On the structure of some p-adic period domains

Chen, Miaofen; Fargues, Laurent; Shen, Xu

doi:10.48550/arxiv.1710.06935

Cited by 12 publications

(46 citation statements)

References 28 publications

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“…Remark 3.9. Recall (for example from [CFS,Lemma 2.4]) the following comparison between parabolic reductions of modifications. Let E, E ′ be two G-bundles on X and assume that E ′ = E x for some x ∈ Gr G (C).…”

Section: Modificationsmentioning

confidence: 99%

“…By Lemma 4.2 above we have a corresponding comparison for the weakly admissible loci in the affine Schubert cell. Hence it is enough to prove the lemma for quasi-split G, in which case it is an immediate consequence of [CFS,Prop. 2.7] together with Lemma 4.3 above.…”

Section: Bymentioning

confidence: 99%

“…By maximality of v, the modification (E b,x ) P × P M of E M bM is weakly admissible (compare the generalities in the proof of [CFS,Lemma 6.3]). Since x is classical, also pr M (x) is classical.…”

Section: Classical Pointsmentioning

confidence: 99%

“…In [Che,Thm. 5.1] Chen observes that the proof of [CFS,Thm. 6.1] yields the following (although the assertion in [CFS] Shen [Sh] proves a variant of the results of [CFS] for non-minuscule µ, but using a different definition of weak admissibility than the one we use.…”

Section: Introductionmentioning

confidence: 99%

“…(2) In [CFS,4] (3) These subsets parametrize the non-empty Newton strata in a given affine Schubert cell, compare Corollary 5.4 below.…”

Section: Introductionmentioning

confidence: 99%

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On Newton strata in the $B_{dR}^+$-Grassmannian

Viehmann¹

2021

Preprint

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We study parabolic reductions and Newton points of G-bundles on the Fargues-Fontaine curve and the Newton stratification on the B + dR -Grassmannian for any reductive group G. Let Bun G be the stack of G-bundles on the Fargues-Fontaine curve. Our first main result is to show that under the identification of the points of Bun G with Kottwitz's set B(G), the closure relations on |Bun G | coincide with the opposite of the usual partial order on B(G). Furthermore, we prove that every non-Hodge-Newton decomposable Newton stratum in a minuscule affine Schubert cell in the B + dR -Grassmannian intersects the weakly admissible locus, proving a conjecture of Chen. On the way, we study several interesting properties of parabolic reductions of G-bundles, and determine which Newton strata have classical points.Contents 18 7. Newton strata in the weakly admissible locus 21 References 28

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Section: Modificationsmentioning

confidence: 99%

Section: Bymentioning

confidence: 99%

Section: Classical Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…(2) In [CFS,4] (3) These subsets parametrize the non-empty Newton strata in a given affine Schubert cell, compare Corollary 5.4 below.…”

Section: Introductionmentioning

confidence: 99%

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On Newton strata in the $B_{dR}^+$-Grassmannian

Viehmann¹

2021

Preprint

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On Irreducible Components of Rapoport–Zink Spaces

Mieda

2018

International Mathematics Research Notices

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Under a mild condition, we prove that the action of the group of self-quasi-isogenies on the set of irreducible components of a Rapoport-Zink space has finite orbits. Our method allows both ramified and non-basic cases. As a consequence, we obtain some finiteness results on the representation obtained from the ℓ-adic cohomology of a Rapoport-Zink tower.Theorem 1.3 (Theorem 4.14) Assume that M is the basic Rapoport-Zink space for GSp 2n , in which case G ′ = G = GSp 2n (Q p ) and J is a quaternion unitary similitude group. For every integers i, r ≥ 0 and every irreducible smooth representation ρ of J, the G-representation Ext r J (H i c (M ∞ ), ρ) Dc-sm has finite length. In particular, the G ′ -representation Hom J (H i c (M ∞ ), ρ) sm has finite length.For the precise notation, see Section 4.2. To prove this theorem, we use Theorem 1.2 for two Rapoport-Zink towers: one is the basic Rapoport-Zink tower for GSp 2n , and the other is a Rapoport-Zink tower for a quaternion unitary similitude group. These two towers are isomorphic at the infinite level, thanks to the duality isomorphism. These finiteness results are very useful when we apply representation theory to the study of H i c (M ∞ ). For example, in [31, §2-3], Theorem 1.2 for the Drinfeld tower, which is well-known, enables the author to apply a result in [6] to investigate H i c (M ∞ ) by using the Lefschetz trace formula. Further, by using Theorem 1.3 for the Drinfeld tower ([31, Corollary 4.3]), we obtained a purely local and geometric proof of the local Jacquet-Langlands correspondence in some cases (see [31, Theorem 6.10]). In [30], the author extends these arguments to the basic Rapoport-Zink space for GSp 4 . For this purpose, Theorem 1.3 plays a crucial role. We also remark that Theorem 1.1 and Theorem 1.3 are indispensable in the study of the relation between Now we can give a proof of Theorem 4.14. By a similar method as in the proof of [31, Lemma 3.1], we can show thatOn the other hand, by Lemma 4.15, we know that Ext r J (V, ρ) Dc-sm is generated by Ext r J (V K , ρ). In particular Ext r J (V, ρ) Dc-sm is finitely generated. Therefore, [40, Théorème VI.6.3] tells us that the G-representation Ext r J (V, ρ) Dc-sm has finite length. This completes the proof.

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Some Results on Affine Deligne–lusztig Varieties

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.2010 Mathematics Subject Classification. 14L05, 20G25.Affine Deligne-Lusztig varieties are schemes locally of finite type over F q . Also the varieties are isomorphic if the element b is replaced by another element b ′ in the same σ-conjugacy class.A major difference between affine Deligne-Lusztig varieties and classical Deligne-Lusztig varieties is that affine Deligne-Lusztig varieties have the second parameter:

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On the structure of some p-adic period domains

Cited by 12 publications

References 28 publications

On Newton strata in the $B_{dR}^+$-Grassmannian

On Newton strata in the $B_{dR}^+$-Grassmannian

On Irreducible Components of Rapoport–Zink Spaces

Some Results on Affine Deligne–lusztig Varieties

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