In the course of earlier investigations [1] on Eisenstein series, it became evident that Eisenstein series on tube domains were in a natural way automorphic forms for certain particular types of arithmetic groups. It seemed appropriate to designate these as special arithmetic groups because of the connection of their localizations with what Bruhat and Tits have called special maximal compact subgroups of simplyconnected, semi-simple, p-adic algebraic groups. The present note represents some preliminary investigations on these. We hope to treat further some of the questions raised here on another occasion.

Many of the results here are straightforward consequences of known facts, but we have not seen them assembled in print elsewhere. The known facts include results on linear, semisimple algebraic groups over an arbitrary field, including their classification, and results of Bruhat and Tits on the buildings associated with reductive p-adic linear groups.

The main references are [5, 7, 8, 11]. We wish to acknowledge here our debt to conversations with H. Hijikata and G. Shimura, as well as to a written communication from J. Tits. All of these have proved most helpful in bringing together the conclusions developed here. The author trusts each of the aforementioned understands his thanks for their help.

Let k be an algebraic number field of finite degree and let G be a connected, semisimple, linear algebraic group defined over k. We generally assume G is simply connected, though for the most elementary definitions this is unnecessary.