The infinite-demensional $p$-adic symplectic group

An analogue of the Fock representation is constructed for the infinite-dimensional $p$-adic Heisenberg group.The restricted symplectic group is defined for an infinite-dimensional symplectic space over the field $\mathbf Q_p$of $p$-adic numbers. For the restricted symplectic group a projective representation is constructed that is compatible with the representation of the Heisenberg group, and an expression for the cocycle of this representation is given in terms of the $p$-adic Maslov index. It is proved that the extension corresponding to this cocycle reduces to a $\mathbf Z_2$-extension.