Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of \(G\)-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness properties of affine Deligne-Lusztig varieties under minimal assumptions on the associated group. We show that affine Deligne-Lusztig varieties are locally of finite type, and prove a global finiteness result related to the natural group action. Similar results have previously been known for special situations.

14G35, 20E42, 20G25

affine Deligne-Lusztig variety, Rapoport-Zink spaces, affine flag variety

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Technische Universität München, Fakultät für Mathematik - M11, Boltzmannstr. 3, 85748 Garching bei München, Germany

Technische Universität München, Fakultät für Mathematik - M11, Boltzmannstr. 3, 85748 Garching bei München, Germany