On the Triple Product Formula: Real Local Calculations

Abstract

We consider a triple of admissible representations π _j for j = 1, 2, 3 of ${\mathrm {GL}}_2(\mathbb R)$ of weights k _j with k ₁ ≥ k ₂ + k ₃. Test vectors are given, and using a formula of Michel-Venkatesh explicit values for local trilinear forms are computed for these vectors. Using this we determine the real archimedean local factors in Ichino’s formula for the triple product L-function. Applications both new and old to subconvexity, quantum chaos and p-adic modular forms are discussed.

Appendix: Normalizing L-Factors

The goal of this appendix is to record the normalizing L-factors for the triple product L-function appearing in (1.2). These factors are determined by applying the local Langlands correspondence relating finite dimensional semisimple representations of the Weil group $W_{\mathbb R}$ to admissible representations of ${\mathrm {GL}}_2(\mathbb R)$ as detailed in [15]. The local factors will be described in terms of

$$\displaystyle \begin{aligned} \Gamma_{\mathbb R}(s) = \pi^{-s/2}\Gamma(s/2),\qquad \mbox{and}\qquad \Gamma_{\mathbb C}(s) = \Gamma_{\mathbb R}(s)\Gamma_{\mathbb R}(s+1)=2(2\pi)^{-s}\Gamma(s). \end{aligned}$$

We recall the following elementary facts:

$$\displaystyle \begin{aligned} \overline{\Gamma(s)} = \Gamma(\overline{s}),\quad \Gamma_{\mathbb R}(1)=1, \quad \Gamma_{\mathbb R}(2) = \frac{1}{\pi}, \quad \Gamma_{\mathbb C}(m) = \frac{(m-1)!}{2^{m-1}\pi^m}. \end{aligned}$$

1.1 Local Langlands Parameters for ${\mathrm {GL}}_2(\mathbb R)$

We recall briefly the local Langlands correspondence for ${\mathrm {GL}}_2(\mathbb R)$. (See [15] for complete details.) Let $W_{\mathbb R}=\mathbb C^\times \cup j\mathbb C^\times$ with j ² = −1 and $jzj^{-1}=\bar {z}$ for $z\in \mathbb C^\times$ be the Weil group. For δ ∈{0, 1} and $t\in \mathbb C$, we have the 1-dimensional representation of $W_{\mathbb R}$ given by

$$\displaystyle \begin{aligned} \rho_1(\delta,t): \begin{array}{c} z \mapsto \lvert z \rvert^t \\ j\mapsto (-1)^\delta. \end{array} \end{aligned}$$

Moreover, if $m\in \mathbb Z$ and $t\in \mathbb C$ we have the 2-dimensional representation

$$\displaystyle \begin{aligned} \rho_2(m,t): \begin{array}{c} re^{i\theta}\mapsto \left( \begin{array}{cc} r^{2t}e^{im\theta} & 0 \\ 0 & r^{2t}e^{-im\theta} \end{array}\right) \\ j \mapsto \left( \begin{array}{cc} 0 & (-1)^m \\ 1 & 0 \end{array}\right) \end{array}, \end{aligned}$$

which is easily checked to be irreducible except when m = 0. The following is a simple exercise.

Lemma A.1

Every semisimple finite-dimensional representation of $W_{\mathbb R}$ is a direct sum of irreducibles of the type ρ ₁ and ρ ₂ as defined above. Under the operations of direct sum and tensor product, the following is a complete set of relations.

$$\displaystyle \begin{aligned} \rho_2(m,t) \simeq & \rho_2(-m,t) \\ \rho_2(0,t) \simeq & \rho_1(0,t)\oplus\rho_1(1,t) \\ \rho_1(\delta_1,t_1)\otimes\rho_1(\delta_2,t_2) \simeq & \rho_1(\delta,t_1+t_2) \qquad \delta\equiv \delta_1+\delta_2\pmod{2}) \\ \rho_1(\delta,t_1)\otimes\rho_2(m,t_2) \simeq & \rho_2(m,t_1+t_2) \\ \rho_2(m_1,t_1)\otimes\rho_2(m_2,t_2) \simeq & \rho_2(m_1+m_2,t_1+t_2)\oplus \rho_2(m_1-m_2,t_1+t_2). \end{aligned} $$

Moreover, if $\widetilde {\rho }$ denotes the contragradient of ρ then

$$\displaystyle \begin{aligned} \widetilde{\rho_1(\delta,t)}\simeq \rho_1(\delta,-t),\quad\mathit{\mbox{ and }}\quad \widetilde{\rho_2(m,t)}\simeq \rho_2(m,-t). \end{aligned}$$

Given an irreducible admissible representation π of ${\mathrm {GL}}_2(\mathbb R)$ we associate to it a representation ρ(π) of $W_{\mathbb R}$. For example, if $\mathcal B(\chi _1,\chi _2)=\mathcal B({\mathrm {sgn}}^{\epsilon _1}\lvert \cdot \rvert ^{s_1},{\mathrm {sgn}}^{\epsilon _2}\lvert \cdot \rvert ^{s_2})$ is irreducible the corresponding representation of $W_{\mathbb R}$ is ρ ₁(𝜖 ₁, s ₁) ⊕ ρ ₁(𝜖 ₂, s ₂). We record how this correspondence works in Table 1 for representations with central character sgn^δ. (We let $\overline {\delta +\epsilon }\in \{0,1\}$ be the reduction of 𝜖 + δ modulo 2.) Note that the third column of the table is calculated using Lemma A.1 and the identity

$$\displaystyle \begin{aligned} {\mathrm{Ad}}(\rho) \simeq \rho\otimes \widetilde{\rho}\ominus\rho_1(0,0). \end{aligned}$$

