  In this paper we consider Deligne-Lusztig varieties and their analogues when
the Frobenius endomorphism is replaced with conjugation by an element in a
group, especially a regular semisimple or regular unipotent one. We calculate
their classes in the Chow group of the flag variety in terms of Schubert
classes. Also we give some sufficient criteria when different elements in the
Weyl group result in the same class.
