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 $\xx^{ \xi}/T$ of the stable part $\xx^{ \xi}$ 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 Poincar\'e series of the quotient $\xx^{ \xi}/T$.
Categories: math.AG math.RT