Central extensions of p-adic algebraic groups by finite p-groups

Some problems on algebraic groups over global fields like the congruence subgroup problem involve the determination of topological central extensions of the adelic group which, in turn, leads naturally to the study of topological central extensions of p-adic Lie groups by finite groups like the group of roots of unity in the p-adic field. Moreover, central extensions of semisimple p-adic Lie groups often come from a subgroup of small rank like SL2 which has the interesting property that abstract central extensions of the locally compact group SL2(k) for a p-adic field k, by a finite group turn out to be automatically topological. Let k be a nonarchimedean local field and let A be a finite, abelian group. We use the corresponding Tits buiding to prove that, for a semisimple, simply connected k-isotropic algebraic group G, the group H2(G(k), A) of abstract central extensions of G(k) by any group A maps injectively under restriction maps, into a direct sum of H2(H(k), A) over k-rank 1 subgroups H when the group A is a finite, abelian p-group, where p is the characteristic of the residue field of k. The aim of this paper is to prove the following theorem. Theorem. Let k be a nonarchimedean local field and A a finite, abelian p-group, where p is the characteristic of the residue field of k. Let G be an absolutely almost simple, simply-connected algebraic group defined over k with k-rank(G) = r ≥ 2. Then there exist semisimple k-subgroups G1, · · · , Gr without kanisotropic factors and, each of k-rank equal to k-rank(G)−1 and semisimple k-subgroups Gij of Gi ∩Gj such that the ‘restriction’ map H(G(k), A) → r ⊕