  We derive a relation for the axial-coupling constants of the nucleon and
spin-3/2 baryons including the $\Delta(1232)$ with the use of an $SU(2)_R\times
SU(2)_L$ invariant Lagrangian. Therein the nucleon belongs to the fundamental
representation of the chiral group, while the spin-3/2 baryons belong to
higher-dimensional representations $(1,\half)\oplus (\half,1)$. The resulting
relation reproduces the one derived from the generalized Adler-Weisberger (GAW)
sum rule for the forward $\pi N$ scattering. We clarify the origin of the GAW
sum rule for the nucleon and $\Delta$ by taking into account both the
group-theoretical aspects and the spontaneous breaking of chiral symmetry. It
turns out that the GAW sum rule for the nucleon and $\Delta$ is a consequence
of the existence of interactions involving a derivative-pion-field and
spontaneous chiral symmetry breaking. This implies that the couplings between
$N$ and $\Delta$ in the GAW sum rule vanish with the change of the chiral
condensate towards chiral restoration.
