# Eva Viehmann: Dimensions of Affine Deligne-Lusztig varieties

Eva Viehmann

## Dimensions of affine Deligne-Lusztig varieties in the affine flag variety – a favorable case

Affine Deligne-Lusztig varieties are locally closed subschemes of the affine flag variety of a reductive group $G$ that are associated with an element $x$ of the affine Weyl group of $G$ and an element $b$ of the loop group of $G$. They play an important role in the description of the special fiber of Shimura varieties and are an analog and a generalization of the famous Deligne-Lusztig varieties in (classical) flag varieties. However, so far we know very little about their geometry, except in the so-called basic case. In this talk I want to give a short introductory overview of affine Deligne-Lusztig varieties. Then I will discuss a group-theoretic condition on the element $x$ (called cordiality) that allows to determine non-emptiness and to compute the dimension of all affine Deligne-Lusztig varieties for this $x$ and all $b$, and further implications on the geometry of the Newton stratification.

This is joint work in progress with E. Milicevic.