On depth in the local Langlands correspondence for tori

Contragredient representations and characterizing the local Langlands correspondence

We consider the question: what is the contragredient in terms of L-homomorphisms? We conjecture that it corresponds to the Chevalley automorphism of the L-group, and prove this in the case of real groups. The proof uses a characterization of the local Langlands correspondence over R. We also consider the related notion of Hermitian dual, in the case of GL(n,R).

On the local Langlands correspondence for non-tempered representations

Let G be a reductive p-adic group. We study how a local Langlands correspondence for irreducible tempered G-representations can be extended to a local Langlands correspondence for all irreducible smooth representations of G. We prove that, under a natural condition involving compatibility with unramified twists, this is possible in a canonical way. To this end we introduce analytic R-groups associated to non-tempered essentially square-integrable representations of Levi subgroups of G. We establish the basic properties of these new R-groups, which generalize Knapp--Stein R-groups.

Transfer relations in essentially tame local Langlands correspondence

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this identity to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results. (i…

Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with appendix by Jessica Fintzen)

Let $E/F$ be a finite and Galois extension of non-archimedean local fields. Let $G$ be a connected reductive group defined over $E$ and let $M: = \mathfrak{R}_{E/F}\, G$ be the reductive group over $F$ obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from $G(E)$ to $M(F)$. We obtain a depth-comparison formula for Weil-restricted groups.

Towards an explicit local Jacquet-Langlands correspondence beyond the cuspidal case

We show how the modular representation theory of inner forms of general linear groups over a non-Archimedean local field can be brought to bear on the complex theory in a remarkable way. Let F be a non-Archimedean locally compact field of residue characteristic p, and let G be an inner form of the general linear group GL(n,F). We consider the problem of describing explicitly the local Jacquet--Langlands correspondence between the complex discrete series representations of G and GL(n,F), in terms of type theory. We show that the congruence properties of the local Jacquet--Langlands corresponden…

The $\ell$-modular local Langlands correspondence and local factors

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell\neq p$ be a prime number, and $\mathrm{W}_F$ the Weil group of $F$. We classify the indecomposable $\mathrm{W}_F$-semisimple Deligne $\overline{\mathbb{F}_\ell}$-representations in terms of the irreducible $\overline{\mathbb{F}_\ell}$-representations of $\mathrm{W}_F$, and extend constructions of Artin-Deligne local factors to this setting. Finally, we define a variant of the $\ell$-modular local Langlands correspondence which satisfies a preservation of local factors statement for generic representations.

有限朗兰兹对应

Finite Langlands correspondence

本文为有限域上的连通约化群构造了有限朗兰兹对应，并探讨其与范畴局部朗兰兹对应的关系。

局部Langlands对应中的最低$K$-型

Lowest $K$-types in the local Langlands correspondence

本文研究实约化群表示的局部Langlands参数化与最低K-型集合的联系，利用细化对应（含分量群覆盖）建立自然关联，并简化描述该细化参数化。

Tempered modules in exotic Deligne-Langlands correspondence

The main purpose of this paper is to identify the tempered modules for the affine Hecke algebra of type $C_n^{(1)}$ with arbitrary, non-root of unity, unequal parameters, in the exotic Deligne-Langlands correspondence in the sense of Kato. Our classification has several applications to the Weyl group module structure of the tempered Hecke algebra modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sign representation of the Weyl group. This last combinatorial classification was expected from the work of Heckman-Opdam and Sloote…

SL_n的驯顺局部朗兰兹对应在有限域上的一个约化

A Reduction over finite fields of the tame local Langlands correspondence for SLn

本研究证明了Vogan关于SL_n的猜想，构造了有限域上SL_n不可约表示到p进域上SL_n驯顺朗兰兹参数惯性等价类的满射，给出其纤维参数化并验证与局部朗兰兹对应的相容性，将GL_n的相关结果推广至SL_n。

Formality in the Deligne-Langlands correspondence

The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra $\mathcal{H}^{\text{aff}}$ by certain perverse sheaves. We show that this can be lifted to an equivalence of triangulated categories. More precisely, we construct for each central character $χ$ of $\mathcal{H}^{\text{aff}}$ an equivalence of triangulated categories between a perfect derived category of dg-modules $D_{\text{perf}}(\mathcal{H}^{\text{aff}}/(\text{ker}(χ)) - \text{dgMod})$ and the triangulated category generated by the corresponding perverse sheaves. The main step in this con…

Twisted geometric Langlands correspondence for a torus

Let T be a split torus over local or global function field. The theory of Brylinski-Deligne gives rise to the metaplectic central extensions of T by a finite cyclic group. The representation theory of these metaplectic tori has been developped to some extent in the works of M. Weissman, G. Savin, W. T. Gan, P. McNamara and others. In this paper we propose a geometrization of this theory in the framework of the geometric Langlands program (in the everywhere nonramified case).

