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

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 $\Phi^{K}_{\rho}$, 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 $\Phi^{K}_{\rho,s}$.