Periods of Deligne-Lusztig Characters associated to Spherical Varieties

TL;DR

This paper computes the periods of Deligne-Lusztig characters linked to spherical varieties and extends existing formulas for symmetric spaces and complexity zero cases, enriching the understanding of these characters.

Contribution

It introduces a new formula for Deligne-Lusztig character periods associated with spherical varieties, generalizing Lusztig's and Reeder's formulas.

Findings

Derived a formula for periods of Deligne-Lusztig characters on spherical varieties.

Extended Lusztig's work on symmetric spaces.

Generalized Reeder's formula in the complexity zero case.

Abstract

In this paper, we calculate the periods of Deligne-Lusztig characters $R_{T,\chi}^G$ associated to $X^F$, where $X=G/H$ 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.