Geometric Representation Theory Seminar | Good position braids, transversal slices and affine Springer fibers

Abstract：

Let G be a reductive group over an algebraically closed field and W be its Weyl group. Using Coxeter elements, Steinberg constructed cross-sections of the adjoint quotient of G which also yield transversal slices of regular unipotent classes. In 2012, He and Lusztig constructed transversal slices using minimal length elements in elliptic conjugacy classes in W, which yield transversal slices of basic unipotent classes. In this talk, we generalize minimal length elements to good position braids in the associated braid monoid of W and use these elements to construct transversal slices of all unipotent classes in G. We shall see these new elements also appear in many other aspects of representation theory, such as affine Springer fibers and the partial order on unipotent classes, etc.