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.

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.

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.

scholarly journals Square integrable highest weight representations

1997 ◽  
Vol 39 (3) ◽  
pp. 296-321 ◽  
Author(s):  
Karl-Hermann Neeb
Keyword(s):  
Discrete Series ◽  
Symmetric Domain ◽  
Bounded Symmetric Domain ◽  
Holomorphic Discrete Series ◽  
Highest Weight ◽  
Discrete Series Representations ◽  
Irreducible Unitary Representations ◽  
Highest Weight Representations ◽  
Highest Weight Modules ◽  
Weight Modules

If G is the group of holomorphic automorphisms of a bounded symmetric domain, then G has a distinguished class of irreducible unitary representations called the holomorphic discrete series of G. These representations have been studied by Harish-Chandra in [7]. On the Lie algebra level, the Harish-Chandra modules corresponding to the holomorphic discrete series representations are highest weight modules. Even for G as above, it turns out that not all the unitary highest weight modules belong to the holomorphic discrete series but there exists a condition on the highest weight which characterizes the holomorphic discrete series among the unitary highest weight representations. They can be defined as those unitary highest weight representations with square integrable matrix coefficients.

scholarly journals A construction by deformation of unitary irreducible representations of SU(1,n) and SU(n + 1)

2019 ◽  
Vol 18 (07) ◽  
pp. 1950125
Author(s):  
Benjamin Cahen
Keyword(s):  
Discrete Series ◽  
Irreducible Representations ◽  
Minimal Realization ◽  
Holomorphic Discrete Series ◽  
Series Representations ◽  
Discrete Series Representations

We recover the holomorphic discrete series representations of [Formula: see text] as well as some unitary irreducible representations of [Formula: see text] by deformation of a minimal realization of [Formula: see text].

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