Unitary Representations Of Group Research Articles

On the Schrödinger–Newton equation and its symmetries: a geometric view

The Schrödinger–Newton (SN) equation is recast on purely geometrical grounds, namely in terms of Bargmann structures over -dimensional Newton–Cartan (NC) spacetimes. Its maximal group of invariance, which we call the SN group, is determined as the group of conformal Bargmann automorphisms that preserve the coupled Schrödinger and NC gravitational field equations. Canonical unitary representations of the SN group are worked out, helping us recover, in particular, a very specific occurrence of dilations with dynamical exponent .

Non-unitary representations of nilpotent groups, I: Cohomologies, extensions and neutral cocycles

Let λ be a finite-dimensional representation of a connected nilpotent group G and U be a unitary representation of G . We investigate the structure of the extensions of λ by U and, correspondingly, the group H 1 ( λ , U ) of 1-cohomologies. A spectral criterion of triviality of H 1 ( λ , U ) is proved and systematically used in the study of various types of decomposition of the extensions. We consider a special type of ( λ , U ) -cocycles – neutral cocycles, which play a crucial role in the theory of J -unitary representations of groups on Pontryagin Π k -spaces.

Endoscopic transfer for unitary groups and holomorphy of AsaiL-functions

The analytic properties of the complete Asai L-functions attached to cuspidal automorphic representations of the general linear group over a quadratic extension of a number field are obtained. The proof is based on the comparison of the Langlands‐Shahidi method and Mok’s endoscopic classification of automorphic representations of quasisplit unitary groups.

A quantization of the harmonic analysis on the infinite-dimensional unitary group

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(∞). A way to describe those measures is to convert them into determinantal point processes on the real line; it turned out that their correlation kernels are computable in explicit form — they admit a closed expression in terms of the Gauss hypergeometric function F12 (Borodin and Olshanski [8]). In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials — the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function ϕ12. A crucial novel ingredient of our approach is an extended version G of the Gelfand–Tsetlin graph (the conventional graph describes the Gelfand–Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G, thus extending previously known results (Gorin [17]).

A descriptive view of unitary group representations

In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if G and H are countable amenable non-type I groups, then the unitary duals of G and H are Borel isomorphic.

On quasi-greedy bases associated with unitary representations of countable groups

On quasi-greedy bases associated with unitary representations of countable groups

Nonlinear Completely Positive Maps and Dilation Theory for Real Involutive Algebras

A real seminormed involutive algebra is a real associative algebra \({\mathcal{A}}\) endowed with an involutive antiautomorphism * and a submultiplicative seminorm p with p(a*) = p(a) for \({a\in \mathcal{A}}\). Then \({\mathtt{ball}(\mathcal{A}, p) := \{ a \in \mathcal{A} \: p(a) < 1\}}\) is an involutive subsemigroup. For the case where \({\mathcal{A}}\) is unital, our main result asserts that a function \({\varphi \: \mathtt{ball}(\mathcal{A},p) \to B(V)}\), V a Hilbert space, is completely positive (defined suitably) if and only if it is positive definite and analytic for any locally convex topology for which \({\mathtt{ball}(\mathcal{A},p)}\) is open. If \({\eta_\mathcal{A} \: \mathcal{A} \to C^{*}(\mathcal{A},p)}\) is the enveloping C*-algebra of \({(\mathcal{A},p)}\) and \({e^{C^{*}(\mathcal{A}, p)}}\) is the c0-direct sum of the symmetric tensor powers \({S^n(C^{*}(\mathcal{A},p))}\), then the above two properties are equivalent to the existence of a factorization \({\varphi = \Phi \circ \Gamma}\), where \({\Phi \: e^{C^{*}(\mathcal{A}, p)} \to B(V)}\) is linear completely positive and \({\Gamma(a) = \sum_{n = 0}^\infty \eta_\mathcal{A}(a)^{\otimes n}}\). We also obtain a suitable generalization to non-unital algebras. An important consequence of this result is a description of the unitary representations of the unitary group \({{\rm U}(\mathcal{A})}\) with bounded analytic extensions to \({\mathtt{ball}(\mathcal{A},p)}\) in terms of representations of the C*-algebra \({e^{C^{*}(\mathcal{A}, p)}}\).

