Rohit Joshi, Steven Spallone
Let G be a connected reductive group over a field of characteristic zero, and consider an orthogonal representation of G. We give a simple criterion for whether the representation lifts to the spin group, in terms of the highest weights of the irreducible constituents of the representation.
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We determine which orthogonal representations V of GL(n,q) lift to the double cover Pin(V ) of the orthogonal group O(V ). We cover all n and prime powers q, except for (n; q) =(3,4).
We prove a generalization of Harish-Chandra's character orthogonality relations for discrete series to arbitrary Harish-Chandra modules for real reductive Lie groups. This result is an analogue of a conjecture by Kazhdan for $\mathfrak p$-adic reductive groups proved by Bezrukavnikov, and Schneider and Stuhler.
We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras.
In this paper, we propose two maximal one-to-one sub-relations $\underlineθ, \overlineθ$ of the Howe correspondence $Θ$ for a finite reductive dual pair consisting of a symplectic group and an orthogonal group. Moreover, we show that both the correspondences $\underlineθ$ and $\overlineθ$ are extensions beyond the stable range of the $η$-correspondence defined by Gurevich-Howe.
Let $(G,G')$ be a reductive dual pair of a symplectic group and an orthogonal group over a finite field of odd characteristic. The Howe correspondence establishes a correspondence between a subset of irreducible characters of $G$ and a subset of irreducible characters of $G'$. The Lusztig correspondence is a bijection between the Lusztig series indexed by the conjugacy class of a semisimple element $s$ in the connected component $(G^*)^0$ of the dual group of $G$ and the set of irreducible unipotent characters of the centralizer of $s$ in $G^*$. In this paper, we prove the commutativity (up to…
Stability of Branching Multiplicities for Orthogonal Gelfand Pairs
本文提出一个框架，研究正交Gelfand对中分支乘法在无穷小特征变化下的行为，通过引入fences划分参数空间，证明乘法函数在每个区域内局部恒定，并统一解释了多个现象。
We continue the study of character sheaves on a not necessarily connected reductive group. We prove orthogonality formulas for certain characteristic functions.
Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of $\mathcal{N}_{\mathfrak{g}^*}$ into smooth locally closed subvarieties (which are indexed by the unipotent classes in the corresponding group over complex numbers) and gives explicit results in types $A$, $C$ and $D$. We discuss type $B$ in this note.
The main theorem of Galois theory states that there are no finite group-subgroup pairs with the same invariants. On the other hand, if we consider complex linear reductive groups instead of finite groups, the analogous statement is no longer true: There exist counterexample group-subgroup pairs with the same invariants. However, it's possible to classify all these counterexamples for certain types of groups. In [16], we provided the classification for connected complex irreducible groups, and, in this paper, for connected complex orthogonal groups, i.e., groups that preserve some non-degenerat…
We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive $p$-adic groups due to Hakim and Murnaghan, as well as the analogous theory for finite reductive groups due to Lusztig. We apply our results to study the representations of ${\rm GL}_n(F)$, with $n$ odd and $F$ a nonarchimedean local field, that are distinguished with respect to an orthogonal group in $n$ variables. In particular, we determine precisely when a supercuspidal representation is distinguished with respect to an orthogonal group and, if so, that the space of distingui…
We construct via generators and relations a generalized Weil representation for the split orthogonal group $\ot_q(2n,2n)$ over a finite field of $q$ elements. Besides, we give an initial decomposition of the representation found. We also show that the constructed representation is equal to the restriction of the Weil representation to $\ot_q(2n,2n)$ for the reductive dual pair $(\Sp_2(\mathbb{F}_q),\ot_q(2n,2n))$ and that the initial decomposition is the same as the decomposition with respect to the action of $\Sp_2(\mathbb{F}_q)$.
