Steinberg lattice of the general linear group and its modular reduction

给出一般线性群的受限斯坦伯格模

Presenting restricted Steinberg modules of general linear groups

研究给出域上斯坦伯格模各种局部限制的生成元和表示，将其与数域中局部对称空间的部分博雷尔 - 塞尔紧化相关联，还厘清相关表示关系，并在附录中关联表示存在性与斯坦伯格VB - 代数的科祖尔性质。

有限典型群的Weil-Steinberg character

The Weil-Steinberg character of finite classical groups

本文计算了定义了Weil特征标的有限典型群，即辛群、酉群和一般线性群中Weil特征标与Steinberg特征标之积的不可约组成部分.结果表明，该积对辛群和一般酉群是无重数的，但对一般线性群不是。作为一个应用，我们证明了这样一个群的Steinberg特征标对稳定自然模中向量的子群的限制是多重性自由的。证明这一结果的酉群使用的观察Brunat，出版作为附录，我们的文件。作为偶特征的辛群的“Weil特征标”，我们使用广义旋量表示的2-模Brauer特征标。它与Steinberg特征标的乘积是投射模的Brauer特征标。我们还确定了它的不可分解的直和项。

Invertible analytic functions on Drinfeld symmetric spaces and universal extensions of Steinberg representations

Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon--Vonk can be computed in terms of $\mathcal{L}$-invariants of the associated Galois representation. The same argument applies …

Steinberg slices and group-valued moment maps

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group G. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of G, which is equipped with the usual symplectic structure in this way. We construct a smooth relative compactification by taking the closure of each centralizer fiber in the wonderful compactification of G. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.

群值动量映射的Whittaker约化之类似

An analogue of Whittaker reduction for group-valued moment maps

本文为半单复Poisson-Lie群的Poisson作用构造了Whittaker约化的类似，该约化沿幂零轨道横截面进行，并在Dirac几何框架下描述其辛叶。

关于仿射 Hecke 代数的两个模几何实现

On two modular geometric realizations of an affine Hecke algebra

本文通过构造三类范畴的幺半等价，将正特征域上仿射 Hecke 代数与 Harish-Chandra 双模、Steinberg 簇等变凝聚层及仿射旗簇构造层联系起来，并据此证明了 Finkelberg-Mirković 关于主块几何实现的猜想。

Blocks of the category of smooth $\ell$-modular representations of $GL(n,F)$ and its inner forms: reduction to level-$0$

Let $G$ be an inner form of a general linear group over a non-archimedean locally compact field of residue characteristic $p$, let $R$ be an algebraically closed field of characteristic different from $p$ and let $\mathscr{R}_R(G)$ be the category of smooth representations of $G$ over $R$. In this paper, we prove that a block (indecomposable summand) of $\mathscr{R}_R(G)$ is equivalent to a level-$0$ block (a block in which every object has non-zero invariant vectors for the pro-$p$-radical of a maximal compact open subgroup) of $\mathscr{R}_R(G')$, where $G'$ is a direct product of groups of …

On the proportion of $p$-elements in a finite group, and a modular Jordan type theorem

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces $\mathbb{C}$ by a field of positive characteristic, due to the presence of large unipotent and/or Lie type subgroups. For this reason, a long-standing problem in group and representation theory has been to find the "correct analogue" of Jordan's theorem in characteristic $p>0$. Progress has been made in a number of different directions, most notably by Braue…

Exactness of the reduction on étale modules

We prove the exactness of the reduction map from étale $(ϕ,Γ)$-modules over completed localized group rings of compact open subgroups of unipotent $p$-adic algebraic groups to usual étale $(ϕ,Γ)$-modules over Fontaine's ring. This reduction map is a component of a functor from smooth $p$-power torsion representations of $p$-adic reductive groups (or more generally of Borel subgroups of these) to $(ϕ,Γ)$-modules. Therefore this gives evidence for this functor---which is intended as some kind of $p$-adic Langlands correspondence for reductive groups---to be exact. We also show that the correspon…

有限简约群的复表示的模降

Modular reduction of complex representations of finite reductive groups

本文研究有限李型群的单幂表示的Brauer-Nesbitt降，通过Weyl模的限制表达，部分验证了Lusztig的猜想，引入了新的基并探讨了导出范畴中的对象及几何解释。

