2017/11/20 by Žunar, Sonja
#11F12 #11F37 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1711.07281
We use Poincaré series of K -finite matrix coefficients of genuine integrable representations of the metaplectic cover of SL2(\mathbb 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 SL2(\mathbb R) , χ is a character of Γ of finite order, and m∈\frac52+\mathbb 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 ΔΓ,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∈\mathbb C\Im(z)>0 and k∈\mathbb Z≥0 . Under some additional conditions on Γ and χ, we provide the Fourier expansion of cusp forms ΔΓ,k,m,ξ,χ and their expansion in a series of classical Poincaré series.