On the Non-Vanishing of Poincaré Series on the Metaplectic Group

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.