A question on splitting of metaplectic covers

TL;DR

This paper investigates the splitting behavior of the 2-fold metaplectic cover of ${\rm GL}_2(E)$ over specific subgroups, advancing understanding of representation restrictions in the context of quadratic extensions.

Contribution

It proves the splitting of the metaplectic cover of ${\rm GL}_2(E)$ over ${\rm GL}_2(F)$ and $D_F^{\times}$, providing foundational results for studying representation restrictions.

Findings

Splitting of the metaplectic cover over ${\rm GL}_2(F)$ established.

Splitting over $D_F^{\times}$ demonstrated.

Results facilitate analysis of representation restrictions in metaplectic groups.

Abstract

Let $E/F$ be a quadratic extension of a non-Archimedian local field. Splitting of the 2-fold metaplectic cover of ${\rm Sp}_{2n}(F)$ when restricted to various subgroups of ${\rm Sp}_{2n}(F)$ 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 ${\rm GL}_{2}(E)$ over the subgroups ${\rm GL}_{2}(F)$ 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 ${\rm GL}_{2}(E)$ to ${\rm GL}_{2}(F)$ and $D_{F}^{\times}$. These results were suggested to the author by Professor Dipendra Prasad.