# The distinction problem for metaplectic case

## TL;DR

This paper investigates the distinction problem for metaplectic groups over quadratic extensions using theta lifts, proposing a conjectural formula for multiplicities in related symplectic pairs.

## Contribution

It introduces a novel approach using theta lifts to analyze distinction problems for metaplectic groups and formulates a conjecture for higher-dimensional cases.

## Key findings

- Established a method for the Mp(2) case over quadratic extensions.
- Proposed a conjectural formula for multiplicities in the (Mp(2n), Sp(2n)) case.
- Connected theta lifts with distinction problems in the metaplectic setting.

## Abstract

We use the theta lifts between Mp(2) and PD to study the distinction problems for the pair (Mp(2,E), SL(2,F )), where E is a quadratic field extension over a nonarchimedean local field F of characteristic zero and D is a quaternion algebra. With a similar strategy, we give a conjectural formula for the multiplicity of distinction problem related to the pair (Mp(2n,E),Sp(2n,F)).