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.

Fourier transforms on a semisimple symmetric space

1997 ◽  
Vol 130 (3) ◽  
pp. 517-574 ◽  
Author(s):  
Erik van den Ban ◽  
Henrik Schlichtkrull
Keyword(s):  
Symmetric Space ◽  
Fourier Transforms ◽  
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.

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