Glasnik matematički, Vol. 53 No. 2, 2018.
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
Publication date:
30.12.2018.
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