Affine Springer fibers and depth zero L-packets
Let $G$ be a connected reductive group over $F=\mathbb F_q((t))$ splitting over $\overline{\mathbb F_q}((t))$. Following [KV], every tamely unramified Langlands parameter $\lambda:W_F\to{}^L G(\overline{\mathbb Q_l})$ in general position gives rise to a finite set $\Pi_{\lambda}$ of irreducible admissible representations of $G(F)$, called the $L$-packet. The goal of this work is to provide a geometric description of characters $\chi_{\pi}$ of all $\pi\in\Pi_{\lambda}$ in terms of homology of affine Springer fibers. As an application, we give a geometric proof of the stability of sum $\chi_{\lambda}^{st}:=\sum_{\pi\in\Pi_{\lambda}}\chi_{\pi}$. Furthermore, as in [KV] we show that the $\chi_{\lambda}^{st}$'s are compatible with inner twistings.