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Automorphic forms on $\operatorname{PGSp}(2)$


Author: Yuval Z. Flicker
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 39-50
MSC (2000): Primary 11F70; Secondary 22E50, 22E55, 22E45
Published electronically: April 23, 2004
MathSciNet review: 2048430
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Abstract: The theory of lifting of automorphic and admissible representations is developed in a new case of great classical interest: Siegel automorphic forms. The self-contragredient representations of PGL(4) are determined as lifts of representations of either symplectic PGSp(2) or orthogonal SO(4) rank two split groups. Our approach to the lifting uses the global tool of the trace formula together with local results such as the fundamental lemma. The lifting is stated in terms of character relations. This permits us to introduce a definition of packets and quasi-packets of representations of the projective symplectic group of similitudes PGSp(2), and analyse the structure of all packets. All representations, not only generic or tempered ones, are studied. Globally we obtain a multiplicity one theorem for the discrete spectrum of the projective symplectic group PGSp(2), a rigidity theorem for packets and quasi-packets, determine all counterexamples to the naive Ramanujan conjecture, and compute the multiplicity of each member in a packet or quasi-packet in the discrete spectrum. The lifting from SO(4) to PGL(4) amounts to establishing a product of two representations of GL(2) with central characters whose product is 1. The rigidity theorem for SO(4) amounts to a strong rigidity statement for a pair of representations of $\operatorname{GL}(2,\mathbb{A} )$.


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Additional Information

Yuval Z. Flicker
Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus, OH 43210-1174
Email: flicker@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S1079-6762-04-00128-3
PII: S 1079-6762(04)00128-3
Keywords: Automorphic representations, symplectic group, liftings, twisted endoscopy, packets, quasi-packets, multiplicity one, rigidity, functoriality, twisted trace formula, character relations
Received by editor(s): March 4, 2004
Published electronically: April 23, 2004
Additional Notes: Partially supported by a Lady Davis Visiting Professorship at the Hebrew University, 2004, and Max-Planck-Institut für Mathematik, Bonn, 2003.
Communicated by: David Kazhdan
Article copyright: © Copyright 2004 American Mathematical Society