Periods of Automorphic Forms Associated to Strongly Tempered Spherical Varieties

TL;DR

This paper computes local relative characters and multiplicities for strongly tempered spherical varieties, establishing a multiplicity formula and proposing a conjecture linking period integrals to automorphic L-functions.

Contribution

It provides explicit calculations of local relative characters and multiplicities, and formulates a new conjecture relating periods to L-values for these varieties.

Findings

Multiplicity sums to 1 over local Vogan L-packets.

Computed local relative characters for 10 spherical varieties.

Proposed an Ichino-Ikeda type conjecture for these models.

Abstract

In this paper, we compute the local relative character for 10 strongly tempered spherical varieties in the unramified case. We also study the local multiplicity for these models. By proving a multiplicity formula, we show that the summation of the multiplicities is always equal to 1 over each local tempered Vogan $L$-packet defined on the pure inner forms of the spherical varieties. Finally, we formulate the Ichino-Ikeda type conjecture on a relation between the period integrals and the central values of certain automorphic $L$-functions for those strongly tempered spherical varieties.