  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.
