Exponential sums for \(GL(n)\) and their applications to base change
This article gives an identity for \(GL(n)\) Kloosterman sums which is related to the relative trace formula. To put this in context, \textit{H. Jacquet} and \textit{Y. Ye} [C. R. Acad. Sci., Paris, Sér. I 311, No. 11, 671-676 (1990; Zbl 0715.11026)] have conjectured a relative trace formula for \(GL(n)\), which characterizes the automorphic representations of \(GL(n)\) which are quadratic base changes as being distinguished with respect to a quasi-split unitary group. This formula should be proved locally. On the one hand, one has Kloosterman sums or local Kloosterman integrals for \(GL(n)\), as in the work of \textit{D. Bump, S. Friedberg} and \textit{D. Goldfeld} [Acta Arith. 50, No. 1, 31-89 (1988; Zbl 0647.10020)], \textit{S. Friedberg} [Math. Z. 196, 165-188 (1987; Zbl 0612.10020)], and \textit{G. Stevens} [Math. Ann. 277, 25-51 (1987; Zbl 0597.12017)]. On the other, there are certain naturally occuring exponential sums (which should perhaps be called relative Kloosterman sums) associated with \(GL(n)\) and a quadratic extension. The fundamental lemma for the relative trace formula states that these sums should be equal, up to a transfer factor. The author proves such a matching for one set of exponential sums, corresponding to the Weyl group element represented by the permutation matrix with the \((n-1)\times(n-1)\) anti-identity matrix in the upper left corner. (In this case the \(GL(n)\) Kloosterman sum reduces to the classical hyper-Kloosterman sum.) When \(n=3\), this identity was previously established by the author in [J. Number Theory 51, No. 2, 275-287 (1995; Zbl 0822.11057)]. The relative trace formula also requires proving a matching of Shalika germs, formulated for \(GL(n)\) by \textit{H. Jacquet} and the author in [Trans. Am. Math. Soc. 348, No. 3, 913-939 (1996; Zbl 0861.11033)]. As a consequence of the matching of exponential sums established here, the author deduces one case of such a matching of Shalika germs.
scientific article; zbMATH DE number 1146323
scientific article