Periods of Automorphic Forms Associated to Strongly Tempered Spherical Varieties

In this paper, we compute the local relative characters for 10 strongly tempered spherical varieties in the unramified case. We also study the local multiplicity for these models. By proving a geometric multiplicity formula, we show that the summation of the multiplicities is always equal to 1 over each local tempered Vogan <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-packet defined on the pure inner forms of the strongly tempered spherical varieties. Finally, we formulate the Ichino–Ikeda type conjecture on a relation between the period integrals and the central values of certain automorphic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions for those strongly tempered spherical varieties.