Fourier expansions of vector-valued automorphic functions with non-unitary twists
Journal article, 2023

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.

twisted automorphic function

generalized automorphic function

Fourier expansion

non-unitary representation


Ksenia Fedosova

University of Freiburg

Anke Pohl

Universität Bremen

Julie Rowlett

Chalmers, Mathematical Sciences, Analysis and Probability Theory

University of Gothenburg

Communications in Number Theory and Physics

1931-4523 (ISSN) 1931-4531 (eISSN)

Vol. 17 1 173-248

Geometric analysis and applications to microbe ecology

Swedish Research Council (VR) (2018-03873), 2019-01-01 -- 2022-12-31.

Subject Categories

Algebra and Logic


Mathematical Analysis



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