On finite group actions on reductive groups and buildings

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Local models of Shimura varieties and a conjecture of Kottwitz (2011, background): “By [51], the building B(G,F ) can be identified with the fixed”
Integral models of Shimura varieties with parahoric level structure (2015, background): “Under the assumption of tameness, by work of Rousseau or [57], the canonical map B(G,K) →֒ B(G, K̃) gives identifications B(G,K) = B(G, K̃) = B(H, K̃); the action of Γ on B(H, K̃) is induced by the action of Γ on H(K̃) given above.”
Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$L-functions (2012, background): “It follows from [79] that M(F )x ⊂ M(F1)x∩M(F ), see [1, Lemma 2.”
Fake projective planes (2005, background): “If v is a nonarchimedean place of k which does not lie over 3, then according to the main theorem of [ PY ], the set of points fixed by g in the Bruhat–Tits building of G(kv) is the Bruhat–Tits building of T(kv) .W e conclude from this that if T is anisotropic at v (i.e., T(kv) is compact), then as ”
Depth-zero supercuspidal L-packets and their stability (2009, result): “Since γs is compact and has finite order prime to p, the results of [46] combined with Remark 7.”