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We prove that the perfect loop functor  LX  of a quasi-projective scheme  X  over a local non-archimedean field  k  satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group  G , the mapL⁢G→L⁢(G/B){LG\rightarrow L(G/B)}is a  v -surjection. This gives a mixed characteristic version (for  v -topology) of an equal characteristic result (in étale topology) of Bouthier–Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne–Lusztig spacesXw⁢(b){X_{w}(b)}attached to unramified  p -adic reductive groups. We show that in various cases these sheaves are ind-representable, thus partially solving a question of Boyarchenko. Finally, we show that the natural covering spacesX˙w˙⁢(b){\dot{X}_{\dot{w}}(b)}are pro-étale torsors over clopen subsets ofXw⁢(b){X_{w}(b)}, and analyze some examples.

Arc-descent for the perfect loop functor and p -adic Deligne–Lusztig spaces