Communications in Number Theory and Physics

Volume 17 (2023)

Number 1

Fourier expansions of vector-valued automorphic functions with non-unitary twists

Pages: 173 – 248



Ksenia Fedosova (Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Germany)

Anke Pohl (Department of Mathematics, Institute for Dynamical Systems, University of Bremen, Germany)

Julie Rowlett (Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden)


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.


Fourier expansion, generalized automorphic function, twisted automorphic function, non-unitary representation

2010 Mathematics Subject Classification

Primary 11F03, 58C40. Secondary 11F30, 33C10, 34L10.

Anke Pohl is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project no. 264148330 and no. 441868048 (Priority Program 2026 “Geometry at Infinity”).

Julie Rowlett is supported by Swedish Research Council Grant 2018-03873.

Received 8 February 2022

Accepted 16 January 2023

Published 23 February 2023