Abstract. Let G be a connected reductive algebraic group over an algebraically closed field k, γ ∈ g(k((ǫ))) a semisimple regular element, we introduce a fundamental domain Fγ for the affine Springer fibers Xγ . We show that the purity conjecture of Xγ is equivalent to that of Fγ via the Arthur-Kottwitz reduction. We then concentrate on the unramified affine Springer fibers for the group GL d . It turns out that their fundamental domains behave nicely with respect to the root valuation of γ. We formulate a rationality conjecture about a generating series of their Poincaré polynomials, and study them in detail for the group GL3. In particular, we pave them in affine spaces and we prove the rationality conjecture.
We introduce a notion of ξ-stability on the affine grassmannian X for the classical groups, this is the local version of the ξ-stability on the moduli space of Higgs bundles on a curve introduced by Chaudouard and Laumon. We prove that the quotient X ξ /T of the stable part X 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 over an algebraic curve. For the group SL d , we calculate the Poincaré series of the quotient X ξ /T .
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