On generalized Fourier Transforms for standard L-functions (with an appendix by Wen-Wei Li)

Abstract: Any generalization of the method of Godement-Jacquet on principal L-functions for GL(n) to other groups as perceived by Braverman-Kazhdan and 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.