Abstract

The notion of affine Springer fiber was introduced by Kazhdan and Lusztig in 1988. It plays a crucial role in the geometric representation theory and the Langlands program. The generalized affine Springer fibers were first studied by Kottwitz and Viehmann for the hyperspecial level structure in 2012 and by Lusztig for arbitrary parahoric level structure in 2015. Many geometric properties for the hyperspecial level structure were further studied by Bouthier and Chi.

In this talk, I will propose a new approach to study the generalized affine Springer fibers. The key observation is that the affine Deligne-Lusztig varieties, in some sense, may be regarded as the ``shadow’’ of generalized affine Springer fibers. I will also explain some ingredients used to deduce some geometric properties (nonemptiness, dimension, irreducible components) of generalized affine Springer fibers from the properties of the affine Deligne-Lusztig varieties. This talk is based a work in progress.