Geometric Waldspurger periods II

Lysenko, Sergey

doi:10.1090/ert/543

Geometric Waldspurger periods II
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Represent. Theory 24 (2020), 235-291 Request permission

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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