FINITE GROUP ACTIONS ON REDUCTIVE CROUPS AND BUILDINGS AND TAMELY-RAMIFIED DESCENT IN BRUHAT-TITS THEORY

摘要

Let K be a discretely valued field with Henselian valuation ring and separably closed (but not necessarily perfect) residue field of characteristic p, H a connected reductive K-group, and Theta a finite group of automorphisms of H. We assume that p does not divide the order of Theta and Bruhat-Tits theory is available for H over K with B(H/K) the Bnihat-Tits building of H(K). We will show that then Bruhat-Tits theory is also available for G := (H-Theta)degrees and B(H/K)(Theta) is the Bruhat-Tits building of G(K). (In case the residue field of K is perfect, this result was proved in a joint paper of the author with Jiu-Kang Yu by a different method.) As a consequence of this result, we obtain that if Bnihat-Tits theory is available for a connected reductive K-group G over a finite tamely-ramified extension L of K, then it is also available for G over K and B(G/K) = B(G/L)(Gal(L/K)). Using this, we prove that if G is quasi-split over L then it is already quasi-split over K.