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 πv is unramified (and so is the group G(Kv)) and has a (local) Whittaker model, and πu is tempered.  (For some groups we also have to assume the characteristic of K is greater than 3.)  Then πw 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 π, 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.

Thursday, September 29, 2022 - 15:00

Thackeray 703 (note the unusual time and location)

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Michael Harris
Columbia University