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.

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.

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.

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].

Reducibility of the Principal Series for (F) over a p-adic Field

2010 ◽  
Vol 62 (4) ◽  
pp. 914-960 ◽  
Author(s):  
Christian Zorn
Keyword(s):  
Symplectic Group ◽  
Principal Series ◽  
Complete List ◽  
Intertwining Operators ◽  
Series Representations ◽  
Principal Series Representations

AbstractLet Gn = Spn(F) be the rank n symplectic group with entries in a nondyadic p-adic field F. We further let be the metaplectic extension of Gn by defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of . In most cases, we will use some techniques developed by Tadić that analyze the Jacquetmodules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters χ with χ2 = 1. Because such representations π are unitary, to show the irreducibility of π, it suffices to show that . We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of . We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in for the π in question.

Symmetric Genuine Spherical Whittaker Functions on

2015 ◽  
Vol 67 (1) ◽  
pp. 214-240 ◽  
Author(s):  
Dani Szpruch
Keyword(s):  
Symplectic Group ◽  
Series Representation ◽  
Principal Series ◽  
Uniqueness Result ◽  
Symplectic Space ◽  
Whittaker Functions ◽  
Series Representations ◽  
Spanning Set ◽  
Double Covers ◽  
Principal Series Representations

AbstractLet F be a p-adic field of odd residual characteristic. Let and be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F, respectively. Let σ be a genuine, possibly reducible, unramified principal series representation of . In these notes we give an explicit formula for a spanning set for the space of Spherical Whittaker functions attached to σ. For odd n, and generically for even n, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of . If n is odd, then each element in the set has an equivariant property that generalizes a uniqueness result proved by Gelbart, Howe, and Piatetski-Shapiro.Using this symmetric set, we construct a family of reducible genuine unramified principal series representations that have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for n even.

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