  This paper treats mathematically some problems in p-adic quantum mechanics.
We first deal with p-adic symplectic group corresponding to the symmetry on the
classical phase space. By the filtrations of isotropic subspaces and almost
self-dual lattices in the p-adic symplectic vector space, we explicitly give
the expressions of parabolic subgroups, maximal compact subgroups and
corresponding Iwasawa decompositions of some symplectic groups. For a triple of
Lagrangian subspaces, we associated it with a quadratic form whose Hasse
invariant is calculated. Next we study the various equivalent realizations of
unique irreducible and admissible representation of p-adic Heisenberg group.
For the Schrodinger representation, we can define Weyl operator and its kernel
function, while for the induced representations from the characters of maximal
abelian subgroups of Heisenberg group generated by the isotropic subspaces or
self-dual lattice in the p-adic symplectic vector space, we calculate the
Maslov index defined via the intertwining operators corresponding to the
representation transformation operators in quantum mechanics.
