A geometric Jacquet functor

Multiplicity free Jacquet modules

Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let P=MU be the corresponding parabolic subgroup. Let J denote the Jacquet functor from representations of G to representations of M (i.e. the functor of coinvariants w.r.t. U). In this paper we prove that J is a multiplicity free functor, i.e. dim Hom(J(π),ρ)<= 1, for any irreducible representations πof G and ρof M. To do that we adapt the classical method of Ge…

An adjunction formula for the Emerton-Jacquet functor

The Emerton-Jacquet functor is a tool for studying locally analytic representations of p-adic Lie groups. It provides a way to access the theory of p-adic automorphic forms. Here we give an adjunction formula for the Emerton-Jacquet functor, relating it directly to locally analytic inductions, under a strict hypothesis that we call non-critical. We also further study the relationship to socles of principal series in the non-critical setting.

Towards the Derived Jacquet-Emerton Module Functor

Let $G$ be a $p$-adic Lie group associated to a connected reductive group over $\mathbb{Q}_{p}$. Let $P$ be a parabolic subgroup of $G$ and let $M$ be a Levi quotient of $P$. In this paper, we define a $δ$-functor $H^{\star}J_{P}$ from the category of admissible locally analytic $G$-representations to the category of essentially admissible locally analytic $M$-representations that extends the Jacquet-Emerton module functor $J_{P}$ defined by Emerton.

Generalized Jacquet modules of parabolic induction

In this paper we study a generalization of the Jacquet module of a parabolic induction and construct a filtration on it. The successive quotient of the filtration is written by using the twisting functor.

光滑Casselman-Jacquet函子的稳定性

Stability of the smooth Casselman-Jacquet functor

本文研究实约化群的光滑Casselman-Jacquet子模与商函子，证明了其正合性、满射性、整体化结果以及Jacquet子空间交的稳定性，并应用于建立中等增长光滑Fréchet表示的实Bernstein-Zelevinsky滤过。

Emerton's Jacquet functors for non-Borel parabolic subgroups

This paper studies Emerton's Jacquet module functor for locally analytic representations of p-adic reductive groups. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for the derived subgroup of M gives rise to essentially admissible locally analytic representations of the torus Z(M), which have a natural interpretation in terms of rigid geometry. We use this to extend Emerton's representation-theoretic construction of eigenvarieties by constructing eigenvarieties interpolating automorphic representations whose local component…

Affine Jacquet functors and Harish-Chandra categories

We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan [KK] on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman [LZ]. The latter provides an extension of the works of Rocha-Caridi, Wallach [RW] and Deodhar, Gabber, Kac [DGK] on block decomposition…

p-adic局部解析表示的Jacquet-Langlands函子

A Jacquet-Langlands functor for $p$-adic locally analytic representations

本文研究了无限层局部Shimura品种的局部解析理论，证明了不同周期sheaves在无限层的局部解析向量与p-adic李群作用无关，并展示了Jacquet-Langlands函子与局部解析向量及李代数中心字符的兼容性。

$p$-adic sheaves on classifying stacks, and the $p$-adic Jacquet-Langlands correspondence

We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze's basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results. Our key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the …

Geometric realization of special cases of local Langlands and Jacquet-Langlands correspondences

Let F be a non-Archimedean local field and let E be an unramified extension of F of degree n>1. To each sufficiently generic multiplicative character of E (the details are explained in the body of the paper) one can associate an irreducible n-dimensional representation of the Weil group W_F of F, which corresponds to an irreducible supercuspidal representation π of GL_n(F) via the local Langlands correspondence. In turn, via the Jacquet-Langlands correspondence, π corresponds to an irreducible representation ρ of the multiplicative group of the central division algebra over F with invariant 1/…

一般线性群上翻译函子与雅凯函子的比较

Comparing Translation and Jaquet functor over general linear groups

本文基于Chan和Wong构造的GL(n,ℂ)哈里什-钱德拉模块到A型分次赫克代数模块的函子，证明其关联了实侧翻译函子与p-adic侧雅凯函子。

关于 Jacquet-Rallis 相对迹公式的几何方面

On the geometric side of the Jacquet-Rallis relative trace formula

本文研究了 Jacquet-Rallis 相对迹公式的几何侧，全局计算了一般线性群上的几何项并证明其由正则轨道积分描述，局部证明了正则轨道积分与酉群半单轨道积分可比较。

Le transfert singulier pour la formule des traces de Jacquet-Rallis

The relative trace formula of Jacquet-Rallis (for unitary groups or general linear groups) is an identity between periods of automorphic representations and geometric distributions. In this paper, we prove the transfer between all geometric terms of unitary groups and those of linear groups. We also show that all geometric terms are in the weak closure of local regular semi-simple orbital integrals. We mention an application to the Gan-Gross-Prasad conjecture for unitary groups.

