Characters and growth of admissible representations of reductive p-adic groups

A short proof of the existence of supercuspidal representations for all reductive $p$-adic groups

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 $G$.

On the existence of admissible supersingular representations of $p$-adic reductive groups

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 $C$.

Cuspidal representations of reductive p-adic groups are relatively injective and projective

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

We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive p-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an affine Hecke algebra with geometric parameters, in the sense of Lusztig (possibly extended by a group of automorphisms of the root datum). It is known that there is a large class of such examples (detailed in the paper). As a consequence, we establish relations between the unitary duals of different groups, in the spirit of endoscopy.

Types and unitary representations of reductive p-adic groups

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

We prove that the Jacquet restriction functor for a parabolic subgroup of a reductive group over a non-Archimedean local field is right adjoint to the parabolic induction functor for the opposite parabolic subgroup, in the generality of smooth group representations on R-modules for any unital ring R in which the residue field characteristic is invertible. Correction: it turned out that the paper contains a mistake (in Lemma 3.4), which the authors have been unable to fix. This renders the proof of the main theorem incomplete.

Wave packets in the Schwartz space of a reductive p-adic symmetric space

We form wave packets in the Schwartz space of a reductive p-adic symmetric space for certain famillies of tempered functions. We show how to construct such families from Eisenstein integrals.

Geometric structure for the principal series of a split reductive $p$-adic group with connected centre

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 $\mathcal{G}$.

Some Unipotent Arthur Packets for Reductive $p$-adic Groups

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…

Resolutions for representations of reductive p-adic groups via their buildings

Schneider-Stuhler and Vigneras have used cosheaves on the affine Bruhat-Tits building to construct natural finite type projective resolutions for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building. We establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein's result that certain subcategories of the category of representations are Serre subcategories. Furthermor…

p进约化群的表示

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。

裂变群与 intertwining 运算符的规范模块

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。

关于约化p进群的中心扩张的表示理论的一个注记

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。

Supercuspidal characters of reductive p-adic groups

We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.

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.

Characters of locally analytic representations of p-adic reductive groups

We propose a definition of characters in the context of Schneider-Teitelbaum's theory of locally analytic representations of p-adic reductive groups. This character will be a function on a compact subgroup of a maximal torus of the reductive group in question. As an example we treat the locally analytic principal series of SL(2,Q_p).

Resolutions of tempered representations of reductive p-adic groups

Let G be a reductive group over a non-archimedean local field and let S(G) be its Schwartz algebra. We compare Ext-groups of tempered G-representations in several module categories: smooth G-representations, algebraic S(G)-modules, bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces which contains all admissible S(G)-modules. We simplify the proofs of known comparison theorems for these Ext-groups, due to Meyer and Schneider-Zink. Our method is based on the Bruhat-Tits building of G and on analytic properties of the Schneider-Stuhler resolutions.

Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups I

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…

约化p-adic群的Schwartz代数的同调代数

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。

Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups II

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…

Local Summability of Characters on $p$-adic Reductive Groups

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…

关于非连通约化p-adic群的容许表示的一些结果

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。

A matrix Paley-Wiener theorem for non-connected $p$-adic reductive groups

Let $F$ be a local non archimedian field of characteristic $0$, and $G$ a non-connected reductive group over $F$. We denote $G^0$ the connected component of the identity and assume the quotient $G/G^0$ is abelian. For $f$ a locally constant compactly supported function on $G$ and $π$ a complex smooth representation of $G$, we define the Fourier transform of $f$ evaluated at $π$ to be $π(f) = \int_{G} f(g) π(g) \, dg$, which is an endomorphism of the underlying vector space of $π$. We give a description of the image of this Fourier transform map : given, for every $π$ in a certain family of ind…

Endomorphism algebras of admissible p-adic representations of p-adic Lie groups

Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible p-adic Banach space (respectively locally analytic) representations of p-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible representation.

关于拟分裂约化p进群的主系列表示

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。

The canonical measure on a reductive p-adic group is motivic

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…

复凝聚reductive p进群的有理性性质

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。

On Ideals Defining Irreducible Representations of Reductive $p$--adic Groups

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.

On Representations of Reductive $p$--adic Groups over $\mathbb Q$--algebras

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.

约化p-adic群的Hecke代数：一个几何猜想

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。

Deligne-Lusztig constructions for unipotent and p-adic groups

In 1979 Lusztig proposed a conjectural construction of supercuspidal representations of reductive p-adic groups, which is similar to the well known construction of Deligne and Lusztig in the setting of finite reductive groups. We present a general method for explicitly calculating the representations arising from Lusztig's construction and illustrate it with several examples. The techniques we develop also provide background for the author's joint work with Weinstein on a purely local and explicit proof of the local Langlands correspondence.

约化 $p$-进群覆盖群表示的纠缠算子

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。

Questions on mod p representations of reductive p-adic groups

This is a list of questions raised by our joint work arXiv:1412.0737 and its sequels.

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.

