Realization of the discrete series of unitary representations of Sl(2,R) in terms of creation and annihilation operators

1992 ◽  
Vol 33 (5) ◽  
pp. 1627-1630
Author(s):  
Samuel W. MacDowell
Keyword(s):  
Discrete Series ◽  
Unitary Representations ◽  
Creation And Annihilation Operators

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.

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q II: THE CONTINUOUS SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 37-47 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Kronecker Product ◽  
Discrete Series ◽  
Unitary Representations ◽  
Continuous Series ◽  
Maximal Dimension ◽  
Cartan Subgroup ◽  
Integrable Functions ◽  
Irreducible Unitary Representations ◽  
Square Integrable Functions ◽  
Vector Part

A class of irreducible unitary representations belonging to the continuous series of SUp,q is explicitly determined in a space composed of the Kronecker product of three spaces of square integrable functions. The continuous series corresponds to a Cartan subgroup whose vector part has maximal dimension. These representations are distinguished by a parameter r = 1, 2, …, p for p ≤ q in SUp,q. For r = 0, one obtains the representations in the discrete series as in [5], and all representations in the continuous series, for r ≠ 0, are obtained explicitly.

IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q I: THE DISCRETE SERIES

2001 ◽  
Vol 12 (01) ◽  
pp. 1-36 ◽  
Author(s):  
RAJ WILSON ◽  
ELIZABETH TANNER
Keyword(s):  
Differential Operators ◽  
Discrete Series ◽  
Compact Subgroup ◽  
Holomorphic Functions ◽  
Irreducible Unitary Representation ◽  
Explicit Construction ◽  
Unitary Representations ◽  
Infinitesimal Generators ◽  
Algebraic Properties ◽  
Irreducible Unitary Representations

A class of irreducible unitary representations in the discrete series of SUp,q is explicitly determined in a space of holomorphic functions of three complex matrices. The discrete series, which is the set of all square integrable representations, corresponds to a compact subgroup of SUp,q. The relevant algebraic properties of the group SUp,q are discussed in detail. For a degenerate irreducible unitary representation an explicit construction of the infinitesimal generators of the Lie algebra [Formula: see text] in terms of differential operators is given.

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.

scholarly journals UNITARY REPRESENTATIONS OF W INFINITY ALGEBRAS

1992 ◽  
Vol 07 (25) ◽  
pp. 6339-6355 ◽  
Author(s):  
SATORU ODAKE
Keyword(s):  
Central Charge ◽  
Discrete Series ◽  
Free Field ◽  
Unitary Representations ◽  
Continuous Series ◽  
Highest Weight ◽  
Free Field Realization ◽  
Highest Weight Representations

We study the irreducible unitary highest weight representations, which are obtained from free field realizations, of W infinity algebras [Formula: see text] with central charges (2, 1, 3, 2M, N, 2M+N). The characters of these representations are computed. We construct a new extended superalgebra [Formula: see text], whose bosonic sector is [Formula: see text]. Its representations obtained from a free field realization with central charge 2M+N, are classified into two classes: continuous series and discrete series. For the former there exists a supersymmetry, but for the latter a supersymmetry exists only for M=N.

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].

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