Continuous unitary projective representations of Polish groups: The BMS-group

In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group \(\mathop{\mathrm{U}}\nolimits (\mathcal{H})\) of a real, complex or quaternionic separable Hilbert space and the subgroup \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})\), 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 \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}\) 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 generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell’s theorem, asserting that the separable unitary representations of \(\mathop{\mathrm{U}}\nolimits (\mathcal{H})\), for a separable Hilbert space \(\mathcal{H}\), are uniquely determined by their restriction to \(\mathop{\mathrm{U}}\nolimits _{\infty }(\mathcal{H})_{0}\). For the 10 classical infinite rank symmetric pairs (G, K) of non-unitary type, such as \((\mathop{\mathrm{GL}}\nolimits (\mathcal{H}),\mathop{\mathrm{U}}\nolimits (\mathcal{H}))\), we also show that all separable unitary representations are trivial.

Continuous wavelet transform and linear time-frequency transforms are coefficients of continuous unitary group representations of the affine and Heisenberg groups. Many properties of these transforms that are important for wideband radar and sonar signal processing follow directly from group representation theory. These properties include volume invariance and variance of narrowband and wideband ambiguity functions and wavelet transform domain implementations of detects and signal estimators. For several radar, sonar, and array processing applications, the basic definition of wavelet and time-frequency representations must be generalized by using unitary representations of other groups and using reproducing kernel Hilbert space (RKHS) inner products in the definition of the linear transforms. The general definition then leads to weighted continuous wavelet transforms where the RKHS is determined by a nonstationary covariance function; generalized wideband ambiguity functions also follow from this general definition along with other important generalizations that arise in wideband array processing and model based signal processing in complex scattering and propagation media. This paper presents the generalized wavelet transform along with the weighted wavelet transforms. The classical narrowband and wideband ambiguity functions are then special cases. The application of generalized transform to multidimensional transforms for space-time processing are also presented, along with the application to conformal groups for detections and estimation of accelerating scatterers.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

In the note, a 30-year old result concerning the von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional continuous complex unitary representations of the group) is corrected and the smallest von Neumann kernel of a connected Lie group (the intersection of kernels of all irreducible finite-dimensional (not necessarily continuous) complex unitary representations of the group) is described.

In this note we show that the representation of the additive group of the Hilbert space L 2 ([0, 1], R) on L 2 ([0, 1], C) given by the multiplication operators (f ) := e if is continuous but its space of smooth vectors is trivial.This example shows that a continuous unitary representation of an infinite dimensional Lie group need not be smooth.

We give a general bosonic construction of oscillator-like unitary irreducible representations (UIR) of non-compact groups whose coset spaces with respect to their maximal compact subgroups are Hermitian symmetric. With the exception of E7(7), they include all the non-compact invariance groups of extended supergravity theories in four dimensions. These representations have the remarkable property that each UIR is uniquely determined by an irreducible representation of the maximal compact subgroup. We study the connection between our construction, the Hermitian symmetric spaces and the Tits-Koecher construction of the Lie algebras of corresponding groups. We then give the bosonic construction of the Lie algebra ofE 7(7) in SU(8), SO(8) and U(7) bases and study its properties. Application of our method toE 7(7) leads to reducible unitary representations.

The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics.We put aside the relativistic symmetry and show that the maximal quantum symmetry that leaves the Heisenberg commutation relations invariant is the projective representations of the conformally scaled inhomogeneous symplectic group. The Weyl-Heisenberg group and noncommutative structure arises directly because the quantum phase requires projective representations.We then consider the relativistic implications of the quantum phase that lead to the Born line element and the projective representations of an inhomogeneous unitary group that defines a noninertial quantum theory. (Understanding noninertial quantum mechanics is a prelude to understanding quantum gravity.) The remarkable properties of this symmetry and its limits are studied.

Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous corepresentations of C*-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is proved that the unitary continuous corepresentations decompose in finite-dimensional irreducible corepresentations.

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus [Formula: see text] must be integral or half integral. Actually he proved that for a system [Formula: see text] of complex numbers of absolute value 1 [Formula: see text] can be an automorphy factor only if [Formula: see text] is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for [Formula: see text] and [Formula: see text], Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.

Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak {g}}} = {{\mathfrak {h}}} \oplus {{\mathfrak {q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal {H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak {q}}},{{\mathfrak {q}}}] \oplus i{{\mathfrak {q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak {h}}} \oplus i{{\mathfrak {q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

With the Bondi‐Metzner‐Sachs (BMS) group in general relativity as the main motivation and example, a theorem is proved which may be described as follows. Let G be a complex semisimple, connected and simply connected Lie group with compact real form K , and let A be a metrizable, complete and locally convex real topological vector space on which there is a continuous G action. Consider the semidirect product topological group Gx s A (which is, in general, infinite‐dimensional) constructed naturally out of G and A . If the set of equivalence classes of irreducible representations of K in A satisfies certain hypotheses, then the second cohomology group of G x s A in the sense of continuous group cohomology is trivial. When G = SL (2, C ) and A is an appropriate function space of real‐valued functions of the 2‐sphere endowed with a specific G action (e.g. A may consist of C k , k ⩾ 3, real‐valued functions defined on the 2‐sphere), the semidirect product group is the universal cover of the BMS group. The theorem implies the existence of lifting of the projective unitary representations of the BMS group to the linear unitary representations of its universal cover. In the quantum context when we consider massless quantum fields at null infinity of a non‐stationary, asymptotically Minkowskian space‐time, in place of the projective unitary representations of the BMS group, there is no loss of generality in considering the linear unitary representations of its universal cover instead.

An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for ℓ 2 ( G ) . An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.

A new approach to the analysis of the physical state space of a theory is\npresented within the general setting of local quantum physics. It also covers\ntheories with long range forces, such as Quantum Electrodynamics. Making use of\nthe notion of charge class, an extension of the concept of superselection\nsector, infrared problems are avoided by restricting the states to observables\nlocalized in a light cone. The charge structure of a theory can be explored in\na systematic manner. The present analysis focuses on simple charges, thus\nincluding the electric charge. It is shown that any such charge has a conjugate\ncharge. There is a meaningful concept of statistics: the corresponding charge\nclasses are either of Bose or of Fermi type. The family of simple charge\nclasses is in one--to--one correspondence with the irreducible unitary\nrepresentations of a compact Abelian group. Moreover, there is a meaningful\ndefinition of covariant charge classes. Any such class determines a continuous\nunitary representation of the Poincar\\'e group or its covering group satisfying\nthe relativistic spectrum condition. The resulting particle aspects are also\nbriefly discussed.\n

Let co be a factor system for the locally compact abelian group G. Then we show that the finite-dimensional unitary irreducible projective representations of G, having factor system co, possess a common dimension d(co).Using a characterisation of d(co) as the index in G of a maximal subgroup on which a> is symmetric we derive a formula for d(co) in the case that G is discrete and finitely generated.

Projective unitary positive-energy representations of diff ( S1)

Unitary Highest Weight Representations in Hilbert Spaces of Holomorphic Functions on Infinite Dimensional Domains