Classification of connected unimodular Lie groups with discrete series

Anh Nguyen Huu

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 1, page 159-192
  • ISSN: 0373-0956

Abstract

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We introduce a new class of connected solvable Lie groups called H -group. Namely a H -group is a unimodular connected solvable Lie group with center Z such that for some ℓ in the Lie algebra h of H , the symplectic for B ℓ on h / z given by ℓ ( [ x , y ] ) is nondegenerate. Moreover, apart form some technical requirements, it will be proved that a connected unimodular Lie group G with center Z , such that the center of G / Rad G is finite, has discrete series if and only if G may be written as G = H S ' , H ∩ S = Z 0 , where H is a H -group with center Z 0 and S ' is a connected reductive Lie group with discrete series such that Cent ( S ) / Z is compact.

How to cite

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Anh Nguyen Huu. "Classification of connected unimodular Lie groups with discrete series." Annales de l'institut Fourier 30.1 (1980): 159-192. <http://eudml.org/doc/74439>.

@article{AnhNguyenHuu1980,
abstract = {We introduce a new class of connected solvable Lie groups called $H$-group. Namely a $H$-group is a unimodular connected solvable Lie group with center $Z$ such that for some $\ell $ in the Lie algebra $h$ of $H$, the symplectic for $B_\ell $ on $h/z$ given by $\ell ([x,y])$ is nondegenerate. Moreover, apart form some technical requirements, it will be proved that a connected unimodular Lie group $G$ with center $Z$, such that the center of $G/\{\rm Rad\}\, G$ is finite, has discrete series if and only if $G$ may be written as $G=HS^\{\prime \}$, $H\cap S=Z^0$, where $H$ is a $H$-group with center $Z^0$ and $S^\{\prime \}$ is a connected reductive Lie group with discrete series such that Cent$(S)/Z$ is compact.},
author = {Anh Nguyen Huu},
journal = {Annales de l'institut Fourier},
keywords = {solvable Lie group; H-group; connected unimodular Lie groups; discrete series},
language = {eng},
number = {1},
pages = {159-192},
publisher = {Association des Annales de l'Institut Fourier},
title = {Classification of connected unimodular Lie groups with discrete series},
url = {http://eudml.org/doc/74439},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Anh Nguyen Huu
TI - Classification of connected unimodular Lie groups with discrete series
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 1
SP - 159
EP - 192
AB - We introduce a new class of connected solvable Lie groups called $H$-group. Namely a $H$-group is a unimodular connected solvable Lie group with center $Z$ such that for some $\ell $ in the Lie algebra $h$ of $H$, the symplectic for $B_\ell $ on $h/z$ given by $\ell ([x,y])$ is nondegenerate. Moreover, apart form some technical requirements, it will be proved that a connected unimodular Lie group $G$ with center $Z$, such that the center of $G/{\rm Rad}\, G$ is finite, has discrete series if and only if $G$ may be written as $G=HS^{\prime }$, $H\cap S=Z^0$, where $H$ is a $H$-group with center $Z^0$ and $S^{\prime }$ is a connected reductive Lie group with discrete series such that Cent$(S)/Z$ is compact.
LA - eng
KW - solvable Lie group; H-group; connected unimodular Lie groups; discrete series
UR - http://eudml.org/doc/74439
ER -

References

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  2. [2] NGUYEN HUU ANH, Classification of unimodular algebraic groups with square integrable representations, Acta Math. Vietnam. (to appear). Zbl0426.22012
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  14. [14] A. WEIL, L'intégration dans les groupes topologiques et ses applications, 2e éd., Act. Sci. Ind., n° 1145, Hermann, Paris, 1953. 
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