On a multiplicity formula for spherical varieties

Abstract

In this paper, we propose a conjectural multiplicity formula for general spherical varieties. For all the cases where a multiplicity formula has been proved, including Whittaker models, Gan–Gross–Prasad models, Ginzburg–Rallis models, Galois models and Shalika models, we show that the multiplicity formulas in our conjecture are the same as the multiplicity formulas that have been proved. We also prove the conjectural multiplicity formula in two new cases.