Dan Barbasch, Birgit Speh
This paper proves the existence of cuspidal automorphic forms for a reductive group, invariant under an automorphism of finite order. The techniques used are a local analysis of orbital integrals and the Arthur-Selberg trace formula.
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We prove that any reductive group G over a non-Archimedean local field has a cuspidal complex representation.
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.
We present a unified parametrisation of $\ell$-blocks of quasi-simple finite groups of Lie type in non-defining characteristic via Lusztig's induction functor in terms of $e$-Jordan-cuspidal pairs and $e$-Jordan quasi-central cuspidal pairs.
Let $G$ be a connected reductive group over a finite field $\mathfrak{f}$ of order $q$. When $q$ is small, we make further assumptions on $G$. Then we determine precisely when $G(\mathfrak{f})$ admits irreducible, cuspidal representations that are self-dual, of Deligne-Lusztig type, or both. Finally, we outline some consequences for the existence of self-dual supercuspidal representations of reductive $p$-adic groups.
In this note we show that all reductive groups are clean in odd characteristic. In characteristic 2 there are two cuspidal local systems (one for $F_4$ and one for $E_8$) which can not be handled by our method.
Let F be a non-archimedean local field of odd residual characteristic p. Let G be a (connected) reductive group that splits over a tamely ramified field extension of F. We show that a construction analogous to Yu's construction of complex supercuspidal representations yields smooth, irreducible, cuspidal representations over an arbitrary algebraically closed field R of characteristic different from p. Moreover, we prove that this construction provides all smooth, irreducible, cuspidal R-representations if p does not divide the order of the Weyl group of G.
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.
On some non-principal locally analytic representations induced by cuspidal Lie algebra representations
本文研究了由cuspidal李代数表示诱导的非主次局部解析表示的构造与性质，证明了其ind-admissible性及超cuspidal性，并在特定情况下证明了拓扑不可约性。
The irreducible characters of a finite reductive group are partitioned into Harish-Chandra series that are labelled by cuspidal pairs. In this note, we describe how one can algorithmically calculate those cuspidal pairs using results of Lusztig.
For reductive symmetric spaces G/H of split rank one we identify a class of minimal parabolic subgroups for which certain cuspidal integrals of Harish-Chandra - Schwartz functions are absolutely convergent. Using these integrals we introduce a notion of cusp forms and investigate its relation with representations of the discrete series for G/H.
We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $\ell$-reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand--Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand--Graev representations attached to cuspidal unipotent classes, as defined and studied in \cite{GM96}.
Let $G$ be a connected split reductive group over a finite field ${\mathbb F}_q$ and $X$ a smooth projective geometrically connected curve over ${\mathbb F}_q$. The $\ell$-adic cohomology of stacks of $G$-shtukas is a generalization of the space of automorphic forms with compact support over the function field of $X$. In this paper, we construct a constant term morphism on the cohomology of stacks of shtukas which is a generalization of the constant term morphism for automorphic forms. We also define the cuspidal cohomology which generalizes the space of cuspidal automorphic forms. Then we sho…
Let $F$ be any non-Archimedean local field with a Galois involution $σ$ and $F_0$ be the fixed field for the action of $σ$. When the residue characteristic of $F_0$ is odd, using the explicit construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of $U(2,1)(F/F_0)$.
We define a new notion of cuspidality for representations of $\GL_n$ over a finite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedean local field $F$ using geometric and infinitesimal induction functors, which involve automorphism groups $G_λ$ of torsion $\Oh$\nobreakdash-modules. When $n$ is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of $\GL_n(F)$. We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear gr…
Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal theta representation is one which is equal to the local twisted theta representation at almost all places. Cuspidal theta representations are known to exist but only for covers of $GL_j$, $j\leq 3$. In this paper we establish necessary conditions for the existence of cuspidal theta representations on the $r$-fold metaplectic cover of the general linear group of arbitrary rank.
We discuss recent developments in the Langlands program for function fields, and in the geometric Langlands program. In particular we explain a canonical decomposition of the space of cuspidal automorphic forms for any reductive group G over a function field, indexed by global Langlands parameters. The proof uses the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence.
