#### Glasnik Matematicki, Vol. 53, No. 2 (2018), 239-264.

### ON POINCARÉ SERIES OF HALF-INTEGRAL WEIGHT

### Sonja Žunar

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

*e-mail:* `szunar@math.hr`

**Abstract.**
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 * 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 S*_{m}(Γ,χ) . Using this last result, we construct a Poincaré series * Δ*_{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 * ξℂ*_{ℑ(z)>0} and * kℤ*_{≥0} . 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.

**2010 Mathematics Subject Classification.** 11F12, 11F37

**Key words and phrases.** Cusp forms of half-integral weight, Poincaré series, metaplectic cover of SL_{2}*(ℝ) *

**Full text (PDF)** (access from subscribing institutions only)
DOI: 10.3336/gm.53.2.03

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