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Gaussian and Chiral Beta-Ensembles, which generalise well known orthogonal(Beta=1), unitary (Beta=2), and symplectic (Beta=4) ensembles of randomHermitian matrices, are considered. Averages are shown to satisfy dualityrelations like {Beta,N,n} <--> {4/Beta,n,N} for all Beta>0, where N and nrespectively denote the number of eigenvalues and products of characteristicpolynomials. At the edge of the spectrum, matrix integrals of the Airy(Kontsevich) type are obtained. Consequences on the integral representation ofthe multiple orthogonal polynomials and the partition function of the formalone-matrix model are also discussed. Proofs rely on the theory of multivariatesymmetric polynomials, especially Jack polynomials.

Duality in random matrix ensembles for all Beta