Verma modules over p-adic Arens-Michael envelopes of reductive Lie algebras

Let K be a locally compact nonarchimedean field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We study the category C_g of coadmissible modules over the nonarchimedean Arens-Michael envelope of U(g). Let p be a parabolic subalgebra of g. The main result identifies a certain explicitly given highest weight category inside C_g with the classical parabolic BGG category of g relative to p. This paper is in final form, replaces and expands the former preprint 'BGG reciprocity for p-adic Arens-Michael envelopes of semisimple Lie algebras' and appears in Journal o…

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。

约化群的可酉化极小主列

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。

计算约化p进群的Hecke代数的代数K-理论的方法

Recipes to compute the algebraic K-theory of Hecke algebras of reductive p-adic groups

该研究利用Farrell-Jones猜想，借助Bruhat-Tits厦性质与等变Atiyah-Hirzebruch谱序列，计算约化p进群Hecke代数的代数K-理论，得出投射类群可表为紧开子群投射类群余极限的结论。

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。

实可约群的表示论的算法

Algorithms for Representation Theory of Real Reductive Groups

给出了一个计算具有正则积分无穷小特征标的实约化群的不可约容许表示的算法。该算法已在计算机上实现，作为李群和表示项目的一部分。

Kazhdan's orthogonality conjecture for real reductive groups

We prove a generalization of Harish-Chandra's character orthogonality relations for discrete series to arbitrary Harish-Chandra modules for real reductive Lie groups. This result is an analogue of a conjecture by Kazhdan for $\mathfrak p$-adic reductive groups proved by Bezrukavnikov, and Schneider and Stuhler.

p-adic 约化群的 mod-$\ell$ 表示的局部特征标

Local characters of mod-$\ell$ representations of a $p$-adic reductive group

本文为 p-adic 约化群的 mod-ℓ 表示定义了提升特征标，推广了 Howe、Harish-Chandra 和 DeBacker 的局部展开结果，并证明 Moeglin-Waldspurger 和 Varma 的退化 Whittaker 模型结论仍成立。

Neighborhoods at infinity and the Plancherel formula for a reductive $p$-adic symmetric space

Yiannis Sakellaridis and Akshay Venkathesh have determined, when the group $G$ is split and the field $\F$ is of characteristic zero, the Plancherel formula for any spherical space $X$ for $G$ modulo the knowledge of the discrete spectrum. The starting point is the determination of good neighborhoods at infinity of $X/J$, where $J$ is a small compact open subgroup of $G$. These neighborhoods are related to "boundary degenerations" of $X$. The proof of their existence is made by using wonderful compactifications. In this article we will show the existence of such neighborhoods assuming that $\F…

Periodic cyclic homology of reductive p-adic groups

Let G be a reductive p-adic group, H(G) its Hecke algebra and S(G) its Schwartz algebra. We will show that these algebras have the same periodic cyclic homology. This might be used to provide an alternative proof of the Baum-Connes conjecture for G, modulo torsion. As preparation for our main theorem we prove two results that have independent interest. Firstly a general comparison theorem for the periodic cyclic homology of finite type algebras and certain Fréchet completions thereof. Secondly a refined form of the Langlands classification for G, which clarifies the relation between the smooth…

Irreducible representations of product of real reductive groups

Let $G_1,G_2$ be real reductive groups and $(π,V)$ a smooth, irreducible, admissible representation of $G_1 \times G_2$. We prove that $(π,V)$ is the completed tensor product of $(π_i,V_i)$, $i=1,2$, where $(π_i,V_i)$ is a smooth,irreducible,admissible representation of $G_i$, $i=1,2$. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in [AG] and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair $H \subset G$ of real reductive groups is equivalent to the usual Gelfand property of the pair …

$p$-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。

A classification of irreducible admissible mod p representations of p-adic reductive groups

Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classificati…

The nonabelian Fourier transform for elliptic unipotent representations of exceptional $p$-adic groups

We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric restriction map agrees with Lusztig's nonabelian Fourier transform for unipotent representations of the finite reductive quotient. This is inspired by recent works of Lusztig on the almost unipotent characters of $p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier transform of odd orthogonal groups.

Morita's Duality for Split Reductive Groups

In this paper, we extend the work in "Morita's Theory for the Symplectic Groups" to split reductive groups. We construct and study the holomorphic discrete series representation and the principal series representation of a split reductive group $\mathrm {G}$ over a $p$-adic field $F$ as well as a duality between certain sub-representations of these two representations.

p-adic约化对称空间中Cartan分解的一个类似

An analogue of the Cartan decomposition for p-adic reductive symmetric spaces

设F是一个非阿基米德局部紧域，其剩余特征不同于2，G是定义在F上的一个连通约化群，s是G的一个对合F-自同构，H是s的不动点群的开F-子群.我们记为G（F）（resp. H（F））是G的F-点群（分别为H）。本文得到了约化对称空间G（F）/H（F）的Cartan分解的一个类似.更精确地说，我们得到了G（F）作为H（F）-陪集的并的一个分解，它与G中极大s-反不变F-分裂环面的H（F）-共轭类有关。当G是F-分裂时，我们得到了一个更精确的结果，涉及到G在F上的Bruhat-Tits建筑的一个特殊点的稳定子。

Distinguished Cuspidal Representations over p-adic and Finite Fields

The author's work with Murnaghan on distinguished tame supercuspidal representations is re-examined using a simplified treatment of Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups. This leads to a unification of aspects of the theories of distinguished cuspidal representations over $p$-adic and finite fields.

约化 p-adic 群的群代数、其表示及其非交换几何

Group algebras of reductive $p$-adic groups, their representations and their noncommutative geometry

本文综述约化 p-adic 群的 Hecke、Schwartz 与约化 C*-代数结构、表示分类及 K-理论，并给出包含挠元素的 $K_*(C_r^*(G))$ 新计算。

The classification of irreducible admissible mod p representations of a p-adic GL_n

Let F be a finite extension of Q_p. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \bar F_p to be supersingular. We then give the classification of irreducible admissible smooth GL_n(F)-representations over \bar F_p in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel-Livne for n = 2. For general split reductive groups we obtain similar results under stronger hypotheses.

Homological duality for covering groups of reductive $p$-adic groups

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre dualit…