Fourier expansions of vector-valued automorphic functions with non-unitary twists
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Publication:6388076
DOI10.4310/CNTP.2023.V17.N1.A5zbMath1509.11027arXiv2201.04454MaRDI QIDQ6388076
Anke D. Pohl, Ksenia Fedosova, Julie Rowlett
Publication date: 12 January 2022
Abstract: We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.
Modular and automorphic functions (11F03) Fourier coefficients of automorphic forms (11F30) Bessel and Airy functions, cylinder functions, ({}_0F_1) (33C10) Spectral theory; eigenvalue problems on manifolds (58C40)
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