Geometric approach to the local Jacquet-Langlands correspondence

In this paper, we give a purely geometric approach to the local Jacquet-Langlands correspondence for GL(n) over a p-adic field, under the assumption that the invariant of the division algebra is 1/n. We use the l-adic etale cohomology of the Drinfeld tower to construct the correspondence at the level of the Grothendieck groups with rational coefficients. Moreover, assuming that n is prime, we prove that this correspondence preserves irreducible representations. This gives a purely local proof of the local Jacquet-Langlands correspondence in this case. We need neither a global automorphic techn…

Les formules des traces relatives de Jacquet-Rallis grossières

We establish the coarse relative trace formulae of Jacquet-Rallis for linear and unitary groups. Both formulae are of the form: a sum of spectral distributions equals a sum of geometric distributions. In order to obtain the spectral decompositions we introduce new truncation operators and we investigate their properties. On the geometric side, by means of the Cayley transform, the decompositions are derived from a procedure of descent to the tangent spaces for which the formulae are known thanks to our previous work.

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.

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。

对称空间的跟踪公式。Guo-Jacquet情况

Une formule des traces pour les espaces symétriques. Le cas de Guo-Jacquet

本文提出了一种对称空间的跟踪公式，用于研究自守谱和L函数特殊值之间的联系，通过谱分布与几何分布的恒等式，结合正则半简单数据和非正则数据的处理方法。

Categorification of Clifford algebra via geometric induction and restriction

We use geometric parabolic induction functors and the adjoint functors for the supergroups Osp(2m+1,2n) (where m and n vary) to categorify the action of the infinite-dimensional Clifford algebra on the Fock space of semi-infinite forms.

抛物几何艾森斯坦级数与常数项函子

Parabolic geometric Eisenstein series and constant term functors

该研究证明惠特克层抛物限制与几何卡塞尔曼-沙利卡等价下表示限制的相容性，建立几何艾森斯坦级数函子的赫克结构并推广相关结果，还通过Koszul对偶联系两类艾森斯坦级数函子，概述几何朗兰兹函子与常数项函子可交换的证明。

几何 Eisenstein 级数 I：有限性定理

Geometric Eisenstein series I: finiteness theorems

本文在 Fargues–Fontaine 曲线的丛栈上建立几何 Eisenstein 级数与常数项函子理论，证明最优有限性及 Bernstein 第二伴随性的几何形式，并实现 Bun_G 层范畴的尖点–Eisenstein 分解。

Betti几何朗兰兹纲领中的自同构粘合函子

Automorphic gluing functor in Betti Geometric Langlands

本文研究曲线结点退化中的幂零层自同构范畴，通过标记点鼓出射影直线解释仿射Hecke算子，并构造连接结点曲线与光滑化自同构范畴的粘合函子。

A geometric Schur functor

We give geometric descriptions of the category C_k(n,d) of rational polynomial representations of GL_n over a field k of degree d for d less than or equal to n, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category C_k(n,d) and the Schur functor.

