  Motivated by geometric Langlands, we initiate a program to study the mirror
symmetry between nilpotent orbit closures of a semisimple Lie algebra and those
of its Langlands dual. The most interesting case is $B_n$ via $C_n$.
Classically, there is a famous Springer duality between special orbits.
Therefore, it is natural to speculate that the mirror symmetry we seek may
coincide with Springer duality in the context of special orbits. Unfortunately,
such a naive statement fails. To remedy the situation, we propose a conjecture
which asserts the mirror symmetry for certain parabolic/induced covers of
special orbits. Then, we prove the conjecture for Richardson orbits and obtain
certain partial results in general. In the process, we reveal some very
interesting and yet subtle structures of these finite covers, which are related
to Lusztig's canonical quotients of special nilpotent orbits. For example,
there is a mysterious asymmetry in the footprint or range of degrees of these
finite covers. Finally, we provide two examples to show that the mirror
symmetry fails outside the footprint.
