Poisson Duality and Zeta Functions

Abstract

This chapter recalls some standard facts about the Poisson summation formula on a euclidean space. It will be applied when the euclidean space is the space of matrices, with the trace scalar product, so we make all the standard formalism explicit in this case. We give two classical applications to the Epstein zeta function, which serve as prototypes. In the extension of the theory to Pos_n, we have to generalize the definition of the Bessel functions to this higher dimensional case, and this will be done in the next chapter. Then we can put all these results together in the study of Eisenstein series.