeta} \) for \(h=me^t \in MA,\) and an equivariant map NEWLINE\[NEWLINE F_{\lambda, \zeta} : Ind_{MA}^G (1 \otimes e^{\zeta t}) \rightarrow \pi_\lambda NEWLINE\]NEWLINE defined by NEWLINE\[NEWLINE F_{\lambda, \zeta} f (s)=\int_{MA} f(g) \pi_\lambda (g) \Theta_{\lambda, \zeta} (s) du(gMA).NEWLINE\]NEWLINE Then, NEWLINE\[NEWLINET_{\lambda_1, \lambda_2,\lambda_3}(f_1,f_2,f_3)= ( F_{-\lambda_3, \lambda_1 -\lambda_2} P_{\lambda_1, \lambda_2}(f_1 \otimes f_2), f_3).NEWLINE\]NEWLINE The authors determine the parameters so that \(T_{\vec{\lambda}} \not= 0\) and they apply their results to show that a convenient unitary principal series occurs as a discrete factor of the tensor product of two unitary principal series.