Descents of unipotent cuspidal representations of finite classical groups

Normalisation des opérateurs d'entrelacement et réductibilité des induites de cuspidales; le cas des groupes classiques p-adiques

On montre comment les conjectures d'Arthur permettent de calculer les points de réductibilité pour les induites de cuspidales des groupes classiques. Les conjectures d'Arthur utilisées portent sur l'existence d'un relèvement faible des représentations automorphes cuspidales d'un groupe classique vers un groupe linéaire convenable. Et les points de réductibilités sont decrits en terme de la géométrie des orbites unipotentes des groupe dual. On en déduit des résultats sur la normalisation des operateurs d'entrelacement. We show how the global conjectures of Arthur allow us to calculate the point…

有限李型群的幂单尖顶表示的Tate上同调

Tate cohomology of unipotent cuspidal representations of finite groups of Lie type

本文针对有限李型群的幂单尖顶表示，利用广义Gelfand-Graev表示与Harish-Chandra诱导构造的投射生成元的重数1性质，证明了特定Tate上同调群的不可约性与消失性。

$Sp_{2n}(F_{q^{2}})$-Invariants In Irreducible Unipotent Representations of $Sp_{4n}(F_{q})$

We show that for any irreducible representation of $Sp_{4n}(F_{q})$, the subspace of all its $Sp_{2n}(F_{q^{2}})$-invariants is at most one-dimensional. In terms of Lusztig symbols, we give a complete list of irreducible unipotent representations of $Sp_{4n}(F_{q})$ which have a nonzero $Sp_{2n}(F_{q^{2}})$-invariant and, in particular, we prove that every irreducible unipotent cuspidal representation has a one-dimensional subspace of $Sp_{2n}(F_{q^{2}})$-invariants. As an application, we give an elementary proof of the fact that the unipotent cuspidal representation is defined over $Q$, which…

Shintani descent for algebraic groups and almost characters of unipotent groups

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_q$. For this, it is essential to treat all the pure inner $\mathbb{F}_q$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the "trace of Frobenius" functi…

Restriction of a character sheaf to conjugacy classes

Let A be a character sheaf on a reductive connected group G over an algebraically closed field. Assuming that the characteristic is not bad, we show that for certain conjugacy classes D in G the restriction of A to D is a local system up to shift; we also give a parametrization of unipotent cuspidal character sheaves on G in terms of restrictions to conjugacy classes. Without restriction on characteristic we define canonical bijections from the set of unipotent character sheaves on G (and from the set of unipotent representations of the corresponding split reductive group over a finite field) …

Supercuspidal unipotent representations: L-packets and formal degrees

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $γ$-factors of L-parameters. This can be regar…

刻画显著幂幺轨道的不等式

Inequalities characterizing distinguished unipotent orbits

针对特征0代数闭域上的连通约化群，本文结合经典群的组合计算与例外群的计算机验证，提出并证明了显著幂幺轨道的新不等式刻画，可服务于L-参数栈上尖点层的后续研究。

一些p-adic分裂F4的unipotent Arthur包

Some unipotent Arthur packets for p-adic split F4

本文研究了p-adic分裂F4中由最小nilpotent轨道确定的unipotent表示，计算了特征环并确定了微Arthur包，验证了弱Arthur包是微Arthur包的并集。

On the unipotent characters of the Ree groups of type G_2

This note is concerned with the unipotent characters of the Ree groups of type G_2. We determine the roots of unity associated by Lusztig and Digne-Michel to unipotent characters of ^2G_2(3^{2n+1}) and we prove that the Fourier matrix of ^2G_2(3^{2n+1}) defined by Geck and Malle satisfies a conjecture of Digne-Michel. Our main tool is the Shintani descent of Ree groups of type G_2.

On character sheaves and characters of reductive groups at unipotent classes

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front se…

Split Spetses for primitive reflection groups

Let $(V,W)$ be an exceptional spetsial irreducible reflection group $W$ on a complex vector space $V$, that is a group $G_n$ for $n \in \{4, 6, 8, 14, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37\}$ in the Shephard-Todd notation. We describe how to determine some data associated to the corresponding (split) "spets", given complete knowledge of the same data for all proper subspetses (the method is thus inductive). The data determined here is the set Uch$(\mathbb G)$ of "unipotent characters" of $\mathbb G$ and the associated set of Frobenius eigenvalues, and its repartition into fami…

