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In this paper, we study the K-finite matrix coefficients of integrable representations of the metaplectic cover of SL 2 (R) and give a result on the non-vanishing of their Poincaré series. We do this by adapting the techniques developed for SL 2 (R) by Muić to the case of the metaplectic group.

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We use Poincaré series of K-finite matrix coefficients of genuine integrable representations of the metaplectic cover of SL 2 (R) to construct a spanning set for the space of cusp forms S m (Γ, χ), where Γ is a discrete subgroup of finite covolume in the metaplectic cover of SL 2 (R), χ is a character of Γ of finite order, and m ∈ 5 2 + Z ≥0 . We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any f ∈ S m (Γ, χ). Using this last result, we construct a Poincaré series ∆ Γ,k,m,ξ,χ ∈ S m (Γ, χ) that corresponds, in the sense of the Riesz representation theorem, to the linear functional f → f (k) (ξ) on S m (Γ, χ), where ξ ∈ C ℑ(z)>0 and k ∈ Z ≥0 . Under some additional conditions on Γ and χ, we provide the Fourier expansion of cusp forms ∆ Γ,k,m,ξ,χ and their expansion in a series of classical Poincaré series.

We use Poincaré series of K-finite matrix coefficients of genuine integrable representations of the metaplectic cover of SL 2 (R) to construct a spanning set for the space of cusp forms Sm(Γ, χ), where Γ is a discrete subgroup of finite covolume in the metaplectic cover of SL 2 (R), χ is a character of Γ of finite order, and m ∈ 5 2 +Z ≥0. We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any f ∈ Sm(Γ, χ). Using this last result, we construct a Poincaré series ∆ Γ,k,m,ξ,χ ∈ Sm(Γ, χ) that corresponds, in the sense of the Riesz representation theorem, to the linear functional f → f (k) (ξ) on Sm(Γ, χ), where ξ ∈ C ℑ(z)>0 and k ∈ Z ≥0. Under some additional conditions on Γ and χ, we provide the Fourier expansion of cusp forms ∆ Γ,k,m,ξ,χ and their expansion in a series of classical Poincaré series. 2010 Mathematics Subject Classification. 11F12, 11F37. Key words and phrases. Cusp forms of half-integral weight, Poincaré series, metaplectic cover of SL 2 (R). The author acknowledges Croatian Science Foundation Grant No. 9364.