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Geometric Waldspurger periods II

Author: Sergey Lysenko
Journal: Represent. Theory 24 (2020), 235-291
MSC (2010): Primary 11R39; Secondary 14H60
Published electronically: July 2, 2020
MathSciNet review: 4127907
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Abstract: In this paper we extend the calculation of the geometric Waldspurger periods from our paper [Compos. Math. 144 (2008), no. 2, 377–438] to the case of ramified coverings. We give some applications to the study of Whittaker coeffcients of the theta-lifting of automorphic sheaves from $\operatorname {PGL}_2$ to the metaplectic group $\widetilde {\operatorname {SL}}_2$; they agree with our conjectures from [Geometric Whittaker models and Eisenstein series for $\mathrm {Mp}_2$, arXiv:1221.1596]. In the process of the proof, we construct some new automorphic sheaves for ${\operatorname {GL}_2}$ in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair $(\widetilde {\operatorname {SL}}_2, \operatorname {PGL}_2)$.

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Additional Information

Sergey Lysenko
Affiliation: Institut Elie Cartan Lorraine, Université de Lorraine, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France
MR Author ID: 614865

Received by editor(s): September 9, 2019
Received by editor(s) in revised form: May 15, 2020
Published electronically: July 2, 2020
Additional Notes: The author was supported by the ANR program ANR-13-BS01-0001-01.
Article copyright: © Copyright 2020 American Mathematical Society