Asymptotics of k dimensional spherical integrals and applications

In this article, we prove that k-dimensional spherical integrals are asymptotically equiv-alent to the product of 1-dimensional spherical integrals. This allows us to generalize several large deviations principles known before only in a one-dimensional case. For example, we prove the uni-versality of the large deviations principle for the law of k extreme eigenvalues of Wigner matrices (resp. Wishart matrices, resp. matrices with general variance profiles) with sharp sub-Gaussian entries, as well as derive large deviations principles for the distribution of extreme eigenvalues of Gaussian Wigner and Wishart matrices with a finite dimensional perturbation.