  We consider a general multivariate affine stochastic recursion and the
associated Markov chain on $\mathbb R^{d}$. We assume a natural geometric
condition which implies existence of an unbounded stationary solution and we
show that the large values of the associated stationary process follow extreme
value properties of classical type, with a non trivial extremal index. We
develop some explicit consequences such as convergence to Fr{\'e}chet's law or
to an exponential law, as well as convergence to a stable law. The proof is
based on a spectral gap property for the action of associated positive
operators on spaces of regular functions with slow growth, and on the
clustering properties of large values in the recursion.
