Cuspidal character sheaves on graded Lie algebras

We show in this paper that in the context of graded Lie algebras, all cuspidal character sheaves arise from a nearby-cycle construction followed by a Fourier–Sato transform in a very specific manner. Combined with results of the last two named authors, this completes the classification of cuspidal character sheaves for Vinberg’s type I graded classical Lie algebras.