An Impossibility Theorem for Linear Symplectic Circle Quotients

We prove that when $d>2$, a $d$-dimensional symplectic quotient at the zero level of a unitary circle representation $V$ such that $V^{\Sp^1}=\{0\}$ cannot be $\Z$-graded regularly symplectomorphic to the quotient of a unitary representations of a finite group.

Wilson loops on Riemann surfaces, Liouville theory and covariantization of the conformal group

The covariantization procedure is usually referred to the translation operator, that is the derivative. Here we introduce a general method to covariantize arbitrary differential operators, such as the ones defining the fundamental group of a given manifold. We focus on the differential operators representing the sl(2,R) generators, which in turn, generate, by exponentiation, the two-dimensional conformal transformations. A key point of our construction is the recent result on the closed forms of the Baker-Campbell-Hausdorff formula. In particular, our covariantization receipt is quite general. This has a deep consequence since it means that the covariantization of the conformal group is {\it always definite}. Our covariantization receipt is quite general and apply in general situations, including AdS/CFT. Here we focus on the projective unitary representations of the fundamental group of a Riemann surface, which may include elliptic points and punctures, introduced in the framework of noncommutative Riemann surfaces. It turns out that the covariantized conformal operators are built in terms of Wilson loops around Poincar\'e geodesics, implying a deep relationship between gauge theories on Riemann surfaces and Liouville theory.

Endoscopic Classification of representations of Quasi-Split Unitary Groups

Introduction Statement of the main theorems Local character identities and the intertwining relation Trace formulas and their stabilization The Standard model Study of critical cases Local classification Nontempered representations Global classification Addendum Bibliography

On the geometry of some unitary Riemann surface braid group representations and Laughlin-type wave functions

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.

WHITTAKER PERIODS, MOTIVIC PERIODS, AND SPECIAL VALUES OF TENSOR PRODUCT -FUNCTIONS

Let${\mathcal{K}}$be an imaginary quadratic field. Let${\rm\Pi}$and${\rm\Pi}^{\prime }$be irreducible generic cohomological automorphic representation of$\text{GL}(n)/{\mathcal{K}}$and$\text{GL}(n-1)/{\mathcal{K}}$, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called aWhittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if${\rm\Pi}$is cuspidal and the weights of${\rm\Pi}$and${\rm\Pi}^{\prime }$are in a standard relative position, the critical values of the Rankin–Selberg product$L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$are essentially algebraic multiples of the product of the Whittaker periods of${\rm\Pi}$and${\rm\Pi}^{\prime }$. We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal${\rm\Pi}$can be given a motivic interpretation, and can also be related to a critical value of the adjoint$L$-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint$L$-functions are compatible with Deligne’s conjecture.

Dihedral group frames which are maximally robust to erasures

Let be a natural number larger than two. Let be the Dihedral group, and an -dimensional unitary representation of acting in as follows. and for . For any representation which is unitarily equivalent to we prove that when is prime, there exists a Zariski open subset of such that for any vector any subset of cardinality of the orbit of under the action of this representation is a basis for . However, when is even, there is no vector in which satisfies this property. As a result, we derive that if is prime, for almost every (with respect to Lebesgue measure) vector in , the -orbit of is a frame which is maximally robust to erasures. We also consider the case where is equivalent to an irreducible unitary representation of the Dihedral group acting in a vector space and we provide conditions under which it is possible to find a vector such that has the Haar property.

Weyl–Pedersen calculus for some semidirect products of nilpotent Lie groups

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl–Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.

Generalized harmonic functions and unitary representations of low dimensional Lie groups

The unitary representations of the three dimensional simple Lie groups are reconsidered from the perspective of harmonic functions acting on certain manifolds related to differential realisations of the groups themselves. By means of contractions of Lie groups, the procedure is also applied to the group E2 of rotations-translations in two dimensions.

