Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\int_{\Gamma \backslash \mathbb{H}} y^{k_i} |\phi(z)|^2 d \mu(z) = 1$. Then Watson's beautiful triple product formula says that, if $k_3=k_1+k_2$, then $$\left |\int_{\Gamma \backslash \mathbb{H}} y^{\frac {k_1+k_2+k_3}{2}} \phi_1(z) \phi_2(z) \overline{\phi_3(z)} d\mu(z) \right |^2 = (constant) \times \frac{\Lambda(1/2, \phi_1 \times \phi_2 \times \phi_3)}{\prod_j \Lambda(1, \mathrm{sym}^2 \phi_j)} .$$ Denote by $C_{k}$ the space of cusp forms of weight $k$, i.e. the space of functions fast decreasing in the cusp that transform by $f(\gamma z) = f(z) \left ( \frac{cz+d}{|cz+d|}\right )^k$, for $\gamma = \begin{pmatrix} * & * \\ c & d\end{pmatrix} \in \Gamma$. We have the Maass raising/lowering operators $K_k: C_k \to C_{k+2}$ and $L_k: C_k \to C_{k-2}$. Let $\{u_j(z)\}$ be a basis for the space of Maass cusp forms and $\{F_{jk}\}$ a Hecke basis for the space of holomorphic cusp forms of weight $k$. Then if $k$ is an even positive integer, it is a well known fact that a orthogonal basis for $C_k$ is given by $$ \{K_{k-2}\dots K_2 K_0 u_j (z)\} \cup \bigcup_{\substack{2 \leq m \leq k \\ m \text{ even}}} \{K_{k-2} \dots K_m (y ^{m/2}F_{jm}(z))\}. $$ In other words, $C_k$ is generated by Maass forms and holomorphic cusp forms of lower and equal weight, under the action of successive rising operators. Motivated by this, it would be nice to have a formula for triple products which takes these shifted elements of the basis into account. So take, for example, $\phi_1, \phi_2, \phi_3$ as in the beginning of weights $k_1, k_2, k_3$ respectively, but now $k_3> k_1 + k_2$. Can we write the triple product $$\left |\int_{\Gamma \backslash \mathbb{H}} (\alpha \cdot K_{k_3-k_2-2} \dots K_{k_1}(y^{\frac {k_1}{2}} \phi_1(z)) y^{\frac{k_2}{2}}\phi_2(z) y^{\frac{k_3}{2}}\overline{\phi_3(z)} d\mu(z) \right |^2 $$ in terms of a central value of $\Lambda(s, \phi_1 \times \phi_2 \times \phi_3)$ and $\Lambda(1, \mathrm{sym}^2 \phi_j)$, similarly as in Watson's formula? Here $\alpha$ is just a normalising constant. In the most optimistic case, the formula would be the same up to some $\Gamma$ factors, since the rising operators commute with Hecke operators and $\phi_i$ and $K_{l} \dots K_{k_i} \phi_i$ have the same $L$-function, but I can imagine this might be far fetched. Thank you, any help or reference is greatly appreciated.
1 Answer
By Ichino's triple product formula, of which Watson's formula is a special case, this integral will be given by the expected ratio of L-values (depending only on the automorphic reps generated by the $\phi_i$) multiplied by a product of local correction terms (which depend on the $\phi_i$ themselves). Since your level is 1, the local integrals at the finite primes are all 1, and the only question is whether the local integral at $\infty$ is zero or not.
I'm not so familiar with the Maass-form setting, but in the case when all $\phi_i$ are holomorphic, I guess your weight-raising operator will be the usual Maass--Shimura derivative $\tfrac{d}{dz} + \tfrac{k}{2\pi i y}$, and in this case it is known that the local integral at $\infty$ is non-zero + can be evaluated explicitly in terms of factorials -- this formula is the key input in many works on algebraicity + p-adic interpolation of triple-product L-values, see e.g. works of Harris--Tilouine and Darmon--Rotger.
