Affine Springer Fibers and the Kazhdan-Lusztig Map

Abstract

Let G be a connected reductive group with Lie algebra g and Weyl group W. Let P ⊂ G((t)) be a parahoric subgroup with Levi quotient Gₚ. Using the topology of Lie P, Kazhdan and Lusztig define a map from nilpotent orbits in Lie Gₚ to conjugacy classes in W. This thesis proves compatibilities between Kazhdan-Lusztig maps associated to different parahoric subgroups, as well as the Kazhdan-Lusztig map for the Langlands dual. These compatibilities come from studying the W-representation on the cohomology of affine Springer fibers. The main tool is Yun’s Global Springer Theory. We give two applications of these compatibilities. The first is an affine analog of the classical picture relating singular supports of IC sheaves on the flag variety with special nilpotent orbits. The second is a resolution of Lusztig’s conjecture that strata can be described by fibers of (parahoric) Kazhdan-Lusztig maps.