Isomorphism theory for orthogonal groups over arbitrary integral domains

Abstract

The isomorphism theory of matrix groups, which began over fields, has in recent years turned to rings. Developments over rings have been primarily in two separate directions: the first is concerned with the classical groups and their congruence groups over integral domains, while the second is concerned with algebraic groups and their arithmetic subgroups. So the first direction considers a very general underlying domain, while the second considers more general groups of matrices. For a historical account of these matters refer to the books of DieudonnC [12] and O’Meara [27] and the survey article by Mostow [22]. For more recent results consult the references of the present article. Regarding the first direction, which is considered here, the theory is complete (with isolated exceptions) for the linear groups in general, the symplectic groups in general, and the unitary groups under additional assumptions of isotropy. As in the classical situation over fields, every isomorphism between such groups is determined in a natural way by a “metric preserving” semilinear isomorphism of the underlying spaces. In sharp contrast, results for the orthogonal groups exist only for arithmetic and local domains, but not for arbitrary integral domains. Refer to O’Meara [26], Hahn [14] and Prasad [31] for example. It is the purpose of this paper to extend the isomorphism theory of the orthogonal groups from these special domains to arbitrary integral domains. One insurmountable obstruction becomes apparent quickly: in any dimension there exist free b-modules with anisotropic quadratic structure, the orthogonal groups of which are trivial (for instance, see 3.1 of [26]); take two such situations in different dimensions; these orthogonal groups are obviously isomorphic since they are trivial; but their spaces cannot be connected with a semilinear isomorphism since their dimensions are not equal. So isotropy assumptions are inevitable in the isomorphism theory of the orthogonal groups over arbitrary integral domains. It is the goal of this paper to prove that isotropy assumptions are also sufficient.