Quantum Bochner’s theorem for phase spaces built on projective representations

Bochner’s theorem gives the necessary and sufficient conditions on a function such that its Fourier transform corresponds to a true probability density function. In the Wigner phase space picture, quantum Bochner’s theorem gives the necessary and sufficient conditions on a function such that it is a quantum characteristic function of a valid (and possibly mixed) quantum state and such that its Fourier transform is a true probability density. We extend this theorem to discrete phase space representations which possess enough symmetry. More precisely, we show that discrete phase space representations that are built on projective unitary representations of abelian groups, with a slight restriction on admissible two-cocycles, enable a quantum Bochner’s theorem.

Endoscopie et conjecture locale raffinée de Gan–Gross–Prasad pour les groupes unitaires

Under endoscopic assumptions about $L$-packets of unitary groups, we prove the local Gan–Gross–Prasad conjecture for tempered representations of unitary groups over $p$-adic fields. Roughly, this conjecture says that branching laws for $U(n-1)\subset U(n)$ can be computed using epsilon factors.

Orientation Maps in V1 and Non-Euclidean Geometry.

In the primary visual cortex, the processing of information uses the distribution of orientations in the visual input: neurons react to some orientations in the stimulus more than to others. In many species, orientation preference is mapped in a remarkable way on the cortical surface, and this organization of the neural population seems to be important for visual processing. Now, existing models for the geometry and development of orientation preference maps in higher mammals make a crucial use of symmetry considerations. In this paper, we consider probabilistic models for V1 maps from the point of view of group theory; we focus on Gaussian random fields with symmetry properties and review the probabilistic arguments that allow one to estimate pinwheel densities and predict the observed value of π. Then, in order to test the relevance of general symmetry arguments and to introduce methods which could be of use in modeling curved regions, we reconsider this model in the light of group representation theory, the canonical mathematics of symmetry. We show that through the Plancherel decomposition of the space of complex-valued maps on the Euclidean plane, each infinite-dimensional irreducible unitary representation of the special Euclidean group yields a unique V1-like map, and we use representation theory as a symmetry-based toolbox to build orientation maps adapted to the most famous non-Euclidean geometries, viz. spherical and hyperbolic geometry. We find that most of the dominant traits of V1 maps are preserved in these; we also study the link between symmetry and the statistics of singularities in orientation maps, and show what the striking quantitative characteristics observed in animals become in our curved models.

On (χ,b)-factors of cuspidal automorphic representations of unitary groups I

Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations σ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter (χ,b) in the global Arthur parameter ψσ associated to σ, by the endoscopic classification of Arthur [Art13,Mok13,KMSW14]. The argument uses the theory of theta correspondence. This can be viewed as a part of the (χ,b)-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of certain L-functions, which was outlined in [Ral91].

Isgur-Wise functions and unitary representations of the Lorentz group: The meson case withj=12light cloud

We pursue the group theoretical method to study Isgur-Wise functions. We apply the general formalism, formerly applied to the baryon case j^P = 0^+ (for \Lambda_b -> \Lambda_c \ell \nu), to mesons with j^P = 1/2^-, i.e. $\overline{B} -> D(D^{(*)})\ell\nu. In this case, more involved from the angular momentum point of view, only the principal series of unitary representations of the Lorentz group contribute. We obtain an integral representation for the IW function xi(w) with a positive measure, recover the bounds for the slope and the curvature of xi(w) obtained from the Bjorken-Uraltsev sum rule method, and get new bounds for higher derivatives. We demonstrate also that if the lower bound for the slope is saturated, the measure is a delta-function, and xi(w) is given by an explicit elementary function. Inverting the integral formula, we obtain the measure in terms of the IW function, allowing to formulate criteria to decide if a given ansatz for the Isgur-Wise function is compatible or not with the sum rule constraints. Moreover, we have obtained an upper bound on the IW function valid for any value of w. We compare these theoretical constraints to a number of forms for \xi(w) proposed in the literature. The "dipole" function \xi(w) = (2/(w+1))^(2c) satisfies all constraints for c \geq 3/4, while the QCD Sum Rule result including condensates does not satisfy them. Special care is devoted to the Bakamjian-Thomas relativistic quark model in the heavy quark limit and to the description of the Lorentz group representation that underlies this model. Consistently, the IW function satisfies all Lorentz group criteria for any explicit form of the meson Hamiltonian at rest.