Parity of the Langlands parameters of conjugate self-dual representations of GL(n) and the local Jacquet-Langlands correspondence

We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of GL(n) over a non-archimedean local field by means of the local Jacquet-Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.

Rectifiers and the local Langlands correspondence: the unramified case

We generalize the rectifier of Bushnell and Henniart, which occurs in the local Langlands correspondence for $GL_n(K)$, to certain Langlands parameters for unramified connected reductive groups.

The local Langlands correspondence for GSp_4 over local function fields

We prove the local Langlands correspondence for GSp_4(F), where F is a non-archimedean local field of positive characteristic with residue characteristic > 2.

A Real Groups Construction of the Tame Local Langlands Correspondence for PGSp(4,F)

In this paper, we continue the work of the first author and give a new construction of the tame local Langlands correspondence for PGSp(4,F), where F is a p-adic field, that is analogous to the construction of the local Langlands correspondence for real groups.

An Approach to the Characterization of the Local Langlands Correspondence

In this paper, we give a method for characterizing the local Langlands conjectures in the vein of Scholze's alternate proof of the local Langlands conjecture for $\mathrm{GL}_n$. More specifically, we show that if a local Langlands correspondence satisfies a Scholze--Shin equation, as in the paper of Scholze and Shin, in addition to the usual desiderata expected of such a correspondence then these properties uniquely characterize the correspondence.

Geometric Langlands correspondence for SL(2), PGL(2) over the pair of pants

We establish the Geometric Langlands correspondence for rank one groups over the projective line with three points of tame ramification.

Local Langlands Correspondence for Even Orthogonal Groups via Theta Lifts

Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover,we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe-Gan for quasi-split even orthogonal groups.

On the modified mod p local Langlands correspondence for GL_2(Q_{\ell})

We give an explicit description of the modified mod p local Langlands correspondence for GL_2(Q_{\ell}) of Emerton-Helm, where p is an odd prime different from \ell.

n-Dimensional global correspondences of Langlands

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…

n-Dimensional global correspondences of Langlands over singular schemes (II)

A rather complete phenomenology of the singularities is developed according to a new algebraic point of view in the frame of Langlands global correspondences. That is to say,a process of: -singularizations and versal deformations of these, -singularizations and monodromies of these, is envisaged on all the sections of sheaves of differentiable (bi)functions on (bi)linear algebraic (semi)groups constituting the n-dimensional representations of the global Weil groups. To get the searched holomorphic and cuspidal representations,it is necessary to consider: -the resolutions of the singularities a…

$\infty$-范畴化广义Langlands对应II：Langlands纲领形式体系

$\infty$-Categorical Generalized Langlands Correspondence II: Langlands Program Formalism

本文通过三种不同函子性扰动推广Langlands纲领，并在局部化设定中利用几何化广义Bernstein中心构造从光滑表示侧到Weil侧的对应。

The Jacquet Langlands correspondence via twisted descent

The existence of the well-known Jacquet-Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg-Rallis-Soudry to a new setting. As a consequence, we recover the classical Jacquet-Langlands correspondence for PGL(2) via a new explicit construction.

简单超尖点表示的局部 Jacquet-Langlands 对应

Local Jacquet-Langlands correspondences for simple supercuspidal representations

本文利用类型理论刻画简单超尖点表示的局部 Jacquet-Langlands 对应，并证明其自同态类在该对应下不变。

A New Construction for the Tame Local Langlands Correspondence for GL(n,F), n a prime

In this paper, we give a new construction of the tame local Langlands correspondence for GL(n,F), n a prime, where F is a p-adic field. In the tame case, supercuspidal representations of GL(n,F) are parameterized by characters of elliptic tori, but the local Langlands correspondence is unnatural because it involves a twist by some character of the torus. Taking the cue from real groups, supercuspidal representations should instead be parameterized by characters of covers of tori. Over the reals, Harish-Chandra described the characters of discrete series restricted to compact tori. They are nat…

