Affine Springer fibers and depth zero L-packets

Let $G$ be a connected reductive group over a field $F=\mathbb{F}_q((t))$ splitting over $\overline{\mathbb{F}}_q((t))$. Following [KV,DR], a tamely unramified Langlands parameter $\lambda:W_F\to{}^L G(\overline{\mathbb{Q}}_{\ell})$ in general position gives rise to a finite set $\Pi_{\lambda}$ of irreducible admissible representations of $G(F)$, called the $L$-packet. The main goal of this work is to provide a geometric description of characters $\chi_{\pi}$ of $\pi\in\Pi_{\lambda}$ and of their endoscopic linear combinations $\chi_{\lambda}^{\kappa}$ in terms of homology of affine Springer fibers, thus establishing an analog of Lusztig conjectures in this case. Furthermore, each $\chi_{\lambda}^{\kappa}$ can be described as the trace of Frobenius function of a conjugation equivariant perverse sheaf on the loop group by the sheaf-function correspondence. As another application, we prove that the sum $\chi_{\lambda}^{st}:=\sum_{\pi\in\Pi_{\lambda}}\chi_{\pi}$ is stable and show that the $\chi_{\lambda}^{st}$'s are compatible with inner twistings. More generally, we prove that each $\chi_{\lambda}^{\kappa}$ is $\mathcal{E}_{\lambda,\kappa}$-stable.