Splittings of the metaplectic covers of some reductive dual pairs

In this paper, we construct a splitting of the metaplectic cover of the red uctive dual pairs of orthogonal and symplectic groups or the reductive dual pairs of unitary groups over a nonarchimedean local field with respect to a generalized lattice model of the Weil representation. We also prove a res ult concerning the splitting that we construct and the theta dichotomy for unitary group. The splitting plays a very crucial role in the study of thet a correspondence for p-adic and finite reductive dual pairs.