On geometrical interpretation of the \(p\)-adic Maslov index
A geometrical interpretation of the \(p\)-adic Maslov index (\(p \neq 2\)) is given in terms of the action of the \(p\)-adic symplectic group \(\text{Sp}({\mathcal V})\) on the space \(\Lambda\) of selfdual lattices. The space \(\Lambda\) is endowed with an integer valued metric \(d\) and a realization of the metric space \((\Lambda,d)\) as a graph \(\Gamma\) is considered. It is shown that \(\Gamma\) consists of cycles of length three and can be derived from the Bruhat-Tits tree by the transformation ``star-triangle''. The symplectic group \(\text{Sp}({\mathcal V})\) acts transitively on sets of vertices. The \(p\)-adic Maslov index is invariant under this action so that the action of \(\text{Sp}({\mathcal V})\) on the set of cycles of length three is not transitive. The main result of the paper is: For any two cycles \([{\mathcal L}_ 1,{\mathcal L}_ 2,{\mathcal L}_ 3]\) and \([{\mathcal L}_ 1', {\mathcal L}_ 2',{\mathcal L}_ 3']\) of length three there is a symplectic transformation \(g \in \text{Sp}({\mathcal V})\) such that \(g {\mathcal L}_ i = {\mathcal L}_ i'\) (\(i = 1,2,3\)) if and only if the \(p\)-adic Maslov indices of these cycles coincide.
scientific article; zbMATH DE number 529863