V. V. Uspenskij
Is the topological group of all motions (including translations) of an infinite-dimensional Hilbert space $H$ isomorphic to a subgroup of the unitary group $U(H)$? This question was asked by Su Gao. We answer the question in the affirmative.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Let $q$ be a power of a prime, $l$ a prime not dividing $q$, $d$ a positive integer coprime to both $l$ and the multiplicative order of $q\mod l$ and $n$ a positive integer. A. Watanabe proved that there is a perfect isometry between the principal $l-$blocks of $GL_n(q)$ and $GL_n(q^d)$ where the correspondence of characters is give by Shintani descent. In the same paper Watanabe also prove that if $l$ and $q$ are odd and $l$ does not divide $GL_n(q^2)|/|U_n(q)|$ then there is a perfect isometry between the principal $l-$blocks of $U_n(q)$ and $GL_n(q^2)$ with the correspondence of characters …
We consider groups $\mathbb{I}$ of isometries of ultrametric Urysohn spaces $\mathbb{U}$. Such spaces $\mathbb{U}$ admit transparent realizations as boundaries of certain $R$-trees and the groups $\mathbb{I}$ are groups of automorphisms of these $R$-trees. Denote by $\mathbb{I}[X]\subset \mathbb{I}$ stabilizers of finite subspaces $X\subset \mathbb{U}$. Double cosets $\mathbb{I}[X]\cdot g\cdot \mathbb{I}[Y]$, where $g\in \mathbb{I}$, are enumerated by ultrametrics on union of spaces $X\cup Y$. We construct natural associative multiplications on double coset spaces $\mathbb{I}[X]\backslash \mat…
As a consequence of Kirchberg's work, Connes' Embedding Conjecture is equivalent to the property that every homomorphism of the group $F_\infty\times F_\infty$ into the unitary group $U(\ell^2)$ with the strong topology is pointwise approximated by homomorphisms with a precompact range. In this form, the property (which we call Kirchberg's property) makes sense for an arbitrary topological group. We establish the validity of the Kirchberg property for the isometry group $Iso(U)$ of the universal Urysohn metric space $U$ as a consequence of a stronger result: every representation of a residuall…
For $C^*$-algebra generated by a finite family of isometries $s_j$, $j=1,\dots,d$ satisfying $q_{ij}$-commutation relations \[ s_j^* s_j = I, \quad s_j^* s_k = q_{ij}s_ks_j^*, \qquad q_{ij} = \bar q_{ji}, |q_{ij}|<1, \ 1\le i \ne j \le d, \] we construct an infinite family of unitarily non-equivalent irreducible representations. These representations are deformations of the corresponding class of representations of the Cuntz algebra $\mathcal O_d$.
Representations of the modular group into the isometries of SL(3,R)/SO(3)
本文研究了模群到SL(3,R)/SO(3)等距群的表示，证明了该空间中某些远离Schwartz表示的表示为Anosov表示，并指出主要结果此前由Schwartz通过不同方法证明。
A unitary (Euclidean) representation of a quiver is given by assigning to each vertex a unitary (Euclidean) vector space and to each arrow a linear mapping of the corresponding vector spaces. We recall an algorithm for reducing the matrices of a unitary representation to canonical form, give a certain description of the representations in canonical form, and reduce the problem of classifying Euclidean representations to the problem of classifying unitary representations. We also describe the set of dimensions of all indecomposable unitary (Euclidean) representations of a quiver and establish t…
In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group $\U(\cH)$ of a real, complex or quaternionic separable Hilbert space and the subgroup $\U_\infty(\cH)$, consisting of those unitary operators $g$ for which $g - \1$ is compact. The Kirillov--Olshanski theorem on the continuous unitary representations of the identity component $\U_\infty(\cH)_0$ asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalize…
We enumerate the number of complex irreducible representations of each degree of general unitary groups of degree 4 over principal ideal rings of length 2.
The main result of this article is an application of the theory of invariant convex cones of Lie algebras to the study of unitary representations of Lie supergroups. It also includes an exposition of recent results of the second author on the classification of irreducible unitary representations of nilpotent Lie supergroups using the method of coadjoint orbits.
We classify the domains of holomorphy of all Harish-Chandra modules of irreducible unitary representations of simple non-compact Lie groups.
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。
We lift the $5$-dimensional characteristic $3$ representation of $M_{11}$ to a complex representation of the amalgam ${\rm GL}(2,3)*_{D_8}S_{4}$, and consider its reduction (mod $p$) for other odd primes.
We continue our work on understanding Howe correspondences by using theta representations from p-adic groups to compact groups. We prove some results for unitary theta representations of compact groups with respect to the induction and restriction functors.
This paper continues the study of the poset of eigenspaces of elements of a unitary reflection group (for a fixed eigenvalue), which was commenced in [6] and [5]. The emphasis in this paper is on the representation theory of unitary reflection groups. The main tool is the theory of poset extensions due to Segev and Webb ([16]). The new results place the well-known representations of unitary reflection groups on the top homology of the lattice of intersections of hyperplanes into a natural family, parameterised by eigenvalue.
