scholarly journals The embeddings of discrete series into principal series for an exceptional real simple Lie group of type $G_2$

1996 ◽  
Vol 36 (3) ◽  
pp. 557-595
Author(s):  
Tetsumi Yoshinaga ◽  
Hiroshi Yamashita
Keyword(s):  
Lie Group ◽  
Discrete Series ◽  
Principal Series

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.

DISCRETE DECOMPOSABLE BRANCHING LAWS AND PROPER MOMENTUM MAPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250021 ◽  
Author(s):  
SALMA NASRIN
Keyword(s):  
Lie Group ◽  
Discrete Series ◽  
Geometric Quantization ◽  
Series Representation ◽  
Coadjoint Orbit ◽  
Irreducible Unitary Representation ◽  
Symmetric Pair ◽  
Holomorphic Discrete Series ◽  
Branching Laws ◽  
Scalar Type

Suppose an irreducible unitary representation π of a Lie group G is obtained as a geometric quantization of a coadjoint orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. Let H be a closed subgroup of G, and we compare the following two conditions. (1) The restriction π|H is discretely decomposable in the sense of Kobayashi. (2) The momentum map [Formula: see text] is proper. In this article, we prove that (1) is equivalent to (2) when π is any holomorphic discrete series representation of scalar type of a semisimple Lie group G and (G, H) is any symmetric pair.

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