GOOD POSITION BRAIDS, TRANSVERSAL SLICES AND AFFINE SPRINGER FIBERS

Abstract

In the study of Iwahori-Hecke algebras, Geck and Pfeiffer introduced good elements inCoxeter groups. These elements played a crucial role in the work of He and Lusztig on generalizing Steinberg’s cross-sections and Steinberg slices. This work yields the transversal slices for basic unipotent conjugacy classes in a reductive group G. We improve this result by introducing some more general braid elements called good position braids. We use them to construct transversal slices for any unipotent conjugacy classes in G. On the other hand, these good position braids also correspond to affine Springer fibers via root valuation strata. The correspondence leads to a reformulation of the dimension formula of affine Springer fibers. We also expect these braid elements to help with a conjecture of Goresky, Kottwitz and MacPherson on the cohomology of affine Springer fibers.