The stable cohomology of the satake compactification of A_g

Charney and Lee have shown that the rational cohomology of the Satake-Baily-Borel compactification A^bb _g of Ag stabilizes as g → ∞ and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebrogeometric proof of their theorem and show that this stable cohomology comes with a mixed Hodge structure of which we determine the Hodge numbers. We find that the mixed Hodge structure on the primitive cohomology in degrees 4r + 2 with r ≥ is an extension of ℚ. (-2r -1) by ℚ.(0); in particular, it is not pure.