域上群代数同构问题的一个约化定理

A reduction theorem for the isomorphism problem of group algebras over fields

本文证明群代数同构问题可约化为素域有限扩张上的群代数，并由此将模同构问题约化为有限模群代数。

对称群中在特征2下与线性特征标成比例的旋量特征标

Spin characters of the symmetric group which are proportional to linear characters in characteristic 2

本文研究对称群双重覆盖在特征2下的表示约化问题，确定不可约旋量表示的2-模Brauer特征标何时与Specht模成比例，并利用归纳和限制函子构造了交换算盘游标的函数。

模情形下广义Witt代数和Jacobson-Witt代数的量子化

Quantizations of generalized-Witt algebra and of Jacobson-Witt algebra in the modular case

利用Song-Su（\cite{GY}）发现的特征为0的广义Witt代数的李双代数结构，构造了广义Witt代数.通过模p约化和模“p-限制性”约化过程，得到Jacobson-Witt代数\mathbf{W}（n; \underline{1}）（对于W型Cartan型单模限制李代数）的限制泛包络代数的2 ^n {-}1族截断p-多项式非余交换变形.我们的结果推广了Grunspan（J. Algebra 280（2004），145- 161，\cite{CG}）在秩n=1，特征为0的情形下的工作.在模化的情况下，一个改进版本的论点遵循模化约简方法（不同于\cite{CG}），并使用了模李代数理论中的一些技术。

Modularity and uniformization of a higher genus algebraic space curve, its distinct arithmetical realizations by cohomology groups and $E_6$, $E_7$, $E_8$-singularities

We prove the modularity for an algebraic space curve $Y$ of genus $50$ in $\mathbb{P}^5$, which consists of $21$ quartic polynomials in six variables, by means of an explicit modular parametrization by theta constants of order $13$. This provides an example of modularity, explicit uniformization and hyperbolic uniformization of arithmetic type for a higher genus algebraic space curve. In particular, it gives a new example for Hilbert's 22nd problem. This gives $21$ modular equations of order $13$, which greatly improve the result of Ramanujan and Evans on the construction of modular equations …

Asymptotic values of modular multiplicities for GL_2

We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of Res_{K/Q_p} GL_2 for K a finite extension of Q_p. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

The Picard group of an order and Külshammer reduction

Let $(K,\mathcal O,k)$ be a $p$-modular system and assume $k$ is algebraically closed. We show that if $Λ$ is an $\mathcal O$-order in a separable $K$-algebra, then $\textrm{Pic}_{\mathcal O}(Λ)$ carries the structure of an algebraic group over $k$. As an application to the modular representation theory of finite groups, we show that a reduction theorem by Külshammer concerned with Donovan's conjecture remains valid over $\mathcal O$.

Rankin-Selberg local factors modulo $\ell$

After extending the theory of Rankin-Selberg local factors to pairs of $\ell$-modular representations of Whittaker type, of general linear groups over a non-archimedean local field, we study the reduction modulo $\ell$ of $\ell$-adic local factors and their relation to these $\ell$-modular local factors. While the $\ell$-modular local $γ$-factor we associate to such a pair turns out to always coincide with the reduction modulo $\ell$ of the $\ell$-adic $γ$-factor of any Whittaker lifts of this pair, the local $L$-factor exhibits a more interesting behaviour; always dividing the reduction modul…

超Loop代数的一维表示

Finite-Dimensional Representations of Hyper Loop Algebras

我们研究超循环代数的有限维表示，即，与复有限维单李代数上的loop代数相关联的具有正特征的代数闭域上的超代数。主要结果是分类的不可约模块，一个版本的斯坦伯格的张量积定理，和建设的积极特征类似物的Weyl模块定义的Chari和Pressley在特征零设置。进一步，我们开始了模p约化的研究，证明了超loop代数的每一个不可约模都可以构造为模p约化过程所得到的模与特征零模的商.我们推测外尔模也可以通过对p进行归约来获得。该推测暗示了外尔模的张量积分解，我们用它来描述底层阿贝尔范畴的块。