Rationality properties of unipotent representations

Descents of unipotent representations of finite unitary groups

Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent representations of unitary groups over finite fields. We give the first descents of unipotent representations explicitly, which are unipotent as well. Our results include both the Bessel case and Fourier-Jacobi case, which are related via theta correspondence.

Unipotent classes and special Weyl group representations

We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group.

Dirac cohomology of unipotent representations of Sp(2n,R) and U(p,q)

In this paper we study the problem of computing the Dirac cohomology of the special unipotent representations of the real groups Sp(2n,R) and U(p,q).

Unipotent Representations Attached to the Principal Nilpotent Orbit

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations.

旋量群的真特殊幂单表示

Genuine special unipotent representations of spin groups

本文确定了实与四元数旋量群的全部真特殊幂单表示，证明其均可酉化，并指出复旋量群无此类表示。

On Extending Unipotent Representations to their Stabilizers

We show that irreducible unipotent representations of split Levi subgroups of finite groups of Lie type extend to their stabilizers inside the normalizer of the given Levi subgroup. For this purpose, we extend the multiplicity-freeness theorem from regular embeddings to arbitrary isotypies.

Representations of unipotent reduction for SO(2n+1), II: endoscopy

For the groups SO(2n+1,F), where F is a p-adic field, we consider the tempered irr{é}ducible representations of unipotent reduction. Lusztig has contructed and parametrized these representations. We prove that they satisfy the expected endoscopic identities which determine the parametrization.

Unipotent Representations and Microlocalization

We develop a theory of microlocalization for Harish-Chandra modules, adapting a construction of Losev (\cite{Losev2011}). We explore the applications of this theory to unipotent representations of real reductive groups. For complex groups, we deduce a formula for the $K$-multiplicities of unipotent representations attached to a nilpotent orbit $\OO$, proving an old conjecture of Vogan (\cite{Vogan1991}) in a large family of cases.

Upper Triangularity for Unipotent Representations

Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary representation of $G$ can be constructed through a sequence of well-understood operations from a finite set of building blocks, called the unipotent representations. These representations are `attached' (in a certain mysterious sense) to the nilpotent orbits of $G$ on the dual space of its Lie algebra. Inside this finite set is a still smaller set, consisting of the u…

Descents of unipotent cuspidal representations of finite classical groups

Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent cuspidal representations of orthogonal and symplectic groups over finite fields.

Formal degrees of unipotent discrete series representations and the exotic Fourier transform

We introduce a notion of elliptic fake degrees for unipotent elliptic representations of a semisimple p-adic group. We conjecture, and verify in some cases, that the relation between the formal degrees of unipotent discrete series representations of a semisimple p-adic group and the elliptic fake degrees is given by the exotic Fourier transform matrix introduced by Lusztig in the study of representations of finite groups of Lie type.

Parametrizing unipotent representations

In a recent paper I defined a new basis for the Grothendieck group of unipotent representations of an almost simple Chevalley group over a finite field. The definition for classical types was different from that for exceptional types. In this paper we reconcile the two definitions.

The Grothendieck group of unipotent representations: a new basis

Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q).

Unipotent Representations and Quantum Induction

In this paper, we construct certain unipotent representations for the real orthogonal group and the metaplectic group in the sense of Vogan. Our construction is based on quantum induction which involves the compositions of even number of theta correspondences. In particular, our results imply that there are irreducible unitary representations attached to each special nilpotent orbit.

例外 Richardson 轨道的幂幺表示

Unipotent Representations of Exceptional Richardson Orbits

本文研究复例外 Richardson 轨道的特殊幂幺表示，并据此验证了 Achar 和 Sommers 提出的相关猜想。

On the characters of unipotent representations of a semisimple p-adic group

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let V be a unipotent representation of G(K) (for example, an Iwahori-spherical representation). We calculate the character of V at compact very regular elements of G(K) at compact very regular elements of G(K).

Extremal unipotent representations for the finite Howe correspondence

We study the Howe correspondence for unipotent representations of irreducible dual pairs $(G',G)=(\text{U}_m(\mathbb{F}_q),\text{U}_n(\mathbb{F}_q))$ and $(G',G)=(\text{Sp}_{2m}(\mathbb{F}_q),\text{O}^ε_{2n}(\mathbb{F}_q))$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements ($q$ odd) and $ε=\pm 1$. We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image of $π$ under the correpondence of a unipotent representation $π$ of $G$.

