On a conjecture of Kottwitz and Rapoport

TL;DR

This paper proves a conjecture by Kottwitz and Rapoport, establishing a converse to Mazur's Inequality for split and quasi-split reductive groups, with implications for affine Deligne-Lusztig varieties.

Contribution

It provides a proof of a conjecture linking group theory and geometric properties of affine Deligne-Lusztig varieties, extending previous results.

Findings

Converse to Mazur's Inequality established for all split and quasi-split groups.

Connected affine Deligne-Lusztig varieties are shown to be non-empty under certain conditions.

The results unify and extend previous partial results in the theory.

Abstract

We prove a conjecture of Kottwitz and Rapoport which implies a converse to Mazur's Inequality for all split and quasi-split (connected) reductive groups. These results are related to the non-emptiness of certain affine Deligne-Lusztig varieties.