On the affineness of distinguished Deligne-Lusztig varieties

We consider Deligne-Lusztig varieties in the variety of parabolic subgroups with a fixed type. In this paper, we give a criterion for the affineness of distinguished Deligne-Lusztig varieties, extending and refining the original Deligne-Lusztig criterion. In particular we show that distinguished Deligne-Lusztig varieties are affine except possibly for small q, and that all distinguished Deligne-Lusztig varieties associated to rank-2 groups are affine.