The goal of this work is to construct a perverse t-structure on the ∞-category of ℓ-adic L G-equivariant sheaves on the loop Lie algebra L g and to show that the affine Grothendieck-Springer sheaf S is perverse. Moreover, S is an intermediate extension of its restriction to the locus of "compact" elements with regular semi-simple reduction. Note that classical methods do not apply in our situation because L G and L g are infinite-dimensional ind-schemes. ContentsPart 2. Sheaves on prestacks and perverse t-structures 4. Categories of sheaves on prestacks 5. Perverse t-structures on topologically placid ∞-stacks 6. Stratified ∞-stacks, semi-small maps, and perversity Part 3. The affine Springer theory 7. The Goresky-Kottwitz-MacPherson stratification 8. Geometry of the affine Grothendieck-Springer fibration 9. Completion of proofs. References

Résumé :Cet article établit une formule de dimension pour les fibres de Springer affines dans le cas des groupes. On suit la méthode initiée par Bezrukavnikov dans le cas des algèbres de Lie. Elle consiste en l'introduction d'un ouvert régulier suffisament gros dont on montre qu'il est de même dimension que la fibre de Springer affine entière. On montre que dans le cas des groupes, un tel ouvert régulier avec des propriétés analogues, existe. Sa construction passe par l'introduction du semi-groupe de Vinberg V G pour lequel nous étudions un morphisme 'polynôme caractéristique' et étendons les résultats précedemment établis par Steinberg pour les groupes. Introduction in EnglishLet k be an algebraically closed field. We consider G a connected algebraic group, semisimple, simply connected over k. Let T be a maximal torus of G and g the Lie algebra of G. We note F = k((π)) and O := k [[π]]. Kazhdan and Lusztig have introduced in [19] the affine Springer fibers for Lie algebras. There are varieties of the formwhere γ ∈ g(F ). They establish that they are k-schemes locally of finite type and of finite dimension if γ is regular semisimple. They also conjecture a dimension formula for these varieties which was later proved by Bezrukavnikov [1]. If we name by g γ , the centralizer of γ in g, the formula is the following :where δ ′ (γ) = val(det(ad(γ) : g(F )/g γ (F ) → g(F )/g γ (F ))) and def(γ) = rg g − rg F (g γ (F )). The first term is the discriminant invariant and the second one is a Galois invariant which mesures the drop of torus rank. In this work, we are interested in the affine Springer fibers for groups :with λ ∈ X * (T ) + a dominant cocharacter and γ ∈ G(F ). These varieties were introduced by Kottwitz-Viehmann [21] An analog dimension formula for affine Deligne-Lusztig was already established by [12] and [43]. The proof of Bezrukavnikov uses crucially a distinguished open subset, called the regular open subset. It consists in the elements g ∈ X γ such that ad(g) −1 (γ) is regular when we reduce mod π. In the case of groups, there is a double difficulty coming from the fact that the condition g −1 γg ∈ G(O)π λ G(O) is non-linear and that we cannot give a sense to the reduction modulo π. To linearize the problem, one way to proceed, is to consider a faithful representation ρ :We note that, if we add a central factor to G, which acts by multiplication by π in End(V ) ⊗ k O, we obtain a similar integrality condition, 2 as in the Lie algebra case. To do that in a uniform way for all groups, there exists a natural envelop, called the Vinberg's semi-group V G , introduced by Vinberg [44] in characteristic zero and Rittatore [36] in arbitrary characteristics. This formulation allows us to define in this context a regular open subset.In the case of Lie algebras, the regular open subset is a torsor under the affine grassmannian of the regular centralizer of γ. Following Ngô [28], the existence of such a regular centralizer comes from the existence of a commutative group scheme J, smooth on the adjoint quotient t/...

Let $X$ be a scheme of finite type over a finite field $k$, and let $\mathcal L X$ denote its arc space; in particular, $\mathcal L X(k) = X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of $\mathcal L X$ in the neighborhood of non-degenerate arcs, we show that a canonical "basic function" can be defined on the non-degenerate locus of $\mathcal L X(k)$, which corresponds to the trace of Frobenius on the stalks of the intersection complex of any finite-dimensional model. We then proceed to compute this function when $X$ is an affine toric variety or an "$L$-monoid". Our computation confirms the expectation that the basic function is a generating function for a local unramified $L$-function; in particular, in the case of an $L$-monoid we prove a conjecture formulated by the second-named author.Comment: Erratum added at the end, to account for a shift in the argument of the L-functio

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