Hidden symmetries of the theory of complex multiplication

(0.1) Let F be a totally real number field of degree d. It is well known that one can associate to any cuspidal Hilbert eigenform f over F of parallel weight 2 a compatible system of two-dimensional l-adic Galois representations Vl(f) of ΓF = Gal(Q/F ) over Ql (having fixed embeddings Q ↪→ C and Q ↪→ Ql). (0.2) On the other hand, the Shimura variety X associated to RF/QGL(2)F has reflex field Q, which means that its étale cohomology groups give rise to l-adic representations of ΓQ = Gal(Q/Q). The action of ΓQ on the intersection cohomology of the Bailey-Borel compactification X∗ of X was determined, up to semi-simplification, by Brylinski and Labesse [Br-La]: non-primitive cohomology (into which we include IH) occurs in even degrees and decomposes as