  Lorentz invariance is one of the basic ingredients of quantum field theories
and violations of it are stringently constrained experimentally. Therefore, the
possibility of Lorentz violation (LV) is usually realized at very high energy
scales, resulting in a strong suppression of it (by the new scale) in
experiments. The Standard-Model Extension (SME) parameterizes LV in a
model-independent way, respecting $SU(2)_L$ gauge invariance. This means, e.g.,
that the neutrino and charged-lepton sectors are linked to each other. Hence,
on the one hand, any modification of neutrino properties simultaneously gives
rise to effects for charged leptons, which is why the tight limits on
flavour-off-diagonal LV for neutrinos imply new bounds on modifications of
charged leptons. On the other hand, LV for left-handed charged leptons implies
LV for neutrinos. Since LV modifications of the charged-lepton sector are, in
general, even more constraining than effects in the flavour-diagonal neutrino
sector, we obtain novel tight bounds on LV in the latter. Subsequently, we
apply the same approach to an analysis of time-of-flight data for neutrinos
(detected by IceCube) and photons from gamma ray bursts where discrepancies
have been observed. Our finding is that an explanation of the arrival time
difference between neutrino and photon events by dim-5 operators in the
neutrino sector would lead to unacceptably large LV effects in the
charged-lepton sector.