On modular Harish-Chandra series of finite unitary groups

In the modular representation theory of finite unitary groups when the characteristic $\ell$ of the ground field is a unitary prime, the $\hat{\mathfrak{sl}}_e$-crystal on level $2$ Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for $\ell$ sufficiently large, the $\mathfrak{sl}_\infty$-crystal on the same level $2$ Fock spaces provides the remaining piece of the puzzle for the …

On Special Unipotent Orbits and Fourier Coefficients for Automorphic Forms on Symplectic Groups

Fourier coefficients of automorphic representations $π$ of $\Sp_{2n}(\BA)$ are attached to unipotent adjoint orbits in $\Sp_{2n}(F)$, where $F$ is a number field and $\BA$ is the ring of adeles of $F$. We prove that for a given $π$, all maximal unipotent orbits, which gives nonzero Fourier coefficients of $π$ are special, and prove, under a well acceptable assumption, that if $π$ is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is $F$-anisotropic as algebraic group over $F$. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and …

On the values of unipotent characters of finite Chevalley groups of type $E_7$ in characteristic 2

Let $G$ be a finite group of Lie type. In order to determine the character table of $G$, Lusztig developed the theory of character sheaves. In this framework, one has to find the transformation between two bases for the space of class functions on $G$, one of them being the irreducible characters of $G$, the other one consisting of characteristic functions associated to character sheaves. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. The problem of specifying these scalars can be reduced to con…

On the computation of character values for finite Chevalley groups of exceptional type

We discuss various computational issues around the problem of determining the character values of finite Chevalley groups, in the framework provided by Lusztig's theory of character sheaves. Some of the remaining open questions (concerning certain roots of unity) for the cuspidal unipotent character sheaves of groups of exceptional type are resolved.

幂幺表示的局部朗兰兹对应

A local Langlands correspondence for unipotent representations

本文为非阿基米德局部域上连通约化群的不可约幂幺表示建立了局部朗兰兹对应，通过研究两侧的仿射Hecke代数并证明比较定理，推广了Lusztig在伴随K-群情形下的工作。

深度零超 cuspidal 表示的固定 pinning 局部 Langlands 对应

A Pinned Local Langlands Correspondence for Depth-Zero Supercuspidal Representations

本文构建了深度零超 cuspidal 表示的固定 pinning 局部 Langlands 对应，通过固定 pinning 分裂的 quasi-split 内形式，得到深度零超 cuspidal 表示与相关 cuspidal 增强深度零 Langlands 参数的双射。

Branching graphs for finite unitary groups in non-defining characteristic

We show that the modular branching rule (in the sense of Harish-Chandra) on unipotent modules for finite unitary groups is piecewise described by particular connected components of the crystal graph of well-chosen Fock spaces, under favourable conditions. Besides, we give the combinatorial formula to pass from one to the other in the case of modules arising from cuspidal modules of defect 0. This partly proves a recent conjecture of Jacon and the authors.

Bounding Harish-Chandra series

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.

On Certain Degenerate Whittaker Models for Cuspidal Representations of $\mathrm{GL}_{k\cdot n}\left(\mathbb{F}_q\right)$

Let $π$ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $π= π_θ$, corresponds to a regular character $θ$ of $\mathbb{F}_{q^{kn}}^{*}$. We consider the twisted Jacquet module of $π$ with respect to a non-degenerate character of the unipotent radical corresponding to the partition $(n^k)$ of $kn$. We show that, as a $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$-representation, this Jacquet module is isomorphic to $π_{θ\upharpoonright_{\mathbb{F}_n^*}} \otimes \mathrm{St}^{k-1}$, where $\mathrm{St}$ is the Steinberg representation of $\mathrm{GL}…

Nonsplit Hecke algebras and perverse sheaves

Let H be a Hecke algebra arising as an endomorphism algebra of the representation of a Chevalley group G over F_q induced by a unipotent cuspidal representation of a Levi quotient L of a parabolic subgroup. We assume that L is not a torus. In this paper we outline a geometric interpretation of the coefficients of the canonical basis of H in terms of perverse sheaves. We illustrate this in detail in the case where the Weyl group of G is of type B_4 and that of L is of type B_2.