The unitary principal series representations of $G=GL(n,\mathbb{C})$ induced from a character of the maximal parabolic subgroup $P=(GL(1,\mathbb{C})\times GL(n-1,\mathbb{C}))\ltimes\mathbb{C}^{n-1}$ attain the minimal Gelfand--Kirillov dimension among all infinite-dimensional unitary representations of $G$. We find the explicit branching laws for the restriction of these representations to symmetric subgroups of $G$.
Let $G$ be $Sp(2n, \mathbb{R})$ or $SO^*(2n)$. We compute the Dirac index of a large class of unitary representations considered by Vogan in Section 8 of [Vog84], which include all weakly fair $A_{\mathfrak{q}}(λ)$ modules and (weakly) unipotent representations of $G$ as two extreme cases. We conjecture that these representations exhaust all unitary representations of $G$ with nonzero Dirac cohomology. In general, for certain irreducible unitary module of an equal rank group, we clarify the link between the possible cancellations in its Dirac index, and the parities of its spin-lowest $K$-type…
We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.
In this paper, we prove that theta correspondence preserves unitarity under certain restrictions.
Given an orthogonal representation of a compact group, we show that any element of the connected component of the isometry group of the orbit space lifts to an equivariant isometry of the original Euclidean space. Corollaries include a simple formula for this connected component, which applies to ``most'' representations.
We continue the study of positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R). We present the full list of these UIRs. We give the Proof of the case osp(1|8,R).
Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,ρ)$ is distinguished if and only if the cuspidal representation $ρ$ is $η^{k-1}$-distinguished, for $η$ the character of $F^*$ with kernel the norms of $K^*$, this actually gives a classification of distinguished unitary representations in terms of distinguished c…
Let A be a finite dimensional central division algebra over a local non-archimedean field F. Fix any parabolic subgroup P of GL(n,A) and a Levi factor M of P. Let πbe an irreducible unitary representation of M and ϕ(not necessarily unitary) character of M. We give an explicit necessary and sufficient condition for the parabolically induced representation Ind(ϕπ) of GL(n,A) to be irreducible.
We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of $\mathrm{SL}(2,\mathbb{R})$.
In this paper we give a classification of the $H$-spherical unitary highest weight representations of a hermitian Lie group $G$.
We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the unitary representations in the category $\OO$ for the rational Cherednik algebra of type A studied in \cite{ESG} correspond to the unitary representations of the corresponding Hecke algebra via the KZ functor defined in \cite{GGOR}.
We present an application of Hodge theory towards the study of irreducible unitary representations of reductive Lie groups. We describe a conjecture about such representations and discuss some progress towards its proof.
In this work we shall generalize the Selberg trace formula to a non-unitary finite-dimensional complex representation $χ:Γ\rightarrow\operatorname{GL}(V)$ of a uniform lattice $Γ$ of a real Lie group $G$.
For an arbitrary unimodular Lie group $G$, we construct strongly continuous unitary representations in the Bergman space of a naturally constructed strongly pseudoconvex neighborhood of $G$ in the complexification of its underlying manifold.
Suppose that $G$ is a locally compact group and $π$ is a (not necessarily irreducible) unitary representation of a closed normal subgroup $N$ of $G$ on a Hilbert space $H$. We extend results of Clifford and Mackey to determine when $π$ extends to a unitary representation of $G$ on the same space $H$ in terms of a cohomological obstruction.
We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split unitary groups. The classification of automorphic representations is given in terms of automorphic representations of general linear groups. The classification of irreducible admissible representations is given in terms of Langlands parameters.
Let G be a locally analytic group. We use deformation theory and Emerton's 'augmented' Iwasawa modules to study the possible liftings of a given absolutely irreducible smooth modular representation of G to a unitary Banach space representation. Final version. Some minor items changed. Appears in Israel J. Math.
We recover the holomorphic discrete series representations of $SU(1,n)$ as well as some unitary irreducible representations of $SU(n+1)$ by deformation of a minimal realization of $sl(n+1, {\mathbb C})$.
In this paper we establish the endoscopic classification of tempered representations of quasi-split unitary groups over local fields, and the endoscopic classification of the discrete automorphic spectrum of quasi-split unitary groups over global number fields. The method is analogous to the work of Arthur on orthogonal and symplectic groups, based on the theory of endoscopy and the comparison of trace formulas on unitary groups and general linear groups.
We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remov…
A number of recent papers treated the representation theory of partially ordered sets in unitary spaces with the so called orthoscalar relation. Such theory generalizes the classical theory which studies the representations of partially ordered sets in linear spaces. It happens that the results in the unitary case are well-correlated with those in the linear case. The purpose of this article is to shed light on this phenomena.
On Arthur representations and the unitary dual
本文提出一个关于Arthur表示与单位元双结构的新猜想，并通过算法和验证支持其完备性。
In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number $-1$.
We explicitly construct the structure of Jacquet modules of parabolically induced representations of even unitary groups and even general unitary groups over a $p$-adic field $F$ of characteristic different than two. As an application, we obtain a classification of strongly positive discrete series representations of those groups.
