Algorithms for Representation Theory of Real Reductive Groups

Spinoriality of orthogonal representations of reductive groups

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.

Representation theory of disconnected reductive groups

We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic $0$ and those in characteristic $p$ (for groups defined over discrete valuation rings). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups.

Guide to the Atlas Software: Computational Representation Theory of Real Reductive Groups

This is an introduction to the Atlas of Lie Groups and Representations software, for computing representation and structure theory of real reductive groups. The user is led through the basic commands of the software, via numerous examples. See Algorithms for Representation Theory of Real Reductive Groups, arXiv number 0807.3093, for a more complete mathematical description of the algorithm

Wave Front Sets of Reductive Lie Group Representations II

In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups.

Some Infinite Dimensional Representations of Reductive Groups with Frobenius Maps

In this paper we construct certain irreducible infinite dimensional representations of algebraic groups with Frobenius maps. In particular, a few classical results of Steinberg and Deligne & Lusztig on complex representations of finite groups of Lie type are extended to reductive algebraic groups with Frobenius maps.

Translation of Dolbeault representations on reductive homogeneous spaces

We adapt techniques used in the study of the cubic Dirac operator on homogeneous reductive spaces to the Dolbeault operator on elliptic coadjoint orbits to prove that cohomologically induced representations have an infinitesimal character, that cohomological induction and Zuckerman translation functor commute and give a geometric interpretation of the Zuchkerman translation functor in this context.

On the tensor square of irreducible representations of reductive Lie superalgebras

For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the alternating or symmetric square representation is irreducible or decomposes into an irreducible representation and a trivial representation.

The Principal Representations of Reductive Algebraic Groups with Frobenius Maps

We introduce the principal representation category $\mathscr{O}({\bf G})$ of reductive algebraic groups with Frobenius maps and put forward a conjecture that this category is a highest weight category. When $\Bbbk$ is complex field $\mathbb{C}$, we provide some evidences of this conjecture. We also study certain kind of bound quiver algebras whose representations are related to the principal representation category $\mathscr{O}({\bf G})$ .

Representations of reductive normal algebraic monoids

The rational representation theory of a reductive normal algebraic monoid (with one-dimensional center) forms a highest weight category, in the sense of Cline, Parshall, and Scott. This is a fundamental fact about the representation theory of reductive normal algebraic monoids. We survey how this result was obtained, and treat some natural examples coming from classical groups.

On the representations of disconnected reductive groups over F_q

Let \hat G be the semidirect product of a connected reductive group G over F_q with a finite cyclic group generated by a quasisemisimple automorphism of G defined over F_q. In this paper we prove a conjecture of G. Malle concerning the extensions of unipotent representations of G(F_q) to \hat G(F_q).

Morita's Duality for Split Reductive Groups

In this paper, we extend the work in "Morita's Theory for the Symplectic Groups" to split reductive groups. We construct and study the holomorphic discrete series representation and the principal series representation of a split reductive group $\mathrm {G}$ over a $p$-adic field $F$ as well as a duality between certain sub-representations of these two representations.

Representations of reductive groups over finite rings

We discuss a cohomological construction of representations of a reductive group over the ring of power series over a finite field modulo the r-th power of the maximal ideal. The case r=1 goes back to Deligne and the author. The case where r is greater than 1 was given without proof in the author's work in 1977. Here we supply the proofs and give some further results.

Characteristic cycles and wave front cycles of representations of reductive Lie groups

Vogan and Barbasch-Vogan attach two similar invariants to representations of a reductive Lie group, one by an algebraic process, the other analytic. They conjectured that the two invariants determine each other in a definite manner. Here we prove the conjecture. Our arguments involve two finer invariants -- the characteristic cycles of representations -- which are interesting in their own right.