Let $G = GL_N$ over an algebraically closed field of odd characteristic, and $θ$ an involutive automorphism on $G$ such that $H = (G^θ)^0$ is isomorphic to $SO_N$. Then $G^{ιθ} = \{ g \in G \mid θ(g) = g^{-1} \}$ is regarded as a symmetric space $G/G^θ$. Let $G^{ιθ}_{uni}$ be the set of unipotent elements in $G^{ιθ}$. $H$ acts on $G^{ιθ}_{uni}$ by the conjugation. As an analogue of the generalized Springer correspondence in the case of reductive groups, we establish in this paper the generalized Springer correspondence between $H$-orbits in $G^{ιθ}_{uni}$ and irreducible representations of var…
The determination of scalars involved in Lusztig's conjecture for finite reductive groups $G(F_q)$ was achieved by Waldspurger in the case of symplectic groups or orthogonal groups, under the condition that $p,q$ are large enough. Here $p$ is the characteristic of the finite field $F_q$. In this paper, we determine the scalars in the case of symplectic groups with $p = 2$, by applying the theory of symmetric spaces over a finite field due to Kawanaka and Lusztig. We also obtain a partial result in the case of special orthogonal groups with $p = 2$.
Symmetry breaking operator for the reductive dual pair $(\mathrm{U}_l,\mathrm{U}_{l'})$
本文研究酉群对偶对在 Weil 表示中的对称破缺算子及其 Weyl 符号，通过两种正交投影计算相互匹配，重新证明重数至多为一并恢复 Howe 对应完整列表。
We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric restriction map agrees with Lusztig's nonabelian Fourier transform for unipotent representations of the finite reductive quotient. This is inspired by recent works of Lusztig on the almost unipotent characters of $p$-adic groups and of Moeglin and Waldspurger on the elliptic Fourier transform of odd orthogonal groups.
The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint $γ$-factor of its $L$-parameter. In this paper, we prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which combined with Arthur's work on the local Langlands correspondence implies the conjecture in full generality.
On the Muić conjecture--the irreducibility of the big theta lift
本文证明了非阿基米德局部域上辛群与偶正交群之间离散系列表示的大theta提升若非零则不可约，从而解决Muić猜想，并进一步完整刻画了调和表示theta提升的不可约性条件。
Let $π, π'$ be irreducible tempered representations of an affine Hecke algebra H with positive parameters. We compute the higher extension groups $Ext_H^n (π,π')$ explicitly in terms of the representations of analytic R-groups corresponding to $π$ and $π'$. The result has immediate applications to the computation of the Euler-Poincaré pairing $EP(π,π')$, the alternating sum of the dimensions of the Ext-groups. The resulting formula for $EP(π,π')$ is equal to Arthur's formula for the elliptic pairing of tempered characters in the setting of reductive p-adic groups. Our proof applies equally wel…
A higher-dimensional Chevalley restriction theorem for orthogonal groups
本文证明了Chen和Ngô猜想中正交群情形的高维Chevalley限制定理，并在特征0下得到交换概形范畴商的整性与正规性，进而推出迹恒等式与Pfaffian的乘法性质。
We use the theory of reductive dual pairs due to Howe to obtain explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra, and of the twisted Yangians corresponding to the symplectic and orthogonal Lie algebras.
Characteristic Classes Of Representations Of Lie Groups
本文研究了李群表示的特征类，通过多项式函数计算陈类和Stiefel-Whitney类的限制，基于最高权重的表达。
We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, inclu…
We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].
Symmetry breaking operators for strongly spherical reductive pairs
本文为强球面对约化对构造了对称破缺算子，证明其全纯依赖并张成交织算子空间，推广至重数一情形，并应用于证明Gross-Prasad猜想早期版本及Shintani函数维数下界。
