Karem Bettaieb, Imed Hichri
The aim of this paper is to give generalizationof the constructionof the Steinberg tempered character on a connected reductive p-adic group. We prove that this character is invariant by the weak restriction of the Jacquet module by analogy to finite reductive groups.
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This is a preliminary version of the first chapter of a book project on the character theory of finite groups of Lie type. It provides the foundations from the general theory of reductive algebraic groups over a finite field.
Let G be a connected reductive group over an algebraic closure of a finite field Fq. In this paper it is proved that the infinite dimensional Steinberg module of kG defined by N. Xi in 2014 is irreducible when k is a field of positive characteristic and char k is not char Fq. For certain special linear groups, we show that the Steinberg modules of the groups are not quasi-finite with respect to some natural quasi-finite sequences of the groups.
Let G be a reductive algebraic group over a field of positive characteristic and denote by C(G) the category of rational G-modules. In this note we investigate the subcategory of C(G) consisting of those modules whose composition factors all have highest weights linked to the Steinberg weight. This subcategory is denoted ST and called the Steinberg component. We give an explicit equivalence between ST and C(G) and we derive some consequences. In particular, our result allows us to relate the Frobenius contracting functor to the projection functor from C(G) onto ST .
We prove an isomorphism for simple perverse sheaves on the affine Grassmannian of a connected reductive algebraic group that is a geometric counterpart (in light of the Finkelberg-Mirković conjecture) of the Steinberg tensor product formula for simple representations of reductive groups over fields of positive characteristic.
The Steinberg Variety and Representations of Reductive Groups
我们给出了一个概述的几何表示论，已被证明通过斯坦伯格品种的一些主要成果。斯坦伯格的洞察力是使用这样的各种三元组，以证明一个由Grothendieck固定公式。斯坦伯格簇后来被用来给施普林格的陈述提供另一种方法，并在卡日丹和卢斯蒂格证明赫克代数的德利涅-朗兰兹猜想中发挥了核心作用。
On the Steinberg character of a semisimple p-adic group
设G是定义在非阿基米德局部域K上的半单几乎单代数群，S是G（K）的Steinberg表示.设t是G（K）的一个非常正则半单元.在本文中，我们给出了一个简单的公式（不是作为一个交替和）的值的特征标S在t;我们表明，这个值是加或减的整数幂的q，基数的剩余域的K。我们还给出了G（K）在Iwahori子群下具有非零不变向量的任何不可约容许表示在分裂非常正则元处的特征标的一个显式公式（在特征标的某些限制下有效）。
Generalized Steinberg representations of split reductive linear algebraic groups
推广了P.Schneider和U. Stuhler对GL_1 +1的推广到一个在非阿基米德局部域K上定义并分裂的约化代数群G。沿着这条线，我们证明了G的系数在任意阿贝尔群中的广义Steinberg表示是循环的.当G是伴随型半单时，只要可能，特别是对于那些最大次数的表示，我们就用G的对位子群给出这些表示的表达式。
Steinberg quotients and Smith-Treumann localization
本文通过涉及Steinberg模的卷积图，统一了仿射Grassmannian上的两种Smith-Treumann局部化，并将Steinberg商范畴化为仿射Grassmannian的Z/pZ不动点上球面奇偶层的茎。
On Extensions of generalized Steinberg Representations
设F是局部非阿基米德域，G是F上约化代数群的F值点群。本文计算了系数在某个自内射环中的光滑G-表示范畴中的广义Steinberg表示的Ext-群。
For a reductive group G defined over an algebraically closed field of positive characteristic, we show that the Frobenius contraction functor of G-modules is right adjoint to the Frobenius twist of the modules tensored with the Steinberg module twice. It follows that the Frobenius contraction functor preserves injectivity, good filtrations, but not semisiplicity.