An Application of Twisted Character on Signature of a finite-dimensional Representation of a Real reductive Lie group

Let $G_\cpl$ be a connected complex reductive Lie group, and $G$ be a real form. Let $(π,V)$ be a finite-dimensional irreducible representation of $G$. Assume $(π,V)$ admits a $G$ invariant hermitian form. {In \cite{Signature-of-a-rep-of-reductive}, an analog of the Weyl dimension formula that instead computes the signature of the invariant form for finite dimensional representations of complex Lie groups is given. The goal of this paper is to give a short alternate proof as an application of a more general twisted character theory of \cite{Dirac-Index-and-Twisted-Characters}.

Cuspidal representations of reductive p-adic groups are relatively injective and projective

Cuspidal representations of a reductive p-adic group G over a field of characteristic different from p are relatively injective and projective with respect to extensions that split by a U-equivariant linear map for any subgroup U that is compact modulo the centre. The category of smooth representations over a field whose characteristic does not divide the pro-order of G is the product of the subcategories of cuspidal representations and of subrepresentations of direct sums of parabolically induced representations.

Topics on global analysis of manifolds and representation theory of reductive groups

Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect representation theoretic properties on function spaces, with focus on multiplicities and spectrum.

Frobenius映射下约化代数群在范畴$\mathscr{X}$中的复表示

Complex representations of reductive algebraic groups with Frobenius maps in the category $\mathscr{X}$

本文引入阿贝尔范畴$\mathscr{X}(\bf G)$研究带Frobenius映射的约化代数群的复表示，分类其单对象并证明其为最高权范畴。

Reductive Lie代数的模表示与相关组合范畴

Modular representation of Reductive Lie algebras and related combinatorial category

本文引入并研究了一个与Reductive Lie代数reduced enveloping algebras表示相关的组合范畴，该范畴与AJS范畴兼容，对研究Lusztig猜想具有重要意义。

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。

A short proof of the existence of supercuspidal representations for all reductive $p$-adic groups

Let $G$ be a reductive $p$-adic group. We give a short proof of the fact that $G$ always admits supercuspidal complex representations. This result has already been established by A. Kret using the Deligne-Lusztig theory of representations of finite groups of Lie type. Our argument is of a different nature and is self-contained. It is based on the Harish-Chandra theory of cusp forms and it ultimately relies on the existence of elliptic maximal tori in $G$.

The Plancherel decomposition for a reductive symmetric space II. Representation theory

We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I (math.RT/0107063). The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized principal series.

Wave Front Sets of Reductive Lie Group Representations III

Let $G$ be a real, reductive algebraic group, and let $X$ be a homogeneous space for $G$ with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of $L^2(X)$. Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on $X$. These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition.

Representations of reductive groups distinguished by symmetric subgroups

Let $G$ be a complex reductive group and $H=G^θ$ be its fixed point subgroup under a Galois involution $θ$. We show that any $H$-distinguished representation $π$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\neq0$) satisfies: 1) $π^θ\simeq\tildeπ$, where $\tildeπ$ is the contragredient representation and $π^θ$ is the twist of $π$ under $θ$. 2) $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.

A notion of rank for unitary representations of reductive groups based on Kirillov's orbit method

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple group is classical, we prove that the new theory is essentially equivalent to Howe's theory of N-rank.

Types and unitary representations of reductive p-adic groups

We prove that for every Bushnell-Kutzko type that satisfies a certain rigidity assumption, the equivalence of categories between the corresponding Bernstein component and the category of modules for the Hecke algebra of the type induces a bijection between irreducible unitary representations in the two categories. This is a generalization of the unitarity criterion of Barbasch and Moy for representations with Iwahori fixed vectors.

On the existence of admissible supersingular representations of $p$-adic reductive groups

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_p$, and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.

Demazure descent and representations of reductive groups

We introduce the notion of Demazure descent data on a triangulated category C and define the descent category for such data. We illustrate the definition by our basic example. Let G be a reductive algebraic group with a Borel subgroup B. Demazure functors form Demazure descent data on the derived category of Rep(B) and the descent category is equivalent to the derived category of Rep(G).

