Smooth transfer of Kloosterman integrals. (English) Zbl 1060.11032

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)].


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
Full Text: DOI


[1] D. Bump, S. Friedberg, and D. Goldfeld, “Poincaré series and Kloosterman sums” in The Selberg Trace Formula and Related Topics (Brunswick, Maine, 1984) , Contemp. Math. 53 , Amer. Math. Soc., Providence, 1986, 39–49. · Zbl 0595.10024
[2] H. Jacquet, The continuous spectrum of the relative trace formula for \(\GL(3)\) over a quadratic extension , Israel J. Math. 89 (1995), 1–59. · Zbl 0818.11025
[3] –. –. –. –., A theorem of density for Kloosterman integrals , Asian J. Math. 2 (1998), 759–778. · Zbl 0963.11026
[4] H. Jacquet and Y. Ye, Relative Kloosterman integrals for \(\GL(3)\) , Bull. Soc. Math. France 120 (1992), 263–295. · Zbl 0785.11032
[5] –. –. –. –., Distinguished representations and quadratic base change for \(\GL(3)\) , Trans. Amer. Math. Soc. 348 (1996), 913–939. JSTOR: · Zbl 0861.11033
[6] –. –. –. –., Germs of Kloosterman integrals for \(\GL(3)\) , Trans. Amer. Math. Soc. 351 (1999), 1227–1255. JSTOR: · Zbl 0919.11038
[7] ——–, Kloosterman integrals over a quadratic extension , to appear in Ann. of Math. (2).
[8] E. Lapid and J. Rogawski, Periods of Eisenstein series: The Galois case , Duke Math. J. 120 (2003), 153–226. · Zbl 1037.11033
[9] B. C. Ngô, Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye , Ann. Sci. École Norm. Sup. (4) 32 (1999), 619–679. · Zbl 1002.11046
[10] –. –. –. –., Le lemme fondamental de Jacquet et Ye en caractéristique positive , Duke Math. J. 96 (1999), 473–520. · Zbl 1047.11517
[11] T. A. Springer, “Some results on algebraic groups with involutions” in Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983) , Adv. Stud. Pure Math. 6 , North-Holland, Amsterdam, 1985, 525–543. · Zbl 0628.20036
[12] G. Stevens, Poincaré series on \(\GL(r)\) and Kloostermann sums , Math. Ann. 277 (1987), 25–51. · Zbl 0597.12017
[13] J. L. Waldspurger, Le lemme fondamental implique le transfert , Compositio Math. 105 (1997), 153–236. · Zbl 0871.22005
[14] Y. Ye, The fundamental lemma of a relative trace formula for \(\GL(3)\) , Compositio Math. 89 (1993), 121–162. · Zbl 0799.11013
[15] –. –. –. –., An integral transform and its applications , Math. Ann. 30 (1994), 405–417. · Zbl 0809.11031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.