Equivariant K-theory of generalized Steinberg varieties
利用广义仿射Hecke代数和广义仿射外尔群中的双陪集，刻画了一类广义Steinberg簇的等变K群，该簇插在约化复代数群的Steinberg簇与其幂零锥之间.作为一个应用，我们使用这个描述来定义Kazhdan-Lusztig“bar”对合和Kazhdan-Lusztig基。
Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures
该研究证明域上连通约化群的Steinberg表示同调在一定范围内消失，推广了前人关于典型分裂群的工作，提出连通性猜想并证明部分特例，改进了$SL_n(\mathbb{Z})$相关同调的已知结果。
Existence of cuspidal representations of p-adic reductive groups
证明了非阿基米德局部域上的任意约化群G都有尖点复表示。
The Steinberg Tensor Product Theorem for General Linear Group Schemes in the Verlinde Category
本文将约化代数群的Steinberg张量积定理推广至Verlinde范畴$\mathrm{Ver}_p$中任意对象$X$上的一般线性群概形$GL(X)$，解决了其模表示的特征标问题。
For two distinguished prime $\ell$ and $p$, we prove a $\ell$-adic version of the Poisson formula for reductive $p$-adic groups. In order to do this we write an identity for the trace of regular representation and orbital integrals. Next we reduce them to orbital integrals for endoscopy groups and look at this as the special value of $L$-function at $s=0$. And finally show that it is equal to the special value of motivic $L$-function at $s=0$.
Koszul duality for generalised Steinberg representations of $p$-adic groups
本文证明了 p-进半单群的广义 Steinberg 表示扩张代数模范畴与 Langlands 参数簇上等变反常层子范畴之间的 Koszul 对偶等价。
Some Infinite Dimensional Representations of Reductive Groups with Frobenius Maps
本文利用Frobenius映射构造了代数群的某些不可约无穷维表示。特别地，将Steinberg和Deligne & Lusztig关于李型有限群的复表示的一些经典结果推广到具有Frobenius映射的约化代数群.
Good product expansions for tame elements of reductive p-adic groups
我们表明，在相当一般的条件下，许多元素的p-adic组可以很好地近似的产品，其因素的属性，有助于执行明确的字符计算。
Let $G$ be a reductive group over a field of prime characteristic. An indecomposable tilting module for $G$ whose highest weight lies above the Steinberg weight has a character that is divisible by the Steinberg character. The resulting "Steinberg quotient" carries important information about $G$-modules, and in previous work we studied patterns in the weight multiplicities of these characters. In this paper we broaden our scope to include quantum Steinberg quotients, and show how the multiplicities in these characters relate to algebraic Steinberg quotients, Weyl characters, and evaluations o…
Cocenters of Hecke algebras $\mathcal H$ play an important role in studying mod $\ell$ or $\mathbb C$ harmonic analysis on connected $p$-adic reductive groups. On the other hand, the depth $r$ Hecke algebra $\mathcal H_{r^+}$ is well suited to study depth $r$ smooth representations. In this paper, we study depth $r$ rigid cocenters $\overline{\mathcal H}^{\mathrm{rig}}_{r^+}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell\neq p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an exp…
Distinction of the Steinberg representation and the dual group of a symmetric space
研究Steinberg表示相对于对称子群的区分问题，通过超图上的调和函数存在性联系，并验证Steinberg表示在p-adic域上的区分性与Langlands参数与对偶群的关系。
Extended quotients in the principal series of reductive p-adic groups
作者在[1，2，3，4]中提出的几何猜想适用于任意约化p-adic群G的光滑对偶Irr（G）。它预言了Irr（G）的Bernstein分解中每个分量的一个确定的几何结构--一个扩展商的结构。本文证明了关于可裂连通约化p-adic群G中主列的几何猜想。通过Springer参数和Langlands参数进行证明。由于这种方法，我们建立了强有力的联系与当地朗兰兹对应。我们的方法的一个重要特点是强调在扩展的仿射外尔群的双边细胞。
Let $G$ be a reductive $p$-adic group. We give a short proof of the fact that $G$ always admits supercuspidal complex representations. This result has already been established by A. Kret using the Deligne-Lusztig theory of representations of finite groups of Lie type. Our argument is of a different nature and is self-contained. It is based on the Harish-Chandra theory of cusp forms and it ultimately relies on the existence of elliptic maximal tori in $G$.
We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.