Graded and Geometric Parabolic Induction for Category $\mathcal{O}$

We prove that the parabolic induction functor on BGG-category $\mathcal{O}$ associated to a complex reductive Lie algebra is gradable, that is, lifts to graded category $\mathcal{O}$ as constructed by Beilinson-Ginzburg-Soergel. Graded category $\mathcal{O}$ is equivalent to a category of stratified mixed Tate motives on a corresponding flag variety as recently defined by Soergel-Wendt. The graded version of parabolic induction is induced by a geometric parabolic induction functor we construct on the level of stratified mixed Tate motives. We also describe the effect of parabolic induction on …

量子Betti几何Langlands函子

Quantum Betti geometric Langlands functor

通过Whittaker系数构造Betti框架下的量子几何Langlands函子，并证明其与2-傅里叶-穆卡伊等价性兼容。

通过纯粹性构建保守的几何函子

Conservative geometric functors via purity

本文通过纯粹性概念，提出判断几何函子联合保守性的准则，并应用于环谱序列极限的两个特定情境。

Geometric Satake, Springer correspondence, and small representations II

For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the fi…

Suzuki functor at the critical level

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings i…

A geometric version of BGP reflection functors

Quiver Grassmannians and quiver flags are natural generalisations of usual Grassmannians and flags. They arise in the study of quiver representations and Hall algebras. In general, they are projective varieties which are neither smooth nor irreducible. We use a scheme theoretic approach to calculate their tangent space and to obtain a dimension estimate similar to one of Reineke. Using this we can show that if there is a generic representation, then these varieties are smooth and irreducible. If we additionally have a counting polynomial we deduce that their Euler characteristic is positive an…

GSp₄的Whittaker函子与Bessel函子

Whittaker and Bessel functors for GSp_4

本研究定义了GSp₄的Whittaker函子并研究其性质，同时探究对应Siegel抛物子群的类似函子及其与Bessel、Waldspurger模型的关联，还定义Waldspurger范畴并证明Waldspurger模型重数一结果的几何版本。

分次gentle代数的Koszul对偶的几何实现

A geometric realization of Koszul duality for graded gentle algebras

本文证明了同调光滑分次gentle代数的Koszul函子可由几何模型中的半旋转实现，并以此证明了涉及该函子的交维数公式。

Geometric theta-lifting for the dual pair GSp_{2n}, GSO_{2m}

Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Consider the dual pair H=GSO_{2m}, G=GSp_{2n} over X, where H splits over an etale two-sheeted covering of X. Write Bun_G and Bun_H for the stacks of G-torsors and H-torsors on X. We show that for m\le n (respectively, for m>n) the theta-lifting functor from D(Bun_H) to D(Bun_G) (respectively, from D(Bun_G) to D(Bun_H)) commutes with Hecke functors with respect to a morphism of the corresponding L-groups involving the SL_2 of Arthur. So, they realize the geometric Langlands functoriality for the corresp…

对偶对SO_{2m}, Sp_{2n}的几何theta提升

Geometric theta-lifting for the dual pair SO_{2m}, Sp_{2n}

本研究在特征大于2的代数闭域光滑射影曲线上，针对对偶对SO_{2m}, Sp_{2n}定义几何theta提升函子，刻画其与Hecke算子的关系，在两种特殊情形下验证几何Langlands函子性，并借助局部结果（Rallis定理几何类比）完成证明。

Theta对应与Whittaker函子的相容性

Compatibility of the Theta correspondence with the Whittaker functors

本研究证明了三类经典李群对的整体几何theta提升函子与Whittaker规范化相容，即theta提升函子与目标群Whittaker函子的复合同构于源群的Whittaker函子，推进了几何表示论中Theta对应理论的发展。

Non-vanishing of geometric Whittaker coefficients for reductive groups

We prove that cuspidal automorphic D-modules have non-vanishing Whittaker coefficients, generalizing known results in the geometric Langlands program from GL_n to general reductive groups. The key tool is a microlocal interpretation of Whittaker coefficients. We establish various exactness properties in the geometric Langlands context that may be of independent interest. Specifically, we show Hecke functors are t-exact on the category of tempered D-modules, strengthening a classical result of Gaitsgory (with different hypotheses) for GL_n. We also show that Whittaker coefficient functors are t…

The geometry of unitary 2-representations of finite groups and their 2-characters

Motivated by topological quantum field theory, we investigate the geometric aspects of unitary 2-representations of finite groups on 2-Hilbert spaces, and their 2-characters. We show how the basic ideas of geometric quantization are `categorified' in this context: just as representations of groups correspond to equivariant line bundles, 2-representations of groups correspond to equivariant gerbes. We also show how the 2-character of a 2-representation can be made functorial with respect to morphisms of 2-representations. Under the geometric correspondence, the 2-character of a 2-representation…