  We extend the results of [GVX] to the setting of a stable polar
representation G|V (G connected, reductive over C), satisfying some mild
additional hypotheses. Given a G-equivariant rank one local system L on the
general fiber of the quotient map f : V --> V/G, we compute the Fourier
transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. Our
main intended application is to the theory of character sheaves for graded
semisimple Lie algebras over C.
