On the eigenvalues of random matrices

Let M be a random matrix chosen from Haar measure on the unitary group U,,. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance 4normal variables. We show that for j = 1, 2 , . . . . Tr(MJ) are independent and distributed as d Z asymptotically as n + m. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups. HAAR MEASURE, ORTHOGONAL GROUPS; SYMPLECTIC GROUPS, SYMMETRIC GROUPS AMS 1991 SUBJECT CLASSIFICATION 15A52

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