  We consider the space of nilpotent Higgs bundles on a weighted projective
line, as a global analog of the nilpotent cone. We show that it is pure,
compute its dimension, and define geometric correspondences between irreducible
components. We prove it by constructing a loop analog of a cristal, in the
spirit of Kashiwara and Saito, for some corresponding loop Kac-Moody algebra.