Given a quaternionic form G of a p-adic classical group (p odd) we classify all cuspidal irreducible representations of G with coefficients in an algebraically closed field of characteristic different from p. We prove two theorems: At first: Every irreducible cuspidal representation of G is induced from a cuspidal type, i.e. from a certain irreducible representation of a compact open subgroup of G, constructed from a beta-extension and a cuspidal representation of a finite group. Secondly we show that two intertwining cuspidal types of G are up to equivalence conjugate under some element of G.
For a cuspidal automorphic representation Πof GL(4,A), H. Kim proved that the exterior square transfer \wedge^2Πis an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations Πfor which \wedge^2Πis cuspidal.
We examine the theory of induced representations for non-connected reductive $p$-adic groups for which $G/G^0$ is abelian. We first examine the structure of those representations of the form $\Ind_{P^0}^G(σ),$ where $P^0$ is a parabolic subgroup of $G^0$ and $\s$ is a discrete series representation of the Levi component of $P^0.$ Here we develop a theory of $R$--groups, extending the theory in the connected case. We then prove some general results in the theory of representations of non-connected $p$-adic groups whose component group is abelian. We define the notion of cuspidal parabolic for $…
Newforms in Cuspidal Representations
本文将 $p$-adic 群尖点表示的 newform 理论推广至 $\ell$-modular 表示，证明了导子与模 $\ell$ 同余的相容性，并在复数与 modular 设定下给出了深度零及 minimax 尖点表示的显式公式。
Level-Rank Dualities for Finite Reductive Groups
本文为有限约化群表示论中循环赫克代数块之间的级秩对偶性提供证据，并在一般线性群情形下用分拆组合学具体表达，重现了Frenkel等人的级秩对偶。
Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: simple supercuspidal representations
本文计算了简单超尖点表示的局部扭曲外平方 gamma 因子，并据此证明了该类表示的局部逆定理。
Let $F$ be a non-Archimedean locally compact field of residue characteristic $p$, $G$ be an inner form of $GL_n(F)$, $n\ge1$, and $\ell$ be a prime number different from $p$. We give a numerical criterion for an integral $\ell$-adic irreducible cuspidal representation $\tildeρ$ of $G$ to have a super\-cuspidal irreducible reduction mod $\ell$, by counting inertial classes of cuspidal representations that are congruent to the inertial class of $\tildeρ$, generalizing results by Vign{é}ras and Dat. In the case the reduction mod $\ell$ of $\tildeρ$ is not super\-cuspidal irreducible, we show that…
We follow Jacquet-Shalika, Matringe and Cogdell-Matringe to define exterior square gamma factors for irreducible cuspidal representations of $\mathrm{GL}_n(\mathbb{F}_q)$. These exterior square gamma factors are expressed in terms of Bessel functions, or in terms of the regular characters associated with the cuspidal representations. We also relate our exterior square gamma factors over finite fields to those over local fields through level zero representations.
Irreducibility of wave-front sets for depth zero cuspidal representations
本文利用已有文献结果，证明了在足够大的剩余特征下，非阿基米德局部域上约化群的深度零不可约尖点表示的波前集的Zariski闭包是不可约簇，从而肯定回答了Moeglin-Waldspurger的相关问题。
Simple cuspidal representations of symplectic groups: Langlands parameter
本文通过显式计算覆盖的 Hecke 代数生成元，确定了非阿基米德局部域上辛群简单尖点表示的 Jordan 集，并在 $p$-进域情形下进一步给出了其朗兰兹参数。
Let $F$ be a non-archimedean local field of residue characteristic $p$, $G$ be the group $GL(n, F)$. In this note, under the assumption $(n, p)=1$, we show a simple cuspidal representation $π$ (that with normalized level $\frac{1}{n}$) of $G$ is determined uniquely up to isomorphism by the local constants of $χ\circ \text{det}\otimes π$ for all characters $χ$ of $F^\times$.
We study irreducible representations of the Hecke algebra of the pair $({\rm PGL}_2 (F[ε] / (ε^2)) , {\rm PGL}_2 (\mathcal{O}[ε] / (ε^2)))$ where $F$ is a local non-Archimedean field of characteristic different than $2$ and $\mathcal{O} \subset F$ is its ring of integers. We expect to apply our analysis to the study of the spectrum of Hecke operators on the space of cuspidal functions on the space of principal ${\rm PGL}_2$-bundles on curves over rings $\mathbb{F}_q [ε] / (ε^2)$.
