Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactification

Abstract

We prove an analogue for Hodge modules of Pink's theorem on the degeneration of l-adic sheaves (Math. Ann. 292). Let j be the open immersion of a Shimura variety M into its Baily-Borel compactification. Its boundary has a natural stratification into locally closed subsets, each of which is itself a Shimura variety (up to taking the quotient by the action of a finite group). Let i be the inclusion of an individual such stratum M'. Saito's formalism gives a functor i^* j_* from the bounded derived category of Hodge modules on M to that of Hodge modules on M'. Our result gives a formula for the effect of i^* j_* on automorphic Hodge modules, i.e., variations of Hodge structure coming from algebraic representations of the group associated to M. This formula is of a purely representation theoretical nature.