\(p\)-adic Heisenberg group and Maslov index (Q1308479)

Properties of the Maslov index \(\mu\) are investigated. It is emphasized that \(\mu\) is a topological characteristic which can be used to study quantization procedures. A ``system of coordinates'' on a set \(\Lambda\) of selfdual \(\mathbb{Z}_ p\)-lattices in a two-dimensional \(p\)-adic symplectic space is introduced. The Heisenberg group of this space is defined and a unitary irreducible representation of this group, depending on a lattice \({\mathcal L}\in\Lambda\) is constructed. The intertwining operator of two such representations is defined and its properties are investigated. Then, the Maslov index \(({\mathcal L}_ 1, {\mathcal L}_ 2,{\mathcal L}_ 3)\) for a triple of self-dual \(\mathbb{Z}_ p\)-lattices is defined and an explicit formula is obtained for it. Finally, the values of \(\mu\) for different triples of lattices are calculated in the coordinates previously introduced.