Discrete Series Representations Research Articles

Spinor-helicity representations of particles of any mass in dS4 and AdS4 spacetimes

The spinor-helicity representations of massive and (partially) massless particles in four-dimensional (anti–)de Sitter (A)dS spacetime are studied within the framework of the dual pair correspondence. We show that the dual groups (also known as “little groups”) of the anti–de Sitter and de Sitter groups are, respectively, O(2N) and O*(2N). For N=1, the generator of the dual algebra so(2)≅so*(2)≅u(1) corresponds to the helicity operator, and the spinor-helicity representation describes massless particles in (A)dS4. For N=2, the dual algebra is composed of two ideals, s and mΛ. The former ideal s≅so(3) fixes the spin of the particle, while the mass is determined by the latter ideal mΛ, which is isomorphic to so(2,1), iso(2), or so(3) depending on the cosmological constant being positive, zero, or negative. In the case of a positive cosmological constant, namely dS4, the spinor-helicity representation contains all massive particles corresponding to the principal series representations and the partially massless particles corresponding to the discrete series representations leaving out only the light massive particles corresponding to the complementary series representations. The zero and negative cosmological constant cases, which had been addressed in earlier references, are also discussed briefly. Finally, we consider the multilinear form of helicity spinors invariant under (A)dS group, which can serve as the (A)dS counterpart of the scattering amplitude, and discuss technical differences and difficulties of the (A)dS cases compared to the flat spacetime case. Published by the American Physical Society 2024

Multivariate Meixner polynomials related to holomorphic discrete series representations of [formula omitted]

We show that Griffiths’ multivariate Meixner polynomials occur as matrix coefficients of holomorphic discrete series representations of the group SU(1,d). Using this interpretation we derive several fundamental properties of the multivariate Meixner polynomials, such as orthogonality relations and difference equations. Furthermore, we also show that matrix coefficients for specific group elements lead to degenerate versions of the multivariate Meixner polynomials and their properties.

Scalar two-point functions at the late-time boundary of de Sitter

We calculate two-point functions of scalar fields of mass m and their conjugate momenta at the late-time boundary of de Sitter with Bunch-Davies boundary conditions, in general d + 1 spacetime dimensions. We perform the calculation using the wavefunction picture and using canonical quantization. With the latter one clearly sees how the late-time field and conjugate momentum operators are linear combinations of the normalized late-time operators αN and βN that correspond to unitary irreducible representations of the de Sitter group with well-defined inner products. The two-point functions resulting from these two different methods are equal and we find that both the autocorrelations of αN and βN and their cross correlations contribute to the late-time field and conjugate momentum two-point functions. This happens both for light scalars m<d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m<\\frac{d}{2}H\\right) $$\\end{document}, corresponding to complementary series representations, and heavy scalars m>d2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left(m>\\frac{d}{2}H\\right) $$\\end{document}, corresponding to principal series representations of the de Sitter group, where H is the Hubble scale of de Sitter. In the special case m = 0, only the βN autocorrelation contributes to the conjugate momentum two-point function in any dimensions and we gather hints that suggest αN to correspond to discrete series representations for this case at d = 3.

An Analogue of Ladder Representations for Classical Groups

Abstract In this paper, we introduce a notion of ladder representations for split odd special orthogonal groups and symplectic groups over a non-archimedean local field of characteristic zero. This is a natural class in the admissible dual, which contains both strongly positive discrete series representations and irreducible representations with irreducible $A$-parameters. We compute Jacquet modules and the Aubert duals of ladder representations, and we establish a formula for describing ladder representations in terms of linear combinations of standard modules.

Notes on gauge fields and discrete series representations in de Sitter spacetimes

In this note we discuss features of the simplest spinning Discrete Series Unitary Irreducible Representations (UIR) of SO(1,4). These representations are known to be realised in the single particle Hilbert space of a free gauge field propagating in a four dimensional fixed de Sitter background. They showcase distinct features as compared to the more common Principal Series realised by heavy fields. Upon computing the 1 loop Sphere path integral we show that the edge modes of the theory can be understood in terms of a Discrete Series of SO(1, 2). We then canonically quantise the theory and show how group theory constrains the mode decomposition. We further clarify the role played by the second SO(4) Casimir in the single particle Hilbert space of the theory.

