Zero-dimensional affine Deligne--Lusztig varieties

Abstract

In this paper, we study the affine Deligne--Lusztig variety $X(\mu,b)_K$ and classify all quadruples $(\mathbf{G}, \mu, b, K)$ with $\dim X(\mu, b)_K=0$. This question was first asked by Rapoport in 2005, who also made an explicit conjecture in the hyperspecial level. We prove that $\dim X(\mu,b)_K=0$ if and only if, up to certain Hodge-Newton decomposition condition, the pair $(\mathbf{G}, \{\mu\})$ is of extended Lubin-Tate type. We also give a combinatorial description of this condition by the essential gap function on $B(\mathbf{G})$ and the $\mu$-ordinary condition for the generic Newton stratum.