  Let V be an 2n-dimensional vector space over an algebraically closed field of
odd characteristic. Let G = GL(V), and H = Sp(V) the symplectic group contained
in G. For a positive integer r > 1, we conisder the variety X = G/H \times
V^{r-1}, on which H acts diagonally. X is called the exotic symmetric space of
level r. Let W_{n,r} be the complex reflection group G(r,1,n). In this paper,
generalizing the result for the case where r = 2, we show that there exists a
natural bijective correspondence (Springer correspondence) between the set of
irreducible representations of W_{n,r} and a certain set of H-equivariant
simple perverse sheaves on X_{uni}$, where X_{uni} is the "unipotent part" of
X. We also consider a similar problem for G \times V^{r-1}, where G = GL(V) for
a finite dimensional vector space V.