We show the existence of integral models for cuspidal representations of GL(2,q), whose reduction modulo p can be identified with the cokernel of a differential operator on F_{q}[X,Y] defined by J-P. Serre. These integral models come from the crystalline cohomology of the projective curve XY^{q}-X^{q}Y-Z^{q+1}=0. As an application, we can extend a construction of C. Khare and B. Edixhoven (2003) giving a cohomological analogue of the Hasse invariant operator acting on spaces of modp modular forms for GL(2).
Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercupidal representations can be used as test functions for this, which kills the continuous spectrum, but also a large class of cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra is used in the recent work [3] to isolate all cuspidal spectrum which provide enough test functions and suitable for the comparison of orbital integrals. These multipliers are then applied to th…
The endoscopic classification via the stable trace formula comparison provides certain character relations between irreducible cuspidal automorphic representations of classical groups and their global Arthur parameters, which are certain automorphic representations of general linear groups. It is a question of J. Arthur and W. Schmid that asks: How to construct concrete modules for irreducible cuspidal automorphic representations of classical groups in term of their global Arthur parameters? In this paper, we formulate a general construction of concrete modules, using Bessel periods, for cuspi…
Cuspidal $\ell$-modular representations of ${\rm GL}_n(F)$ distinguished by a Galois involution, II
本文研究了GL_n(F)的cuspidal ℓ-模不可约表示的Galois反演性质，指出当ℓ≠2时，区分性等价于提升为ℓ-adic表示，而当ℓ=2时，区分性等价于共轭自对偶。
Let $G$ be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of Moeglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a Ramification Theorem for $G$, giving a bijection between the set of endo-parameters for $G$ and the set of restrictions to wild inertia of discrete Langlands parameters …
On relative cuspidality
本文在对称对(reductive groups)条件下，证明了存在强相对紧致表示，验证了Kato和Takano的预期。
Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks
本文研究了 $p$-adic 有限简约群的 Hecke 代数与伯恩斯坦块的局部朗兰兹对应关系，通过构建增强的 $L$-参数集并应用到各种群上，证明了某些猜想并简化了深度零情况。
Generalized Springer Theory for D-modules on a Reductive Lie Algebra
本文建立了约化李代数上等变 D-模范畴的广义 Springer 理论，将其正交分解为以尖点数据为指标的分块，并证明每个分块等价于中心上相对 Weyl 群等变 D-模范畴，核心方法是发展抛物诱导与限制函子及相应单子理论。
Let $E/F$ be a quadratic extension of non archimedean local fields of odd residual characteristic. We prove a conjecture of Prasad and Takloo-Bighash, in the case of cuspidal representations of depth zero of $\mathrm{GL}(2m,F)$. This conjecture characterizes distinction for the pair $(\mathrm{GL}(2m,F),\mathrm{GL}(m,E))$ with respect to a character $μ\circ \mathrm{det}$ of $\mathrm{GL}(m,E)$, in terms of certain conditions on Langlands paremeters, including an epsilon value. We also compute the multiplicity of the involved equivariant linear forms when $E/F$ is unramified, and also when $μ$ is…
This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the F-coinvariants of the component group of the center of G has prime order and if p is a good prime for G, then the relative McKay conjecture holds for G at the prime p. In particular, this conjecture is true for G^F in defining characteristic for G a simple and simply-connected group of type B_n, C_n, E_6 and E_7. Our main tools are the theory of Gelfand-Grae…
A new approach to Jiu-Kang Yu's construction of tame supercuspidal representations of $p$-adic reductive groups is presented. Connections with the theory of cuspidal Deligne-Lusztig representations of finite groups of Lie type are also discussed.
This is a translation in English of version 5 of the article arXiv:1404.3998, which is itself an introduction to arXiv:1209.5352. We explain all the ideas of the proof of the following theorem. For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the "automorphic to Galois" direction. Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at a…
Types and collapse for cuspidal representations of groups acting on trees
本文针对作用于树的群的尖点表示，在之前秩一p进群超尖点表示的几何局部化方法基础上，通过几何极小K型描述诱导数据、证明等变层坍缩为内射层，将相关概念推广至高秩群并提出猜想。
We give a description of all the cuspidal representations of $\mathrm{GL}_4(\mathfrak{o}_2)$, where $\mathfrak{o}_2$ is a finite ring coming from the ring of integers in a local field, modulo the square of its maximal ideal $\mathfrak{p}$. This shows in particular the existence of representations which are cuspidal, yet are not strongly cuspidal, that is, do not have orbit with irreducible characteristic polynomial mod $\mathfrak{p}$. It has been shown by Aubert, Onn, and Prasad that this phenomenon cannot occur for $\mathrm{GL}_n$, when $n$ is prime.
