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 $w\in \tilde{W}$.