On the Braverman-Kazhdan Proposal for Local Factors: Spherical Case

Abstract

In this paper, 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 $S_{\rho }(G,K)$ and the $\rho$-Fourier transform kernel ${\Phi }_{\rho }^K$, and verify several conjectures in [BK00] in this situation. In the archimedean case, we study the asymptotic of the basic function $1_{\rho ,s}$ and the $\rho$-Fourier transform kernel ${\Phi }_{\rho ,s}^K$.

Accepted for publication in the Pacific Journal of Mathematics