Naslov
On Poincaré series of half-integral weight

Autori
Žunar, Sonja

Izvornik
Glasnik matematički (0017-095X) 53 (2018), 2; 239-264

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
Cusp forms of half-integral weight, Poincaré series, metaplectic cover of SL_2(R)

Sažetak
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 m is in 5/2+ℤ_{; ; ; ≥0}; ; ; . We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any f 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 ↦ f^{; ; ; (k)}; ; ; (ξ) on S_m(Γ, χ), where ξ is in ℂ_{; ; ; ℑ(z)>0}; ; ; and k is in ℤ_{; ; ; ≥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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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