  We report the first detection of high-energy X-ray (E>10 keV) emission from
the Galactic Center non-thermal filament G359.89-0.08 (Sgr A-E) using data
acquired with the Nuclear Spectroscopic Telescope Array (NuSTAR). The bright
filament was detected up to ~50 keV during a NuSTAR Galactic Center monitoring
campaign. The featureless power-law spectrum with a photon index of ~2.3
confirms a non-thermal emission mechanism. The observed flux in the 3-79 keV
band is ~ 2.0e-12 erg/cm^2/s, corresponding to an unabsorbed X-ray luminosity
of ~2.6e34 erg/s assuming a distance of 8.0 kpc. Based on theoretical
predictions and observations, we conclude that Sgr A-E is unlikely to be a
pulsar wind nebula (PWN) or supernova remnant-molecular cloud (SNR-MC)
interaction, as previously hypothesized. Instead, the emission could be due to
a magnetic flux tube which traps TeV electrons. We propose two possible TeV
electron sources: old PWNe (up to ~100 kyr) with low surface brightness and
radii up to ~30 pc or molecular clouds (MCs) illuminated by cosmic rays (CRs)
from CR accelerators such as SNRs or Sgr A*.
