Spherical varieties and integral representations of $L$-functions

Abstract

We present a conceptual and uniform interpretation of the methods of integral representations of $L$-functions (period integrals, Rankin–Selberg integrals). This leads to (i) a way to classify such integrals, based on the classification of certain embeddings of spherical varieties (whenever the latter is available), (ii) a conjecture that would imply a vast generalization of the method, and (iii) an explanation of the phenomenon of “weight factors” in a relative trace formula. We also prove results of independent interest, such as the generalized Cartan decomposition for spherical varieties of split groups over $p$-adic fields (following an argument of Gaitsgory and Nadler).