# Fine Deligne-Lusztig varieties and Arithmetic Fundamental Lemmas

## TL;DR

This paper establishes a character formula for fine Deligne-Lusztig varieties, applies it to Shimura varieties, and proves the arithmetic fundamental lemma in the minuscule case without residual characteristic restrictions.

## Contribution

It introduces a new character formula for fine Deligne-Lusztig varieties and applies it to prove the arithmetic fundamental lemma in a broad setting.

## Key findings

- Proved a character formula for certain Deligne-Lusztig varieties.
- Computed fixed points for varieties from Shimura basic loci.
- Established the arithmetic fundamental lemma without residual characteristic assumptions.

## Abstract

We prove a character formula for some closed fine Deligne-Lusztig varieties. We apply it to compute fixed points for fine Deligne-Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport-Zink spaces arising from the arithmetic Gan-Gross-Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.