Quotient of Deligne–Lusztig varieties

We study the quotient of parabolic Deligne–Lusztig varieties by a finite unipotent group UF where U is the unipotent radical of a rational parabolic subgroup P=LU. We show that in some particular cases the cohomology of this quotient can be expressed in terms of “smaller” parabolic Deligne–Lusztig varieties associated to the Levi subgroup L.