Topological central extensions of semi-simple groups over local fields. I, II
A central extension \(U\) of a group \(G\) is called universal if any central extension \(E\) of \(G\) factors uniquely through a homomorphism \(U\to E\) over \(G\); such an extension exists if and only if \(G\) is equal to its commutator subgroup \(G'=(G,G)\) and then it is unique up to a \(G\)-isomorphism. In this case, such a universal central extension is described from a free group \(F\) with image \(G\) as the commutator group of \(F/(K,F)\) with \(K\) the kernel of \(F\to G\), and the kernel of the projection of this extension onto \(G\) is naturally the second homology group \(H_2(G,{\mathbb{Z}})\). When \(G\) is a topological group, one has to consider the topological central extensions; when the group is second countable, the classes of central extensions of \(G\) by a commutative group are classified by the group \(H^2(G,C)\) of measurable classes of 2-cocycles on \(G\) with values in \(C\). For the group \(G=SL_ 2(F)\) over a field \(F\), the restriction to \(F^{\times}\) embedded in \(G\) by the co-root \(t\mapsto\left(\begin{smallmatrix} t&0\\ 0&t^{-1}\end{smallmatrix}\right)\) of a 2-cocycle on \(G\) gives a 2-cocycle on \(F^{\times}\), and using the presentation given by R. Steinberg, C. Moore and H. Matsumoto described all the co-cycles obtained, using a short co-root and also those coming from the case when \(F\) is a local field and the extensions are topological. For a non-archimedean local field, this gives \(H_ 2(G,{\mathbb{Z}})\) as the group \(\mu(F)\) of roots of unity in \(F\), and \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) as its dual \(\mu(F)^{\wedge}\) sent in \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) through the norm residue symbol and the above restriction from \(G\) to \(F^{\times}\). H. Matsumoto proved that the same result holds when \(G\) is the group of \(F\)-points, \(F\) a non-archimedean local field, of a connected, simply connected, simple, split algebraic group over \(F\), and the case of quasi-split groups was done by V. V. Deodhar and P. Deligne. These methods are based on the presentation given by R. Steinberg of the Chevalley groups. In this article, the authors deal with the group \(G\) of rational points in a non-archimedean local field \(F\) of a connected, simply-connected, absolutely simple, isotropic algebraic group over \(F\). They use the Bruhat-Tits building of this group on which it acts simplicially to reduce the computation of the cohomology of \(G\) to the one of the parahoric subgroups. They observe that \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) is in fact \(H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}})\) with the discrete topology on \({\mathbb{Q}}/{\mathbb{Z}}\), and they decompose \({\mathbb{Q}}/{\mathbb{Z}}\) in its \(p\)-primary component -- \(p\) is the residual characteristic of the field -- and its \(p'\)-primary component. This latter gives \(H^ 2(G,({\mathbb{Q}}/{\mathbb{Z}})_{p'})\) as the dual of \(\mu (F)_{p'}\), the group of roots of unity of order prime to \(p\); the former leads to \(H^ 2(G,({\mathbb{Q}}/{\mathbb{Z}})_ p)\) as the dual of \(\mu(F)_ p\) or to a subgroup of index two in \(\mu(F)^{\wedge}_ p\), and this part is technically the most difficult one of the paper. As a consequence of the determination of \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\), the authors prove that \(G\) is equal to its commutator subgroup, that \(G\) admits a universal central extension in the topological sense with fundamental group the dual of \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\). They also sketch a proof of the Kneser-Tits conjecture, proved by V. P. Platonov by a case by case check, which asserts that \(G\) is generated by the unipotent radicals of its \(F\)-parabolic subgroups.
connected, simply-connected absolutely simple isotropic algebraic groups
scientific article; zbMATH DE number 3879147