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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 https://www.math.harvard.edu/media/clausen.pdf