Extension theorems for reductive group schemes

Reductive group schemes over the Fargues-Fontaine curve

For an arbitrary non-archimedean local field we classify reductive group schemes over the corresponding Fargues-Fontaine curve by group schemes over the category of isocrystals. We then classify torsors under such reductive group schemes by a generalization of Kottwitz' set B(G). In particular, we extend a theorem of Fargues on torsors under constant reductive groups to the case of equal characteristic.

Representations of reductive groups over finite rings and extended Deligne-Lusztig varieties

In a previous paper it was shown that a certain family of varieties suggested by Lusztig, is not enough to construct all irreducible complex representations of reductive groups over finite rings coming from the ring of integers in a local field, modulo a power of the maximal ideal. In this paper we define a generalisation of Lusztig's varieties, corresponding to an extension of the maximal unramified extension of the local field. We show in a particular case that all irreducible representations appear in the cohomology of some extended variety. We conclude with a discussion about reformulation…

$p$-adic 有限维单重群的几何引理对偶与第二对偶定理

Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups

本文研究了$p$-adic有限维单重群中几何引理对偶与第二对偶定理的关系，并推广了Bezrukavnikov-Kazhdan关于第二对偶性显式描述的结果。

Covering groups and their integral models

Given a reductive group $\boldsymbol{\mathrm{G}}$ over a base scheme $S$, Brylinski and Deligne studied the central extensions of a reductive group $\boldsymbol{\mathrm{G}}$ by $\boldsymbol{\mathrm{K}}_2$, viewing both as sheaves of groups on the big Zariski site over $S$. Their work classified these extensions by three invariants, for $S$ the spectrum of a field. We expand upon their work to study "integral models" of such central extensions, obtaining similar results for $S$ the spectrum of a sufficiently nice ring, e.g., a DVR with finite residue field or a DVR containing a field. Milder re…

On a classification of irreducible admissible modulo $p$ representations of a $p$-adic split reductive group

We give a classification of irreducible admissible modulo $p$ representations of a split $p$-adic reductive group in terms of supersingular representations. This is a generalization of a theorem of Herzig.

$\mathbb{C}((t))$ 上函数域上约化群的 Galois 上同调

Galois Cohomology of Reductive Groups over Function Fields over $\mathbb{C}((t))$

本文扩展 Kottwitz 的 $B(F,G)$ 理论，在附加曲线假设下，显式描述了 $\mathbb{C}((t))$ 上曲线函数域 $F$ 中约化群 Galois 上同调的局部化映射。

Intermediate extensions and crystalline distribution algebras

Let G be a connected split reductive group over a complete discrete valuation ring of mixed characteristic. We use the theory of intermediate extensions due to Abe-Caro and arithmetic Beilinson-Bernstein localization to classify irreducible modules over the crystalline distribution algebra of G in terms of overconvergent isocrystals on locally closed subspaces in the (formal) flag variety of G. We treat the case of SL(2) as an example.

导出的Emerton--Gee栈的经典性II：广义约化群

Classicality of derived Emerton--Gee stack II: generalised reductive groups

该研究利用坦纳坎形式定义一般群的Emerton--Gee栈，证明其为Z_p上局部有限展示的形式代数栈，构造带G结构的Laurent F-晶体导出栈并证明其在连通及广义约化群情形下的经典性，完善了局部Langlands参数导出栈的经典性结论。

抛物群方案的简化

Reductification of parahoric group schemes

本文研究抛物群方案在有限伽罗华扩张下的可约性，并证明在良好余特征下，一般平凡抛物 torsors 的平凡性。

分裂约化群上乘积的高斯普适性与Satake同构

Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism

该研究利用Satake同构与Hall-Littlewood多项式，刻画非阿基米德域上分裂约化群乘积的奇异数与角元，推广已有结果并证明其最小分离性，推导相关概率极限定理，建立代数分解与概率极限的统一框架。

常群下的泛平凡主丛

Generically trivial torsors under constant groups

本文解决了任意基域上的 Grothendieck-Serre 问题，通过发展伪完全和拟约化群下 torsor 的纯性与扩展定理，结合群概型结构理论，证明了泛平凡 torsor 的半局部平凡性。

Managing Metaplectiphobia: Covering p-adic groups

Brylinski and Deligne have provided a framework to study central extensions of reductive groups by K2 over a field F. Such central extensions can be used to construct central extensions of p-adic groups by finite cyclic groups, including the metaplectic groups. Particularly interesting is the observation of Brylinski and Deligne that a central extension of a reductive group by K2, over a p-adic field, yields a family of central extensions of reductive groups by the multiplicative group over the residue field, indexed by the points of the building. These algebraic groups over the residue field …

