Cordial elements and dimensions of affine Deligne-Lusztig varieties

The affine Deligne-Lusztig variety $X_w(b)$ in the affine flag variety of a reductive group $\mathbb G$ depends on two parameters: the $\sigma$-conjugacy class $[b]$ and the element $w$ in the Iwahori-Weyl group $\tilde W$ of $\mathbb G$. In this paper, for any given $\sigma$-conjugacy class $[b]$, we determine the nonemptiness pattern and the dimension formula of $X_w(b)$ for most $w \in \tilde W$.