Complements on disconnected reductive groups

On unitary representations of disconnected real reductive groups

Let $G$ be the real reductive group and let $G_0$ be the identity component. Let us assume that the unitary dual $\hat{G_0}$ is known. In this paper (in Section 5) the unitary dual $\hat{G}$ is constructed. Automorphisms of $G_0$ generated by elements of $G$ are the main ingredient of the construction. If the automorphism is outer, one has to consider the corresponding intertwining operators $S$. Operators $S$ and their properties are analyzed in Section 4. Automorphisms of $g_0$ are closely related to automorphisms of $G_0$. They are investigated in Section 3. Automorphisms of so(4,4)$ are an…

非连通约化群：分类与表示

Disconnected Reductive Groups: Classification and Representations

本文对代数闭域上的非连通约化群按同构分类，得出固定维数和分支数时这类群数量有限的结论，还取得其结构与表示论的新成果。

On Unipotent Supports of Reductive Groups with a Disconnected Centre

Let $G$ be a simple algebraic group defined over a finite field of good characteristic, with associated Frobenius endomorphism $F$. In this article we extend an observation of Lusztig, (which gives a numerical relationship between an ordinary character of $G^F$ and its unipotent support), to the case where $Z(G)$ is disconnected. We then use this observation in some applications to the ordinary character theory of $G^F$.

有限域上非连通约化群表示的内窥

Endoscopy for representations of disconnected reductive groups over finite fields

本文基于 Lusztig 和 Yun 的工作，将有限域上连通约化群不可约表示的 Jordan 分解典范化，并推广到具有约化中性分量的光滑群概形，对有理级数与几何级数建立与内窥群幂幺表示之间的典范等价。

Character sheaves on disconnected groups, I

We begin the study of character sheaves on a not necessarily connected reductive group, extending the known theory for connected groups.

Character sheaves on disconnected groups, V

We continue the study of character sheaves on a not necessarily connected reductive group. We prove orthogonality formulas for certain characteristic functions.

Character sheaves on disconnected groups, VI

We define the notion of character sheaf on a possibly disconnected reductive group. We show that the restriction functor carries a character sheaf to a direct sum of character sheaves.

Character sheaves on disconnected groups, II

The main theme of this paper is establishing the "generalized Springer correspondence" in complete generality that is, for not necessarily connected reductive algebraic groups.

关于离散L-参数的精细局部朗兰兹猜想：对于准分裂不连通实可约群的内形式

On the refined local Langlands conjecture for discrete $L$-parameters of inner forms of quasi-split disconnected real reductive groups

本文针对准分裂不连通实可约群的内形式，构造了离散L-参数对应的不可约离散系列表示包，并证明了这些包满足能斯特字符恒等式，从而验证了精细局部朗兰兹对应猜想。

Character sheaves on disconnected groups, X

We classify the unipotent character sheaves on a fixed connected component of a reductive algebraic group under a mild hypothesis on the characteristic of the ground field.

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…

Character sheaves on disconnected groups, III

We continue the attempt to develop a theory of character sheaves on a not necessarily connected reductive algebraic group. In this paper we introduce and study the generalized Green functions.

Generic character sheaves on disconnected groups and character values

We relate a generic character sheaf on a disconnected reductive group with a character of a representation of the rational points of the group over a finite field extending a result known in the connected case.

本质上一元的超 cuspidal 表示的局部朗兰兹对应（对于不连通的半单群）

The local Langlands correspondence of essentially unipotent supercuspidal representations for disconnected reductive groups

本文在刚性内形式框架下构造了本质上一元的超 cuspidal 表示的局部朗兰兹对应，并证明了某种泛函性和兼容性，该结果比 [Sol20] 中的类似结果更强，因为后者不考虑内扭的刚性化。同时，本文将这种对应推广到不连通的半单群。

Character sheaves on disconnected groups, IV

Let G be a possibly disconnected reductive group over a finite field with Frobenius map F. The main result of this paper is that the characteristic functions af "admissible complexes" A on G such that F^*A is isomorphic to A form a basis of the space of functions on G^F that are constant on G^{0F}-conjugacy classes.

