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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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