scholarly journals An explicit construction of the K-finite vectors in the discrete series for an isotropic semisimple symmetric space

1984 ◽  
Vol 1 ◽  
pp. 157-199 ◽  
Author(s):  
Mogens Flensted-Jensen ◽  
Kiyosato Okamoto
Keyword(s):  
Symmetric Space ◽  
Discrete Series ◽  
Explicit Construction ◽  
Semisimple Symmetric Space

scholarly journals Relative Discrete Series Representations for Two Quotients ofp-adic GLn

2018 ◽  
Vol 70 (6) ◽  
pp. 1339-1372 ◽  
Author(s):  
Jerrod Manford Smith
Keyword(s):  
Symmetric Space ◽  
Local Field ◽  
Galois Extension ◽  
Symmetric Spaces ◽  
Discrete Series ◽  
Explicit Construction ◽  
Series Representations ◽  
Discrete Series Representations ◽  
Square Integrability ◽  
Residual Characteristic

AbstractWe provide an explicit construction of representations in the discrete spectrum of twop-adic symmetric spaces. We consider GLn(F) × GLn(F)\GL2n(F) and GLn(F)\GLn(E), whereEis a quadratic Galois extension of a nonarchimedean local fieldFof characteristic zero and odd residual characteristic. The proof of the main result involves an application of a symmetric space version of Casselman’s Criterion for square integrability due to Kato and Takano.

Harmonic Spinors on Hyperbolic Space

1993 ◽  
Vol 36 (3) ◽  
pp. 257-262 ◽  
Author(s):  
Pierre-Yves Gaillard
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Representation Theory ◽  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Short Note ◽  
Harmonic Forms ◽  
Semisimple Symmetric Space ◽  
Invariant Differential ◽  
Harmonic Spinors

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

SOME VECTOR-VALUED SINGULAR AUTOMORPHIC FORMS ON U(2, 2) AND THEIR RESTRICTION TO Sp(1, 1)

2012 ◽  
Vol 23 (10) ◽  
pp. 1250104
Author(s):  
ATSUO YAMAUCHI ◽  
HIRO-AKI NARITA
Keyword(s):  
Hyperbolic Space ◽  
Symplectic Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Discrete Series ◽  
Theta Series ◽  
Explicit Construction ◽  
Discrete Series Representations ◽  
Vector Valued ◽  
Real Symplectic Group

In this paper we provide a construction of theta series on the real symplectic group of signature (1,1) or the 4-dimensional hyperbolic space. We obtain these by considering the restriction of some vector-valued singular theta series on the unitary group of signature (2,2) to this indefinite symplectic group. Our (vector-valued) theta series are proved to have algebraic Fourier coefficients, and lead to a new explicit construction of automorphic forms generating quaternionic discrete series representations and automorphic functions on the hyperbolic space.

scholarly journals An example of rank two symmetric Osserman space

1997 ◽  
Vol 56 (3) ◽  
pp. 517-521 ◽  
Author(s):  
Zoran Rakić
Keyword(s):  
Symmetric Space ◽  
Riemannian Manifolds ◽  
Explicit Construction ◽  
Quaternionic Structure ◽  
Locally Symmetric Space ◽  
Rank 2

Recently, Blažić, Bokan and Rakić, obtained some classes of 4-dimensional Osserman pseudo-Riemannian manifolds. One of these is the class of rank 2 locally symmetric space endowed with an integrable para-quaternionic structure. In this paper we give an explicit construction of an example of a space of that kind.

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.

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