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

TL;DR

This paper explores the relationships among smooth integral models of unitary, orthogonal, and symplectic groups over local fields, identifying conditions under which these relations hold or fail, inspired by classical Lie group facts.

Contribution

It establishes whether classical relations among Lie groups extend to their smooth integral models over local fields and characterizes properties of hermitian forms affecting these relations.

Findings

Identifies conditions for relations among integral models of classical groups

Determines when classical group equalities extend to integral models

Highlights properties of hermitian forms influencing group relations

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.