The orthogonal group of a Form Hilbert space

Abstract

Form Hilbert spaces are constructed over fields that are complete in a non-archimedean valuation. They share with classical Hilbert spaces the basic property expressed by the Projection Theorem. However, there appear some remarkable geometric features which are unknown in Euclidean geometry. In fact, due to the so-called type condition there are only a few orthogonal straight lines containing vectors of the same length, so these non-archimedean spaces are utmost inhomogeneous. In the paper we consider a typical Form Hilbert space $(E, < , >)$ and we show that this geometric feature has a strong impact on the group $\mathcal{O}(E)$ of all isometries $T:E\longrightarrow E$ and on the lattice $\mathcal{L}$ of all normal subgroups of $\mathcal{O}$. In particular, we describe some remarkable sublattices of $\mathcal{L}$ which have no analogue in the classical orthogonal groups.