  We reduce the local limit theorem for a non-compact semisimple Lie group
acting on its symmetric space to establishing that a natural operator
associated to the measure is quasicompact. Under strong Diophantine assumptions
on the underlying measure, we deduce the necessary spectral results for the
operator in question. We thereby give the first examples of finitely supported
measures satisfying such a local limit theorem. Moreover, quantitative error
rates for the local limit theorem are proved under additional assumptions.
