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Foliations in deformation spaces of local $$G$$-shtukas. (English) Zbl 1316.14089
Let $$\mathbb{F}_q$$ be the finite field with $$q$$ elements and $$S$$ be an integral noetherian $$\mathbb{F}_q$$-scheme. For a split connected reductive group $$G$$ over $$\mathbb{F}_q$$, let $$LG$$ be the loop group of $$G$$. The first main theorem of the paper is the following:
Theorem. Let $$g \in LG(S)$$ be such that the $$\sigma$$-conjugacy classes of $$g_s$$ in the geometric points $$s$$ of $$S$$ all coincide. Let $$x$$ be a $$P$$-fundamental alcove associated with this $$\sigma$$-conjugacy class where $$P$$ is a parabolic subgroup of $$G$$ containing a Borel subgroup. Then there are morphisms $$\beta: \tilde{S} \to S'$$ and $$\alpha : S' \to S$$ with $$S'$$ integral, $$\alpha$$ finite surjective and $$\beta$$ an étale covering, and a bounded element $$\tilde{h} \in LG(\tilde{S})$$ such that $$\tilde{h}^{-1}g_{\tilde{S}}\sigma^*(\tilde{h}) \in I(\tilde{S})xI(\tilde{S})$$ where $$I$$ is the standard Iwahori subgroup scheme of $$K_0$$ with the property that $$K_0(\text{Spec}(R)) = G(R[[z]])$$.
As an corollary of the theorem, the authors show that if $$\underline{\mathcal{G}}$$ is a local $$G$$-shtuka over $$S$$ whose quasi-isogeny class is constant in all geometric points of $$S$$, then after a suitable finite base change and after passing to an étale covering, $$\underline{\mathcal{G}}$$ is isogenous to a local $$G$$-shtuka which is completely slope divisible.
The second main result studies the analogs of Oort’s foliations and central leaves in this situation. Now let $$S$$ be a noetherian $$\mathbb{F}$$-scheme where $$\mathbb{F}/\mathbb{F}_q$$ is a field extension. The central leaf corresponding to $$\underline{\mathbb{G}}$$ in $$S$$ is a geometrically irreducible component of $$\mathcal{C}_{\underline{\mathbb{G}},S} \subset S$$ where $$\mathcal{C}_{\underline{\mathbb{G}},S}$$ is defined by $\mathcal{C}_{\underline{\mathbb{G}},S} = \{s \in S : \underline{\mathbb{G}}_k \cong \underline{\mathcal{G}}_s \otimes_{k(s)} k, k = \bar{k}, k \supset k(s)\}$ It can be shown that $$\mathcal{C}_{\underline{\mathbb{G}},S}$$ defines a reduced locally closed subscheme which is closed in the Newton stratum of $$\underline{\mathbb{G}}$$.
The closed affine Deligne-Lusztig variety associated with $$b \in LG(k)$$ and a dominant $$\mu \in X_*(T)$$ is the reduced closed subscheme $$X_{\preceq \mu}(b)$$ of the affine Grassmannian $$LG/K_0$$ with $X_{\preceq \mu}(b)(k) = \{g \in LG(k)/K_0(k) : g^{-1}b\sigma^*(g) \in K_0(k)z^{\mu'}K_0(k) \; \text{for some} \; \mu' \preceq \mu \}.$ Theorem. Let $$\underline{\mathbb{G}} = (K_{0,k}, b\sigma^*)$$ be a local $$G$$-shtuka over $$k$$ bounded by a dominant $$\mu \in X_*(T)$$ and with Newton point $$\nu$$. Let $$\mathcal{N}_{\nu}$$ be the Newton stratum of $$[b]$$ in the universal deformation space of $$\underline{\mathbb{G}}$$ bounded by $$\mu$$. Let $$x$$ be a $$P$$-fundamental alcove associated with $$[b]$$. Then there is a reduced scheme $$S$$ and a finite surjective morphism $$S \to \mathcal{N}_{\nu}$$ which factorizes into finite surjective morphisms $$S \to X_{\preceq \mu}(b)^{\wedge} \widehat{\times}_k \mathcal{I}^{\wedge} \to \mathcal{N}_{\nu}$$. Here $$X_{\preceq \mu}(b)^{\wedge}$$ denotes the completion of $$X_{\preceq \mu}(b)$$ at $$1$$ and $$\mathcal{I}^{\wedge}$$ denotes the completion of $$\mathbb{A}^{\langle 2\rho, \nu \rangle}$$ at $$0$$. Furthermore, $$\mathcal{C}_{\underline{\mathbb{G}}, \mathcal{N}_{\nu}}$$ is geometrically irreducible and equal to the image of $$\{1\} \widehat{\times}_k \mathcal{I}^{\wedge}$$ in $$\mathcal{N}_{\nu}$$.
The last theorem essential implies that the Newton stratum is up to a finite surjective morphism a product of a closed affine Deligne-Lusztig variety and a central leaf which is an affine space. Using this theorem, the authors obtain lower bounds on the dimension of each irreducible components of the affine Deligne-Lusztig varieties, and in the case of affine Grassmannian case, this implies that they are equidimensional of some dimension which is previously known.
Reviewer: Xiao Xiao (Utica)

##### MSC:
 14L05 Formal groups, $$p$$-divisible groups 14M15 Grassmannians, Schubert varieties, flag manifolds 20G25 Linear algebraic groups over local fields and their integers 11G09 Drinfel’d modules; higher-dimensional motives, etc.
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