Représentations de réduction unipotente pour SO(2n+1), III: exemples de fronts d'onde

Let G be a group SO(2n+1) defined over a p-adic field. We compute the wave front set of the anti-tempered irreducible representations of G(F) which are of unipotent reduction. It is the orthogonal orbit dual to the symplectic orbit appearing in the Arthur's parametrization of the representation.

Non-unipotent representations and categorical centres

Let G be a connected reductive group defined over a finite field F_q. We give a parametrization of the irreducible representations of G(F_q) in terms of (twisted) categorical centres of various monoidal categories associated to G. (Results of this type were known earlier for unipotent representations. Analogous results for the parametrization of character sheaves on G were also known earlier.)

Irreducible representations of the group of unipotent matrices of order $4$ over integers

We study a coarse moduli space of irreducible representations of the group of unipotent matrices of order $\mathbb{4}$ over the ring of integers which have finite weight. All such representations are known to be monomial. To describe a coarse moduli space of such representations, we need to study pairs of subgroups and their characters, which induce non-isomorphic irreducible representations. We obtain a full classification of such pairs and, respectively, a coarse moduli space.

幂幺超尖点表示的波前集

The Wavefront Sets of Unipotent Supercuspidal Representations

针对内型分裂的半单$p$进群，研究计算了Lusztig幂幺表示范畴中其不可约超尖点表示的代数与典范非分歧波前集。

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…

Unipotent Ideals for Spin and Exceptional Groups

In the monograph arXiv:2108.03453, we define the notion of a unipotent representation of a complex reductive group. The representations we define include, as a proper subset, all special unipotent representations in the sense of Barbasch-Vogan and form the (conjectural) building blocks of the unitary dual. In arXiv:2108.03453 we provide combinatorial formulas for the infinitesimal characters of all unipotent representations of linear classical groups. In this paper, we establish analogous formulas for spin and exceptional groups, thus completing the determination of the infinitesimal character…

单幂表示：将 q 变为 -q

Unipotent representations: changing q to -q

本文针对有限域上 Chevalley 群（Weyl 群最长元素中心）构造单幂表示的对合，使次数多项式在符号意义下通过 q 变 -q 相互关联。

Classification of unipotent representations of simple p-adic groups,II

Let G(K) be the group of K-rational points of a connected adjoint simple algebraic group defined over a non-archimedean local field K. In this paper we classify the unipotent representations of G(K) in terms of the geometry of the Langlands dual group. (This was known earlier in the special case where G(K) is an inner form of a split group.) We also determine which representations are tempered or square integrable.

Categorical actions on unipotent representations I. Finite unitary groups

Using Harish-Chandra induction and restriction, we construct a categorical action of a Kac-Moody algebra on the category of unipotent representations of finite unitary groups in non-defining characteristic. We show that the decategorified representation is naturally isomorphic to a direct sum of level 2 Fock spaces. From our construction we deduce that the Harish-Chandra branching graph coincide with the crystal graph of these Fock spaces, solving a recent conjecture of Gerber-Hiss-Jacon. We also obtain derived equivalences between blocks, yielding Broué's abelian defect groups conjecture for …

Categorical actions on unipotent representations of finite classical groups

We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given by the authors. Then, we extend this construction to finite reductive groups of types B or C, in non-defining characteristic. We show that the decategorified representation is isomorphic to a direct sum of level 2 Fock spaces. We deduce that the Harish-Chandra branching graph coincides with the crystal graph of these Fock spaces. We also obtain derived equivalences between blocks, yielding Broue's abelian defect group conjecture for unipoten…

On formal degrees of unipotent representations

Let G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda conjectured that the formal degree of a square-integrable G-representation $π$ can be expressed in terms of the adjoint $γ$-factor of the enhanced L-parameter of $π$. A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations. We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $γ$-factors.

