The $\xi$-stability on the affine grassmannian

Abstract

We introduce a notion of $\xi$-stability on the affine grassmannian $\xx$ for the classical groups, this is the local version of the $\xi$-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient $\xx^ξ/T$ of the stable part $\xx^ξ$ by the maximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reduction process analogous to the Harder-Narasimhan reduction for vector bundles. For the group ${\mathrm{SL}}_d$, we calculate the Poincaré series of the quotient $\xx^ξ/T$.

arXiv comment23 pages, Published version on Mathematische Zeitschrift