A note on Galois representations valued in reductive groups with open image

Let $G$ be a split reductive group with $\dim Z(G) \leq 1$. We show that for any prime $p$ that is large enough relative to $G$, there is a finitely ramified Galois representation $ρ\colon Γ_{\mathbb Q} \to G(\mathbb Z_p)$ with open image. We also show that for any given integer $e$, if the index of irregularity of $p$ is at most $e$ and if $p$ is large enough relative to $G$ and $e$, then there is a Galois representation $Γ_{\mathbb Q} \to G(\mathbb Z_p)$ ramified only at $p$ with open image, generalizing a theorem of A. Ray. The first type of Galois representation is constructed by lifting a…

约化群的三角线簇

The trianguline variety for reductive groups

研究分裂连通约化群的三角线簇，推广Breuil-Hellmann-Schraen定理，证明三角化参数正则性条件决定的局部光滑性及某些非光滑点的正规性，并给出带G结构的(φ,Γ_K)-模的结晶性判据。

分布与正规性定理

Distributions and normality theorems

本文推导特征$p$下环路群分布代数的Serre表示，并据此为无$p$-挠基本群的约化群的抛物仿射格拉斯曼流形中Schubert簇正规性提供新证明。

The L-group of a covering group

We incorporate nonlinear covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. This L-group is an extension of the absolute Galois group of a local or global field $F$ by a complex reductive group. The L-group depends on an extension of a quasisplit reductive $F$-group by $\mathbf{K}_2$, a positive integer $n$ (the degree of the cover), an injective character $ε\colon μ_n \rightarrow {\mathbb C}^\times$, and a separable closure of $F$. Our L-group is consistent with previous work on covering groups, and its construction is contravaria…

关于 $p$-adic 红uctive 组局部可积性结果的研究

On local integrability results for $p$-adic reductive groups

本文基于局部特征展开法，给出了哈里-查纳关于 $p$-adic 红uctive 组复特征局部可积性的著名定理的简短证明，展示了适用于某些非复系数的代数情形，并验证了文献未覆盖的局部可积性情况。

本质上一元的超 cuspidal 表示的局部朗兰兹对应（对于不连通的半单群）

The local Langlands correspondence of essentially unipotent supercuspidal representations for disconnected reductive groups

本文在刚性内形式框架下构造了本质上一元的超 cuspidal 表示的局部朗兰兹对应，并证明了某种泛函性和兼容性，该结果比 [Sol20] 中的类似结果更强，因为后者不考虑内扭的刚性化。同时，本文将这种对应推广到不连通的半单群。

On special representations of $p$-adic reductive groups

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\infty}…

On two theorems for flat, affine group schemes over a discrete valuation ring

We include short and elementary proofs of two theorems characterizing reductive group schemes over a discrete valuation ring, in a slightly more general context.

Stable vectors in Moy-Prasad filtrations

Let k be a finite extension of Q_p, let G be an absolutely simple split reductive group over k, and let K be a maximal unramified extension of k. To each point in the Bruhat-Tits building of G_K, Moy and Prasad have attached a filtration of G(K) by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy-Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. Our work extends earlier results by Reeder and Yu, who gave a classification in the case when p is sufficiently large. By passing to a finite unramif…

关于p进约化群的p进巴拿赫主系列的不可约性

On the irreducibility of $p$-adic Banach principal series of $p$-adic reductive groups

研究p进约化群在p进巴拿赫空间上主系列的不可约性，通过多种方法，对酉诱导表示获最优准则，对GL_n(F)等获施耐德猜想变体，还将问题归结为光滑诱导表示和特定G。

On the Moy-Prasad filtration

Let K be a maximal unramified extension of a nonarchimedean local field with arbitrary residual characteristic p. Let G be a reductive group over K which splits over a tamely ramified extension of K. We show that the associated Moy-Prasad filtration representations are in a certain sense independent of p. We also establish descriptions of these representations in terms of explicit Weyl modules and as representations occurring in a generalized Vinberg-Levy theory. As an application, we use these results to provide necessary and sufficient conditions for the existence of stable vectors in Moy-Pr…

On extensions of characters of affine pro-$p$ Iwahori--Hecke algebra

Let $K$ be a non-discrete non-Archimedean local field with residue characteristic $p$. Let $G$ be the group of $K$ rational points of a algebraic connected reductive group defined over $K$. In this article we compute the extensions between characters of affine pro-$p$ Iwahori--Hecke algebra $\mathcal{H}^{\text{aff}}$ over an algebraically closed field $R$ of characteristic $p$. In rank one case we deduce the relation between the blocks and $L$-packets.