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

Zhilin Luo

Vol. 300 (2019), No. 2, 431–471

DOI: 10.2140/pjm.2019.300.431

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 ${\mathcal{S}}_{\rho }\left(G,K\right)$ 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

Milestones

Received: 14 March 2018

Revised: 13 September 2018

Accepted: 13 September 2018

Published: 30 July 2019