  We investigate supersymmetric hybrid inflation in a realistic model based on
the gauge symmetry $SU(4)_c \times SU(2)_L \times SU(2)_R$. The minimal
supersymmetric standard model (MSSM) $\mu$ term arises, following Dvali,
Lazarides, and Shafi, from the coupling of the MSSM electroweak doublets to a
gauge singlet superfield which plays an essential role in inflation. The
primordial monopoles are inflated away by arranging that the $SU(4)_c \times
SU(2)_L \times SU(2)_R$ symmetry is broken along the inflationary trajectory.
The interplay between the (above) $\mu$ coupling, the gravitino mass, and the
reheating following inflation is discussed in detail. We explore regions of the
parameter space that yield gravitino dark matter and observable gravity waves
with the tensor-to-scalar ratio $r \sim 10^{-4}-10^{-3}$.