Signatures for finite-dimensional representations of real reductive Lie groups

We present a closed formula, analogous to the Weyl dimension formula, for the signature of an invariant Hermitian form on any finite-dimensional irreducible representation of a real reductive Lie group, assuming that such a form exists. The formula shows in a precise sense that the form must be very indefinite. For example, if an irreducible representation of $GL(n,R)$ admits an invariant form of signature $(p,q)$, then we show that $(p-q)^2 \le p+q$. The proof is an application of Kostant's computation of the kernel of the Dirac operator.

Characters and growth of admissible representations of reductive p-adic groups

We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.

Lifting representations of finite reductive groups: a character relation

Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\varepsilon$ of $\tilde{G}$ of finite order, let $G$ denote the connected part of the group of $\varepsilon$-fixed points. Then there exists a lifting from packets of representations of $G(k)$ to packets for $\tilde{G}(k)$. In the case of Deligne-Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.

Irreducible representations of product of real reductive groups

Let $G_1,G_2$ be real reductive groups and $(π,V)$ a smooth, irreducible, admissible representation of $G_1 \times G_2$. We prove that $(π,V)$ is the completed tensor product of $(π_i,V_i)$, $i=1,2$, where $(π_i,V_i)$ is a smooth,irreducible,admissible representation of $G_i$, $i=1,2$. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in [AG] and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair $H \subset G$ of real reductive groups is equivalent to the usual Gelfand property of the pair …

On Bernstein's presentation of Iwahori-Hecke algebras and representations of split reductive groups over non-Archimedean local fields

This article gives conceptual statements and proofs relating parabolic induction and Jacquet functors on split reductive groups over a non-Archimedean local field to the associated Iwahori-Hecke algebra as tensoring from and restricting to parabolic subalgebras. The main tool is Bernstein's presentation of the Iwahori-Hecke algebra.

Resolutions for representations of reductive p-adic groups via their buildings

Schneider-Stuhler and Vigneras have used cosheaves on the affine Bruhat-Tits building to construct natural finite type projective resolutions for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building. We establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein's result that certain subcategories of the category of representations are Serre subcategories. Furthermor…

Convexity properties of gradient maps associated to real reductive representations

Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson. A similar result is proved for the gradient map associated to the natural $G$ action on P(V). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real …

Infinite-dimensional reductive monoids associated to highest weight representations of Kac-Moody groups

Starting with a highest weight representation of a Kac-Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highes…

Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups II

The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations in terms of the Deligne-Langlands-Lusztig correspondence. Our formula generalizes the main result of arXiv:2112.14354, where this formula was obtained in the Iwahori-spherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the…

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。

On Imprimitive Representations of Finite Reductive Groups in Non-defining Characteristic

In this paper, we begin with the classification of Harish-Chandra imprimitive representations in non-defining characteristic. We recall the connection of this problem to certain generalizations of Iwahori-Hecke algebras and show that Harish-Chandra induction is compatible with the Morita equivalence by Bonnafé and Rouquier, thus reducing the classification problem to quasi-isolated blocks. Afterwards, we consider imprimitivity of unipotent representations of certain classical groups. In the case of general linear and unitary groups, our reduction methods then lead to results for arbitrary Lusz…

Polar orthogonal representations of real reductive algebraic groups

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.

p-adic 场上连通可约群的Gelfand-Graev表示的Iwahori分量

Iwahori component of the Gelfand--Graev representation for reductive groups

研究p-adic域上连通可约群的Gelfand-Graev表示的Iwahori分量，证明其与Hecke代数和符号表示的同构关系，扩展了Chan-Savin定理的应用范围。

Infinite-dimensional reductive monoids associated to highest weight representations of Kac-Moody groups

Starting with a highest weight representation of a Kac-Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac-Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J-irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highes…

Filtrations in Modular Representations of Reductive Lie Algebras

Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with $p$-character $χ$ of standard Levi-form associated with an index subset $I$ of simple roots. With aid of support variety theory we prove a theorem that a $U_χ(\ggg)$-module is projective if and only if it is a strong "tilting" module, i.e. admitting both $\cz_Q$- and $\cz^{w^I}_Q$-filtrations (to see Theorem \ref{THMFORINV}). Then by analogy of the arguments in \cite{…