Local Jacquet-Langlands correspondence for regular supercuspidal representations

We prove that Kaletha's local Langlands correspondence for regular supercuspidal representations gives the classical local Jacquet--Langlands correspondence due to Deligne--Kazhdan--Vigneras and Badulescu. As in a former joint paper with Oi, where a similar result is proved for the local Langlands correspondence for the general linear group, the key ingredients in our proof are the work of Bushnell--Henniart explicitly describing the local Jacquet--Langlands correspondence for essentially tame supercuspidal representations and its reinterpretation due to Tam in terms of Langlands--Shelstad's \…

Typical representations, parabolic induction and the inertial local Langlands correspondence

We prove a result which provides a link between the decomposition of parabolically induced representations and the Bushnell--Kutzko theory of typical representations. As an application, we show that there exists a well-defined inertial Langlands correspondence which respects the monodromy action of L-parameters, under some standard conjectures regarding the local Langlands correspondence. To allow for potential applications of this inertial Langlands correspondence, we also provide a complete construction of the set of typical representations, giving a parametrization of these in terms of the …

The local Langlands correspondence for inner forms of $SL_n$

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $SL_n (F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for $SL_n (F)$ enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of $SL_n (F)$. An analogous result is shown in the archimedean case. To settle the case where F has positive characteristic, we employ the method of close fields. We prove that this me…

Geometrization of the local Langlands correspondence: an overview

This article is an overview of the geometrization conjecture for the local Langlands correspondence formulated by the author.

Depth preserving property of the local Langlands correspondence for non-quasi-split unitary groups

In this paper, we extend our result on a depth preserving property of the local Langlands correspondence for quasi-split unitary groups (arXiv:1804.10901) to non-quasi-split unitary groups by using the local theta correspondence. The key ingredients are a depth preserving property of the local theta correspondence proved by Pan and a description of the local theta correspondence via the local Langlands correspondence established by Gan--Ichino. To combine them, we compare splittings for metaplectic covers of unitary groups constructed by Kudla with those constructed by Pan.

Shtukas for reductive groups and Langlands correspondence for function fields

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.

Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations

In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).

Local Theta correspondence of Tempered Representations and Langlands parameters

In this paper, we give an explicit determination of the theta lifting for symplectic-orthogonal and unitary dual pairs over a nonarchimedean field $F$ of characteristic $0$. We determine when theta lifts of tempered representations are nonzero, and determine the theta lifts in terms of the local Langlands correspondence.

The inertial Jacquet-Langlands correspondence

We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet-Langlands correspondence. Along the way, we prove a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes.

Some endoscopic properties of the essentially tame Jacquet-Langlands correspondence

Let $F$ be a a non-Archimedean local field of characteristic 0 and $G$ be an inner form of the general linear group $G^*=\mathrm{GL}_n$ over $F$. We show that the rectifying character appearing in the essentially tame Jacquet-Langlands correspondence of Bushnell and Henniart for $G$ and $G^*$ can be factorized into a product of some special characters, called zeta-data in this paper, in the theory of endoscopy of Langlands and Shelstad. As a consequence, the essentially tame local Langlands correspondence for $G$ can be described using admissible embeddings of L-tori.

p-adic Jacquet-Langlands correspondence and patching

We describe two candidates for a local p-adic Jacquet-Langlands correspondence and using patching we show that they are in fact isomorphic. We then study locally algebraic vectors of the given correspondence.

Local Langlands correspondence for regular supercuspidal representations of GL(n)

In this paper, we prove the coincidence of Kaletha's recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris--Taylor's one in the case of general linear groups. The keys are Bushnell--Henniart's essentially tame local Langlands correspondence and Tam's result on Bushnell--Henniart's rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha's and Tam's $χ$-data.

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。

On the local Langlands correspondence for quasi-split even orthogonal groups

In this paper, we highlight and state precisely the local Langlands correspondence for quasi-split O_{2n} established by Arthur. We give two applications: Prasad's conjecture and Gross--Prasad conjecture for O_{n}. Also, we discuss the Arthur conjecture for O_{2n}, and establish the Arthur multiplicity formula for O_{2n}.

Notes on modular representations of $p$-adic groups, and the Langlands correspondence

These are expanded notes of some lectures given by the author for a workshop held at the Indian Statistical Institute, Bangalore in June, 2010, giving an exposition on the modular representations of finite groups of Lie type and $p$-adic groups, and the modular Langlands correspondence. These notes give an overview of the subject with several examples.

关于局部朗兰兹对应的几何化

On the geometrization of the local Langlands correspondence

本文综述了 Fargues-Scholze 对局部朗兰兹对应几何化的研究。

Geometric Langlands Correspondence Near Opers

We extend the categorical geometric Langlands correspondence from the locus of opers in the stack of de Rham local systems on a smooth projective algebraic curve to the formal neighborhood of opers (for any semi-simple complex algebraic group).

