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 \(\Phi^{K}_{\rho}\), 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 \(\Phi^{K}_{\rho,s}\).