On the symmetric square: applications of a trace formula

Abstract

In this paper we prove the existence of the symmetric-square lifting of admissible and of automorphic representations from the group SL ( 2 ) {\text {SL}}(2) to the group PGL ( 3 ) {\text {PGL}}(3) . Complete local results are obtained, relating the character of an SL ( 2 ) {\text {SL}}(2) -packet with the twisted character of self-contragredient PGL ( 3 ) {\text {PGL}}(3) -modules. Our global results relate packets of cuspidal representations of SL ( 2 ) {\text {SL}}(2) with a square-integrable component, and self-contragredient automorphic PGL ( 3 ) {\text {PGL}}(3) -modules with a component coming from a square-integrable one. The sharp results, which concern SL ( 2 ) {\text {SL}}(2) rather than GL ( 2 ) {\text {GL}}(2) , are afforded by the usage of the trace formula. The surjectivity and injectivity of the correspondence implies that any self-contragredient automorphic PGL ( 3 ) {\text {PGL}}(3) -module as above is a lift, and that the space of cuspidal SL ( 2 ) {\text {SL}}(2) -modules with a square-integrable component admits multiplicity one theorem and rigidity ("strong multiplicity one") theorem for packets (and not for individual representations). The techniques of this paper, based on the usage of regular functions to simplify the trace formula, are pursued in the sequel [ VI \text {VI} ] to extend our results to all cuspidal SL ( 2 ) {\text {SL}}(2) -modules and self-contragredient PGL ( 3 ) {\text {PGL}}(3) -modules