#### Vol. 6, No. 4, 2012

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Spherical varieties and integral representations of $L$-functions

### Yiannis Sakellaridis

Vol. 6 (2012), No. 4, 611–667
##### 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).

##### Keywords
automorphic $L$-functions, spherical varieties, Rankin–Selberg, periods of automorphic forms
##### Mathematical Subject Classification 2000
Primary: 11F67
Secondary: 22E55, 11F70
##### Milestones
Received: 31 March 2010
Revised: 4 July 2011
Accepted: 1 August 2011
Published: 25 July 2012
##### Authors
 Yiannis Sakellaridis Department of Mathematics and Computer Science Rutgers University 101 Warren Street Smith Hall 216 Newark, NJ 07102 United States