Generalization of Watson's triple product

Question

In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean:

If $\phi_n$'s are orthonormal Hecke-Maass eigenforms in some arithmetic congruence (compact/noncompact) manifold $M=\Gamma\backslash G$, i.e. normalized eigenforms of Laplacian and Hecke operators in corresponding manifold, then for $I\subset\mathbb{N}$, $$\left(\int_M\prod_{n\in I}\phi_ndvol_M\right)^2=C\frac{L\left(\frac12,\bigotimes\limits_{n\in I}\phi_n\right)}{\prod\limits_{n\in I}L(1,sym^2\phi_n)},$$ for some constant $C$.

$\textbf{Question}:$

1) Is the above (or some modified version) true?

2) If not, why is this only true for THREE eigenforms?

Thanks. Any help or reference is highly appreciated.

Answer 1

The reason it works for three (but not other numbers) is the uniqueness of trilinear functionals. Let $G = \mathrm{GL}_2(F)$, $F$ a local field and let $\pi_i$ ($1\leq i\leq 3$) be three irreducible admissible representations. Then there is at most one $G$-invariant functional on $\pi_1 \otimes \pi_2 \otimes \pi_3$ (this is mostly due to D. Prasad). Now let $\pi_i$ instead be global automorphic representations, and $G$ the adelic group. Then one such functional is given by the integral on the LHS of Watson's formula, so by the uniqueness it must factor as a product of local functionals, and this "explains" why the integral is Eulerian.

For the general question of when one should expect such integral representations for L-functions see the papers of Sakellaridis and Sakellaridis–Venkatesh.