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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.

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