Jeffrey Adams’s research while affiliated with University of Maryland, College Park and other places

Let G be a reductive group over an algebraically closed field and let W be its Weyl group. In a series of papers, Lusztig introduced a map from the set [W] of conjugacy classes of W to the set [Gu] of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes [We] of W, is injective. In this paper, we show that Lusztig's map [We]→[Gu] is order-reversing, with respect to the natural partial order on [We] arising from combinatorics and the natural partial order on [Gu] arising from geometry.

Let G be a reductive group over an algebraically closed field and let W be its Weyl group. In a series of papers, Lusztig introduced a map from the set [W] of conjugacy classes of W to the set $[G_u]$ of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of W, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry.

Suppose G is a connected complex semisimple group and W is its Weyl group. The lifting of an element of W to G is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of W to the set of semisimple conjugacy classes of G. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.