Generalized Jantzen filtration, hyperbolic restriction, and characteristic cycles

Abstract

In this paper we give a geometric description of the behavior of analytic intertwining operators between parabolically induced representations of $\mathrm{GL}_n(\mathrm{F})$, where $\mathrm{F}$ is a local non-archimedean field, in terms of hyperbolic localization functors of Braden. As a consequence, we can show that the image of an intertwining operator is always semi-simple and give a lower bound on the order of its pole, which is conjectured to be an equality as well as an upper bound in terms of the singular support of certain perverse sheaves. Moreover, we are able to reduce the conjecture of Lapid and M\'inguez on the shape of irreducible subrepresentations of induced representations to a computation of characteristic cycles. The theory of mixed Hodge modules plays a crucial role in the proofs.