  We prove that the Drinfeld halfspace is essentially the only Deligne-Lusztig
variety which is at the same time a period domain over a finite field. This is
done by comparing a cohomology vanishing theorem for DL-varieties, due to
Digne, Michel, and Rouquier, with a non-vanishing theorem for PD, due to the
first author. We also discuss an affineness criterion for DL-varieties.
