On finite group actions on reductive groups and buildings (Q1608558)

Let \(H\) be a connected reductive group over a non-Archimedean local field \(k\), and let \(F\subset \text{Aut}_k(H)\) be a finite group of order \(f\) not divisible by the residual characteristic of \(k\). Let \(G\) be the identity component of the subgroup of \(H\) consisting of points fixed by \(F\). The authors prove that \(G\) is reductive (this is actually proved for any field \(k\), provided \(\operatorname {char} k\) does not divide \(f\)), and that the Bruhat-Tits building \(\mathcal B(G)\) of \(G\) can be identified with the set of \(F\)-fixed points of \(\mathcal B(H)\). The last result was recently proved by several authors for a number of special cases. A similar problem for spherical buildings is also considered.