A variant of the induction theorem for Springer representations

Let G be a simple algebraic group over C with the Weyl group W. For a unipotent element u∈G, let Bu be the variety of Borel subgroups of G containing u. Let L be a Levi subgroup of a parabolic subgroup of G with the Weyl subgroup WL of W. Assume that u∈L and let be a similar variety as Bu for L. For a certain choice of L, u∈L and e⩾1, we describe the W-modules for k=0,…,e−1, in terms of the WL-module with some additional data, which is a refinement of the induction theorem due to Lusztig. As an application, we give an explicit formula for the values of Green functions at root of unity, in the case where u is a regular unipotent element in L.