Doubling constructions: Global functoriality for non-generic cuspidal representations
针对辛群等群与$GL_k$的表示对，研究广义加倍方法，分析局部积分极点并证明扭曲的整体完备$L$-函数的整性，结合逆定理给出不依赖迹公式的弱函子转移新证明。
Modular functoriality for finite groups
本文推广Deligne--Lusztig理论至paraductive群，计算其Tate上同调下界，并应用模函子性计算非奇异深度0尖点表示的L-参数。
Character Sheaves on Reductive Lie Algebras in Positive Characteristic
本文在正特性下研究李代数上的特征层，证明了其微局部刻画及可允许性与特征层等价性的几何证明。
For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial …
Let $G_n$ denote either the group $Sp(2n, F)$ or $SO(2n+1, F)$ over a non-archimedean local field $F$. We determine the composition series of representations of $G_n$ induced from cuspidal and ladder representations such that the minimal exponent in the cuspidal support of the ladder representation is greater than or equal to $\frac{1}{2}$.
We study a notion of cusp forms for the symmetric spaces G/H with G = SL(n,R) and H = S(GL(n-1,R) x GL(1,R)). We classify all minimal parabolic subgroups of G for which the associated cuspidal integrals are convergent and discuss the possible definitions of cusp forms. Finally, we show that the closure of the direct sum of the discrete series of representations of G/H coincides with the space of cusp forms.
Let $E/F$ be a quadratic extension of non-archimedean local fields, and let $\ell$ be a prime number different from the residual characteristic of $F$. For a complex cuspidal representation $π$ of $GL(n,E)$, the Asai $L$-factor $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is $GL(n,F)$-distinguished. In this paper we solve the problem of characterising the occurrence of a pole at $X=1$ of $L^+(X,π)$ when $π$ is an $\ell$-modular cuspidal representation of $GL(n,E)$: we show that $L^+(X,π)$ has a pole at $X=1$ if and only if $π$ is a relatively banal distinguished representation; namely $π$…
Let n be a positive integer, F be a non-Archimedean locally compact field of odd residue characteristic p and G be an inner form of GL(2n,F). This is a group of the form GL(r,D) for a positive integer r and division F-algebra D of reduced degree d such that rd=2n. Let K be a quadratic extension of F in the algebra of matrices of size r with coefficients in D, and H be its centralizer in G. We study selfdual cuspidal representations of G and their distinction by H from the point of view of type theory. Given such a representation pi of G, we compute the value at 1/2 of the epsilon factor of the…
In this paper, we study the l-adic cohomology of the Rapoport-Zink tower for GSp(4). We prove that the smooth representation of GSp_4(Q_p) obtained as the i-th compactly supported l-adic cohomology of the Rapoport-Zink tower has no quasi-cuspidal subquotient unless i=2,3,4. Our proof is purely local and does not require global automorphic methods.
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.
In this paper we study the subcategory of cuspidal modules of the category of weight modules over the Lie algebra sl(n+1). Our main result is a complete classification and explicit description of the indecomposable cuspidal modules.
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。
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).
Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on $G$ induced from unramified cusp forms on finite-dimensional Levi subgroups of maximal parabolic subgroups. Under a natural condition on maximal parabolic subgroups, we prove that the cuspidal Eisenstein series are entire on the full complex plane.
We construct an endoscopic decomposition for local L-packets associated to irreducible cuspidal Deligne-Lusztig representations. Moreover, the obtained decomposition is compatible with inner twistings.
We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility properties of the refined local Langlands correspondence.
the program of Langlands is studied here on the basis of: a)new concepts of global class field theory related to the explicit construction of global class fields and of reciprocity laws; b)the representations of the reductive algebraic groups GL(n) constituting the n-dimensional representations of the associated global Weil groups; c)a toroidal compactification of the conjugacy classes of these reductive algebraic groups whose analytic representations constitute the cuspidal representations of these groups GL(n) in the context of harmonic analysis. This leads us to build two types of n-dimensi…
In the present paper we continue the project of systematic construction of invariant differential operators on the example of representations of the conformal algebra induced from the maximal cuspidal parabolic.
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