On the Non-Vanishing of Poincar\'{e} Series on the Metaplectic Group

TL;DR

This paper investigates the non-vanishing properties of Poincaré series associated with K-finite matrix coefficients of integrable representations of the metaplectic cover of SL(2,R), adapting techniques from classical SL(2,R) analysis.

Contribution

It extends Muić's methods to the metaplectic group, providing new results on the non-vanishing of Poincaré series in this setting.

Findings

Established non-vanishing criteria for Poincaré series on the metaplectic group.

Adapted classical techniques to the metaplectic cover context.

Provided insights into the structure of integrable representations.

Abstract

In this paper, we study the $ K $-finite matrix coefficients of integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb R) $ and give a result on the non-vanishing of their Poincar\'{e} series. We do this by adapting the techniques developed for $ \mathrm{SL}_2(\mathbb R) $ by Mui\'{c} to the case of the metaplectic group.