附加于抛物子群的酉表示：阿贝尔幂零根基的情形

Unitary representations attached to parabolic subgroups: the case of abelian unipotent radical

本文研究了在几何参数空间源于 Levi 子群作用于抛物子代数阿贝尔幂零根基的情形下，对 Lusztig 幂零表示范畴中具有整数无穷小特征的酉表示进行分类，并将其组织成微局部 Arthur 包，作为验证相关酉表示猜想描述的测试用例。

Representations of unipotent reduction for SO(2n+1), I: an involution

We consider a group SO(2n+1) over a p-adic field, and tempered irreducible representations of this group, of unipotent reduction. We use the construction due to Lusztig of these representations. In an old paper with Moeglin, we have defined an involution in the complex vector space generated by those representations which are elliptic. It is strongly related to another involution defined by Lusztig for finite groups. We give a new definition of our involution and we prove it commutes, in some sense, with Jacquet functor.

幂单表示：将 q 变为 -q，II

Unipotent representations: changing q to -q, II

本文研究有限域上 Chevalley 群中，将 q 变为 -q 对半单元素处幂单表示特征值多项式的影响。

A型单线性李群的特殊幂幺表示

Special unipotent representations of simple linear Lie groups of type A

本文针对A型单线性李群（实、复、四元数特殊线性群及特殊酉群），完整分类了Arthur与Barbasch-Vogan意义下的特殊幂幺表示，并证明这些表示均可酉化。

Generic Representation Theory of the Unipotent Upper Triangular Groups

It is generally believed (and for the most part is probably true) that Lie theory, in contrast to the characteristic zero case, is insufficient to tackle the representation theory of algebraic groups over prime characteristic fields. However, in this paper we show that, for a large and important class of unipotent algebraic groups (namely the unipotent upper triangular groups $U_n$), and under a certain hypothesis relating the characteristic $p$ to both $n$ and the dimension $d$ of a representation (specifically, $p \geq \text{max}(n,2d)$), Lie theory is completely sufficient to determine the …

On the definition of unipotent representations

Let G be a connected reductive group over a finite field F_q. A map from irreducible representations of G(F_q) to semisimple classes in the dual group was defined by a cohomological method by Deligne and the author in 1976. Here we show that when q is large enough this map has an alternative (non-cohomological) definition.

Explicit Hilbert spaces for certain unipotent representations II

We construct an explicit realization of a minimal representation of G, where G is the conformal group of a real Jordan algebra N. We characterize spherical vectors for these representation and prove that they are closely related to the Bessel K-function $K_τ(z)$. The resulting construction can be used to study tensor powers of the minimal representation and establish an extension of the Howe duality correspondence to some exceptional groups.

On unitary unipotent representations of $p$-adic groups and affine Hecke algebras with unequal parameters

We determine the unitary dual of the geometric graded Hecke algebras with {unequal} parameters which appear in Lusztig's classification of unipotent representations for {exceptional} $p$-adic groups. The largest such algebra is of type $F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's o…

Unipotent representations as a categorical centre

Let G(F_q) be the group of rational points of a split connected reductive group G defined over the finite field F_q. In this paper we show that the category of representations of G(F_q) which are finite direct sums of unipotent representations in a fixed two-sided cell is equivalent to the centre of a certain monoidal category of sheaves on the product of two copies of the flag manifold of G. We also consider a version of this for nonsplit groups.

The nonabelian Fourier transform for elliptic unipotent representations of exceptional $p$-adic groups

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.

On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups

In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig's nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G_2.

温和幂幺表示的非阿贝尔傅里叶变换

A nonabelian Fourier transform for tempered unipotent representations

本文定义分裂 $p$-进群内扭形的紧温和幂幺表示空间上的对合，提出其与 Lusztig 非阿贝尔傅里叶变换关系的精确猜想，并给出 $\mathsf{SL}_n$ 和 $\mathsf{PGL}_n$ 的证明。

The orbit method for unipotent groups over finite field

We apply the orbit method to obtain formula for multiplicities of certain representations of unipotent groups over the finite field.

