On generalized Igusa local zeta functions associated to simple Chevalley K-groups under the adjoint representation

Abstract

The purpose of this paper is to provide the reader with an introduction to the theory of generalized Igusa local zeta functions associated to irreducible matrix groups. As an application, the generalized Igusa local zeta function $Z_K\left(s\right)$ associated to the simple Chevalley $K$-groups of type $B_l,$$C_l,$$D_l,$$E_6,$$E_7,$$E_8,$$F_4$ and $G_2$ under the adjoint representation is explicitly exhibited as arational function satisfying a certain functional equation.