The Holomorphic Discrete Series of an Affine Symmetric Space and Representations with Reproducing Kernels

1991 ◽  
Vol 326 (1) ◽  
pp. 385 ◽  
Author(s):  
G. 'Olafsson ◽  
B. Orsted
Keyword(s):  
Symmetric Space ◽  
Discrete Series ◽  
Reproducing Kernels ◽  
Holomorphic Discrete Series

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.

-types common to a Borel–de Siebenthal discrete series and its associated holomorphic discrete series

2012 ◽  
Vol 350 (23-24) ◽  
pp. 1007-1009
Author(s):  
Pampa Paul ◽  
K.N. Raghavan ◽  
Parameswaran Sankaran
Keyword(s):  
Discrete Series ◽  
Holomorphic Discrete Series

Tensor Products of Analytic Continuations of Holomorphic Discrete Series

1997 ◽  
Vol 49 (6) ◽  
pp. 1224-1241 ◽  
Author(s):  
Bent Ørsted ◽  
Genkai Zhang
Keyword(s):  
Tensor Product ◽  
Analytic Continuation ◽  
Discrete Series ◽  
Tensor Products ◽  
Holomorphic Discrete Series ◽  
Irreducible Decomposition

AbstractWe give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of SU(2, 2) with its conjugate.

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