  A population of globular clusters (GCs) has been recently established by the
Fermi-LAT telescope as a new class of GeV $\gamma$-ray sources. Leptons
accelerated to TeV energies, in the inner magnetospheres of MSPs or in their
wind regions, should produce $\gamma$-rays through the inverse Compton
scattering in the dense radiation field from the huge population of stars. We
have conducted deep observations of the globular cluster M15 with the MAGIC
telescopes and used 165 hrs in order to search for $\gamma$-ray emission. A
strong upper limit on the TeV $\gamma$-ray flux $<3.2\times
10^{-13}\mathrm{cm^{-2}s^{-1}}$ above 300 GeV ($<0.26\%$ of the Crab nebula
flux) has been obtained. We interpret this limit as a constraint on the
efficiency of the acceleration of leptons in the magnetospheres of the MSPs. We
constrain the injection rate of relativistic leptons, $\eta_{\rm e}$, from the
MSPs magnetospheres and their surrounding. We conclude that $\eta_{\rm e}$ must
be lower than expected from the modelling of high energy processes in MSP inner
magnetospheres. For leptons accelerated with the power law spectrum in the MSP
wind regions, $\eta_{\rm e}$ is constrained to be much lower than derived for
the wind regions around classical pulsars. These constraints are valid for the
expected range of magnetic field strengths within the GC and for the range of
likely energies of leptons injected from the inner magnetospheres, provided
that the leptons are not removed from the globular cluster very efficiently due
to advection process. We discuss consequences of these constraints for the
models of radiation processes around millisecond pulsars.

High Energy Astrophysical Phenomena