On the coherence conjecture of Pappas and Rapoport. (English) Zbl 1300.14042

Local models were introduced by M. Rapoport and Th. Zink [Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies. 141. Princeton, NJ: Princeton Univ. Press. (1996; Zbl 0873.14039)] in order to study the PEL-type Shimura varieties. The so-called “naive” local model, which is a scheme over the ring of integers of the reflex field, is unfortunately not flat in general. To remedy this defect, a common strategy is to take the flat closure of the generic fiber; however, the moduli interpretation is lost in this procedure.
It turns out that questions about the special fibers of these flat models become tractable if one admits the Coherence Conjecture of Pappas and Rapoport, which was motivated by the case of ramified unitary groups. To state this conjecture, take \(k\) to be an algebraically closed field and set \(F = k((t))\), \(\mathcal{O} = k[[t]]\). Let \(G\) be a connected reductive \(F\)-group that splits over a tamely ramified extension \(\tilde{F}/F\). Choose a maximal torus \(T\) containing a maximal split torus.
To define local models, we consider a geometric conjugacy class of homomorphisms \(\mu: \mathbb{G}_{m, \tilde{F}} \to G \times_F \tilde{F}\). If \(\mu\) is a minuscule coweight, we may define the corresponding partial flag variety \(X(\mu) = H/P(\mu)\) where \(H\) is the Chevalley group attached to \(G\). Let \(\mathcal{L}(\mu)\) stand for the ample generator of \(\mathrm{Pic} X(\mu)\) and put \(h_\mu(a) = \dim_k \Gamma(X(\mu), \mathcal{L}(\mu)^a)\). In general, if \(\mu = \mu_1 + \cdots + \mu_r\) is a sum of minuscule coweights, we put \(h_\mu = h_{\mu_1} \cdots h_{\mu_r}\).
On the other hand, let \(Y\) be a subset of a chosen set of simple affine roots of \((G, T, \cdots)\). One may define the \(k\)-variety \(\mathcal{A}^Y(\mu)^\circ\) which is a union of certain Schubert varieties for the loop group \(LG_{\mathrm{sc}}\) indexed by the admissible set \(\mathrm{Adm}^Y(\mu)\). The set \(Y\) also gives rise to the Schubert variety \(\mathrm{Fl}^Y := LG/L^+ G_{\sigma_Y}\), with \(\sigma_Y\) being the corresponding facet in the Bruhat-Tits building. Let \(\mathcal{L}\) be an ample line bundle on \(\mathrm{Fl}^Y\). In its modified form stated in this paper, the Coherence Conjecture asserts that \[ \dim_k \Gamma(\mathcal{A}^Y(\mu)^{\circ}, \mathcal{L}^a) = h_\mu(c(\mathcal{L})a) \] where \(c(\mathcal{L})\) is the central charge of \(\mathcal{L}\) (think in terms of Kac-Moody groups).
The proof is too involved to be summarized here. One important ingredient is the construction of the global Grassmannian \(\mathrm{Gr}_{\mathcal{G}}\) and the “global Schubert variety” \(\overline{\mathrm{Gr}_{\mathcal{G}, \mu}}\), over a suitable curve \(C\) and its ramified cover \(\tilde{C} \to C\). A crucial step is to show that the central charge of a line bundle on \(\mathrm{Gr}_{\mathcal{G}}\) is constant along the curve \(C(k)\).
This paper corrects several mistakes in [G. Pappas and M. Rapoport, Adv. Math. 219, No. 1, 118–198 (2008; Zbl 1159.22010)] and [J. Heinloth, Math. Ann. 347, No. 3, 499–528 (2010; Zbl 1193.14014)].


14L05 Formal groups, \(p\)-divisible groups
14M15 Grassmannians, Schubert varieties, flag manifolds
14G20 Local ground fields in algebraic geometry
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