  We study a simple Dirac fermion dark matter model in U(1)$_{L_\mu-L_\tau}$
theory. The new light gauge boson $X$ plays important roles in both dark matter
physics and the explanation for the muon $g-2$ anomaly. The observed dark
matter relic density is realized by a large U(1)$_{L_\mu-L_\tau}$ charge
without introducing a resonance effect of the $X$ boson. As a by-product of the
model, characteristic neutrino signatures from sub-GeV dark matter $\psi$ are
predicted depending on the mass spectrum. We formulate the analysis of
$\psi\bar\psi\to \nu\bar\nu$, and of $\psi\bar\psi\to XX$ followed by
$X\to\nu\bar\nu$, in a model independent way. The energy spectrum of neutrinos
in the former process is monochromatic while in the latter process is
bowl-shape. We also evaluate sensitivity at Super-Kamiokande and future
Hyper-Kamiokande detectors. The analysis is finally applied to the
U(1)$_{L_\mu-L_\tau}$ dark matter model.
