# Classification of connected unimodular Lie groups with discrete series

Annales de l'institut Fourier (1980)

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

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topAnh 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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- [9] C.C. MOORE, The Plancherel formula for non unimodular groups, Abs. Int. Cong. on Func. Analysis, Univ. of Maryland, 1971.
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- [12] Séminaire Sophus Lie, 1954-1955, ENS, Secr. math., Paris, 1955.
- [13] N. TATSUUMA, The Plancherel formula for non unimodular locally compact groups, J. Math. Kyoto Univ., 12 (1972), 179-261. Zbl0241.22017MR45 #8777
- [14] A. WEIL, L'intégration dans les groupes topologiques et ses applications, 2e éd., Act. Sci. Ind., n° 1145, Hermann, Paris, 1953.
- [15] J.A. WOLF and C.C. MOORE, Square integrable representations of nilpotent groups, Trans. A.M.S., 185 (1973), 445-462. Zbl0274.22016MR49 #3033
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