On Generalized Fourier Transforms for Standard L-Functions

  • Freydoon ShahidiEmail author
Conference paper
Part of the Simons Symposia book series (SISY)


Any generalization of the method of Godement–Jacquet on principal L-functions for GL(n) to other groups as perceived by Braverman–Kazhdan/Ngo requires a Fourier transform on a space of Schwartz functions. In the case of standard L-functions for classical groups, a theory of this nature was developed by Piatetski-Shapiro and Rallis, called the doubling method. It was later that Braverman and Kazhdan, using an algebro-geometric approach, different from doubling method, introduced a space of Schwartz functions and a Fourier transform, which projected onto those from doubling method. In both methods a normalized intertwining operator played the role of the Fourier transform. The purpose of this paper is to show that the Fourier transform of Braverman–Kazhdan projects onto that of doubling method. In particular, we show that they preserve their corresponding basic functions. The normalizations involved are not the standard ones suggested by Langlands, but rather a singular version of local coefficients of Langlands–Shahidi method. The basic function will require a shift by 1/2 as dictated by doubling construction, reflecting the global theory, and begs explanation when compared with the work of Bouthier–Ngo–Sakellaridis. This matter is further discussed in an appendix by Wen-Wei Li.


Braverman-Kazhdan Schwartz spaces and Fourier transforms Doubling method Normalized intertwining operators 



Confusions stemming from the shift \(s+\frac 12\), which did not seem to agree with Ngo’s shift [N3, BNS], did result in a number of communications with Erez Lapid for which I like to thank him. Similar gratitude is owed to Dihua Jiang, Wee Teck Gan, Shunsuke Yamana, David Kazhdan, and Jayce Getz. I also like to thank Werner Müller, Sug Woo Shin, and Nicolas Templier for their invitation to the Simons Symposium at Elmau, Germany, in April of 2016 and for the present proceedings. Parts of this paper were presented as a series of lectures at University of Minnesota where author was invited as an Ordway Distinguished Visitor during the Fall of 2016 and for which thanks are due to Dihua Jiang. Last but not least, I like to thank Wen-Wei Li for a numbers of helpful comments and communications after the first version of this manuscript was distributed, which in particular led to his insightful appendix [Li2] to this paper.


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Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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