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

In this short note, we compute higher extension groups for all irreducible representations and deduce the multiplicity formula for finite length representations in triple product case.

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