关于不连通群的模广义Springer对应

On the modular generalized Springer correspondence for disconnected groups

本文研究不连通复约化代数群上模广义Springer对应的构造问题，拓展了经典Springer对应理论。

关于非连通群的奥伯特和伯恩斯坦对合

The Aubert and Bernstein involutions for disconnected groups

研究将奥伯特和伯恩斯坦对偶性扩展到非连通约化\(p\)-adic群，证明相关函子性质，通过示例应用得到斯坦伯格表示定义、计算其特征并讨论扭曲内窥镜性质。

Character sheaves on disconnected groups, VIII

In this paper we continue the study of character sheaves on a reductive group G. To each subset of the set of simple reflections in the Weyl group we associate an algebra of the same kind as an Iwahori-Hecke algebra with unequal parameters in terms of parabolic character sheaves. We also prove a Mackey type formula for character sheaves. We define a duality operation for character sheaves. We also prove a quasi-rationality property for character sheaves.

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。

轨道方法与非连通实代数群 tempered 表示的特征公式

The Orbit Method and Character Formulas for Tempered representations of a Nonconnected Reductive Real Algebraic Group

本文提出了一种基于 Kirillov 公式的方法，用于参数化非连通实代数群的 tempered 表示特征，适用于线性连通群的最终基本参数。

On semisimple classes and semisimple characters in finite reductive groups

In this article, we study the elements with disconnected centralizer in the Brauer complex associated to a simple algebraic group G defined over a finite field with corresponding Frobenius map F and derive the number of F-stable semisimple classes of G with disconnected centralizer when the order of the fundamental group has prime order. We also discuss extendibility of semisimple characters to their inertia group in the full automorphism group. As a consequence, we prove that "twisted" and "untwisted" simple groups of type E_6 are "good" in defining characteristic, which is a contribution to …

Counting p'-characters in finite reductive groups

This article is concerned with the relative McKay conjecture for finite reductive groups. Let G be a connected reductive group defined over the finite field F_q of characteristic p>0 with corresponding Frobenius map F. We prove that if the F-coinvariants of the component group of the center of G has prime order and if p is a good prime for G, then the relative McKay conjecture holds for G at the prime p. In particular, this conjecture is true for G^F in defining characteristic for G a simple and simply-connected group of type B_n, C_n, E_6 and E_7. Our main tools are the theory of Gelfand-Grae…

The principal series of $p$-adic groups with disconnected centre

Let G be a split connected reductive group over a local non-archimedean field. We classify all irreducible complex G-representations in the principal series, irrespective of the (dis)connectedness of the centre of G. This leads to a local Langlands correspondence for principal series representations, which satisfies all expected properties. We also prove that the ABPS conjecture about the geometric structure of Bernstein components is valid throughout the principal series of G.

On Lusztig's conjecture for connected and disconnected exceptional groups

Lusztig conjectured that the almost characters of a finite reductive group are up to a scalar the same as the characteristic functions of the rational character sheaves defined on the corresponding algebraic group. We propose in this paper to verify this conjecture for the uniform almost characters of the Suzuki and the Ree groups and to show that a conjecture of the same type holds for the disconnected case of type B_2. We then state a conjecture on the possible values of the uniform almost characters on unipotent elements in the disconnected case of type F_4.