Let $(G,H)$ be a reductive spherical pair and $P\subset H$ a parabolic subgroup such that $(G,P)$ is spherical. The triples $(G,H,P)$ with this property are called multiplicity free systems and they are classified in this paper. Denote by $π^{H}_μ=\mathrm{ind}_{P}^{H}μ$ the Borel-Weil realization of the irreducible $H$-representation of highest weight $μ\in P^{+}_{H}$ and consider the induced representation $\mathrm{ind}_{P}^{G}χ_μ=\mathrm{ind}_{H}^{G}π^{H}_μ$, a multiplicity free induced representation. Some properties of the spectrum of the multiplicity free induced representations are discu…
Given a supercuspidal representation $σ$ of a parabolic subgroup $P$ of reductive group $G$, we discover a universal hierarchical structure of reducibility of the parabolic induction $Ind^G_P(σ)$, i.e. always irreducible from some Levi-level up. As its applications, we provide a new simple proof of the generic irreducibility property of parabolic induction, and prove Clozel's finiteness conjecture of special exponents under some conditions. Indeed, those conditions are predicted by two conjectures of Shahidi which in some sense are proved for classical groups by Arthur in his monumental book--…
Orthogonal and symplectic orbits in the affine flag variety of type A
本文研究A型仿射旗簇中正交群或辛群的轨道，通过建立双陪集与扭仿射对合之间的显式双射，首次给出这些轨道的组合描述。
Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for $G = \mathrm{GL}_n$, we consider polynomial representations of $G_r T$ for an arbitrary closed reductive subgroup scheme $G \subseteq \mathrm{GL}_n$ and a maximal torus $T$ of $G$ in positive characteristic. We give sufficient conditions on $G$ making a classification of simple polynomial $G_r T$-modules similar to the case $G = \mathrm{GL}_n$ possible and apply this to recover the corresponding result for $\mathrm{GL}_n$ with a different proof, extendi…
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。
A $GL_d$-pseudocharacter is a function from a group $Γ$ to a ring $k$ satisfying polynomial relations which make it "look like" the character of a representation. When $k$ is an algebraically closed field, Taylor proved that $GL_d$-pseudocharacters of $Γ$ are the same as degree-$d$ characters of $Γ$ with values in $k$, hence are in bijection with equivalence classes of semisimple representations $Γ\rightarrow GL_d(k)$. Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group $H$ over an algebraically closed field $k$ of characteristic 0 and for any grou…
When are the natural embeddings of classical invariant rings pure?
本文研究正特征域上经典不变量环自然嵌入的纯性，证明若该嵌入为纯嵌入，则不变量环正则或群线性约化。
On sharpness in Local Converse Theorems for classical groups and $G_2$
该研究针对非阿基米德局部域上的分裂约化群，证明了经典群与$G_2$的局部逆定理的尖锐性结果，给出了相关改进的猜想，并提供了反例。
Given a connected simply connected semisimple group G and a connected spherical subgroup K we determine the generators of the extended weight monoid of G/K, based on the homogeneous spherical datum of G/K. Let H be a reductive subgroup of G and let P be a parabolic subgroup of H for which G/P is spherical. A triple (G,H,P) with this property is called multiplicity free system and we determine the generators of the extended weight monoid of G/P explicitly in the cases where (G,H) is strictly indecomposable. The extended weight monoid of G/P describes the induction from H to G of an irreducible …
An Explicit Construction of CAP Representations of $Sp_{4n}(\mathbb A)$ associated to Non-trivial Automorphic Characters of Orthogonal Groups $O_{2n}(\mathbb A)$
本文利用 theta 对应和 Howe 对偶性，通过构造正交群的非平凡自守特征并提升，显式构造了 $Sp_{4n}$ 的一族 CAP 表示，并刻画了其局部分量。
Let $G$ be a symplectic or special orthogonal group, let $H$ be a connected reductive subgroup of $G$, and let $X$ be a flag variety of $G$. We classify all triples $(G,H,X)$ such that the natural action of $H$ on $X$ is spherical. For each of these triples, we determine the restrictions to $H$ of all irreducible representations of $G$ realized in spaces of sections of homogeneous line bundles on $X$.