Cuspidal components of Siegel modular forms for large discrete series representations of $$\textrm{Sp}_4({\mathbb {R}})$$

Cuspidal components of Siegel modular forms for large discrete series representations of $$\textrm{Sp}_4({\mathbb {R}})$$

The SL(2,R) Group Representations on Spaces of Holomorphic Functions on the Unit Disc

We can realise the representations of the group SL(2,R) on the unit disc. This is due to an isomorphism between the group SL(2,R) and the group SU(1,1). The discrete series representations for the group SL(2,R)given by\pi_{n}(g)\varphi(z)=\varphi (\frac{d z-b}{a-cz} )(a-c z)^{-n}, where n is an integer number,is on the Bergman space where n&gt;2 .Lang studies the discrete series on the group in the upper half-plane and on the unit disc. For n=1, the SL(2,R) representation is called the mock discrete series. The representation space of the mock discrete series is the Hardy space.In this article we describe the SL(2,R) representation on the Dirichlet space.

Restriction of irreducible unitary representations of 𝑆𝑝𝑖𝑛(𝑁,1) to parabolic subgroups

In this paper, we obtain explicit branching laws for all irreducible unitary representations of G = Spin ⁡ ( N , 1 ) G=\operatorname {Spin}(N,1) when restricted to a parabolic subgroup P P . The restriction turns out to be a finite direct sum of irreducible unitary representations of P P . We also verify Duflo’s conjecture for the branching laws of discrete series representations of G G with respect to P P . That is to show: in the framework of the orbit method, the branching law of a discrete series representation is determined by some geometric behavior of the moment map for the corresponding coadjoint orbit.

Formal degrees of genuine Iwahori-spherical representations

Abstract For irreducible genuine Iwahori-spherical discrete series representations of central covers, we verify the Hiraga–Ichino–Ikeda formula for their formal degrees.

Character formulas from matrix factorizations

abstract: In this paper I present a new and unified method of proving character formulas for discrete series representations of connected Lie groups by applying a Chern character-type construction to the matrix factorizations of Freed-Hopkins-Teleman. In the case of a compact group I recover the Kirillov formula, thereby exhibiting the work of Freed-Teleman as a categorification of the Kirillov correspondence. In the case of a real semisimple group I recover the Rossman character formula with only a minimal amount of analysis. The appeal of this method is that it relies almost entirely on highest-weight theory, which is a far more ubiquitous phenomenon than the varied techniques that were previously used to prove such formulas.

On a family of Siegel Poincaré series

Let [Formula: see text] be a congruence subgroup of [Formula: see text]. Using Poincaré series of [Formula: see text]-finite matrix coefficients of integrable discrete series representations of [Formula: see text], we construct a spanning set for the space [Formula: see text] of Siegel cusp forms of weight [Formula: see text]. We prove the non-vanishing of certain elements of this spanning set using Muić’s integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of [Formula: see text], we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of [Formula: see text]. Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.

Ornstein-Uhlenbeck operators and holomorphic representations for a class of bounded symmetric domains

Let G be the group of holomorphic automorphisms of a bounded symmetric domain D. We discuss the correspondence between unitarizing measures for scalar-valued holomorphic discrete series representations of G and Ornstein-Uhlenbeck operators on D. We complement the results obtained for the domains in [1] and [6] by showing that for Cartan domain II of n×n symmetric complex matrices and domain III of skew-symmetric complex matrices Z such that ZZ⁎<I, the vector fields in the infinitesimal holomorphic representation are the entries of Hua matrices. Similar results stay valid for some Kähler submanifolds M of D, assuming that the entries of the matrices in M depend linearly on fixed independent entries and M is isomorphic to a product of Cartan domains.

On Petersson norms of generic cusp forms and special values of adjoint L -functions for GSp4

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${\rm GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations. Our formula expresses the Petersson norms in terms of special values of adjoint $L$-functions and some elementary constants depending only on local representations.

