BRANCHING RULES OF SINGULAR UNITARY REPRESENTATIONS WITH RESPECT TO SYMMETRIC PAIRS (A2n-1, Dn)

2013 ◽  
Vol 24 (04) ◽  
pp. 1350011
Author(s):  
HIDEKO SEKIGUCHI
Keyword(s):  
Discrete Series ◽  
Unitary Representations ◽  
Local Analysis ◽  
Symmetric Pair ◽  
Indefinite Metric ◽  
Holomorphic Discrete Series ◽  
Branching Rule ◽  
Branching Rules ◽  
Discrete Series Representations ◽  
Multiplicity Free

The irreducible decomposition of scalar holomorphic discrete series representations when restricted to semisimple symmetric pairs (G, H) is explicitly known by Schmid [Die Randwerte holomorphe funktionen auf hermetisch symmetrischen Raumen, Invent. Math.9 (1969–1970) 61–80] for H compact and by Kobayashi [Multiplicity-Free Theorems of the Restrictions of Unitary Highest Weight Modules with Respect to Reductive Symmetric Pairs, Progress in Mathematics, Vol. 255 (Birhäuser, 2007), pp. 45–109] for H non-compact. In this paper, we deal with the symmetric pair (U(n, n), SO* (2n)), and extend the Kobayashi–Schmid formula to certain non-tempered unitary representations which are realized in Dolbeault cohomology groups over open Grassmannian manifolds with indefinite metric. The resulting branching rule is multiplicity-free and discretely decomposable, which fits in the framework of the general theory of discrete decomposable restrictions by Kobayashi [Discrete decomposability of the restriction of A𝔮(λ) with respect to reductive subgroups II — micro-local analysis and asymptotic K-support, Ann. Math.147 (1998), 709–729].

scholarly journals ON THE CHIRAL RINGS IN N=2 AND N=4 SUPERCONFORMAL ALGEBRAS

1993 ◽  
Vol 08 (02) ◽  
pp. 301-324 ◽  
Author(s):  
MURAT GÜNAYDIN
Keyword(s):  
Discrete Series ◽  
Special Property ◽  
Unitary Representations ◽  
Holomorphic Discrete Series ◽  
Triple Systems ◽  
Chiral Rings ◽  
Coset Spaces ◽  
Discrete Series Representations ◽  
Jordan Triple Systems ◽  
Infinite Set

We study the chiral primary rings of N=2 and N=4 superconformal algebras (SCA’s) constructed over triple systems. The chiral primary states of N=2 SCA’s realized over Hermitian Jordan triple systems are given. Their coset spaces G/H are Hermitian-symmetric and can be compact or noncompact. In the noncompact case under the requirement of unitarity of the representations of G, we find an infinite discrete set of chiral primary states associated with the holomorphic discrete series representations of G and their analytic continuation. A further requirement that the corresponding N=2 module be unitary truncates this infinite set to a finite subset. There are no chiral primary states associated with the other unitary representations of noncompact groups. Remarkably, the only noncompact groups G that admit holomorphic discrete series unitary representations are such that their quotients G/H with their maximal compact subgroups H are Hermitian-symmetric. The chiral primary states of N=2 SCA’s constructed over the Freudenthal triple systems are also studied. These algebras have the special property that they admit an extension to N=4 superconformal algebras with the gauge group SU(2) ⊗ SU(2) ⊗ U(1). We then generalize the concept of chiral rings to these maximal N=4 superconformal algebras. We find four different rings associated with each sector (left- or right-moving). Inclusion of both sectors gives 16 different rings. We also show that our analysis yields all the possible rings of N=4 SCA’s.

On the Super-Unitarity of Discrete Series Representations of Orthosymplectic Lie Superalgebras

1998 ◽  
Vol 10 (04) ◽  
pp. 467-497
Author(s):  
Amine M. El Gradechi
Keyword(s):  
Discrete Series ◽  
Series Representation ◽  
Lie Superalgebras ◽  
Point Of View ◽  
Low Rank ◽  
Holomorphic Discrete Series ◽  
Computational Point ◽  
First Order ◽  
Series Representations ◽  
Discrete Series Representations

