Smooth transfer of Kloosterman integrals. (English) Zbl 1060.11032

Jacquet, Hervé

Duke Math. J. 120, No. 1, 121-152 (2003).

Let \(E/F\) be a quadratic extension of local non-Archimedean fields. Let \(G= \text{GL}(m)\), \(A\) be the diagonal torus of \(G\) and \(N\) be the unipotent radical of the standard Borel subgroup. Let \(\psi : \;F \to C^\times\) be a non trivial character. For \(n\in N(F)\), define \(\theta(n)=\psi(\sum_{i} n_{i,i+1})\). For any smooth function \(\Phi\) of compact support on \(G(F)\), the (diagonal) Kloosterman orbital integral is defined by \[ \Omega(\Phi,a)=\int_{N(F)\times N(F)} \Phi(\;^tn_1 an_2)\theta(n_1n_2)\,dn_1\,dn_2 \] for \(a\in A(F)\). Let \(S(F)\) be the subset of Hermitian matrices in \(\text{GL}(m,E)\). For \(n\in N(E)\), define a character \(\theta\) by \(\theta(n\bar{n})= \psi(\sum_{i} n_{i,i+1}+\bar{n}_{i,i+1})\). The non-trivial element of \(\text{Gal}(E/F)\) is denoted by a bar. For any smooth function \(\Psi\) of compact support on \(S(F)\), the (diagonal) relative Kloosterman orbital integral is defined by \[ \Omega(\Psi,a)=\int_{N(E)} \Psi( ^ t \bar{n}an)\theta(n\bar{n})\, dn \] for \(a\in A(F)\). The functions \(\Phi\) et \(\Psi\) match if for every \( a\in A(F)\) \[ \Omega(\Phi,a)=\gamma(a) \Omega(\Psi,a). \] Here \(\gamma\) is a transfer factor defined explicity by \(\gamma(a)=\eta(a_1) \ldots \eta(a_1a_2\ldots a_{m-1})\) where \(\eta\) is the quadratic character of \(F^\times\).
The main result of this paper is that for all \(\Phi\) there is a \(\Psi\) (and vice-versa) such that the functions \(\Phi\) and \(\Psi\) match. The method of the author is to linearize the problem by replacing \(G(F)\) (resp. \(S(F)\)) by the vector space of (resp. hermitian) matrices. Then he proves an inversion formula between the orbital integrals of a function and its Fourier transform. As a consequence, if the functions \(\Phi\) and \(\Psi\) match then their Fourier transforms also match. This provides many matching functions and is used to prove the existence of the transfer. This is in strong analogy with the work of J.-L. Waldspurger on the Langlands-Shelstad transfer [Compos. Math. 105, No. 2, 153–236 (1997; Zbl 0871.22005)].
The matching here is relevant to a relative trace formula which may be used to study the quadratic base change (cf. for GL(3) [H. Jacquet and Y. Ye, Trans. Am. Math. Soc. 348, No. 3, 913–939 (1996; Zbl 0861.11033)]). In this context, there is also a fundamental lemma. B. C. Ngô has established it for function fields [Duke Math. J. 96, No. 3, 473–520 (1999; Zbl 1047.11517)]. The author uses the previous matching to prove the fundamental lemma for number fields (in this article for \(m=2,3\) and for all \(m\) in [Kloosterman identities over a quadratic extension, Ann. Math. (2) 160, No. 2, 755–779 (2004; Zbl 1071.11026)].

Reviewer: Pierre-Henri Chaudouard (Cachen)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11F72	Spectral theory; trace formulas (e.g., that of Selberg)
11L05	Gauss and Kloosterman sums; generalizations

Keywords:

base change; Kloosterman integral; relative trace formula

Citations:

Zbl 0871.22005; Zbl 0861.11033; Zbl 1047.11517; Zbl 1071.11026

References:

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