Discrete Series Representations Of SU Research Articles

Coherent states and geometry on the Siegel–Jacobi disk

The coherent state representation of the Jacobi group [Formula: see text] is indexed with two parameters, [Formula: see text], describing the part coming from the Heisenberg group, and k, characterizing the positive discrete series representation of SU(1,1). The Ricci form, the scalar curvature and the geodesics of the Siegel–Jacobi disk [Formula: see text] are investigated. The significance in the language of coherent states of the transform which realizes the fundamental conjecture on the Siegel–Jacobi disk is emphasized. The Berezin kernel, Calabi's diastasis, the Kobayashi embedding and the Cauchy formula for the Siegel–Jacobi disk are presented.

Matrix coefficients of the middle discrete series of [formula omitted] II, the explicit asymptotic expansion

We obtain the explicit asymptotic expansion of the matrix coefficients of the middle discrete series representation of SU ( 2 , 2 ) by means of degenerate Gaussian hypergeometric series.

Generalized Orthogonality Relations and SU(1,1)-Quantum Tomography

We present a mathematically precise derivation of some generalized orthogonality relations for the discrete series representations of SU(1,1). These orthogonality relations are applied to derive tomographical reconstruction formulas. Their physical interpretation is also discussed.

Certain components of Springer fibers and associated cycles for discrete series representations of 𝑆𝑈(𝑝,𝑞)

An explicit geometric description of certain components of Springer fibers for S L ( n , C ) SL(n,C) s given in this article. These components are associated to closed S ( G L ( p ) × G L ( q ) ) S(GL(p)\times GL(q)) -orbits in the flag variety. The geometric results are used to compute the associated cycles of the discrete series representations of S U ( p , q ) SU(p,q) . A discussion of an alternative, and more general, computation of associated cycles is given in the appendix.

Contractions ofSU(1, n) andSU(n+1) via Berezin quantization

We establish contractions of discrete series representations ofSU(1,n) and of unitary irreducible representations ofSU(n+1) to the unitary irreducible representations of the (2n+1)-dimensional Heisenberg group by use of the Berezin calculus on the coadjoint orbits associated to these representations by the Kirillov-Kostant method of orbits.

On the Clebsch–Gordan problem for SU(1,1): Coupling nonstandard representations

The Clebsch–Gordan coefficients coupling two unitary, irreducible, positive discrete series representations of SU(1,1), are constructed. In contrast to the Clebsch–Gordan coefficients obtained a long time ago by Holman and Biedenharn [Ann. Phys. (N.Y.) 39, 1 (1966)], the derived coefficients are valid even when coupling nonstandard representations such as those for which the corresponding Bargmann indices k may be k=14 and/or 34, values associated with the “two-photon” realization of the su(1,1) Lie algebra, the corresponding representations covering the even and odd number states, respectively, of the single-mode boson system. These nonstandard cases are actually representations associated with the covering group SU¯(1,1). The results are extended to the coupling of three positive discrete series and the corresponding SU(1,1) Racah coefficients are obtained.

Laguerre functions and representations of su(1,1)

Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra su(l, 1). The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of su(l,l) are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and functions as overlap coefficients, we obtain a product formula for the Laguerre polynomials, given by an integral over Laguerre functions, Jacobi functions and continuous dual Hahn polynomials.

Realizations of coupled vectors in the tensor product of representations of [formula omitted] and [formula omitted

Using the realization of positive discrete series representations of su(1,1) in terms of a complex variable z, we give an explicit expression for coupled basis vectors in the tensor product of ν+1 representations as polynomials in ν+1 variables z 1,…, z ν+1 . These expressions use the terminology of binary coupling trees (describing the coupled basis vectors), and are explicit in the sense that there is no reference to the Clebsch–Gordan coefficients of su(1,1) . In general, these polynomials can be written as (terminating) multiple hypergeometric series. For ν=2, these polynomials are triple hypergeometric series, and a relation between the two binary coupling trees yields a relation between two triple hypergeometric series. The case of su(2) is discussed next. Also here the polynomials are determined explicitly in terms of a known realization; they yield an efficient way of computing coupled basis vectors in terms of uncoupled basis vectors.

