Relations among smooth integral models associated to quadratic, symplectic and hermitian lattices

Abstract

This work is motivated by an investigation into whether, and if so how, certain well known facts about Lie groups manifest in the context of group schemes over rings of integers of local fields. There are the following well-known relations among unitary, orthogonal and symplectic groups: U(n)=O(2n) \cap GL(n, C)=Sp(2n) \cap GL(n, C). Therefore, it is natural to ask whether or not there exist such relations among smooth integral models of unitary, orthogonal and symplectic groups defined over a local field. Moreover, if there do not exist such relations, it would still be worthwhile if one can identify the properties of a hermitian form that lead to failure. We answer all these questions in this paper.