Cuspidal representations of a reductive p-adic group G over a field of characteristic different from p are relatively injective and projective with respect to extensions that split by a U-equivariant linear map for any subgroup U that is compact modulo the centre. The category of smooth representations over a field whose characteristic does not divide the pro-order of G is the product of the subcategories of cuspidal representations and of subrepresentations of direct sums of parabolically induced representations.
Unitary equivalences for reductive p-adic groups
在类型论的框架下，我们建立了约化p-adic群的光滑表示范畴的Bernstein分支到相应的Hecke代数的酉性的转移，只要Hecke代数是具有几何参数的仿射Hecke代数，在Lusztig意义下（可能由根数据的自同构群扩展）.众所周知，有一大类这样的例子（在论文中详细介绍）。因此，我们在内窥镜检查的精神下，建立了不同群体之间的整体性联系。
Classify all representation which contains a Steinberg in its hyperspecial subgroup
本文研究了不可约光滑表示的分类，其固定空间包含Steinberg表示，证明此类表示必为Iwahori球形，且每个主系列贡献一个唯一子商包含Steinberg表示。
We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.
The Second Adjointness Theorem for reductive p-adic groups
在R-模上光滑群表示的一般性下，证明了非阿基米德局部域上约化群的抛物子群的Jacquet限制函子与其相反的抛物子群的抛物归纳函子是右伴随的，其中R的剩余域特征是可逆的.更正：原来这篇论文包含一个错误（在引理3.4中），作者一直无法修复。这使得主要定理的证明不完整。
Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.
Wave packets in the Schwartz space of a reductive p-adic symmetric space
我们在约化p-adic对称空间的施瓦兹空间中对某些调和函数族形成波包。我们将展示如何从爱森斯坦积分构建这样的家庭。
Whittaker functions for Steinberg representations of $GL(n)$ over a $p$-adic field
本文通过确定Iwahori Hecke代数特征标在特定元素上的取值，完整计算了p-进域上$GL(n)$的广义Steinberg表示的Whittaker函数，并证明该表示限制到$SL_n(F)$上仍不可约，推广了Baruch-Purkait关于$GL_2(F)$的结果。
Let $\mathcal{G}$ be a split reductive $p$-adic group with connected centre. We show that each Bernstein block in the principal series of $\mathcal{G}$ admits a definite geometric structure, namely that of an extended quotient. For the Iwahori-spherical block, this extended quotient has the form $T//W$ where $T$ is a maximal torus in the Langlands dual group of $\mathcal{G}$ and $W$ is the Weyl group of $\mathcal{G}$.
The homology of the Steinberg variety and Weyl group coinvariants
设G是复连通约化代数群，具有外尔群W和Steinberg簇Z.证明了Z的分次Borel-Moore同调同构于W的共不变代数与W的群代数的Smash积。
Let $k$ be a $p$-adic field and let $\mathbf{G}(k)$ be the $k$-points of a connected reductive group, inner to split. The set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form an Arthur packet for $\mathbf{G}(k)$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets which we call \emph{weak} Arthur packets and describe their constituents in terms of thei…
The Steinberg representation is irreducible
本文证明了无限域上连通约化群的 Steinberg 表示是不可约的，将 Steinberg 和 Curtis 关于有限域的经典定理推广至无限域。
Resolutions for representations of reductive p-adic groups via their buildings
Schneider-Stuhler和Vigneras利用仿射Bruhat-Tits建筑物上的余层构造了特征不等于p的约化p-adic群的容许表示的自然有限型投射分解.我们利用具有某些性质的表示的幂等元自同态系统构造了建筑物上的余层和层.我们建立，这些都是非循环的，并计算这些系数的同调和上同调。这意味着伯恩斯坦的结果，某些子范畴的范畴的表示是塞尔子范畴。此外，我们还得到了建筑物的凸子复形的结果。根据科曼的工作，这导致了可接受的陈述的迹公式。
Representations of $p$-adic reductive groups
本书介绍p进约化群的表示理论，从基础内容出发，经多章研究，包括建立伯恩斯坦分解定理等，最终证明朗兰兹分类定理，前四章在更一般框架下展开，还有相关回顾与附录内容。
Standard modules and intertwining operators for reductive p-adic groups
