  The recent analyses of the Fermi Large Area Telescope data show an extended
GeV $\gamma$-ray excess on top of the expected diffuse background in the
Galactic center region, which can be explained with annihilating dark matter or
a population of millisecond pulsars (MSPs). We propose to observe the very high
energy $\gamma$-rays for distinguishing the MSP scenario from the dark matter
scenario. The GeV $\gamma$-ray MSPs should release most energy to the
relativistic $e^{\pm}$ wind, which will diffuse in the Galaxy and radiate TeV
$\gamma$-rays through inverse Compton scattering and bremsstrahlung processes.
By calculating the spectrum and spatial distribution, we show that such
emission is detectable with the next generation very high energy $\gamma$-ray
observatory, the Cherenkov Telescope Array (CTA), under reasonable model
parameters. It is essential to search for the multi-wavelength counterparts to
the GeV $\gamma$-ray excess for solving this mystery in the high energy
universe.
