Germs of Kloosterman integrals for \(GL(3)\). (English) Zbl 0919.11038
A detailed conjecture describing the image of base change of automorphic forms from a unitary group of a quadratic extension \(E/F\) to \(GL(n,E)\), as those forms with non zero \(GL(n,F)\)-periods, was made in Y. Flicker [J. Reine Angew. Math. 418, 139-172 (1991; Zbl 0725.11026)]. The authors make an analogous conjecture reversing the roles of \(GL(n,F)\) and the unitary group. This relates to base change for \(GL(n)\) and a quadratic extension \(E/F\), which is known for any cyclic extension \(E/F\) using the trace formula ([J. Arthur and L. Clozel, Simple algebras, base change…, Ann. Math. Studies 120, (1989; Zbl 0682.10022)] in general and [Y. Flicker, Ann. Inst. Fourier 40, 1-30 (1990; Zbl 0706.11031)] simple proof in a special case). Attempting to reprove this quadratic base change in analogy with the trace formula, the authors compare orbital integrals arising from a consideration of a double coset \(N\backslash G/N\), \(G=GL(n,F)\), \(N\) the unipotent upper triangular subgroup, with similar ones on \(GL(n,E)\), but only for \(n=3\) – a restriction imposed by the brute force nature of the computations.
Reviewer: Y.Flicker (Columbus/Ohio)
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11R39 | Langlands-Weil conjectures, nonabelian class field theory |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
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