Generalized Ramanujan conjecture for GL(n)

Let $F$ be a number field with adele ring $\mathbb A_F$, and let $n\ge2$. Let $\pi=\bigotimes_v'\pi_v$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_F)$ whose central character is unitary.

Each local component $\pi_v$ is irreducible, generic and unitary. By the Langlands classification it is a fully induced representation

$$\pi_v\cong\operatorname{Ind}_{P}^{\mathrm{GL}_n(F_v)}\bigl(\tau_1|\det|_v^{\sigma_1}\otimes\cdots\otimes\tau_r|\det|_v^{\sigma_r}\bigr),$$

where $P$ is a standard parabolic subgroup with Levi factor $\mathrm{GL}_{n_1}\times\cdots\times\mathrm{GL}_{n_r}$, each $\tau_j$ is a tempered representation of $\mathrm{GL}_{n_j}(F_v)$, and $\sigma_j\in\mathbb R$. The representation $\pi_v$ is tempered exactly when every $\sigma_j=0$. Set

$$\theta(\pi_v)=\max_j|\sigma_j| .$$

At a finite place $v$ where $\pi_v$ is unramified with Satake parameters $\alpha_{1,v},\dots,\alpha_{n,v}$, this says $\max_i\bigl|\log_{q_v}|\alpha_{i,v}|\bigr|=\theta(\pi_v)$, where $q_v$ is the size of the residue field.

Conjecture (generalized Ramanujan conjecture for $\mathrm{GL}_n$). For every number field $F$, every $n\ge2$, every such $\pi$, and every place $v$ of $F$, archimedean or not, $\pi_v$ is tempered, that is, $\theta(\pi_v)=0$. At unramified finite places this means

$$|\alpha_{i,v}|=1\qquad(1\le i\le n).$$

The conjecture is known for large classes of $\pi$ that are regular algebraic, listed below. It is open in general, and open at every place for non-algebraic $\pi$, including Maass forms on $\mathrm{GL}_2$ over $\mathbb Q$ [1, 2].

Background

Ramanujan conjectured in 1916 that the coefficients of the discriminant form $\Delta(z)=\sum\tau(m)q^m$ satisfy $|\tau(p)|\le2p^{11/2}$. Petersson extended this to holomorphic cusp forms, and Satake recast it as temperedness of local components [1]. The archimedean case on $\mathrm{GL}_2$ over $\mathbb Q$ includes Selberg's eigenvalue conjecture: every nonconstant Maass cusp form for a congruence subgroup has Laplace eigenvalue $\lambda\ge1/4$ [1].

The restriction to cuspidal representations of $\mathrm{GL}_n$ is essential. Howe and Piatetski-Shapiro found cuspidal representations of $\mathrm{Sp}_4$ whose local components are almost all non-tempered [3]. For other groups, temperedness is expected only for representations whose Arthur parameter is generic [1].

Bounds toward the conjecture are used throughout analytic number theory, for example in subconvexity, equidistribution and sieve problems [4]. Langlands observed that functoriality of all tensor powers $\pi\mapsto\pi^{\otimes k}$ would imply the conjecture [1].