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Distinguished representations and quadratic base change for \(GL(3)\). (English) Zbl 0861.11033

This paper is part of a series concerned with developing the relative trace formula for quadratic extensions of \(GL(3)\) and applying it to study automorphic representations. The authors prove the Theorem: Suppose that \(E/F\) is a quadratic extension of number fields, such that every archimedean place of \(F\) splits in \(E\). Let \(\Pi\) be a cuspidal automorphic representation of \(GL(3,E_A)\) which is a base change. Then \(\Pi\) is distinguished for the unitary group. Here distinguished means that there exists \(\varphi\) in the space of \(\Pi\) whose period over the unitary group is nonzero. (There is more than one unitary group in this context, but they are all isomorphic.) The converse is also true, by an argument of Harder, Langlands and Rapoport.
The Theorem is obtained by means of the relative trace formula. This formula requires a matching of the orbital integrals of test functions. The matching on the Hecke algebra was proved in previous work of the authors [Bull. Soc. Math. Fr. 120, 263-295 (1992; Zbl 0785.11032)] (the unit element) and of the second author [Compos. Math. 89, 121-162 (1993; Zbl 0799.11013)] (the full Hecke algebra), while the continuous spectrum was analyzed by the first author in [Isr. J. Math. 89, 1-59 (1995; Zbl 0818.11025)]. The remaining piece is the matching of arbitrary test functions at a finite number of places. This is carried out for finite places in this paper. The matching is accomplished by a computation of the asymptotics of the orbital integrals in question. To do this, the authors first introduce the new and important notion of a system of Shalika germs for such integrals. They do so in the context of \(GL(n)\). They show the existence of a system of Shalika germs in general, and its uniqueness in an appropriate sense. They then compute them in the case at hand, and obtain the desired matching.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
11R39 Langlands-Weil conjectures, nonabelian class field theory
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11L05 Gauss and Kloosterman sums; generalizations
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