THE TITS BUILDING AND AN APPLICATION TO ABSTRACT CENTRAL EXTENSIONS OF p-ADIC ALGEBRAIC GROUPS BY FINITE p-GROUPS

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 nonArchimedean 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 H2(G(k), A) of abstract central extensions injects into a finite direct sum of H2(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. Introduction and statement of main theorem Certain 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. Thus, it may be of some interest to look at abstract central extensions of p-adic Lie groups by finite groups. Let k be a non-Archimedean local field and let A be a finite, abelian group. Consider a connected, semisimple, simply connected algebraic group G defined over k. For the trivial action of G(k) on A, one has the group H(G(k), A) of abstract central extensions of the locally compact group G(k) by the finite group and its subgroup H top(G(k), A) of topological central extensions. If G is quasi-split, these two groups coincide; this was noticed in [Su93]. The equality, in general, is a question posed by Gopal Prasad. In [PR84], Gopal Prasad and M.S. Raghunathan have proved, among several other things, that for a semisimple, simply connected k-isotropic algebraic group G, the group H top(G(k), A) of topological central extensions of G(k) by any group A maps injectively under restriction maps into a Received by the editors November 19, 2009 and, in revised form, June 11, 2010. 2010 Mathematics Subject Classification. Primary 20G25, 20G10.