  Recent analyses of multiwavelength light curves of gamma-ray bursts
afterglows point to values of the magnetic turbulence well below the canonical
$\sim1\,$\% of equipartition, in agreement with theoretical expectations of a
micro-turbulence generated in the shock precursor, which then decays downstream
of the shock front through collisionless damping. As a direct consequence, the
Compton parameter $Y$ can take large values in the blast. In the presence of
decaying micro-turbulence and/or as a result of the Klein-Nishina suppression
of inverse Compton cooling, the $Y$ parameter carries a non-trivial dependence
on the electron Lorentz factor, which modifies the spectral shape of the
synchrotron and inverse Compton components. This paper provides detailed
calculations of this synchrotron-self-Compton spectrum in this large $Y$
regime, accounting for the possibility of decaying micro-turbulence. It
calculates the expected temporal and spectral indices $\alpha$ and $\beta$
customarily defined by $F_\nu\,\propto\,t_{\rm obs}^{-\alpha}\nu^{-\beta}$ in
various spectral domains. This paper also makes predictions for the very high
energy photon flux; in particular, it shows that the large $Y$ regime would
imply a detection rate of gamma-ray bursts at $>10\,$GeV several times larger
than currently anticipated.
