On Poincaré series of half-integral weight

Abstract

We use Poincaré series of $ K $-finite matrix coefficients of genuine integrable representations of the metaplectic cover of $ \mathrm{SL}_2(\mathbb 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 $ \mathrm{SL}_2(\mathbb R) $, $ χ$ is a character of $ Γ$ of finite order, and $ m\in\frac52+\mathbb Z_{\geq0} $. We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any $ f\in S_m(Γ,χ) $. Using this last result, we construct a Poincaré series $ Δ_{Γ,k,m,ξ,χ}\in S_m(Γ,χ) $ that corresponds, in the sense of the Riesz representation theorem, to the linear functional $ f\mapsto f^{(k)}(ξ) $ on $ S_m(Γ,χ) $, where $ ξ\in\mathbb C_{\Im(z)>0} $ and $ k\in\mathbb Z_{\geq0} $. Under some additional conditions on $ Γ$ and $ χ$, we provide the Fourier expansion of cusp forms $ Δ_{Γ,k,m,ξ,χ} $ and their expansion in a series of classical Poincaré series.