2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/133046We 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.16 pages, reference added to the paper of X. He (arXiv:0707.0259) in which the affineness conjecture for DL-varieties is provedAlgebraic GeometryRepresentation Theory20G40; 14M15;20C33text