We investigate the notion of super-unitarity from a functional analytic point of view. For this purpose we consider examples of explicit realizations of a certain type of irreducible representations of low rank orthosymplectic Lie superalgebras which are super-unitary by construction. These are the so-called superholomorphic discrete series representations of osp (1/2,ℝ) and osp (2/2,ℝ) which we recently constructed using a ℤ2–graded extension of the orbit method. It turns out here that super-unitarity of these representations is a consequence of the self-adjointness of two pairs of anticommuting operators which act in the Hilbert sum of two Hilbert spaces each of which carrying a holomorphic discrete series representation of su (1,1) such that the difference of the respective lowest weights is [Formula: see text]. At an intermediate stage, we show that the generators of the considered orthosymplectic Lie superalgebras can be realized either as matrix-valued first order differential operators or as first order differential superoperators. Even though the former realization is less convenient than the latter from the computational point of view, it has the advantage of avoiding the use of anticommuting Grassmann variables, and is moreover important for our analysis of super-unitarity. The latter emphasizes the fundamental role played by the atypical (or degenerate) superholomorphic discrete series representations of osp (2/2,ℝ) for the super-unitarity of the other representations considered in this work, and shows that the anticommuting (unbounded) self-adjoint operators mentioned above anticommute in a proper sense, thus connecting our work with the analysis of supersymmetric quantum mechanics.

scholarly journals MORITA'S THEORY FOR THE SYMPLECTIC GROUPS

2011 ◽  
Vol 07 (08) ◽  
pp. 2115-2137 ◽  
Author(s):  
ZHI QI ◽  
CHANG YANG
Keyword(s):  
Symplectic Group ◽  
Discrete Series ◽  
Principal Series ◽  
Symplectic Groups ◽  
Holomorphic Discrete Series ◽  
Algebraic Interpretation ◽  
Series Representations ◽  
Discrete Series Representations ◽  
Principal Series Representations

We construct and study the holomorphic discrete series representations and the principal series representations of the symplectic group Sp (2n, F) over a p-adic field F as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for SL (2, F) defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.

scholarly journals ON THE GEOMETRY OF SIEGEL–JACOBI DOMAINS

2011 ◽  
Vol 08 (08) ◽  
pp. 1783-1798 ◽  
Author(s):  
S. BERCEANU ◽  
A. GHEORGHE
Keyword(s):  
Discrete Series ◽  
Unitary Representations ◽  
Fock Spaces ◽  
Holomorphic Discrete Series ◽  
Orthonormal Bases ◽  
Jacobi Group ◽  
Representation Spaces

We study the holomorphic unitary representations of the Jacobi group based on Siegel–Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel–Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel–Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.

Tensor Products of Holomorphic Discrete Series Representations

1979 ◽  
Vol 31 (4) ◽  
pp. 836-844 ◽  
Author(s):  
Joe Repka
Keyword(s):  
Discrete Series ◽  
Tensor Products ◽  
Holomorphic Functions ◽  
Verma Modules ◽  
Holomorphic Discrete Series ◽  
Spaces Of Holomorphic Functions ◽  
Highest Weight ◽  
Generalized Verma Modules ◽  
Series Representations ◽  
Discrete Series Representations

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.

The Jacobi group and its holomorphic discrete series representations on Siegel–Jacobi domains

2019 ◽  
pp. 345-353
Author(s):  
Stefan Berceanu ◽  
Alexandru Gheorghe
Keyword(s):  
Discrete Series ◽  
Holomorphic Discrete Series ◽  
Series Representations ◽  
Jacobi Group ◽  
Discrete Series Representations ◽  
Jacobi Groups

This is the summary of a part of the talk delivered at the workshop held at the Tambov University in September 2012, reporting several results on Jacobi groups and its holomorphic representations published by the authors.

DISCRETE DECOMPOSABLE BRANCHING LAWS AND PROPER MOMENTUM MAPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250021 ◽  
Author(s):  
SALMA NASRIN
Keyword(s):  
Lie Group ◽  
Discrete Series ◽  
Geometric Quantization ◽  
Series Representation ◽  
Coadjoint Orbit ◽  
Irreducible Unitary Representation ◽  
Symmetric Pair ◽  
Holomorphic Discrete Series ◽  
Branching Laws ◽  
Scalar Type

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions. (1) The restriction π|H is discretely decomposable in the sense of Kobayashi. (2) The momentum map [Formula: see text] is proper. In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

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