We prove a Large Deviation Principle for the random spectral measure associated to the pair (HN , e) where HN is sampled in the GUE and e is a fixed unit vector (and more generally in the β 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

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... value in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in those cases the expression of the rate function is not explicit.