# Finiteness properties of affine Deligne-Lusztig varieties

## TL;DR

This paper establishes fundamental finiteness properties of affine Deligne-Lusztig varieties, showing they are locally of finite type and have global finiteness under minimal assumptions, advancing understanding of their geometric structure.

## Contribution

It proves that affine Deligne-Lusztig varieties are locally of finite type and globally finite under minimal assumptions, generalizing previous special case results.

## Key findings

- Affine Deligne-Lusztig varieties are locally of finite type.
- They exhibit global finiteness related to group actions.
- Results hold under minimal assumptions on the group.

## Abstract

Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of $G$-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness properties of affine Deligne-Lusztig varieties under minimal assumptions on the associated group. We show that affine Deligne-Lusztig varieties are locally of finite type, and prove a global finiteness result related to the natural group action. Similar results have previously been known for special situations.