Families of Canonical Local Periods on Spherical Varieties

Abstract

In this paper, we consider the variation of canonical local periods on spherical varieties proposed by Sakellaridis-Venkatesh in families. We formulate conjectures for the meromorphic property of these canonical local periods. As examples, we discuss these conjectures for the triple product case and the Gan-Gross-Prasad case. 3.4. Families of spherical characters 12 3.5. Homological multiplicities 13 4. The Triple product case 17 4.1. Jacquet-Langlands-Shimizu lifting and PSR zeta integral in families 17 4.2. Triple product local period in families 23 5. The Gan-Gross-Prasad case 24 5.1. Rankin-Selberg and Asai zeta integral in families 24 5.2. GGP spherical character in families 27 Appendix A. Co-Whittaker families 30 Appendix B. Bernstein variety and rigidity of automorphic type 31 References 32

Higher Ext-groups and Relatively Supercuspidal Spectra

“…Finally, we note that with the local constancy of Euler-Poincare number (see [6,Proposition 3.18]), Theorem 1.3 implies the local constancy for the dimension of the Hom-space for families in the triple product case, which was originally deduced from the meromorphy property of the canonical local periods in families (See [6, Theorem 4.8, Corollary 4.9]).…”