| Résume | Let G be a connected reductive group. The Springer correspondence, discovered by Springer in 1976, is an embedding of the category of representations of the Weyl group of G into the category Perv(N/G) of conjugation-equivariant perverse sheaves on the nilpotent cone of G. It was generalized by Lusztig in 1984 into a full description of Perv(N/G) by considering smaller relative Weyl groups attached to so-called cuspidal data. These constructions were upgraded to derived equivalences in 2017 by Rider and Russell through algebraic methods, and separately by Gunningham in the setting of D-modules.
In this talk, we geometrically construct a Weyl group-equivariant structure on the functor of parabolic induction from a cuspidal block. This provides another realization of these derived equivalences, thereby generalizing the case of induction from a torus done in 2020 by Laumon and Letellier. Our approach is expected to give a geometric proof of the orthogonality formula for generalized Green functions appearing in the character theory of finite reductive groups. |
