Spherical varieties and integral representations of L-functions

By Y. Sakellaridis. 2009

56 citations recorded by the source corpus.

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 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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ae5672563e9f49a1e9c780bb9f06dacbcef9ff80

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In Geometry and Topology

1. A. Bouthier
2. Ngo Bao Chau
3. Y. Sakellaridis

In Mathematical Physics

1. A. Bouthier
2. Ngo Bao Chau
3. Y. Sakellaridis

Citations in the corpus

• Spherical functions on spherical varieties (2009, methodology): “An alternate proof was given by Gaitsgory and Nadler in [GN10], which was adapted to the p-adic case, under assumptions which hold at almost every place if the variety is defined over a global field, in [Sa12].”
• On the formal arc space of a reductive monoid (2014, methodology): “This expectation was stated in [19] in order to give a conceptual explanation to the Rankin-Selberg method, but the idea draws from the work of Braverman and Kazhdan who studied the Schwartz space of the basic affine space [8], and from relevant work in the geometric Langlands program [6, 5].”