Periods of Deligne-Lusztig Characters associated to Spherical Varieties

In this paper, we calculate the periods of Deligne-Lusztig characters $R_{T,\chi }^G$ associated to $X^F$, where $X=G\mathrm{/}H$ for a connected spherical subgroup $H$ of $G$. As an application, we deduce a formula that partially generalizes Lusztig’s work on symmetric spaces and extends Reeder’s formula in the complexity $0$ case.