Unitary Representations Of Group Research Articles

Dirac cohomology, K -characters and branching laws

Inspired by work of Enright and Willenbring, we prove a generalized Littlewood’s restriction formula in terms of Dirac cohomology. Our approach is to use a character formula of irreducible unitary lowest weight modules instead of the Bernstein-Gelfand-Gelfand resolution, and the proof is much simpler. We also show that our branching formula is equivalent to the formula of Enright and Willenbring in terms of nilpotent Lie algebra cohomology. This follows from the close relationship between the Dirac cohomology and the corresponding nilpotent Lie algebra cohomology for unitary representations of semisimple Lie groups of Hermitian type, which was established by Huang, Pandzic, and Renard.

Berezin Quantization and Holomorphic Representations

Let G be a quasi-Hermitian Lie group and let {\pi} be a unitary highest weight representation of G realized in a reproducing kernel Hilbert space of holomorphic functions. We study the Berezin symbol map S and the corresponding Stratonovich–Weyl map W which is defined on the space of Hilbert–Schmidt operators acting on the space of {\pi} , generalizing some results that we have already obtained for the holomorphic discrete series representations of a semi-simple Lie group. In particular, we give explicit formulas for the Berezin symbols of the representation operators {\pi} (\kern 1pt g) (for g\in G ) and d{\pi} (X) (for X in the Lie algebra of G ) and we show that S provides an adapted Weyl correspondence in the sense of [B. Cahen, Weyl quantization for semidirect products, Differential Geom. Appl. 25 (2007), 177–190]. Moreover, in the case when G is reductive, we prove that W can be extended to the operators d{\pi} (X) and we give the expression of W(d{\pi} (X)) . As an example, we study the case when {\pi} is a generic unitary representation of the diamond group.

Residual automorphic forms and spherical unitary representations of exceptional groups

Arthur has conjectured that the unitarity of a number of representations can be shown by nding appropriate automorphic realizations. This has been veried for classical groups by Mglin and for the exceptional Chevalley group G2 by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups E6, E7, E8, and F4. In particular, we prove Arthur’s conjecture that the spherical constituent of an unramied principal series of a Chevalley group over any local eld of characteristic zero is unitarizable if its Langlands parameter coincides with half the weighted marking of a coadjoint nilpotent orbit of the Langlands dual Lie algebra.

On the classification of homogeneous $2$-spheres in complex Grassmannians

In this paper we discuss a classification problem of homogeneous 2-spheres in the complex Grassmann manifold $G(k + 1, n + 1)$ by theory of unitary representations of the 3-dimensional special unitary group $\mathit{SU}(2)$. First we observe that if an immersion $x\colon S^{2} \to G(k + 1, n + 1)$ is homogeneous, then its image $x(S^{2})$ is a 2-dimensional $\rho(\mathit{SU}(2))$-orbit in $G(k + 1, n + 1)$, where $\rho\colon \mathit{SU}(2) \to U(n + 1)$ is a unitary representation of $\mathit{SU}(2)$. Then we give a classification theorem of homogeneous 2-spheres in $G(k + 1, n + 1)$. As an application we describe explicitly all homogeneous 2-spheres in $G(2, 4)$. Also we mention about an example of non-homogeneous holomorphic 2-sphere with constant curvature in $G(2, 4)$.

A CLT Plancherel representations of the infinite-dimensional unitary group

We study the asymptotics of traces of (noncommutative) monomials formed by the images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge to (commutative) moments of a Gaussian process that can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian free fields. The limiting process has previously arisen via the global scaling limit of submatrices of Wigner Hermitian random matrices. This note is an announcement, proofs will appear elsewhere. Bibliography: 14 titles.

The symmetry groups of noncommutative quantum mechanics and coherent state quantization

We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in (2+1)-space-time dimensions and the two-fold extension of the group of translations of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^4$\end{document}R4. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.

A new set of generators and a physical interpretation for the $$SU (3)$$ finite subgroup $$D(9,1,1;2,1,1)$$

After 100 years of effort, the classification of all the finite subgroups of SU(3) is yet incomplete. The most recently updated list can be found in P.O. Ludl, J. Phys. A: Math. Theor. 44 255204 (2011), where the structure of the series (C) and (D) of SU(3)-subgroups is studied. We provide a minimal set of generators for one of these groups which has order 162. These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones-Kauffman version of SU(2) Chern-Simons theory at level 4. In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product Z_9 \times Z_3 \rtimes S_3 with respect to conjugation.

Harmonic reconstruction systems

This paper considers group reconstruction systems (GRS's), for finite dimensional real or complex Hilbert spaces H, that are associated with unitary representations of finite abelian groups. The relation between these GRS's and the generalized Fourier matrix is established. A special type of Parseval GRS, called harmonic reconstruction system (HRS), is defined. It is shown that there exist HRS's that present maximal robustness to erasures given characterizations of certain families.

The Koszul complex of a moment map

Let $K \to \operatorname{U}(V)$ be a unitary representation of the compact Lie group $K$. Then there is a canonical moment mapping $\rho \colon V \to {\mathfrak k}^*$. We have the Koszul complex $\mathcal{K}(\rho, \mathcal{C}^\infty(V))$ of the component functions $\rho_1, \dots, \rho_k$ of $\rho$. Let $G=K_{\mathbb {C}}$, the complexification of $K$. We show that the Koszul complex is a resolution of the smooth functions on $\rho ^{-1}(0)$ if and only if $G \to \operatorname{GL}(V)$ is $1$-large, a concept introduced in [11,12]. Now let $M$ be a symplectic manifold with a Hamiltonian action of $K$. Let $\rho$ be a moment mapping and consider the Koszul complex given by the component functions of $\rho$. We show that the Koszul complex is a resolution of the smooth functions on $Z= \rho ^{-1}(0)$ if and only if the complexification of each symplectic slice representation at a point of $Z$ is $1$-large.