G2的显式局部朗兰兹对应 II：特征标公式与稳定性

The explicit Local Langlands Correspondence for $G_2$ II: character formulas and stability

本文通过写出G2表示的特征标公式，并利用L-包络的稳定性唯一确定Aubert-Xu构造的局部朗兰兹对应，从而证明其唯一性。

论 $p$-进局部朗兰兹对应的几何化

On the geometrization of the $p$-adic local Langlands correspondence

本文综述了作者将 ${\mathrm{GL}}_2({\mathbf Q}_p)$ 的 $p$-进局部朗兰兹对应的一部分进行几何化的相关结果。

Explicit local Jacquet-Langlands correspondence: the non-dyadic wild case

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with $p\neq 2$. Let $n$ be a power of $p$ and let $G$ be an inner form of the general linear group $\text{\rm GL}_n(F)$. We give a transparent parametrization of the irreducible, totally ramified, cuspidal representations of $G$ of parametric degree $n$. We show that the parametrization is respected by the Jacquet-Langlands correspondence, relative to any other inner form. This expresses the Jacquet-Langlands correspondence for such representations within a single, compact formula.

Affine Deligne-Lusztig varieties of higher level and the local Langlands correspondence for $GL_2$

In the present article we define coverings of affine Deligne-Lusztig varieties attached to a connected reductive group over a local field of characteristic $p > 0$. In the case of $\GL_2$, the unramified part of the local Langlands correspondence is realized in the $\ell$-adic cohomology of these varieties. We show this by giving a detailed comparison with the realization of local Langlands via cuspidal types by Bushnell-Henniart. All proofs are purely local.

雅科夫-兰加尔斯对应关系的一个算术不变量

An Arithmetic Invariant of the Jacquet-Langlands correspondence

本文研究雅科夫-兰加尔斯对应关系下局部-整体兼容性及Tamagawa测度的保持，提出模块密度概念并证明其在主算术群上的保持性。

刚性内扭曲的局部theta对应与朗兰兹参数

Local theta correspondences and Langlands parameters for rigid inner twists

本文提出用刚性内扭曲的局部朗兰兹对应描述局部theta对应（含四元数对偶对对应）的猜想，并验证其在特定情况下成立。

Local Langlands correspondence in rigid families

We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. The novelty is we allow non-classical points, possibly non-étale over weight space. More precisely we interpolate the local Langlands correspondence for GL(n) across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Scholze from his new approach to the local Langlands conjecture.

一个弗兰克滤波器的 functorial 细化及其对于自守形式空间的雅各布-朗格兰斯对应关系

A Functorial Refinement of the Franke Filtration and the Jacquet--Langlands Correspondence for Spaces of Automorphic Forms

本文通过扩展弗兰克滤波器的functorial细化，建立了自守形式空间中一般线性群及其内形式之间的雅各布-朗格兰斯对应关系，扩展了Badulescu的对应关系并包含完整functorial提升。

等晶体与刚性局部朗兰兹对应的极限

Isocrystals and limits of rigid local Langlands correspondences

本文研究非阿基米德局部域上连通约化群的刚性精细化局部朗兰兹对应，通过构造中心扩张投射系取极限的方法，将中心连通情形的结果推广到一般连通约化群，并证明该对应与等晶体对应等价，推广了已有$p$进情形的工作。

n-Dimensional geometric-shifted global bilinear correspondences of Langlands on mixed motives III

This third paper,devoted to global correspondences of Langlands,bears more particularly on geometric-shifted bilinear correspondences on mixed (bi)motives generated under the action of the products,right by left,of differential elliptic operators.The mathematical frame,underlying these correspondences,deals with the categories of the Suslin-Voevodsky mixed (bi)motives and of the Chow mixed (bi)motives which are both in one-to-one correspondence with the functional representation spaces of the shifted algebraic bilinear semigroups.A bilinear holomorphic and supercuspidal spectral representation…

A congruence property of the local Langlands correspondence

Let $F$ be a non-Archimedean local field of residual characteristic $p$, and $\ell$ a prime number, $\ell \neq p$. We consider the Langlands correspondence, between irreducible, $n$-dimensional, smooth representations of the Weil group of $F$ and irreducible cuspidal representations of $\text{\rm GL}_n(F)$. We use an explicit description of the correspondence from an earlier paper, and otherwise entirely elementary methods, to show that it respects the relationship of congruence modulo $\ell$. The $\ell$-modular correspondence thereby becomes as effective as the complex one.