Higgs algebra of curves and loop crystals

Abstract

We define the Higgs algebra ${\mathcal{H}}_{\P }1$ of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone ${\underline{\Lambda }}_{\P }1$, a lagrangian substack of the Higgs bundle $T^*\Coh_\P1$, where $\Coh_\P1$ is the stack of coherent sheaves on $\P 1$. We prove that ${\mathcal{H}}_{\P }1$ is isomorphic to (some completion of) $U^{+}({\widehat{sl}}_2)$. We use this geometric realization to define a semicanonical basis of $U^{+}({\widehat{sl}}_2)$, indexed by irreducible components of ${\underline{\Lambda }}_{\P }1$. We also construct a combinatorial data on this set of irreducible components in the spirit of \cite{KS}, which is an affine analog of a crystal. We call it a loop crystal and give some of its properties.