Uncertainty Principles and Differential Operators on the Weighted Bergman Space

We classify self-adjoint first-order differential operators on weighted Bergman spaces on the unit disc and answer questions related to uncertainty principles for such operators. Our main tools are the discrete series representations of $$\textrm{SU}(1,1)$$ . This approach has the promise to generalize to other bounded symmetric domains.

Spectral correspondences for rank one locally symmetric spaces: the case of exceptional parameters

In this paper we complete the program of relating the Laplace spectrum for rank one compact locally symmetric spaces with the first band Ruelle-Pollicott resonances of the geodesic flow on its sphere bundle. This program was started in [FF03] by Flaminio and Forni for hyperbolic surfaces, continued in [DFG15] for real hyperbolic spaces and in [GHW21] for general rank one spaces. Except for the case of hyperbolic surfaces (see also [GHW18]) a countable set of exceptional spectral parameters always remained untreated since the corresponding Poisson transforms are neither injective nor surjective. We use vector valued Poisson transforms to treat also the exceptional spectral parameters. For surfaces the exceptional spectral parameters lead to discrete series representations of SL(2,ℝ) (see [FF03, GHW18]). In general, the resulting representations turn out to be the relative discrete series representations for associated non-Riemannian symmetric spaces.

The product of lattice covolume and discrete series formal dimension: [formula omitted]-adic [formula omitted

Let F be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in PGL(2,F) and the formal dimension of a discrete series representation of GL(2,F). The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.

A geometrical point of view for branching problems for holomorphic discrete series of conformal Lie groups

This paper is devoted to branching problems for holomorphic discrete series representations of a conformal group [Formula: see text] of a tube domain [Formula: see text] over a symmetric cone [Formula: see text] restricted to the conformal group [Formula: see text] of a tube domain [Formula: see text] holomorphically embedded in [Formula: see text]. The goal of this work is the explicit construction of the symmetry breaking and holographic operators in this geometrical setting. We answer this program with the introduction of another functional model for holomorphic discrete series representations. This model leads to a geometrical interpretation and a close relation to the theory of orthogonal polynomials for such branching problems. This program is illustrated by three cases. First, we consider the [Formula: see text]-fold tensor product of holomorphic discrete series of the universal covering of [Formula: see text]. Then, it is tested on the restrictions of a member of the scalar-valued holomorphic discrete series of the conformal group [Formula: see text] to the subgroup [Formula: see text], and finally to the subgroup [Formula: see text].

The generalized injectivity conjecture

We prove a conjecture of Casselman and Shahidi stating that the unique irreducible generic subquotient of a standard module is necessarily a subrepresentation for a large class of connected quasi-split reductive groups, in particular for those which have a root system of classical type (or product of such groups). To do so, we prove and use the existence of strategic embeddings for irreducible generic discrete series representations, extending some results of Moeglin.

Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Let pi _{alpha } be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space A^2_{alpha } (Omega ) on a bounded symmetric domain Omega , of formal dimension d_{pi _{alpha }} > 0. It is shown that if the Bergman kernel k^{(alpha )}_z is a cyclic vector for the restriction pi _{alpha } |_{Gamma } to a lattice Gamma le G (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma } is a frame for A^2_{alpha }(Omega )), then {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} le |Gamma _z|^{-1}. The estimate {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} ge |Gamma _z|^{-1} holds for k^{(alpha )}_z being a p_z-separating vector (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma / Gamma _z} being a Riesz sequence in A^2_{alpha } (Omega )). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for G ={mathrm {PSU}}(1, 1).

On sufficient density conditions for lattice orbits of relative discrete series

This note provides new criteria on a unimodular group G and a discrete series representation (pi , mathcal {H}_{pi }) of formal degree d_{pi } > 0 under which any lattice Gamma le G with {{,mathrm{vol},}}(G/Gamma ) d_{pi } le 1 (resp. {{,mathrm{vol},}}(G/Gamma ) d_{pi } ge 1) admits g in mathcal {H}_{pi } such that pi (Gamma ) g is a frame (resp. Riesz sequence). The results apply to all projective discrete series of exponential Lie groups.