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We introduce a notion of $\xi$-stability on the affine grassmannian $\xx$ forthe classical groups, this is the local version of the $\xi$-stability on themoduli 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 themaximal torus $T$ exists as an ind-$k$-scheme, and we introduce a reductionprocess analogous to the Harder-Narasimhan reduction for vector bundles. Forthe group $\mathrm{SL}_{d}$, we calculate the Poincar\'e series of the quotient$\xx^{\xi}/T$.

The $\xi$-stability on the affine grassmannian