We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3[2] manifolds with polarisation of degree 2d and split type is of general type if d≥12.

In this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standard L-functions.

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.