On the symmetric square: applications of a trace formula

Flicker, Yuval Z.

doi:10.1090/S0002-9947-1992-1041045-0

On the symmetric square: applications of a trace formula
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by Yuval Z. Flicker

Trans. Amer. Math. Soc. 330 (1992), 125-152

DOI: https://doi.org/10.1090/S0002-9947-1992-1041045-0

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Abstract:

In this paper we prove the existence of the symmetric-square lifting of admissible and of automorphic representations from the group ${\text {SL}}(2)$ to the group ${\text {PGL}}(3)$. Complete local results are obtained, relating the character of an ${\text {SL}}(2)$-packet with the twisted character of self-contragredient ${\text {PGL}}(3)$-modules. Our global results relate packets of cuspidal representations of ${\text {SL}}(2)$ with a square-integrable component, and self-contragredient automorphic ${\text {PGL}}(3)$-modules with a component coming from a square-integrable one. The sharp results, which concern ${\text {SL}}(2)$ rather than ${\text {GL}}(2)$, are afforded by the usage of the trace formula. The surjectivity and injectivity of the correspondence implies that any self-contragredient automorphic ${\text {PGL}}(3)$-module as above is a lift, and that the space of cuspidal ${\text {SL}}(2)$-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 [$\text {VI}$] to extend our results to all cuspidal ${\text {SL}}(2)$-modules and self-contragredient ${\text {PGL}}(3)$-modules

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