A question on splitting of metaplectic covers

Abstract

Let $E/F$ be a quadratic extension of a non-Archimedian local field. Splitting of the 2-fold metaplectic cover of ${\mathrm{Sp}}_{2n}(F)$ when restricted to various subgroups of ${\mathrm{Sp}}_{2n}(F)$ plays an important role in application of the Weil representation of the metaplectic group. In this paper we prove the splitting of the metaplectic cover of ${\mathrm{GL}}_2(E)$ over the subgroups ${\mathrm{GL}}_2(F)$ and $D_F^{\times }$, where $D_F$ is the quaternion division algebra with center $F$, as a first step in our study of the restriction of representations of metaplectic cover of ${\mathrm{GL}}_2(E)$ to ${\mathrm{GL}}_2(F)$ and $D_F^{\times }$. These results were suggested to the author by Professor Dipendra Prasad.

9 pages