Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: simple supercuspidal representations
Rongqing Ye, Elad Zelingher
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。
我们计算了简单超尖点表示的局部扭曲外平方 gamma 因子，并借此证明了简单超尖点表示的局部逆定理。
We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.
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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.
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.
Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$, and let $σ$ denote its non-trivial automorphism. Let $R$ be an algebraically closed field of characteristic different from $p$. To any cuspidal representation $π$ of ${\rm GL}_n(F)$, with coefficients in $R$, such that $π^σ\simeqπ^{\vee}$ (such a representation is said to be $σ$-selfdual) we associate a quadratic extension $D/D_0$, where $D$ is a tamely ramified extension of $F$ and $D_0$ is a tamely ramified extension of $F_0$, together with a quadratic character of $D_0^{\times…
Parametrization of supercuspidal representations of depth zero for some simple adjoint groups
本文为简单伴随群构造深度为零的超尖表示与增强L-参数间的满射映射，并在双射情形下证明形式度猜想成立。
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.
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.
Local Langlands functoriality for Yu's supercuspidals I: Kaletha's parametrization
本文为两篇系列论文之首，受Kaletha工作启发，为Yu构造的所有尖点表示构造任意系数的半单惯性L-参数，并基于函子性性质刻画对应，为后续计算Fargues–Scholze对应奠定基础。
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.
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。
Local Langlands functoriality for Yu's supercuspidals II: Fargues--Scholze's parametrization
本文通过计算Yu构造尖点表示的Tate上同调，结合模函子性，比较Fargues--Scholze与Kaletha参数化，证明前者具有有限纤维且满射到惯性L-参数。
On Speh representations for level zero supercuspidal representations and Ginzburg-Kaplan gamma factors
本文建立了有限域与非阿基米德局部域上 $\mathrm{GL}_n$ 的 Speh 表示之间的联系，通过交换图证明零阶超尖点表示对应的 Speh 表示及其 Whittaker 模型相关联，并将局部 Ginzburg-Kaplan 积分与有限域情形相联系。
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。
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。
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.
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…
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。
This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal repr…
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…
In this article, we consider the representations of the general linear group over a non-archimedean local field obtained from the vanishing cycle cohomology of the Lubin-Tate tower. We give an easy and direct proof of the fact that no supercuspidal representation appears as a subquotient of such representations unless they are obtained from the cohomology of the middle degree. Our proof is purely local and does not require Shimura varieties.
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
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…
Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. We classify all smooth irreducible representations of GL(m,D) with coefficients in R, in terms of multisegments, generalizing works by Zelevinski, Tadic and Vignéras. We prove that any irreducible R-representation of GL(m,D) has a unique supercuspidal support, and thus get two classifications: one by supercuspidal multisegments, classifying representations with a given supercuspidal …
Reducibility points and characteristic $p$ local fields I- Simple supercuspidal representations of symplectic groups
本文研究特征为奇素数p的非阿基米德局部域上辛群的简单超尖点表示，推广Moeglin可约性准则至任意正特征域，通过显式构造给出其到一般线性群表示的转移，推进了正特征下辛群泛型尖点表示的朗兰兹对应研究。
We define epsilon factors for irreducible representations of finite general linear groups using Macdonald's correspondence. These epsilon factors satisfy multiplicativity, and are expressible as products of Gauss sums. The tensor product epsilon factors are related to the Rankin-Selberg gamma factors, by which we prove that the Rankin-Selberg gamma factors can be written as products of Gauss sums. The exterior square epsilon factors relate the Jacquet-Shalika exterior square gamma factors and the Langlands-Shahidi exterior square gamma factors for level zero supercuspidal representations. We p…
Let $π$ be an irreducible admissible representation of $GL_m(F)$, where $F$ is a non-archimedean local field of characteristic zero. We follow the method developed by Cogdell and Piatetski-Shapiro to complete the computation of the local exterior square $L$-function $L(s,π,\wedge^2)$ in terms of $L$-functions of supercuspidal representations via an integral representation established by Jacquet and Shalika in $1990$. We analyze the local exterior square $L$-functions via exceptional poles and Bernstein and Zelevinsky derivatives. With this result, we show the equality of the local analytic $L$…
Let $\mathbf{G}$ be an unramified quasi-split unitary group over a p-adic field of odd residual characteristic. The goal of this paper is to describe the supercuspidal representations within certain L-packets of $\mathbf{G}$, which are classified by Arthur and Mok using the theory of endoscopy. The description is given in terms of the cuspidal types constructed by Bushnell-Kutzko and Stevens. As a starting example, we require the parameters of our packets to satisfy certain regularity conditions, such that these packets consist of very cuspidal representations in the sense of Adler and Reeder.…
The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig), is a rational function of the cardinality q of the residue field. The irreducible factors of this rational function are $q$ and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of Mark Reeder, and for the remaining exceptional cases this …
Bruhat-Tits buildings, representations of $p$-adic groups and Langlands correspondence
该研究将Bruhat-Tits理论相关的超尖点表示结果推广到任意$p$-进约化群表示，证明半单尖点支集的增强$L$-参数的Springer对应性质，并在特定条件下建立其与非奇异超尖点支集表示的局部Langlands对应，得出复合$L$-包的相关结论。
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。
In this paper, we extend Ginzburg-Rallis' integral representation for the exterior cube automorphic $L$-function of ${\rm GL}_6\times {\rm GL}_1$ to that of the quasi-split unitary similitude group ${\rm GU}_6$ and establish its analytic properties to determine the poles of this $L$-function. Furthermore, we introduce the automorphic induction for automorphic representations of ${\rm GU}_n$ and then show that the weak Langlands functorial lift for the automorphic induction exists for generic cuspidal automorphic representations. By using this automorphic induction, we give a conjectural criter…
We construct all cuspidal l-modular representations of a unitary group in three variables attached to an unramified extension of local fields of odd residual characteristic p with l\neq p. We describe the l-modular principal series and show that the supercuspidal support of an irreducible l-modular representation is unique up to conjugacy.
