Abstract
We study the quotient of parabolic Deligne-Lusztig varieties by a finite unipotent group U^F where U is the unipotent radical of a rational parabolic subgroup P = L U. 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.
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