Table 1 Representations of $W_{\mathbb R}$ attached to admissible unitary representations of ${\mathrm {GL}}_2(\mathbb R)$

1.2 Triple Product and Adjoint L-Factors

We associate to each of ρ ₁(δ, t) and ρ ₂(m, t) the L-functions

$$\displaystyle \begin{aligned} L(s,\rho_1(\delta,t)) = \Gamma_{\mathbb R}(s+\delta+t), \qquad L(s,\rho_2(m,t)) = \Gamma_{\mathbb C}(s+\textstyle{\frac{m}{2}}+t). \end{aligned} $$

(A.1)

More generally, given ρ ≃ ρ ₁ ⊕ ρ ₂ ⊕⋯ ⊕ ρ _r a (semisimple) representation of $W_{\mathbb R}$ with ρ _j irreducible we define

$$\displaystyle \begin{aligned} L(s,\rho) = \prod_{j=1}^r L(s,\rho_j). \end{aligned}$$

Using this definition it follows, setting L(s, π, Ad) = L(s, Ad(ρ(π))) and combining (A.1) with Table 1, that

$$\displaystyle \begin{aligned} L(1,\pi,{\mathrm{Ad}}{})) = \begin{cases} \Gamma_{\mathbb R}(2)\Gamma_{\mathbb C}(k) & \mbox{ if }\pi=\pi_{\mathrm{dis}}^{k}, \vspace{2pt} \\ \displaystyle{\frac{\Gamma(\frac{1+\delta}{2}+\nu)\Gamma(\frac{1+\delta}{2}-\nu)}{\pi^{1+\delta}}} & \mbox{ if }\pi=\pi_{\delta,\epsilon}(\textstyle{\frac 12}+\nu).\end{cases} \end{aligned} $$

(A.2)

Recall that we are considering admissible representations π ₁, π ₂, π ₃ of ${\mathrm {GL}}_2(\mathbb R)$ such that Π = π ₁ ⊗ π ₂ ⊗ π ₃ has trivial central character. This means that we may assume without loss of generality that the central character of each π _j is of the form ${\mathrm {sgn}}^{\delta _j}$ with δ ₁ + δ ₂ + δ ₃ ≡ 0 (mod 2).

Proposition A.2

Consider Π = π ₁ ⊗ π ₂ ⊗ π ₃ a triple product of admissible ${\mathrm {GL}}_2(\mathbb R)$ representations. Let

$$\displaystyle \begin{aligned} &L(s,\Pi) = L(s,\rho(\pi_1)\otimes \rho(\pi_2)\otimes \rho(\pi_3)), \\ & L(s,\Pi,{\mathrm{Ad}}{}) = L(s,{\mathrm{Ad}}(\rho(\pi_1))\otimes {\mathrm{Ad}}(\rho(\pi_2) \otimes{\mathrm{Ad}}(\rho(\pi_3)). \end{aligned} $$

The normalizing factors relating I to I ^′ in (1.2) for

$$\displaystyle \begin{aligned} \Pi^1 & = \pi_{0,\epsilon}(\textstyle{\frac 12}+\nu_1)\otimes \pi_{\delta,\epsilon'}(\textstyle{\frac 12}+\nu_2) \otimes \pi_{\delta,\epsilon'}(\textstyle{\frac 12}+\nu_3), \\ \Pi^2 & = \pi_{\mathrm{dis}}^{k}\otimes \pi_{\delta_2,\epsilon_2}(\textstyle{\frac 12}+\nu_2) \otimes \pi_{\delta_3,\epsilon_3}(\textstyle{\frac 12}+\nu_3), \\ \Pi^3 & = \pi_{\mathrm{dis}}^{k_1}\otimes \pi_{\mathrm{dis}}^{k_2} \otimes \pi_{\delta,\epsilon}(\textstyle{\frac 12}+\nu) \qquad\qquad\qquad \mathit{\mbox{ (with }} k_1\geq k_2+\delta), \\ \Pi^4 & = \pi_{\mathrm{dis}}^{k_1}\otimes \pi_{\mathrm{dis}}^{k_2}\otimes \pi_{\mathrm{dis}}^{k_3} \qquad\qquad\qquad\qquad\quad \mathit{\mbox{ (with }}k_1\geq k_2+k_3) \end{aligned} $$

are given by Table 2.

Table 2 Normalizing factors for triple product L-function at a real place

Proof

A simple exercise in applying Lemma A.1 gives the following.

$$\displaystyle \begin{aligned} \rho(\Pi^1) & = \bigg( \bigoplus_{\gamma_j\in\pm} \rho_1(\epsilon,\gamma_1\nu_1+\gamma_2(\nu_2+\nu_3)) \bigg) \oplus \bigg( \bigoplus_{\gamma_j\in\pm}\rho_1(\overline{\epsilon+\delta},\gamma_1\nu_1+\gamma_2(\nu_2-\nu_3)) \bigg) \\ \rho(\Pi^2) & = \bigoplus_{\gamma_j\in \pm} \rho_2(k-1,\gamma_2\nu_2+\gamma_3\nu_3) \\ \rho(\Pi^3) & = \rho_2(k_1+k_2-2,\nu) \oplus \rho_2(k_1+k_2-2,\nu) \oplus \rho_2(k_1-k_2,-\nu) \oplus \rho_2(k_1-k_2,-\nu) \\ \rho(\Pi^4) & = \rho_2(k_1+k_2+k_3-3,0) \oplus \rho_2(k_1+k_2-k_3-1,0) \\ & \qquad \oplus \rho_2(k_1-k_2+k_3-1,0) \oplus \rho_2(k_1-k_2-k_3+1,0) \end{aligned} $$

Combining each of these with the appropriate factors for L(1, π, Ad) from (A.2) together with $\Gamma _{\mathbb R}(2)^2$ gives the result. □