On the Divisibility of Degrees of Representations of Lie Algebras
本文研究约化李代数不可约表示次数不被给定正整数整除的自然密度，证明在一般线性李代数情形下该密度随维数增大指数衰减至零，并将结果推广到单李代数、李群及自对偶与正交表示。
Assume $G$ is a connected reductive algebraic group defined over $\bar{\mathbb{F}_p}$ such that $p$ is good prime for $G$. Furthermore we assume that $Z(G)$ is connected and $G/Z(G)$ is simple of classical type. Let $F$ be a Frobenius endomorphism of $G$ admitting an $\mathbb{F}_q$-rational structure $G^F$. This paper is one of a series whose overall goal is to compute explicitly the multiplicity $< D_0745664 {G^F}(Γ_u),χ>$ where: $χ$ is an irreducible character of $G^F$, $D_{G^F}(Γ_u)$ is the Alvis--Curtis dual of a generalised Gelfand--Graev representation of $G^F$ and $u \in G^F$ is contain…
Splitting the difference: Computations of the Reynolds operator in classical invariant theory
本文提出了一种计算经典不变理论中雷诺算子的算法，适用于通用线性群、特殊线性群、正交群和辛群的经典表示。
We adapt the conjectural local Langlands parameterization to split metaplectic groups over local fields. When $\tilde G$ is a central extension of a split connected reductive group over a local field (arising from the framework of Brylinski and Deligne), we construct a dual group $\mathbf{\tilde G}^\vee$ and an L-group ${}^L \mathbf{\tilde G}^\vee$ as group schemes over ${\mathbb Z}$. Such a construction leads to a definition of Weil-Deligne parameters (Langlands parameters) with values in this L-group, and to a conjectural parameterization of the irreducible genuine representations of $\tilde…
Opérateurs d'entrelacement et algèbres de Hecke avec paramètres d'un groupe réductif $p$-adique - le cas des groupes classiques
本文研究了p-adic简约群的缠结算子与Hecke代数的参数化问题，通过计算诱导项目生成元的终等代数，证明其同构于Hecke代数与有限群代数的半直接积。
Let $G$ be a real or $p$-adic reductive group. We consider the tempered dual of $G$, and its connected components. For real groups, Wassermann proved in 1987, by noncommutative-geometric methods, that each connected component has a simple geometric structure which encodes the reducibility of induced representations. For $p$-adic groups, each connected component of the tempered dual comes with a compact torus equipped with a finite group action, and we prove that a version of Wassermann's theorem holds true under a certain geometric assumption on the structure of stabilizers for that action. We…
Comparing the Kirwan and noncommutative resolutions of quotient varieties
本文比较了商变种的Kirwan解析与非交换解析，证明非交换解析的导出范畴可嵌入到Kirwan解析的导出范畴中，并扩展为半正交分解。
Let G be a reductive complex Lie group with Lie algebra g. We call a subgroup H of G {\bf cramped} if there is an integer b(G,H) such that each finite dimensional representation of G has a non-trivial invariant subspace of dimension less than b(G,H). We show that a subgroup is cramped if and only if the moment map from T^*(K/L) to k^* is surjective, where K and L are compact forms of G and H. We will use this in conjunction with sufficient conditions for crampedness given by Willenbring and Zuckerman (2004) to prove a geometric lemma on the intersections between adjoint orbits and Killing orth…
Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $Φ$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the representation space. The map $Φ$ identifies the field $Q(G)$ of rational functions on $G$ with an algebraic extension of the field $Q(\g)$ of rational functions on $\g$. For the spin representation of $\on{Spin}(V)$ the map $Φ$ essentially coincides with the classical Cayley transform. In general, properties of $Φ$ are established and these properties are applied to deal…
Matchings of theta lifts associated to Non-trivial Automorphic Characters of Odd Orthogonal Groups
本文推广 Snitz 关于 $\widetilde{SL}_2$ 全局 theta 提升同构的结果，研究辛群或 metaplectic 群情形下，满足局部必要条件的二次空间与非平凡自守二次特征何时给出匹配的提升。
We introduce the `Fourier transform' F_C on the isotropic cone C associated to an indefinite quadratic form of signature (n_1,n_2) on R^n (n=n_1+n_2: even). This transform is in some sense the unique and natural unitary operator on L^2(C), as is the case with the Euclidean Fourier transform. Inspired by recent developments of algebraic representation theory of reductive groups, we shed new light on classical analysis on the one hand, and give the global formulas for the L^2-model of the minimal representation of the simple Lie group G=O(n_1+1,n_2+1) on the other hand. The transform F_C expands…
On the complementary Arthur representations and unitary dual for p-adic classical groups
本文针对特征零的非阿基米德局部域上连通约化群的酉对偶结构猜想，给出辛群和分裂奇特殊正交群互补 Arthur 表示的刻画，得到\(\mathrm{GL}_N\)（\(N = 2,3\)）不可约自对偶尖点自守表示局部分量的限制。
