The nonvanishing hypothesis at infinity for Rankin-Selberg convolutions (Q2826778)

The author solves a long standing problem which arises in the study of special values of L-functions called the non-vanishing hypothesis. He reinterprets and reduces to the question to showing that a certain map between \(({\mathfrak{g}},K)\)-cohomologies of \({\mathrm{GL}}_n({\mathbb{R}})\) or \({\mathrm{GL}}_n({\mathbb{C}})\) is nonzero. Finally he proves that the map is nonzero and thus solving the problem.NEWLINENEWLINEFor the ease of explaining, we will restrict to the case when the group is \({\mathrm{GL}}_n({\mathbb{R}})\) and \(n\) is odd. Let \(F_\mu\) be a complex finite dimensional irreducible representation of \({\mathrm{GL}}_n({\mathbb{R}})\) with highest weight \(\mu = (\mu_1, \mu_2, \ldots, \mu_n)\). Let \(\Omega(\mu)\) be the set of irreducible Casselman-Wallach representations \(\pi\) of \({\mathrm{GL}}_n({\mathbb{R}})\) whose restriction to \({\mathrm{SL}}_n^\pm({\mathbb{R}})\) is tempered and NEWLINE\[NEWLINE H^*({\mathfrak{g}}l_n({\mathbb{C}}), {\mathrm{GO}}(n)^\circ; F^\vee_\mu \otimes \pi) NEWLINE\]NEWLINE is nonzero. Here \({\mathrm{GO}}(n) = \{ g \in {\mathrm{GL}}_n({\mathbb{C}}) | g^\top g \text{ is a scalar matrix} \}\). We warn that all the finite length representations of \({\mathrm{GL}}_n({\mathbb{R}})\) considered below are Casselman-Wallach representations rather than their \(({\mathfrak{g}},K)\)-modules. It is known that \(\Omega(\mu)\) is non-empty if and only if \(\mu\) is pure, i.e. NEWLINE\[NEWLINE \mu_1 + \mu_n = \mu_2 + \mu_{n-1} = \ldots = \mu_n + \mu_1. NEWLINE\]NEWLINE In this case \(\Omega(\mu)\) consists of a single irreducible representation \(\pi_\mu\) of \({\mathrm{GL}}_n({\mathbb{R}})\) obtained by cohomological induction.NEWLINENEWLINENext we consider the group \({\mathrm{GL}}_{n-1}({\mathbb{R}})\) and a pure weight \(\nu\) of it. In this case \(\Omega(\nu) = \{ \pi_\nu, \pi_\nu \otimes {\mathrm{sgn}} \}\) consists of two irreducible representations of \({\mathrm{GL}}_{n-1}({\mathbb{R}})\).NEWLINENEWLINEWe turn to L-functions. An element \(\frac{1}{2} + j\) where \(j\) is an integer is called a critical place of \(\pi_\mu \times \pi_\nu\) if it is not a pole of the local L-function \({\mathrm{L}}(s,\pi_\mu \times \pi_\nu)\) and \({\mathrm{L}}(1-s,\pi_\mu \times \pi_\nu)\). Let \(G = {\mathrm{GL}}_n({\mathbb{R}}) \times {\mathrm{GL}}_{n-1}({\mathbb{R}})\) and let \(H\) be the diagonal subgroup \({\mathrm{GL}}_{n-1}({\mathbb{R}})\). Let \({\mathfrak{g}}\) and \({\mathfrak{h}}\) be their complexified Lie algebras.NEWLINENEWLINEFrom now on, we will suppose that \(F_\xi := F_\mu \otimes F_\nu\) contains \(\det_H^{-j}\), i.e. there is a nonzero map NEWLINE\[NEWLINE \phi_F : F_\xi^\vee \rightarrow \det{}_H^j. NEWLINE\]NEWLINE Then it is known that \(\frac{1}{2} + j\) is a critical place. Using the Rankin-Selberg integrals for \(\pi_\mu \times \pi_\nu\) at \(\frac{1}{2}+j\), we have a nonzero homomorphism NEWLINE\[NEWLINE \phi_\pi : \pi_\xi \rightarrow |\det{}_H^{-j}| NEWLINE\]NEWLINE of \(H\)-modules where \(\pi_\xi := \pi_\mu \hat{\otimes} \pi_\nu\) is a Casselman-Wallach representation of \(G\). Tensoring the two nonzero maps gives a nonzero \(H\)-module homomorphism NEWLINE\[NEWLINE \phi_F \otimes \phi_\pi : F_\xi^\vee \otimes \pi_\xi \rightarrow \det{}_H^j \otimes |\det{}_H^j| = {\mathrm{sgn}}_H^j. NEWLINE\]NEWLINE This in turn induces a map between the following cohomological spaces: NEWLINE\[NEWLINE \phi_b : H^b({\mathfrak{g}}, {\mathrm{GO}}(n)^\circ \times {\mathrm{GO}}(n-1)^\circ ; F_\xi^\vee \otimes \pi_\xi) \rightarrow H^b({\mathfrak{h}}, {\mathrm{SO}}(n);{\mathrm{sgn}}_H^j) NEWLINE\]NEWLINE where \(b = \frac{1}{2} n(n-1) = \dim({\mathfrak{h}}/{\mathfrak{so}}(n))\). It is known that the domain of \(\phi_b\) has dimension 2 and its codomain has dimension 1. In particular, both spaces are nonzero. The main theorem of this paper states that the map \(\phi_b\) is nonzero.NEWLINENEWLINEThe proof is divided into two main steps. First the author considers the special cases of those \(\mu\) and \(\nu\) such that \(\pi_\xi\) is unitarizable and tempered. Then he uses the translation principle to prove the main theorem for the remaining cases.