There has recently been interest in relating properties of matrices drawn atrandom from the classical compact groups to statistical characteristics ofnumber-theoretical L-functions. One example is the relationship conjectured tohold between the value distributions of the characteristic polynomials of suchmatrices and value distributions within families of L-functions. Theseconnections are here extended to non-classical groups. We focus on an explicitexample: the exceptional Lie group G_2. The value distributions forcharacteristic polynomials associated with the 7- and 14-dimensionalrepresentations of G_2, defined with respect to the uniform invariant (Haar)measure, are calculated using two of the Macdonald constant term identities. Aone parameter family of L-functions over a finite field is described whosevalue distribution in the limit as the size of the finite field grows isrelated to that of the characteristic polynomials associated with the7-dimensional representation of G_2. The random matrix calculations extend toall exceptional Lie groups
展开▼
