Affine Springer fibers and depth zero L-packets

Abstract

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 $\hat{I}»:W_F\to {}^{L}G({\overline{\mathbb{Q}}}_{\ell })$ in general position gives rise to a finite set ${\hat{I}}_{\hat{I}}»$ 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 $\ddot{I}{‡}_{\ddot{I}}€$ of $\ddot{I}€\in {\hat{I}}_{\hat{I}}»$ and of their endoscopic linear combinations $\ddot{I}{‡}_{\hat{I}}{»}^{\hat{I}}º$ in terms of homology of affine Springer fibers, thus establishing an analog of Lusztig conjectures in this case. Furthermore, each $\ddot{I}{‡}_{\hat{I}}{»}^{\hat{I}}º$ 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 $\ddot{I}{‡}_{\hat{I}}{»}^{st}:={\sum }_{\ddot{I}€\in {\hat{I}}_{\hat{I}}»}\ddot{I}{‡}_{\ddot{I}}€$ is stable and show that the $\ddot{I}{‡}_{\hat{I}}{»}^{st}$'s are compatible with inner twistings. More generally, we prove that each $\ddot{I}{‡}_{\hat{I}}{»}^{\hat{I}}º$ is ${\mathcal{E}}_{\hat{I}»,\hat{I}º}$-stable.

v.3, 96 pages, minor changes in abstract and introduction