字符的固定约当分解与零深度海 kè 算代数

Pinned Jordan Decomposition of Characters and Depth-Zero Hecke Algebras

本文研究了有限缩减群中相关对偶中心化子可能不连通时的固定约当分解构造，通过Pinning和Deligne-Lusztig公式建立了Lusztig系列与单因子的对应关系，同时探讨了具有阿贝尔组件群的可能不连通缩减群的约当分解扩展。

A note on perfect isometries between finite general linear and unitary groups at unitary primes

Let $q$ be a power of a prime, $l$ a prime not dividing $q$, $d$ a positive integer coprime to both $l$ and the multiplicative order of $q\mod l$ and $n$ a positive integer. A. Watanabe proved that there is a perfect isometry between the principal $l-$blocks of $GL_n(q)$ and $GL_n(q^d)$ where the correspondence of characters is give by Shintani descent. In the same paper Watanabe also prove that if $l$ and $q$ are odd and $l$ does not divide $GL_n(q^2)|/|U_n(q)|$ then there is a perfect isometry between the principal $l-$blocks of $U_n(q)$ and $GL_n(q^2)$ with the correspondence of characters …

有限经典群的波前集与下降法

Rational wavefront sets and the descent method for finite classical groups

本文通过下降法完成有限域上经典群不可约表示波前集中有理幂零轨道的显式描述，并建立单幂表示波前集与Lusztig典范商的关系及Alvis-Curtis对偶。

Actions of relative Weyl groups I, II

In the first part, we construct a new isomorphism between the endomorphism algebra of an induced cuspidal character sheaf and the group algebra of the relative Weyl group involved. We show it differs from the isomorphism of Lusztig by a linear character, and we relate this linear character to some stabilizers. Some consequences for characteristic functions of character sheaves are obtained. In the second part, we compute explicitly this linear character whenever the cuspidal local system is supported by the regular unipotent class and, as an application of these methods, we obtain a refinement…

哈里-钱德拉诱导与特征标Jordan分解

Harish-Chandra Induction and Jordan Decomposition of Characters

研究证明有限连通可约群中总能选择Jordan分解与哈里-钱德拉诱导相容，并推广了中心连通群的经典结果。

Double Descent in Classical Groups

Let ${\bf A}$ be the ring of adeles of a number field $F$. Given a self-dual irreducible, automorphic, cuspidal representation $τ$ of $\GL_n(\BA)$, with trivial central characters, we construct its full inverse image under the weak Langlands functorial lift from the appropriate split classical group $G$. We do this by a new automorphic descent method, namely the double descent. This method is derived from the recent generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan \cite{CFGK17}, which represent the standard $L$-functions for $G\times \GL_n$. Our results are valid also for …

Tate上同调的Jacquet模与基变更提升

Jacquet modules of Tate cohomology and base change lifting

本文在非阿基米德局部域的连通约化群框架下，证明了Treumann-Venkatesh关于Tate上同调群为有限长表示的猜想，给出GL_n深度0尖点表示基变更提升情形的显式计算，拓展至非尖点情形并研究了Sp_4的相关Tate上同调。

Descent Construction for Gspin Groups: Main Results and Applications

The purpose of this note is to announce an extension of the descent method of Ginzburg, Rallis and Soudry to the setting of essentially self dual representations. This extension of the descent construction provides a complement to recent work of Asgari and Shahidi on the generic transfer for general Spin groups as well as to the work of Asgari and Raghuram on cuspidality of the exterior square lift for representations of GL4. Complete proofs of the results announced in the present note will appear in our forthcoming articles.

与极大抛物子群相关的扭Jacquet模及$GL(n, q)$的尖表示

Twisted Jacquet modules associated to maximal parabolic subgroups and cuspidal representations of $GL(n, q)$

本文研究有限域上一般线性群尖表示关于极大抛物子群的扭Jacquet模的结构，通过建立扭Jacquet函子的范畴等价并构造具有泛性质的表示，直接解决了所有参数情形下的模结构问题。

有限域上Howe对偶III：完整计算与Gurevich-Howe猜想

Howe duality over finite fields III: Full computation and the Gurevich-Howe conjectures

本文研究有限域上Howe对偶的完整计算，探讨对称群和正交群的振荡表示限制，并证明Gurevich-Howe猜想。

GL_n(q)中辉煌Rickard等价的下降

Descent of splendid Rickard equivalences in ${\rm GL}_n(q)$

本文证明Broué阿贝尔亏群猜想的精细版本对GL_n(q)的幂零ℓ-块成立，并给出一般ℓ-块满足该猜想的一个充分条件，同时举例说明该条件并非普遍成立。

有限域上的Howe双射I：两种稳定的范围

Howe duality over finite fields I: The two stable ranges

本文研究有限域上Howe双射的I型，探讨辛群振子表示对辛与正交群乘积的限制，通过两个稳定范围构造对应关系，解决张量对多重性、Gurevich-Howe秩猜想C型及递归字符公式问题。

