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We prove effective decay of certain multiple correlation coefficients for measure preserving, mixing Weyl chamber actions of semidirect products of semisimple groups with G-vector spaces. These estimates provide decay for actions in split semisimple groups of higher rank.

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We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families B t of certain Borel sets. Our main result is that for almost every general symplectic lattice, the discrepancy D(Λ, B t ) of the lattice point count with respect to the volumes is O(vol(B t ) −δ ). This extends work of W. Schmidt who gave similar discrepancy bounds for the space of all lattices in R n .

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