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Summary: Compositio Math. 142 (2006) 541550
doi:10.1112/S0010437X06001904
Generic transfer from GSp(4) to GL(4)
Mahdi Asgari and Freydoon Shahidi
Abstract
We establish the Langlands functoriality conjecture for the transfer from the generic spec-
trum of GSp(4) to GL(4) and give a criterion for the cuspidality of its image. We apply this
to prove results toward the generalized Ramanujan conjecture for generic representations
of GSp(4).
1. Introduction
Let k be a number field and let G denote the group GSp(4, Ak). The (connected component of the)
L-group of G is GSp(4, C), which has a natural embedding into GL(4, C). Langlands functoriality
predicts that associated to this embedding there should be a transfer of automorphic representations
of G to those of GL(4, Ak) (see [Art04]). Langlands' theory of Eisenstein series reduces the proof
of this to unitary cuspidal automorphic representations. We establish functoriality for the generic
spectrum of GSp(4, Ak). More precisely (cf. Theorem 2.4), we prove the following.
Let be a unitary cuspidal representation of GSp(4, Ak), which we assume to be globally
generic. Then has a unique transfer to an automorphic representation of GL(4, Ak).
The transfer is generic (globally and locally) and satisfies = 2
and . Here,
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