TY - JOUR
T1 - Livshitz theorem for the unitary frame flow
AU - Katok, Svetlana
AU - Foth, Tatyana
N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2004/2
Y1 - 2004/2
N2 - Let Γ be a lattice in SU(n, 1). For each loxodromic element γ0 ∈ Γ we define a closed curve {γ 0} on Γ\SU(n, 1) that projects to the closed geodesic on the factor of the complex hyperbolic space Γ\ℍℂ n associated with γ0. We prove that the cohomological equation Script D F = f has a solution if f is the lift of a holomorphic cusp form to SU(n, 1) under the following condition: for each restriction of f to {γ0} a finite number of Fourier coefficients vanish, and this finite number grows linearly with the length of the curve. This is a generalization of the classical Livshitz theorem for SU (1, 1) (A. Livshitz. Mat. Zametki 10 (1971), 555-564) where the curves are the closed geodesies themselves and the vanishing of the integrals of / over them, i.e. the zeroth Fourier coefficients, is both necessary and sufficient. An application of our result to the construction of spanning sets for spaces of holomorphic cusp forms on complex hyperbolic spaces is given in Appendix A.
AB - Let Γ be a lattice in SU(n, 1). For each loxodromic element γ0 ∈ Γ we define a closed curve {γ 0} on Γ\SU(n, 1) that projects to the closed geodesic on the factor of the complex hyperbolic space Γ\ℍℂ n associated with γ0. We prove that the cohomological equation Script D F = f has a solution if f is the lift of a holomorphic cusp form to SU(n, 1) under the following condition: for each restriction of f to {γ0} a finite number of Fourier coefficients vanish, and this finite number grows linearly with the length of the curve. This is a generalization of the classical Livshitz theorem for SU (1, 1) (A. Livshitz. Mat. Zametki 10 (1971), 555-564) where the curves are the closed geodesies themselves and the vanishing of the integrals of / over them, i.e. the zeroth Fourier coefficients, is both necessary and sufficient. An application of our result to the construction of spanning sets for spaces of holomorphic cusp forms on complex hyperbolic spaces is given in Appendix A.
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U2 - 10.1017/S0143385703000403
DO - 10.1017/S0143385703000403
M3 - Article
AN - SCOPUS:1542403812
VL - 24
SP - 127
EP - 140
JO - Ergodic Theory and Dynamical Systems
JF - Ergodic Theory and Dynamical Systems
SN - 0143-3857
IS - 1
ER -