本文研究了裂变群G的规范模块在Galois群作用下的稳定性，并给出了Levi子群表示的μ函数显式公式。
Supercuspidal characters of reductive p-adic groups
我们计算了约化p-adic群的许多超尖点表示的特征标。具体来说，我们处理表示，通过余的建设，从满足一定的紧凑性条件的数据。每个字符都表示为一个更小的群的深度为零的字符，出现在余的建设，傅里叶变换的轨道积分，和某些符号和基数，明确描述的数据相关的表示和元素的字符进行评估。
Resolutions of tempered representations of reductive p-adic groups
设G是非阿基米德局部域上的约化群，S（G）是它的Schwartz代数.本文比较了LF-空间上的光滑G-表示、代数S（G）-模、代数S（G）-模和包含所有容许S（G）-模的正合S（G）-模范畴中的调和G-表示的Ext-群。我们简化了证明已知的比较定理，这些外群，由于迈耶和施奈德-辛克。我们的方法是基于G的Bruhat-Tits建筑和Schneider-Stuhler决议的分析性质。
A note on the representation theory of central extensions of reductive $p$-adic groups
本文验证了约化p进群表示理论中的若干基本结果可推广到其有限中心扩张，为相关表示论研究提供了更一般的适用框架。
The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an…
Homological algebra for Schwartz algebras of reductive p-adic groups
设G是非阿基米德局部域上的约化群。则从G的光滑调和表示的导出范畴到G的所有光滑表示的导出范畴的正则函子是完全忠实的。在这里，我们考虑的表示在拓扑向量空间。因此，如果V和W是两个调和不可约表示，并且如果V或W是平方可积的，则Ext_G^n（V，W）对所有n>0为零。我们用它来证明在充分的一般性公式的形式尺寸平方可积表示由于施耐德和Stuhler。
On a classification of irreducible admissible modulo $p$ representations of a $p$-adic split reductive group
利用超奇异表示给出了分裂$p$ -adic可约群的不可约容许模$p$表示的分类.这是Herzig定理的一个推广。
Characters of locally analytic representations of p-adic reductive groups
我们提出了一个定义的字符的上下文中的施奈德-泰特尔鲍姆的局部解析表示的p-adic约化群的理论。这个特征将是一个功能的一个紧凑的子群的极大环面的约化群。作为例子，我们处理SL（2，Q_p）的局部解析主列.
The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations in terms of the Deligne-Langlands-Lusztig correspondence. Our formula generalizes the main result of arXiv:2112.14354, where this formula was obtained in the Iwahori-spherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the…
In this paper we study the complex representations of reductive groups over local non-Archimedean fields. We use the building of the reductive group to give upper-bounds for the absolute value of the character of an admissible representation and for the Weyl integration formula for certain regular elements. The upper-bound for the character of a representation is based on the alternative description, depending on the building, of the character as given by R. Meyer and M. Solleveld [MS12]. Once the character and the Weyl integration formula are related to the building, the upper-bounds will fol…
Some results on the admissible representations of non-connected reductive p-adic groups
我们研究了非连通约化$p$ -adic群的诱导表示理论，其中$G/G^0$是交换的。我们首先研究的结构，这些表示的形式$\Ind_{P^0}^G(σ),$其中$P^0$是一个抛物子群的$G^0$和$\s$是一个离散的系列表示的列维组件的$P^0.$在这里，我们开发了一个理论的$R$-组，扩展理论的连接的情况下。然后，我们证明了一些一般性的结果表示的非连通$p$ -进群的组成部分是阿贝尔的理论。我们定义的尖点抛物$G$的概念，以便为这一讨论提供一个背景。交织运营商的非连接的情况下进行检查，并定义了超尖点和离散系列的概念。最后，我们研究抛物诱导从尖点抛物子群的$G.$在这里，我们还开发了一个理论的$R$ -组，并表明，这些群体parameterized诱导表示的方式是一致的连接的情况下，以及与第一组的结果。
Introductory and pedagogical treatmeant of the article : P. Broussous "Distinction of the Steinberg representation", with an appendix by François Courtès, IMRN 2014, no 11, 3140-3157. To appear in Proceedings of Chaire Jean Morlet, Dipendra Prasad, Volker Heiermann Ed. 2017. Contains modified and simplified proofs of loc. cit. This article is written in memory of François Courtès who passed away in september 2016.
