On certain varieties attached to braid representatives of a Weyl group element

Abstract: Let $G$ be a connected reductive group over an algebraically closed field with Weyl group $W$. The analogy between Lusztig varieties and Deligne-Lusztig varieties associated to minimal length elements in elliptic conjugacy classes of $W$ was studied by He and Lusztig. In this paper, we study the parabolic version of above varieties associated to some good braid elements for any conjugacy class of $W$. This also leads to an alternative version of Lusztig's map in general case.