Some questions on global distinction for $\mathrm{SL}(n)$

Let $E/F$ be a quadratic extension of number fields and let $π$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished cuspidal automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Using an unfolding argument, we prove that an element of the $\mathrm{L}$-packet of $π$ is distinguished if and only if it is $ψ$-generic for a non-degenerate character $ψ$ of $N_n(\mathbb{A}_E)$ trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. We then use this result to analyze the non-vanishing of the period integral on different realizations …

Wavefront sets and descent method for finite symplectic groups

In \cite{JZ1}, D. Jiang and L. Zhang proposed a conjecture which related the wavefront sets and the descent method in the local fields case. Recently, in \cite{JLZ}, they and D. Liu define the arithmetic wavefront set of certain irreducible admissible representation $π$ of a classical group $G(k)$ defined over local field $k$, which is a subset of $k$-rational nilpotent orbits of the Lie algebra of $G(k)$, by the arithmetic structures of the enhanced L-parameter of $π$. These arithmetic structures are based on the rationality of the local Langlands correspondence and the local Gan-Gross-Prasad…

Springer 对应的推广与尖点 Langlands 参数

Generalizations of the Springer correspondence and cuspidal Langlands parameters

本文为约化 p-进群引入增强 Langlands 参数的尖点性概念及 Bernstein 分量，揭示其空间为扭曲扩展商并集，并将 Lusztig 广义 Springer 对应推广到不连通复约化群，从而将局部 Langlands 对应问题归结为 Levi 子群的超尖点表示情形。

A computational approach to the Frobenius-Schur indicators of finite exceptional groups

We prove that the finite exceptional groups $F_4(q)$, $E_7(q)_{\mathrm{ad}}$, and $E_8(q)$ have no irreducible complex characters with Frobenius-Schur indicator $-1$, and we list exactly which irreducible characters of these groups are not real-valued. We also give an exact list of complex irreducible characters of the Ree groups ${^2 F_4}(q^2)$ which are not real-valued, and we show the only character of this group which has Frobenius-Schur indicator $-1$ is the cuspidal unipotent character $χ_{21}$ found by M. Geck.

Graded Hecke algebras for disconnected reductive groups

We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G. We develop the representation theory of the algebras H. obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come d…

Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

Let $F$ be a local non-archimedian field and $G$ be the group of $F$-points of a split connected reductive group over $F$. In a previous aricle we defined an algebra $\mathcal J(G)$ of functions on $G$ which contains the Hecke algebra $\mathcal H(G)$ and is contained in the Harish-Chandra Schwartz algebra $\mathcal C(G)$. We consider $\mathcal J(G)$ as an algebraic analog the algebra $\mathcal C(G)$. Given a parabolic subgroup $P$ of $G$ with a Levi subgroup $M$ and the unipotent radical $U_P$ we write $X_P:=G/U_P$. In this paper we study two versions of the Schwartz space of $X_P$. The first …

有限群的模函子性

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。

Fourier coefficients of automorphic forms, character variety orbits, and small representations

We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL(n), Sp(4), and G_2, we perform an analysis of the expansions on split real forms of E_6 and E_7 where simplifications take place for automorphic realizations of real representations which have small Gelfand-Kirillov dimension. Though the character varieties are more complicated for exceptional groups, we explain how the nonvanishing Fourier coefficients for small representations behave analogously to Fou…

关于辛群的剩余自守表示与周期积分

On residual automorphic representations and period integrals for symplectic groups

本文构造了辛群离散自守谱中的新不可约分支，将具有线性周期的 GL_{2n} 尖点自守表示提升为任意秩 k≥2n 辛群的剩余表示，并证明其存在非零不变线性型，推广了下降法中的经典构造。

$G_2$ 的显式局部 Langlands 对应

The Explicit Local Langlands Correspondence for $G_2$

本文提出一种基于 Hecke 代数和 Levi 子群约化的一般策略，并据此构造了 $p$-进域上 $G_2$ 的显式局部 Langlands 对应，给出显式 $L$-包、匹配关系及奇异超尖点表示的存在性证明。