Local models in the ramified case. III Unitary groups

Pappas, G.; Rapoport, Michael

doi:10.1017/s1474748009000139

Cited by 38 publications

(78 citation statements)

References 25 publications

(113 reference statements)

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“…This gives the embedding . Here, we follow the original approach of [PR08], [PR09] to construct another line bundle L I , which is more explicit.…”

Section: Fix the Isomorphisms λmentioning

confidence: 99%

“…Therefore, nowadays the (local) models defined in [RZ96] are usually called the naive models. In a series of papers ([PR03], [PR05], [PR09]), Pappas and Rapoport investigated the corrected definition of flat local models. The easiest definition of these local models is by taking the flat closures of the generic fibers in the naive local models.…”

Section: Introductionmentioning

confidence: 99%

“…Usually, an integral model defined in this way is not useful since the moduli interpretation is lost and therefore it is very difficult to study the special fiber, etc. (In fact, a considerable part [PR03], [PR05], [PR09] is devoted in an attempt to cutting out the correct closed subschemes inside the naive models by strengthening the original moduli problem of [RZ96].) Indeed, most investigations of local models so far used these strengthened moduli problems in a way or another.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, two questions arise: which Schubert varieties appear in the special fibers (of the flat models) and whether the special fibers are reduced. These questions are reduced to the coherence conjecture (see [PR08], [PR09], at least in the case the group splits over a tamely ramified extension), which characterizes the dimension of the spaces of global sections of certain ample line bundles on certain union of Schubert varieties. Therefore, we will have a fairly good understanding of the local models even if we do not know the moduli problem they represent, provided we can prove the coherence conjecture.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, to such I, one can canonically associate a subset Y ⊂ S (S is the set of vertices in the local Dynkin diagram of GU(V , φ )) such that (ii) As pointed out in [PR08], [PR09], if n = 2m + 1, then P {0} and P {m} are the special parahoric group schemes, and if n = 2m, then P {m} , P {m } are the special parahoric group schemes. We further point out the following:…”

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confidence: 99%

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On the coherence conjecture of Pappas and Rapoport

Zhu

2014

Ann. Math.

100

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We prove the (generalized) coherence conjecture proposed by Pappas and Rapoport. As a corollary, one of their theorems, which describes the geometry of the special fibers of the local models for ramified unitary groups, holds unconditionally. Our proof is based on the study of the geometry (in particular, certain line bundles and -adic sheaves) of the global Schubert varieties, which are the equal characteristic counterparts of the local models.

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“…This gives the embedding . Here, we follow the original approach of [PR08], [PR09] to construct another line bundle L I , which is more explicit.…”

Section: Fix the Isomorphisms λmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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On the coherence conjecture of Pappas and Rapoport

Zhu

2014

Ann. Math.

100

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Kottwitz’s nearby cycles conjecture for a class of unitary Shimura varieties

Rostami

2016

Sel. Math. New Ser.

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This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori-Hecke algebras. This is commonly referred to as Kottwitz's Conjecture. The reductive groups associated to the PEL local models under consideration are unramified unitary similitude groups with even dimension. The proof follows the method of [HN02]. Upon completion of the first version of this paper, Pappas and Zhu released a preprint (now published as [PZ13]) which contained within its scope the main theorem of this paper. However, the methods of [PZ13] are very different and some of the proofs from this paper have been useful in forthcoming work of Haines-Stroh.

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Topological flatness of orthogonal local models in the split, even case. I

Smithling

2010

Math. Ann.

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Local models are schemes, defined in terms of linear algebra, that were introduced by Rapoport and Zink to study the étale-local structure of integral models of certain PEL Shimura varieties over p-adic fields. A basic requirement for the integral models, or equivalently for the local models, is that they be flat. In the case of local models for even orthogonal groups, Genestier observed that the original definition of the local model does not yield a flat scheme. In a recent article, Pappas and Rapoport introduced a new condition to the moduli problem defining the local model, the so-called spin condition, and conjectured that the resulting "spin" local model is flat. We prove a preliminary form of their conjecture in the split, Iwahori case, namely that the spin local model is topologically flat. An essential combinatorial ingredient is the equivalence of μ-admissibility and μ-permissibility for two minuscule cocharacters μ in root systems of type D.

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Local models in the ramified case. III Unitary groups

Cited by 38 publications

References 25 publications

On the coherence conjecture of Pappas and Rapoport

On the coherence conjecture of Pappas and Rapoport

Kottwitz’s nearby cycles conjecture for a class of unitary Shimura varieties

Topological flatness of orthogonal local models in the split, even case. I

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