  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
$\xx^{\xi}/T$.
