  We prove a Large Deviation Principle for the random spec- tral measure
associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is
a fixed unit vector (and more generally in the $\beta$- extension of this
model). The rate function consists of two parts. The contribution of the
absolutely continuous part of the measure is the reversed Kullback information
with respect to the semicircle distribution and the contribution of the
singular part is connected to the rate function of the extreme eigenvalue in
the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but
in thoses cases the expression of the rate function is not so explicit.