La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic $0$ and let $G=U(n)$, $H=U(m)$ be unitary groups of hermitian spaces $V$ and $W$. Assume that $V$ contains $W$ and that the orthogonal complement of $W$ is a quasisplit hermitian space (i.e. whose unitary group is quasisplit over $F$). Let $\pi$ and $\sigma$ be smooth irreducible representations of $G(F)$ and $H(F)$ respectively. Then Gan, Gross and Prasad have defined a multiplicity $m(\pi,\sigma)$ which for $m=n-1$ is just the dimension of $Hom_{H(F)}(\pi,\sigma)$. For $\pi$ and $\sigma$ tempered, we state and prove an integral formula for this multiplicity. As a consequence, assuming some expected properties of tempered $L$-packets, we prove a part of the local Gross-Prasad conjecture for tempered representations of unitary groups. This article represents a straight continuation of recent papers of Waldspurger dealing with special orthogonal groups.

Noncommutativity in (2+1)-dimensions and the Lorentz group

In this article we considered models of particles living in a three-dimensional space-time with a nonstandard noncommutativity induced by shifting canonical coordinates and momenta with generators of a unitary irreducible representation of the Lorentz group. The Hilbert space gets the structure of a direct product with the representation space, where we are able to construct operators which realize the algebra of Lorentz transformations. We study the modified Landau problem for both Schr\"odinger and Dirac particles, whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of the dynamical variables from the ones of the usual problem in the normal space. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters. We find no constraint between the parameters referring to noncommutativity in coordinates and momenta but they rather play similar roles. Since the representation space of the unitary irreducible representations $SL(2,\mathbb{R})$ can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.

On Representations of Classical Groups Over Principal Ideal Local Rings of Length Two

We study the complex irreducible representations of special linear, symplectic, orthogonal, and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. The case for general linear groups has already been proved by the author.

Tilting, deformations and representations of linear groups over Euclidean algebras

We study irreducible unitary representations of linear groups over Dynkinian and Euclidean algebras, i.e. finite dimensional algebras derived equivalent to the path algebra of a Dynkin or Euclidean quiver. In particular, we describe such generic representations: we prove that there exists an open dense subset of the space of irreducible unitary representations isomorphic to the product of dual spaces of full linear groups and, perhaps, one more (explicitly described) space. The proof uses the technique of bimodule categories, deformations and representations of quivers.

Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method

Let \(\overline {S{p_{2n}}({\rm{ F }})} \) be the metaplectic double cover of F where F is a local field of characteristic 0. We use the Uniqueness of Whittaker model to define a metaplectic analog to Shahidi local coefficients and we use these coefficients to define gamma factors. We show that these gamma factors are multiplicative and satisfy the crude global functional equation. Then, we compute these factors in various cases and obtain explicit formulas for Plancherel measures. These computations are then used to prove some irreducibility theorems for parabolic induction on the metaplectic group over p-adic fields. In particular, we show that all principal series representations induced from unitary characters are irreducible. We also prove that parabolic induction from unitary supercuspidal representation of the Siegel parabolic sub group is irreducible if and only if a certain parabolic induction on F is irreducible.

On the essential spectrum of phase-space anisotropic pseudodifferential operators

AbstractA phase-space anisotropic operator in=L2(ℝn) is a self-adjoint operator whose resolvent family belongs to a naturalC*-completion of the space of Hörmander symbols of order zero. Equivalently, each member of the resolvent family is norm-continuous under conjugation with the Schrödinger unitary representation of the Heisenberg group. The essential spectrum of such a phase-space anisotropic operator is the closure of the union of usual spectra of all its “phase-space asymptotic localizations”, obtained as limits over diverging ultrafilters of ℝn×ℝn-translations of the operator. The result extends previous analysis of the purely configurational anisotropic operators, for which only the behavior at infinity in ℝnwas allowed to be non-trivial.

Unitary cocycle representations of the Galilean line group: Quantum mechanical principle of equivalence

We present a formalism of Galilean quantum mechanics in non-inertial reference frames and discuss its implications for the equivalence principle. This extension of quantum mechanics rests on the Galilean line group, the semidirect product of the real line and the group of analytic functions from the real line to the Euclidean group in three dimensions. This group provides transformations between all inertial and non-inertial reference frames and contains the Galilei group as a subgroup. We construct a certain class of unitary representations of the Galilean line group and show that these representations determine the structure of quantum mechanics in non-inertial reference frames. Our representations of the Galilean line group contain the usual unitary projective representations of the Galilei group, but have a more intricate cocycle structure. The transformation formula for the Hamiltonian under the Galilean line group shows that in a non-inertial reference frame it acquires a fictitious potential energy term that is proportional to the inertial mass, suggesting the equivalence of inertial mass and gravitational mass in quantum mechanics.

The Genuine Omega-regular Unitary Dual of the Metaplectic Group

Abstract We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.

Spectra of elements in the range of strongly continuous unitary representations of Lie groups

The main result proved in this work is that, for a strongly continuous unitary representation of a Lie group, we have the following dichotomy: The representation is norm continuous or on an open neighborhood of the unit (which is the whole group for a weakly exponential Lie group), there is a residual set of elements whose range admits the entire torus as spectrum.

Path integral junctions

We propose path integral description for quantum mechanical systems on compact graphs consisting of N segments of the same length. Provided the bulk Hamiltonian is segment-independent, scale-invariant boundary conditions given by the self-adjoint extension of a Hamiltonian operator turn out to be in one-to-one correspondence with N × N matrix-valued weight factors on the path integral side. We show that these weight factors are given by N-dimensional unitary representations of the infinite dihedral group.

GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.