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

Viehmann, Eva

doi:10.48550/arxiv.2101.07510

Cited by 3 publications

(7 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and each connected component Bun α G admits a unique semistable point. By a recent result of Viehmann [Vie21], the map | Bun G | → B(G) is a homeomorphism. This had previously been proved for G = GL n by Hansen [Han17] based on [BFH + 17]; that argument was extended to some classical groups in unpublished work of Hamann.…”

Section: Introductionmentioning

confidence: 95%

Geometrization of the local Langlands correspondence

Fargues,

Scholze

2021

Preprint

View full text Add to dashboard Cite

Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues-Fontaine curve. In particular, we define a category of -adic sheaves on the stack BunG of G-bundles on the Fargues-Fontaine curve, prove a geometric Satake equivalence over the Fargues-Fontaine curve, and study the stack of L-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define L-parameters associated with irreducible smooth representations of G(E), a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of L-parameters on the category of -adic sheaves on BunG. ContentsChapter I. Introduction I.1. The local Langlands correspondence I.2. The big picture I.3. The Fargues-Fontaine curve I.4. The geometry of Bun G I.5. -adic sheaves on Bun G I.6. The geometric Satake equivalence I.7. Cohomology of moduli spaces of shtuka I.8. The stack of L-parameters I.9. Construction of L-parameters I.10. The spectral action I.11. The origin of the ideas I.12. Acknowledgments I.13. Notation Chapter II. The Fargues-Fontaine curve and vector bundles V.1. Classifying stacks V.2. Étale sheaves on strata V.3. Local charts V.4. Compact generation V.5. Bernstein-Zelevinsky duality V.6. Verdier duality V.7. ULA sheaves Chapter VI. Geometric Satake VI.1. The Beilinson-Drinfeld Grassmannian VI.2. Schubert varieties VI.3. Semi-infinite orbits VI.4. Equivariant sheaves VI.5. Affine flag variety VI.6. ULA sheaves VI.7. Perverse Sheaves VI.8. Convolution VI.9. Fusion VI.10. Tannakian reconstruction VI.11. Identification of the dual group VI.12. Cartan involution Chapter VII. D (X) VII.1. Solid sheaves VII.2. Four functors VII.3. Relative homology VII.4. Relation to D ét VII.5. Dualizability VII.6. Lisse sheaves VII.7. D lis (Bun G ) Chapter VIII. L-parameter VIII.1. The stack of L-parameters VIII.2. The singularities of the moduli space VIII.3. The coarse moduli space VIII.4. Excursion operators VIII.5. Modular representation theory Chapter IX. The Hecke action IX.1. Condensed ∞-categories IX.2. Hecke operators IX.3. Cohomology of local Shimura varieties IX.4. L-parameter IX.5. The Bernstein center IX.6. Properties of the correspondence IX.7. Applications to representations of G(E) CONTENTS Chapter X. The spectral action X.1. Rational coefficients X.2. Elliptic parameters X.3. Integral coefficients Bibliography CHAPTER I

show abstract

Section: Introductionmentioning

confidence: 95%

Geometrization of the local Langlands correspondence

Fargues,

Scholze

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…This question naturally arises in the study of p-adic flag varieties and the B + dR -Grassmanninans. We hope that our main results will lead to a complete classification of modifications of a given vector bundle on the Fargues-Fontaine curve, and will consequently yield a concise description of how the Harder-Narasimhan strata and the Newton strata on the B + dR -Grassmanninans intersect, building upon the work of Shen [She19], Viehmann [Vie21], and Nguyen-Viehmann [NV21].…”

Section: Introductionmentioning

confidence: 99%

Extensions of vector bundles on the Fargues-Fontaine curve II

Hong¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…The condition (i) is in fact equivalent to having b ′ in the generalized Kottwitz set considered by Chen-Fargues-Shen [CFS21] and Viehmann [Vie21]. When b is basic, the condition (i) also implies the conditions (ii) and (iii).…”

mentioning

confidence: 96%

“…The Newton stratification of minuscule Schubert cells was originally introduced in the aforementioned work of Caraiani-Scholze [CS17] as a key tool for studying the fibers of the Hodge-Tate period map. It has also been used as a pivotal tool for studying the p-adic period domain by many authors, such as Chen-Fargues-Shen [CFS21], Shen [She19], Chen [Che20], Viehmann [Vie21], Nguyen-Viehmann [NV21], and Chen-Tong [CT22].…”

mentioning

confidence: 99%

“…For the trivial element b = 1, a result of Rapoport [Rap18] shows that the Newton stratum Gr b ′ G,µ,b is nonempty if and only if b ′ is an element of the set B(G, −µ) defined by Kottwitz [Kot85]. When b is basic, meaning that E b is semistable, Chen-Fargues-Shen [CFS21] and Viehmann [Vie21] extends the result of Rapoport to parametrize all nonempty Newton strata by a generalized Kottwitz set. However, for a general element b ∈ B(G), no explicit parametrization is known for nonempty Newton strata in an arbitrary Schubert cell.…”

mentioning

confidence: 99%

See 1 more Smart Citation

On nonemptiness of Newton strata in the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GL}_n$

Hong¹

2022

Preprint

View full text Add to dashboard Cite

We study the Newton stratification in the B + dR -Grassmannian for GLn associated to an arbitrary (possibly nonbasic) element of B(GLn). Our main result classifies all nonempty Newton strata in an arbitrary minuscule Schubert cell. For a large class of elements in B(GLn), our classification is given by some explicit conditions in terms of Newton polygons. For the proof, we proceed by induction on n using a previous result of the author that classifies all extensions of two given vector bundles on the Fargues-Fontaine curve.Contents 14 3.3. An explicit classification of nonempty Newton strata 15 References 20

show abstract

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

Cited by 3 publications

References 10 publications

Geometrization of the local Langlands correspondence

Geometrization of the local Langlands correspondence

Extensions of vector bundles on the Fargues-Fontaine curve II

On nonemptiness of Newton strata in the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GL}_n$

Contact Info

Product

Resources

About