Geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group

Question

What is the geometric meaning of inducing a representation from a parabolic subgroup of a Weyl group? Could Springer theory of Weyl group representations be used to obtain such a geometric meaning?

Answer 1

Thanks to colleagues, it turns out that the Springer correspondence is a functor which associates representations of the Weyl group to sheaves on the nilpotent cone, and this functor maps induction from a parabolic subgroup $W_L$ to the Weyl group $W_G$ to parabolic induction of sheaves on the nilpotent cone defined by the pull-push in the diagram $G \leftarrow P \to L$. The reference for this fact is Theorem 1.3 in Clausen - The Springer correspondence.