Spherical varieties and integral representations of L-functions

Details Spherical varieties and integral representations of L-functions Spherical varieties and integral representations of L-functions

2009-05-26

arxiv.org/abs/0905.4245v2

Mathematics

Number Theory

54 pages, conjectural part shortened, theorems on hyperspecial and Iwahori orbits strengthened, submitted for publication

Scientific paper

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 of such integrals, based on the classification of certain embeddings of spherical varieties (whenever the latter is available), (ii) a conjecture which 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).

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