Higgs algebra of curves and loop crystals

We define the Higgs algebra $\mathcal{H}_\P1$ of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone $\underlineΛ_\P1$, a lagrangian substack of the Higgs bundle $T^*\Coh_\P1$, where $\Coh_\P1$ is the stack of coherent sheaves on $\P1$. We prove that $\mathcal{H}_\P1$ is isomorphic to (some completion of) $U^+(\hat{sl}_2)$. We use this geometric realization to define a semicanonical basis of $U^+(\hat{sl}_2)$, indexed by irreducible components of $\underlineΛ_\P1$. 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.

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