A matrix Paley-Wiener theorem for non-connected $p$-adic reductive groups
设$F$是特征$0$的局部非阿基米德域，$G$是$F$上的非连通约化群.我们将$G^0$表示为恒等式的连通分支，并假设商$G/G^0$是阿贝尔的。对于$f$是$G$上的一个局部常数紧支撑函数，$π$是$G$的一个复光滑表示，我们定义$f$在$π$上的傅里叶变换为$π(f) = \int_{G} f(g) π(g) \, dg$，它是$π$的基础向量空间的一个自同态.我们给出了一个描述的图像的傅立叶变换映射：给定的，每个$π$在一定的家庭诱导表示的PDMATHAMO，一个自同态$φ(π)$的基础向量空间，我们提供了必要和充分条件下，存在一个功能$f$（必然是唯一的），使$π(f) = φ(π)$的所有$π$的家庭。
On principal series representations of quasi-split reductive p-adic groups
针对非阿基米德局部域上拟分裂约化群G，为其主系列不可约光滑复表示建立基于惠特克数据选择的典范局部朗兰兹对应，通过单射参数化映射关联增强L-参数集并验证其多项预期性质。
Let $G$ be a connected reductive group over a non-Archimedean local field. We prove that its parahoric subgroups are definable in the Denef-Pas language, which is a first-order language of logic used in the theory of motivic integration developed by Cluckers and Loeser. The main technical result is the definability of the connected component of the Néron model of a tamely ramified algebraic torus. As a corollary, we prove that the canonical Haar measure on $G$, which assigns volume $1$ to the particular \emph{canonical} maximal parahoric defined by Gross, is motivic. This result resolves a tec…
Rationality properties of complex representations of reductive p-adic groups
研究复表示与代数闭包Qbar表示的有理性，证明椭圆表示可实现于Qbar当且仅当其中心字符取值于Qbar，并探讨复表示的自同构群作用。
Deligne-Lusztig constructions for unipotent and p-adic groups
在1979年Lusztig提出了一个约化p-adic群的超尖点表示的构造，它类似于Deligne和Lusztig在有限约化群的背景下的著名构造。我们提出了一个一般的方法，明确计算所产生的Lusztig的建设表示，并说明它与几个例子。我们开发的技术也提供了背景，作者的联合工作与温斯坦在一个纯粹的本地和明确的证明当地朗兰兹对应。
Intertwining operators for representations of covering groups of reductive $p$-adic groups
本文研究约化 $p$-进群覆盖群的抛物诱导表示间的纠缠算子，证明其伴随关系，并据此给出 Harish-Chandra $μ$-函数的显式公式及极点定位方法。
Let $G$ be a reductive $p$--adic group. Assume that $L\subset G$ is an open--compact subgroup, and $\mathcal H_L$ is the Hecke algebra of $L$--biinivariant complex functions on $G$. It is a well--known and standard result on how to prove existence of a complex smooth irreducible $G$--module out of a maximal left ideal $I\subset \mathcal H_L$. Using theory on Bernstein center we make this construction explicit. This leads us to some very interesting questions.
In this paper we study certain category of smooth modules for reductive $p$--adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a $\mathbb Q$--algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations proving by reduction to the complex case that if the space of $K$--invariants is finite dimensional in an irreducible smooth unramified representation that the representation is admissible.
The Hecke algebra of a reductive p-adic group: a geometric conjecture
设H（G）是约化p-adic群G的Hecke代数。我们对H（G）的Bernstein分解中的理想提出了一个猜想.这个猜想说每个理想在几何上等价于一个代数簇。我们的猜想是密切相关的Lusztig猜想的渐近Hecke代数。我们证明了SL（2）和GL（n）的猜想。我们还证明了关于群PGL（n）和SO（5）的Iwahori理想的猜想的第（1）部分。
Unitarizable minimal principal series of reductive groups
本文主要讨论特征为0的实域或p-adic域上约化群的极小主列的么量化Langlands导数的确定问题。
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