On the Ramanujan Conjecture for Quasisplit Groups

Early experiences with classical (holomorphic) cusp forms, which initially started with the Ramanujan t-function, and later extended to even Maass cusp forms (cf. [70];[78], last paragraph) on the upper half plane, suggested that their Fourier coefficients ap at a prime p must be bounded by 2p(k-1)/2, where k is the weight (cf. [25, 96]). This is what is classically called the Ramanujan-Petersson conjecture. Its archimedean counterpart, the Selberg conjecture [79], states that the positive eigenvalues of the hyperbolic Laplacian on the space of cuspidal functions (functions vanishing at all the cusps) on a hyperbolic Riemann surface parametrized by a congruence subgroup must all be at least 1/4 (cf. [76, 79, 94]). While for the holomorphic modular cusp forms, this is a theorem ([25], also see [8, 12]), the case of Maass forms is far from resolved and both conjectures are yet unsettled and out of reach. Satake [78] was the first to observe that both conjectures can be uniformly formulated. More precisely, if one considers the global cuspidal representation attached to a given cuspidal eigenfunction, then all its local components must be tempered. This means that their matrix coefficients must all belong to $L2+(PGL2(Qp))$ for all $e \gt 0$ and every prime p of Q. We note that here we are allowing p = 8 and letting $Q_{/inf} = R$. It is now generally believed that the conjecture in its general form should be valid for GLn over number fields to the effect that all the local components of an irreducible (unitary) cuspidal representation of GLn(AF ) must be tempered (modulo center).

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Published 1 January 2004