  We consider a mirror symmetry of simple elliptic singularities. In
particular, we construct isomorphisms of Frobenius manifolds among the one from
the Gromov--Witten theory of a weighted projective line, the one from the
theory of primitive forms for a universal unfolding of a simple elliptic
singularity and the one from the invariant theory for an elliptic Weyl group.
As a consequence, we give a geometric interpretation of the Fourier
coefficients of an eta product considered by K. Saito.