3nj-coefficients of su(1,1) as connection coefficients between orthogonal polynomials in n variables

In the tensor product of n+1 positive discrete series representations of su(1,1), a coupled basis vector can be described by a certain binary coupling tree. To every such binary coupling tree, polynomials Rl(k)(x) and Rl(k)(x) are associated. These polynomials are n-variable Jacobi and continuous Hahn polynomials, and are orthogonal with respect to a weight function. The connection coefficients expressing such a polynomial associated with a given binary coupling tree in terms of those polynomials associated with another binary coupling tree are proportional to 3nj-coefficients of su(1,1).

Composition series for analytic continuations of holomorphic discrete series representations of SU p, q

A family of holomorphic discrete series representations of SU p, q is studied, and analytic continuation in terms of the reproducing kernel parameter ν is discussed. The composition series of the module of all polynomials on the hermitian symmetric domain D z of SU p, q is determined for those values of ν for which the representations are reducible.

Orthogonality Relations and Harmonic Forms for Semisimple Lie Groups

Let G be a connected semisimple Lie group with finite center, and suppose G contains a compact Cartan subgroup T. Certain irreducible unitary representations of G arise as spaces of harmonic forms associated to Dolbeault cohomology of line bundles over the complex homogeneous space G/T. In this work the unitary structures of these realizations are directly related to the orthogonality relations for the matrix coefficients of these representations. Using this connection, we exhibit unitary realizations of certain limits of discrete series representations of SU(2, 1) as spaces of harmonic forms.

The expressions of the Harish-Chandra C-functions of semisimple Lie groups Spin(n,1), SU(n,1)

We give the recursion formula of the Harish-Chandra C-function with respect to the highest weight of the representations of K. Using this formula, we get the explicit expressions of the Harish-Chandra C-functions for Spin(n,1) and SU(n,1). As an application, by using these expressions, we get the realizations of discrete series representations of SU(n,1) as subquotients of nonunitary principal series representations. We shall also get the decompositions of holomorphic and antiholomorphic discrete series when restricted to U(n-1,1). By using the structures of K-spectra of discrete series representations, we can concretely construct the invariant subspaces of the representation spaces of holomorphic and antiholomorphic discrete series.

Holomorphic realization of [formula omitted]-cohomology and constructions of representations

A language is developed for ∂̄-cohomology, which is different from both the Dolbeault and the Čech descriptions, and involves only holomorphic objects. This language is then illustrated in certain cases of interest to representation theory. This makes possible a new geometric construction of the ladder representations for SU(2, p) and the non-holomorphic discrete series representations of SU(2, 1). The constructions are closely related to Penrose transforms.

Poincare contraction of SU(1,1) Fock-Bargmann structure

Some aspects of the contraction process SO0(1,2) to Poincare are studied. The starting point is the choice of a suitable parametrization for the de Sitterian phase space SO0(1,2)/SO(2) approximately=SU(1,1)/U(1). The authors show that the contraction to Poincare must be realized by restricting the Fock-Bargmann space to a specific subspace. This constraint is necessary to make the divergent terms disappear. In particular, the classical result according to which the discrete series representation of SU(1,1) contracts onto the Wigner representation P(m) is described at a global level.

Representations of Osp(2, 1) and the metaplectic representation

Shift operator techniques are used to treat the irreducible representations of the superalgebra Osp(2, 1). Apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact SU(2) algebra, the case when the star conditions on the even part are those satisfied by the noncompact SU(1, 1) algebra is also treated. In this case no gradestar representations arise, and the star representations are found to consist of the direct sum of two discrete series representations of SU(1, 1). One of these representations can be realized in terms of functions of a single complex variable, and turns out to be a simple example of a metaplectic representation.

Composition Series for Analytic Continuations of Holomorphic Discrete Series Representations of SU(n, n)

Composition Series for Analytic Continuations of Holomorphic Discrete Series Representations of SU(n, n)

Composition series for analytic continuations of holomorphic discrete series representations of 𝑆𝑈(𝑛,𝑛)

We study a certain family of holomorphic discrete series representations of the semisimple Lie group G = S U ( n , n ) G\, = \,SU(n,\,n) and the corresponding analytic continuation in the inducing parameter λ \lambda . At the values of λ \lambda where the representations become reducible, we compute the composition series in terms of a Peter-Weyl basis on the Shilov boundary of the Hermitian symmetric space for G.