  The H.E.S.S. collaboration has reported a high-energy spherically symmetric
diffuse gamma-ray emission in the inner 50 pc of the Milky Way, up to ~ 50 TeV.
Here we propose a leptonic model which provides an alternative to the hadronic
scenario presented by the H.E.S.S. collaboration, and connects the newly
reported TeV emission to the Fermi-LAT Galactic center GeV excess. Our model
relies on a combination of inverse Compton emission from a population of
millisecond pulsars---which can account for the GeV excess---and a supermassive
black hole-induced spike of heavy (~ 60 TeV) dark matter particles annihilating
into electrons with a sub-thermal cross-section. With an up-to-date
interstellar radiation field, as well as a standard magnetic field and
diffusion set-up, our model accounts for the spectral morphology of the
detected emission. Moreover, we show that the dark matter induced emission
reproduces the spatial morphology of the H.E.S.S. signal above ~ 10 TeV, while
we obtain a slightly more extended component from pulsars at lower energies,
which could be used as a prediction for future H.E.S.S. observations.