关于不连通环面离散自守表示的重数公式

On the multiplicity formula for discrete automorphic representations of disconnected tori

研究不连通环面离散自守表示，通过对其局部朗兰兹对应进行内在重新解释，建立了自守重数公式。

p进约化群的表示

Representations of $p$-adic reductive groups

本书介绍p进约化群的表示理论，从基础内容出发，经多章研究，包括建立伯恩斯坦分解定理等，最终证明朗兰兹分类定理，前四章在更一般框架下展开，还有相关回顾与附录内容。

扩展幂幺特征的合理性

Rationality of extended unipotent characters

本文确定了在 Frobenius 映射下作为不连通约化群不动点产生的有限约化群的幂幺特征的合理性，证明中结合了 Deligne--Lusztig 簇的 $\ell$-adic 上同调群中的特征实现与块理论考虑。

The 2-parameter Green functions for 8-dimensional spin groups

The 2-parameter Green functions occur as a crucial ingredient in the character formula for Lusztig induction in finite reductive groups. Still, very little is known about these functions, in particular in the case of groups arsing from algebraic groups with disconnected centre. We collect some basic properties and then apply these, together with some explicit computations, to determine all 2-parameter Green functions for 8-dimensional spin groups in odd characteristic, whose centre is disconnected of order~4.

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。

Dimension Data, Local and Global Conjugacy in Reductive Groups

Let G be a connected reductive group (over $\mathbb{C}$) and H a connected semisimple subgroup. The dimension data of H (realative to its given embedding in G) is the collection of the numbers $\{{\rm dim} V^{H}\}$, where V runs over all the finite dimensional representations of G. By a Theorem of Larsen-Pink ([L-P90]), the dimension data determines H up to isomorphism, and if G = GL (n) even up to conjugacy. Professor Langlands raised the question as to whether the strong (conjugacy) result holds for arbitrary G. In this paper We provided the following (negative) answer: If H is simple of typ…

特征簇的态射

Morphisms of character varieties

本文将特征簇态射的有限性结果从域推广到任意局部诺特概形，回答Dat的问题，并利用Bruhat-Tits理论建立离散赋值环上约化群积分模型的新结果。

On the endomorphism algebra of generalised Gelfand-Graev representations

Let $G$ be a connected reductive algebraic group defined over the finite field $\F_q$, where $q$ is a power of a good prime for $G$, and let $F$ denote the corresponding Frobenius endomorphism, so that $G^F$ is a finite reductive group. Let $u \in G^F$ be a unipotent element and let $Γ_u$ be the associated generalised Gelfand-Graev representation of $G^F$. Under the assumption that $G$ has a connected centre, we show that the dimension of the endomorphism algebra of $Γ_u$ is a polynomial in $q$, with degree given by $\dim C_G(u)$. When the centre of $G$ is disconnected, it is impossible, in ge…

On equivariant bijections relative to the defining characteristic

This paper is a contribution to the general program introduced by Isaacs, Malle and Navarro to prove the McKay conjecture in the representation theory of finite groups. We develop new methods for dealing with simple groups of Lie type in the defining characteristic case. Using a general argument based on the representation theory of connected reductive groups with disconnected center, we show that the inductive McKay condition holds if the Schur multiplier of the simple group has order 2. As a consequence, the simple groups \Orth_{2m+1}(p^n) and PSp_{2m}(p^n) are "good" for p>2 and the simple …

On pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups

Let $F$ be locally compact field with residue characteristic $p$, and $\mathbf{G}$ a connected reductive $F$-group. Let $\mathcal{U}$ be a pro-$p$ Iwahori subgroup of $G = \mathbf{G}(F)$. Fix a commutative ring $R$. If $π$ is a smooth $R[G]$-representation, the space of invariants $π^{\mathcal{U}}$ is a right module over the Hecke algebra $\mathcal{H}$ of $\mathcal{U}$ in $G$. Let $P$ be a parabolic subgroup of $G$ with a Levi decomposition $P = MN$ adapted to $\mathcal{U}$. We complement previous investigation of Ollivier-Vignéras on the relation between taking $\mathcal{U}$-invariants and va…

Lectures on the Geometry and Modular Representation Theory of Algebraic Groups

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

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。

Vanishing theorems for representation homology and the derived cotangent complex

Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := π_*{\mathcal O}[{\rm DRep}_G(X)] $ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, l…