The Tits building and an application to abstract central extensions of $p$-adic algebraic groups by finite $p$-groups

Abstract:

For a connected, semisimple, simply connected algebraic group $G$ defined and isotropic over a field $k$, the corresponding Tits building is used to study central extensions of the abstract group $G(k)$. When $k$ is a non-Archimedean local field and $A$ is a finite, abelian $p$-group where $p$ is the characteristic of the residue field of $k$, then with $G$ of $k$-rank at least $2$, we show that the group $H^2(G(k),A)$ of abstract central extensions injects into a finite direct sum of $H^2(H(k),A)$ for certain semisimple $k$-subgroups $H$ of smaller $k$-ranks. On the way, we prove some results which are valid over a general field $k$; for instance, we prove that the analogue of the Steinberg module for $G(k)$ has no nonzero $G(k)$-invariants.