Supercuspidal unipotent representations: L-packets and formal degrees

Let K be a non-archimedean local field and let G be a connected reductive K-group which splits over an unramified extension of K. We investigate supercuspidal unipotent representations of the group G(K). We establish a bijection between the set of irreducible G(K)-representations of this kind and the set of cuspidal enhanced L-parameters for G(K), which are trivial on the inertia subgroup of the Weil group of K. The bijection is characterized by a few simple equivariance properties and a comparison of formal degrees of representations with adjoint $γ$-factors of L-parameters. This can be regar…

On a uniqueness property of cuspidal unipotent representations

The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig), is a rational function of the cardinality q of the residue field. The irreducible factors of this rational function are $q$ and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of Mark Reeder, and for the remaining exceptional cases this …

实经典群的特殊幂幺表示：构造与酉性

Special unipotent representations of real classical groups: construction and unitarity

本文研究实经典群附属于对偶群幂零轨道的特殊幂幺表示，通过 theta 提升完成构造与分类，证明其可酉化并确定相伴循环结构。

Stable combinations of special unipotent representations

The set of special unipotent representations of a semisimple Lie group was defined by Barbasch and Vogan. According to predictions of Arthur (established by Adams, Barbasch, and Vogan), this set is an overlapping union of Arthur packets. Each packet gives rise to a canonical virtual representation whose character is stable in the sense of Langlands and Shelstad. But these stable characters seldom span the space of all stable characters of virtual special unipotent representations. Under natural hypotheses, we give a canonical basis of this latter space in terms of rational forms of special pie…

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.

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…

实典型群的特殊幂零表示：计数与约化

Special unipotent representations of real classical groups: counting and reduction

本文研究实典型群（含实亚辛群）中附着于对偶幂零轨道的特殊幂零表示，给出其数目的精确计数，并将构造问题约化为轨道解析偶的情形。

温和单式表示的非阿贝尔傅里叶变换，II

A nonabelian Fourier transform for tempered unipotent representations, II

将非阿贝尔傅里叶变换从椭圆单式特征扩展到紧单式特征，研究其与极大紧开子群分支的关系，证明与某些群的抛物限制兼容，并通过抛物诱导联系紧与椭圆情形。

Orbites Nilpotentes Sphériques et Représentations unipotentes associées : Le cas $SL_n$

Let $G$ be a real simple Lie group, $\got g$ its Lie algebra. Given a nilpotent adjoint $G$-orbit $O$, the question is to determine the irreducible unitary representations of $G$ that we can associate to $O$, according to the orbit method. P.Torasso, in [22], gave a method to solve this problem if $O$ is minimal. In this paper, we study the case where $O$ is any spherical nilpotent orbit of $sl_n({\mathbb R})$, we construct, from $O$, a family of representations of the two-sheeted covering of $SL_n({\mathbb R})$ with Torasso's method and, finally, we show that all these representations are ass…

具有半整数无穷小特征标的特殊幂幺表示

Special Unipotent Representations with Half Integral Infinitesimal Characters

本文针对特殊幂零轨道对应的特殊幂幺表示，在无穷小特征标中半单元素的一半为非整数情形下，完整证明了 Barbasch-Vogan 理论中此前仅在整数情形下已知的若干性质。

有限单移位海森堡型群的不可约表示数目

On the number of irreducible representations for finite unipotent Heisenberg-type groups

本文研究了有限单移位海森堡型群的不可约表示数目，通过轨道方法分类，并证明该数目为非负整系数多项式，符合Isaacs猜想。

Strong density of spherical characters attached to unipotent subgroups

We prove the following result in relative representation theory of a reductive p-adic group $G$: Let $U$ be the unipotent radical of a minimal parabolic subgroup of $G$, and let $ψ$ be an arbitrary smooth character of $U$. Let $S \subset Irr(G)$ be a Zariski dense collection of irreducible representations of $G$. Then the span of the Bessel distributions $B_π$ attached to representations $π$ from $S$ is dense in the space $\mathcal S^*(G)^{U\times U,ψ\times ψ}$ of all $(U\times U,ψ\times ψ)$-equivariant distributions on $G.$ We base our proof on the following results: 1. The category of smooth…

Explicit Hilbert spaces for certain unipotent representations III

We consider the groups G which arise from real semisimple Jordan algebras via the Tits-Koecher-Kantor construction. Such a G is characterized by the fact that it admits a parabolic subgroup P=LN which is conjugate to its opposite, and for which the nilradical N is abelian. In this situation, the Levi component L has a finite number of orbits on N; and each orbit carries a measure which transforms by a character under L. By Mackey theory the space of L2-functions on each orbit carries a natural irreducible unitary representation of P, and we consider the following two problems: (1) Extend this …