Original scientific paper

https://doi.org/10.3336/gm.53.2.03

On Poincaré series of half-integral weight

Sonja Žunar ; Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia

Abstract

We use Poincaré series of K -finite matrix coefficients of genuine integrable representations of the metaplectic cover of SL2(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(ℝ) , χ is a character of Γ of finite order, and m5/2+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 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 ξℂℑ(z)>0 and kℤ≥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.

Keywords

Cusp forms of half-integral weight; Poincaré series; metaplectic cover of SL2(ℝ)

Hrčak ID:

214474

URI

https://hrcak.srce.hr/214474

Publication date:

30.12.2018.

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