Shintani lifts for Weil representations of unitary groups over finite fields
该研究通过几何构造给出有限域上酉群的扩展 Weil 表示，并证明其构成相应 Weil 表示的 Shintani 提升。
On $A$-parameters containing unitary lowest weight representations of $\mathrm{U}(p, q)$
本文利用Barbasch-Vogan参数化和Trapa算法，确定了实酉群U(p, q)包含不可约酉最低权表示的所有Arthur包，证明每个包最多含一个此类表示，并给出其最低K型的显式公式。
We present modules for the extended affine Lie algebra $\widetilde{\frak{gl}_{3}(\bc_q)}$ by using the idea of free fields. A necessary and sufficient condition for the modules being unitary is given.
In this paper, we establish the theory of local newforms for irreducible tempered generic representations of unramified odd unitary groups over a non-archimedean local field. For the proof, we prove an analogue of the fundamental lemma for our compact groups.
On corank 4 unitary representations of classical groups
本文通过分类协秩4的Arthur表示，验证了 Hazeltine-Jiang-Liu-Lo-Zhang 提出的酉对偶猜想，完成了非阿基米德局部域上辛群或分裂奇特殊正交群的协秩4酉表示的分类。
In this note we introduce the concept of a semi-bounded unitary representations of an infinite-dimensional Lie group $G$. Semi-boundedness is defined in terms of the corresponding momentum set in the dual $\g'$ of the Lie algebra $\g$ of $G$. After dealing with some functional analytic issues concerning certain weak-$*$-locally compact subsets of dual spaces, called semi-equicontinuous, we characterize unitary representations which are bounded in the sense that their momentum set is equicontinuous, we characterize semi-bounded representations of locally convex spaces in terms of spectral measu…
Let $\Goo$ be a semisimple real Lie group with unitary dual $\Ghat$. The goal of this note is to produce new upper bounds for the multiplicities with which representations $π\in \Ghat$ of cohomological type appear in certain spaces of cusp forms on $\Goo$.
We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the weakly fair range. This extends a result of J.-S. Li in the case of discrete series representations with sufficiently regular infinitesimal character, whose theta lifts can be expressed as cohomologically induced representations in the good range.
We construct projective unitary representations of the smooth Deligne cohomology group of a compact oriented Riemannian manifold of dimension 4k+1, generalizing positive energy representations of the loop group of the circle. We also classify such representations under a certain condition. The number of the equivalence classes of irreducible representations is finite, and is determined by the cohomology of the manifold.
In this paper, the Moyal star-product quantization is used to construct the unitary irreducible representations of the Euclidean motion group on 3-dimensions. These unitary representations will come from the representation of its Lie algebra whose operators are defined by the left Moyal star-product multiplication. In fact, these representations of the Lie algebra is the infinitisimal representation. Hence, the exponentiation of these operators gives rise to unitary operators that defines the desired unitary representations.
We juxtapose two approaches to the representations of the super-Heisenberg group. Physical one, sometimes called concrete approach, based on the super-wave functions depending on the anti-commuting variables, yielding the harmonic superanalysis and recently developed strict theory of unitary representations of the nilpotent super Lie groups covering the unitary representations of the super-Heisenberg group.
For an infinite dimensional Lie group $G$ modelled on a locally convex Lie algebra $\mathfrak{g}$, we prove that every smooth projective unitary representation of $G$ corresponds to a smooth linear unitary representation of a Lie group extension $G^{\sharp}$ of $G$. (The main point is the smooth structure on $G^{\sharp}$.) For infinite dimensional Lie groups $G$ which are 1-connected, regular, and modelled on a barrelled Lie algebra $\mathfrak{g}$, we characterize the unitary $\mathfrak{g}$-representations which integrate to $G$. Combining these results, we give a precise formulation of the co…
For a broad class of Frechet-Lie supergroups we prove that there exists a correspondence between positive definite smooth superfunctions and matrix coefficients of unitary representations. We also give a characterization of linear functionals on the corresponding universal enveloping algebra which correspond to matrix coefficients of analytic unitary representations.
We show that the unitary conjugacy relation for unitary representations of a second countable locally compact group on a separable Hilbert space is a Borel equivalence relation.
We study the unitary boundary representation of a strongly transitive group acting on a right-angled hyperbolic building. We show its irreducibility. We do so by associating to such a representation a representation of a certain Hecke algebra, which is a deformation of the classical representation of a hyperbolic reflection group. We show that the associated Hecke algebra representation is irreducible.
Governed by locality, we explore a connection between unitary braid group representations associated to a unitary $R$-matrix and to a simple object in a unitary braided fusion category. Unitary $R$-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get rep…
We give conditions for unitarizability of Harish-Chandra super modules for Lie supergroups and superalgebras.
Strong convergence of unitary and permutation representations of discrete groups
本文综述了离散群酉表示与置换表示强收敛性的研究项目，并补充了未公开的细节。
Unitary representations and von Neumann's continuous geometries
本文证明了冯·诺依曼意义下非离散不可约连续环的幺群不具有任何连续于强算子拓扑的非平凡幺正表示。
We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of infinite matrices over a residue ring modulo $p^k$. Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。