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…

非连通约化群的Mackey公式

Mackey formula for disconnected reductive groups

本文证明了源自非连通约化群的李型有限群表示的Mackey公式。

Resolutions of tempered representations of reductive p-adic groups

Let G be a reductive group over a non-archimedean local field and let S(G) be its Schwartz algebra. We compare Ext-groups of tempered G-representations in several module categories: smooth G-representations, algebraic S(G)-modules, bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces which contains all admissible S(G)-modules. We simplify the proofs of known comparison theorems for these Ext-groups, due to Meyer and Schneider-Zink. Our method is based on the Bruhat-Tits building of G and on analytic properties of the Schneider-Stuhler resolutions.

The $\ell$-modular representation of reductive groups over finite local rings of length two

Let $\mathcal{O}_2$ and $\mathcal{O}'_2$ be two distinct finite local rings of length two with residue field of characteristic $p$. Let $\mathbb{G}(\mathcal{O}_2)$ and $\mathbb{G}(\mathcal{O}'_2)$, be the group of points of any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G} \times \mathbb{F}_q$. We prove that there exists an isomorphism of group algebra $K[\mathbb{G}(\mathcal{O}_2)] \cong K[\mathbb{G}(\mathcal{O}'_2)]$, where $K$ is a sufficiently large field of characteristic different from $p$.

Wavefront Sets of Unipotent Representations of Reductive $p$-adic Groups I

The wavefront set is a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Deligne-Langlands-Lusztig parameters. In particular, for the Iwahori-spherical representations with real infinitesimal character, we deduce that the algebraic wavefront set is a singleton, as conjectured by Moeglin and Waldspurger. As a corollary, we obtain an…

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。

Wave Front Sets of Reductive Lie Group Representations

If $G$ is a Lie group, $H\subset G$ is a closed subgroup, and $τ$ is a unitary representation of $H$, then the authors give a sufficient condition on $ξ\in i\mathfrak{g}^*$ to be in the wave front set of $\operatorname{Ind}_H^Gτ$. In the special case where $τ$ is the trivial representation, this result was conjectured by Howe. If $G$ is a real, reductive algebraic group and $π$ is a unitary representation of $G$ that is weakly contained in the regular representation, then the authors give a geometric description of $\operatorname{WF}(π)$ in terms of the direct integral decomposition of $π$ int…

复凝聚reductive p进群的有理性性质

Rationality properties of complex representations of reductive p-adic groups

研究复表示与代数闭包Qbar表示的有理性，证明椭圆表示可实现于Qbar当且仅当其中心字符取值于Qbar，并探讨复表示的自同构群作用。

On the analogy between real reductive groups and Cartan motion groups. II: Contraction of irreducible tempered representations

Attached to any reductive Lie group $G$ is a "Cartan motion group" $G_0$ $-$ a Lie group with the same dimension as $G$, but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of $G$ and the unitary irreducible representations of $G_0$, whose existence had been suggested by Mackey in the 1970s, has recently been described by the author. In the present notes, we use the existence of a family of groups interpolating between $G$ and $G_0$ to realize the bijection as a deformation: for every irreducible tempered representation $π$ of G, …

裂变群与 intertwining 运算符的规范模块

Standard modules and intertwining operators for reductive p-adic groups

本文研究了裂变群G的规范模块在Galois群作用下的稳定性，并给出了Levi子群表示的μ函数显式公式。

紧致表示在可约齐性空间上的平方可积性

Square integrability of regular representations on reductive homogeneous spaces

本文研究了可约齐性空间上$L^2(G/H)$作为$G$的单位亚表示的条件，并推导出不存在离散级数的结论。

Special Unipotent Arthur Packets For Real Reductive Groups

We compute special unipotent Arthur packets for real reductive groups in many cases. We list the cases that lead to incomplete answers, and in those cases, provide a suitable set of representations that could lead to a complete description of the special Arthur packet. In the process of achieving this goal we classify theta forms of a given even complex nilpotent orbit, and find methods to compute the associated varieties of irreducible group representations.