Character Sheaves for Graded Lie Algebras: Stable Gradings

In this paper we construct full support character sheaves for stably graded Lie algebras. Conjecturally these are precisely the cuspidal character sheaves. Irreducible representations of Hecke algebras associated to complex reflection groups at roots of unity enter the description. We do so by analysing the Fourier transform of the nearby cycle sheaves constructed in [GVX2].