For a pair $(G,G')=(O(n+1,1), O(n,1))$ of reductive groups, we investigate intertwining operators (symmetry breaking operators) between principal series representations $I_δ(V,λ)$ of $G$, and $J_ε(W,ν)$ of the subgroup $G'$. The representations are parametrized by finite-dimensional representations $V,W$ of $O(n)$ respectively of $O(n-1)$, characters $δ$, $\varepsilon$ of O(1), and $λ, ν\in C$. The multiplicty [V:W] of W occurring in the restriction $V|_{O(n-1)}$ is either 0 or 1. If $[V:W] \ne 0$ then we construct a holomorphic family of symmetry breaking operators and prove that dim $Hom_{G'…
Let $k$ be a field, let $G$ be a reductive algebraic group over $k$, and let $V$ be a linear representation of $G$. Geometric invariant theory involves the study of the $k$-algebra of $G$-invariant polynomials on $V$, and the relation between these invariants and the $G$-orbits on $V$, usually under the hypothesis that the base field $k$ is algebraically closed. In favorable cases, one can determine the geometric quotient $V//G = Spec(Sym(V^*))^G$ and can identify certain fibers of the morphism $V \rightarrow V/G$ with certain $G$-orbits on $V$. In this paper, we study the analogous problem wh…
Distinction of the Steinberg representation with respect to a symmetric pair
本文将 Steinberg 表示实现为调和上链空间，研究其关于对称对固定子群的区分空间维数，给出可达上界并应用于一般线性群与特殊正交子群完成分类。
Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is the power of an odd prime, and let $\mathrm{GSp}(2n, \mathbb{F}_q)$ and $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ denote the symplectic and orthogonal groups of similitudes over $\mathbb{F}_q$, respectively. We prove that every real-valued irreducible character of $\mathrm{GSp}(2n, \mathbb{F}_q)$ or $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if $\boldsymbol{G}$ is a classical connected…
For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ ξ^P_λ: Rep^\mathbb{C}_{λ-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{λ-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(λ)$ of G with highest weight $λ$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\leq k\leq n$ (with the choice of $V(λ) $ to be the defining representation $V(ω_1) $ in $\…
Let $F$ be a non-Archimedean local field of characteristic $0$, let $G$ be the group of $F$-rational points of a connected reductive group defined over $F$ and let $G'$ be the group of $F$-rational points of its quasi-split inner form. Given standard modules $I(τ,ν)$ and $I(τ',ν')$ for $G$ and $G'$ respectively with $τ'$ a generic tempered representation, such that the Harish-Chandra's $μ$-functions of a representation in the supercuspidal support of $τ$ and of a generic essentially square-integral representation in some Jacquet module of $τ'$ agree (after a suitable identification of the unde…
We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $G$ be a reductive group over a number field $F$ which admits discrete series representations at infinity. Let $^{L}G=\hat G \rtimes \mathrm{Gal}(\bar F/F)$ be the associated $L$-group and $r:{}^L G\to \mathrm{GL}(d,\mathbb{C})$ a continuous homomorphism which is irreducible and does not factor through $\mathrm{Gal}(\bar F/F)$. The families under consideration consist of discrete automorphic representations of $G(\mathbb{A}_F)$ of given weight and level a…
Partial character tables for $\mathbb{Z}_\ell$-spetses
本文为单连通 $\mathbb{Z}_\ell$-spets 提出了计算单幂特征标和主块特征标值的公式，扩展了 Lie 型群的 Curtis-Schewe 型公式，并验证了块正交性及相关猜想。
Let g be a semi-simple Lie algebra and let h be a reductive subalgebra of maximal rank in g. Given any irreducible representation of g, consider its tensor product with the spin representation associated to the orthogonal complement of h in g. Gross, Kostant, Ramond, and Sternberg recently proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of h, called a multiplet. Kostant then constructed a formal h-equivariant Dirac operator on such product representation…
Intertwining Algebras and Affine Hecke Algebras for Finite Central Extensions of Classical $p$-adic Groups with Application to Metaplectic Groups
本文研究经典 p-adic 群有限中心扩张的交织代数，证明其为仿射 Hecke 代数与扭曲有限群代数的扭曲半直积，并应用于 metaplectic 群，证明其真表示范畴与相应特殊正交群表示范畴的等价。
An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "$F$-zips with a $G$-structure", for an arbitrary reductive linear algebraic group $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F…
Derived Satake morphisms for $p$-small weights in characteristic $p$
本文在特征 $p$ 下，针对 $p$-小且充分正则的权，利用 Heyer 左伴随证明了复形的正交分裂，由此构造了导出 Satake 态射，推广了 Ronchetti 的模 $p$ Satake 同态，并给出了标准抛物子群的部分结果。
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