  The nature of the insulating phases of the SU($N$)-generalization of the
one-dimensional Kondo lattice model is investigated by means of
non-perturbative approaches. By extending the Lieb-Schultz-Mattis (LSM)
argument to multi-component fermion systems with translation and global SU($N$)
symmetries, we derive two indices which depend on the filling and the
``SU($N$)-spin'' (representation) of the local moments. These indices strongly
constrain possible insulating phases; for instance, when the local moments
transform in the $N$-dimensional (defining) representation of SU($N$), a
featureless Kondo insulator is possible only at filling $f= 1-1/N$. To obtain
further insight into the insulating phases suggested by the LSM argument, we
derive low-energy effective theories by adding an antiferromagnetic Heisenberg
exchange interaction among the local moments [the SU($N$) Kondo-Heisenberg
model]. A conjectured global phase diagram of the SU($N$) Kondo lattice model
as a function of the filling and the Kondo coupling is then obtained by a
combination of different analytical approaches.
