Digital Object Identifier (DOI) 10.1007/s002220100182 Invent. math. 147, 545–560 (2002) On finite group actions on reductive groups and buildings Gopal Prasad 1 , , Jiu-Kang Yu 2, 1 Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (e-mail: gprasad@math.lsa.umich.edu) 2 Department of Mathematics, University of Maryland, College Park, MD 20742, USA (e-mail: yu@math.umd.edu) Oblatum 8-III-2001 & 25-VII-2001 Published online: 19 November 2001 –  Springer-Verlag 2001 Dedicated to Jacques Tits Introduction Let H be a connected reductive group over a non-archimedean local field k and let F ⊂ Aut k ( H ) be a finite group of order not divisible by p, the residual characteristic of k. Let G = ( H F ) ◦ be the identity component of the subgroup of H consisting of points fixed by F . The main theorem of this paper asserts that the Bruhat-Tits building B(G ) of G can be identified with the set of F -fixed points of B( H ). Several special cases of this theorem have been known previously. When E/k is a finite totally ramified Galois extension, H = Res E/k G, and F = Gal( E/k), the condition p  # F is simply that E/k is tamely ramified. In this case, our main theorem is a well-known but unpublished theorem of G. Rousseau. Recently, one of us (G.P.) found a simple proof of this theorem [P]. When G is a classical group, realized in the standard way as the identity component of the group of fixed points of an involution φ of a general linear group H , Bruhat and Tits [BT4] have given a description of B(G ) as a sub- set of B( H ). In particular, it follows from their description that B(G ) = B( H ) φ when p  = 2. Recently, this fact has been rediscovered by J. Kim and A. Moy [KM], and independently, a simple proof has been given by [GY]. Let H be the split form of Spin(8). Then one can choose an outer automorphism φ ∈ Aut( H ) of order 3 such that G = H φ is of type G 2 . From a joint work [GY] by W.T. Gan and one of us (J.Y.), it is known that B(G ) = B( H ) φ is true for arbitrary p.  Partially supported by a Guggenheim Foundation Fellowship and an NSF grant  Partially supported by grant DMS 9801633 from the National Science Foundation