Vol. 300, No. 2, 2019

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On the Braverman–Kazhdan proposal for local factors: spherical case

Zhilin Luo

Vol. 300 (2019), No. 2, 431–471
Abstract

We study the Braverman–Kazhdan proposal for the local spherical situation. In the $p$-adic case, we give a definition of the spherical component of conjectural space ${\mathsc{S}}_{\rho }\left(G,K\right)$ and the $\rho$-Fourier transform kernel ${\Phi }_{\rho }^{K}$, and verify several conjectures of Braverman and Kazhdan (GAFA 2000 special volume, part I, 237–278) in this situation. In the archimedean case, we study the asymptotic of the basic function ${1}_{\rho ,s}$ and the $\rho$-Fourier transform kernel ${\Phi }_{\rho ,s}^{K}$.

Keywords
L-function, Fourier transform, Schwartz space
Mathematical Subject Classification 2010
Primary: 11F70, 22E45
Secondary: 22E35