  We prove three new results about the global Springer action defined in
\cite{GSI}. The first one determines the support of the perverse cohomology
sheaves of the parabolic Hitchin complex, which serves as a technical tool for
the next results. The second one (the Endoscopic Decomposition Theorem) links
certain direct summands of the parabolic Hitchin complex of $G$ to the
endoscopic groups of $G$. This result generalizes Ng\^o's geometric
stabilization of the trace formula in \cite{NgoFL}. The third result links the
stable parts of the parabolic Hitchin complexes for Langlands dual groups, and
establishes a relation between the global Springer action on one hand and
certain Chern class action on the other. This result is inspired by the mirror
symmetry between dual Hitchin fibrations. Finally, we present the first
nontrivial example in the global Springer theory.

Algebraic Geometry;Representation Theory