Parabolic subgroups of the full linear group over a Dedekind ring of arithmetical type

Suppose K is a global field, S a finite set of valuations of K containing all Archimedean valuations, and R the ring of S-integral elements of K. Assume that card S2, R is generated by its invertible elements, and the ideal of R generated by the differences –1 for all invertible coincides with R. Under these assumptions, the parabolic subgroups of GL(n, R) are described. Namely, for each parabolic subgroup P there exists a unique net of ideals of R (Ref. Zh. Mat., 1977, 2A280) such that e()PG(), where G is the net subgroup of () and E() is the subgroup generated by the transvections in G(). It is shown that E() is a normal subgroup of G(). The factor group G(/E()) is studied. The case of the special linear group is also considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 66–79, 1977.In conclusion,the author would like to thank Z. I. Borevich for posing the problem and for his continued interest, and A. A. Suslin for useful discussions leading to simplifications of some of the proofs.