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A Question on Splitting of Metaplectic Covers

Let F be a non-Archimedian local field. Splitting of twofold metaplectic cover of Sp2n(F) restricted to various subgroups of Sp2n(F) is important in application of the Weil representation of the metaplectic group. Let E/F be a quadratic extension. In this paper, we prove the splitting of the metaplectic cover of GL2(E) restricted to the subgroup , where DF is the quaternion division algebra with center F, as a first step in our study of the restriction of representations of metaplectic cover of GL2(E) to GL2(F) and . These results were suggested to the author by Professor Dipendra Prasad.