  In this paper, we study the cohomology of semisimple local systems in the
spirit of classical Hodge theory. On the one hand, we establish a
generalization of Hodge-Riemann bilinear relations. For a semisimple local
system on a smooth projective variety, we define a canonical isomorphism from
the complex conjugate of its cohomology to the cohomology of the dual local
system, which is a generalization of the classical Weil operator for pure Hodge
structures. This isomorphism establishes a relation between the twisted
Poincar\'e pairing, a purely topological object, and a positive definite
Hermitian pairing. On the other hand, we prove a global invariant cycle theorem
for semisimple local systems.
  As an application, we give a new and geometric proof of Sabbah's
Decomposition Theorem for the direct images of semisimple local systems under
proper algebraic maps, without using the category of polarizable twistor
D-modules.
