Higgs algebra of curves and loop crystals

TL;DR

This paper constructs a geometric realization of the positive part of the affine Lie algebra 0, defines a semicanonical basis indexed by irreducible components of a nilpotent cone, and introduces a new combinatorial structure called a loop crystal.

Contribution

It introduces the Higgs algebra of the projective line, establishes its isomorphism with a completion of U^+(0), and defines a novel loop crystal structure on irreducible components.

Findings

Higgs algebra  of 0 is isomorphic to a completion of U^+(0)

A semicanonical basis indexed by irreducible components is constructed

A new combinatorial structure called a loop crystal is introduced

Abstract

We define the Higgs algebra $\mathcal{H}_\P1$ of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone $\underline{\Lambda}_\P1$, 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 $\underline{\Lambda}_\P1$. 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.