In this paper we characterize irreducible generic representations of $\SO_{2n+1}(k)$ where $k$ is a $p$-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of $\SO_{2n+1}({\Bbb A})$ (where ${\Bbb A}$ is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem);and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of $\SO_{2n+1}(k)$.
On the Local Langlands Functorial Transfer from $\text{SO}(5)$ to $\text{GL}(4)$
针对特征零的非阿基米德局部域F，研究GL(4,F)超尖点表示从SO(5,F)到GL(4,F)的局部朗兰兹函子转移，给出其等价判定条件，推进了局部朗兰兹纲领相关研究。
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。
Let $k$ be an algebraically closed field of characteristic $l\neq p$. We construct maximal simple cuspidal $k$-types of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$, when $F$ is a non-archimedean locally compact field of residual characteristic $p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$ is unique up to $\mathrm{M}'$-conjugation, when $F$ is either a finite field of characteristic $p$ or a non-archimedean locally compact field of residual characteristic $p$.
The local exterior square and Asai $L$-functions for $GL(n)$ in odd characteristic
本文研究了奇特征下 $GL_m(F)$ 不可约容许表示的局部外平方、Bump-Friedberg 及 Asai $L$-函数，通过积分表示理论与局部到全局论证，证明了它们在局部 Langlands 对应下等于相应的 Artin $L$-函数。
We calculate a $\mathrm{G}_2$-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group $\mathrm{E}_6$, and relate this period to the Ginzburg-Rallis period of cusp forms on $\mathrm{GL}_6$. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of $\mathrm{GL}_6$
Twisted Jacquet modules and induction from Speh representations
本文研究非阿基米德局部域上的扭曲雅克比模，证明其与斯佩表示诱导的相关性质，确定例外参数集，并给出辛二重覆盖及大特征有限域上的对应结果。
Finite period vectors and Gauss sums
本文通过显式计算与 Deligne–Kazhdan 理论，建立有限域及局部域上多种 gamma 因子与 Gauss 和、Artin 因子的关系，并刻画相关周期向量。
Periods and nonvanishing of central L-values for GL(2n)
本文证明Guo-Jacquet关于GL(2n)中心L值非消失性与GL(n,E)周期非消失性关系的猜想，并应用于超尖点表示，部分证实Prasad-Takloo-Bighash猜想。
On Swan exponents of symmetric and exterior square Galois representations
本文研究了局部非Archimedes域上Galois表示经张量运算后的Swan指数关系，在剩余特征奇偶不同时分别确定或界定了对称与外平方表示的Swan指数，并应用于局部Langlands对应。
On classical doubling method gamma factors for certain depth zero representations
本文研究了经典有限李型群不可约表示的倍增方法伽马因子，证明其乘法性，并通过Deligne-Lusztig数据给出非共轭对偶特征情况下的显式公式，同时将构造与Lapid-Rallis的局部构造联系起来。
A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations
本文研究了扭曲海丘代数及其与温特表示的克莱因瓶模型的联系，并展示了伯恩斯坦谱中点的同构性。
Local transfer for quasi-split classical groups and congruences mod l
本文研究拟分裂 p 进经典群上具有同构模 l 约化的两个尖点不可约 l 进表示，证明其局部转移的模 l 约化虽可能不同构但具有同构的超尖点支撑，并在非 SO(2) 情形下共享唯一公共泛型分量。
A two variable Rankin-Selberg integral for $\mathrm{GU}(2,2)$ and the degree 5 $L$-function of $\mathrm{GSp}_4$
本文提出了一种双变量 Rankin-Selberg 积分，用于表示 exterior square $L$-functions 的乘积，并作为 residue 得到了被二次特征扭曲的 $\mathrm{GSp}_4$ 5 次 $L$-函数的积分表示。
Using linear periods on the mirabolic subgroup of $GL(n,F)$, for $F$ a non archimedean local field, we give a list of the maximal Levi subgroups of $GL(n,F)$ which can distinguish a discrete series, and a generic representation. We also obtain the functional equation of the local exterior-square $L$-function of a generic representation of $GL(n,F)$ when $n=2m$ is even. Then we discuss the relation between Shalika models and local models for representations of $GL(2m,F)$. Finally we give a necessary and sufficient condition on the cuspidal representation $ρ$ for a generalised Steinberg represen…
On a refined local converse theorem for SO(4)
本文针对SO(4)的局部逆定理问题，通过引入额外的扭曲伽马因子，证明了广义超 cuspidal 表示的唯一性。
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