On the generalized Ramanujan conjecture over function fields

Abstract. This is a report on joint work with Dan Ciubotaru.  We consider a cuspidal automorphic representation π of a semisimple group G over a function field K.  When G = PGL(n), the Ramanujan conjecture, which asserts that every local component of π is tempered, was proved in connection with the global Langlands correspondence by L. Lafforgue.  For general G the direct generalization of the Ramanujan conjecture is known to be false, but conjectures of Arthur and Shahidi suggest that if π is globally generic — that it, π has a Whittaker model — then every local component is tempered.   We prove a related statement.  Suppose there are two places v and u of K such that π_v is unramified (and so is the group G(K_v)) and has a (local) Whittaker model, and π_u is tempered.  (For some groups we also have to assume the characteristic of K is greater than 3.)  Then π_w is tempered for every w at which π is unramified.  The proof is based on the application of Deligne's theory of Frobenius weights to V. Lafforgue's global parameter attached to $\pi$, and on the classification of generic spherical representations of unramified p-adic groups, due to Barbasch and Ciubotaru.  We also need a result on the local (Genestier-Lafforgue) parameters attached to discrete series representations; this is proved in a joint paper with Gan and Sawin, and in turn depends on a new globalization result due to Beuzart-Plessis.