Classification of connected unimodular Lie groups with discrete series

Anh Nguyen Huu

Annales de l'Institut Fourier, Tome 30 (1980), p. 159-192 / Harvested from Numdam

Résumé

On introduit la classe des $H$-groupes, intermédiaire entre celle des groupes de Lie connexes nilpotents et celle des groupes de Lie connexes résolubles. Soit $G$ un groupe de Lie connexe unimodulaire, de centre $Z$, tel que $G/{\rm Rad}\, G$ ait un centre fini. A quelques restrictions techniques près, on montre qu’un tel groupe $G$ a une série discrète de représentations si et seulement s’il se représente sous la forme $G=HS$ avec les hypothèses suivantes : a) $H$ est un $H$-groupe de centre $Z^0$; b) $S$ est un groupe de Lie réductif connexe, possédant une série discrète; c) Cent$(S)/Z$ est compact; d) on a $H\cap S=Z^0$.

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.

Détail

Publié le : 1980-01-01

MR 82a:22016 | Zbl 0418.22010 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.779

Bibliographie

[1] Nguyen Huu Anh, Lie groups with square integrable representations, Annals of Math., 104 (1976), 431-458. | MR 55 #5802 | Zbl 0359.22007

[2] Nguyen Huu Anh, Classification of unimodular algebraic groups with square integrable representations, Acta Math. Vietnam. (to appear). | Zbl 0426.22012

[3] Nguyen Huu Anh and V.M. Son, On square integrable factor representations of locally compact groups, Acta Math. Vietnam. (to appear). | Zbl 0466.22007

[4] L. Auslander and B. Kostant, Polarization and unitary representations of solvable Lie groups, Invent. Math., 14 (1971), 255-354. | MR 45 #2092 | Zbl 0233.22005

[5] C. Chevalley, Théorie des groupes de Lie, Vol. 3, Act. Sci. Ind., n° 1226, Hermann, Paris, 1955.

[6] Harish-Chandra, The discrete series for semisimple Lie groups II. Explicit determination of the characters, Acta Math., 116 (1966), 1-111. | MR 36 #2745 | Zbl 0199.20102

[7] Harish-Chandra, Invariant eigendistributions on a semisimple Lie group, Trans. A.M.S., 119 (1965), 457-508. | MR 31 #4862d | Zbl 0199.46402

[8] G.W. Mackey, Unitary representations of group extensions I, Annals of Math., 99 (1958), 265-311. | MR 20 #4789 | Zbl 0082.11301

[9] C.C. Moore, The Plancherel formula for non unimodular groups, Abs. Int. Cong. on Func. Analysis, Univ. of Maryland, 1971.

[10] J. Rosenberg, Square integrable factor representations of locally compact groups, Preprint, Univ. of Calif., Berkeley. | Zbl 0412.22003

[11] I. Satake, Classification theory of semisimple algebraic groups, Lecture notes in Pure and Appl. Math., Dekker Inc., New York, 1971. | MR 47 #5135 | Zbl 0226.20037

[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. | MR 45 #8777 | Zbl 0241.22017

[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. | MR 49 #3033 | Zbl 0274.22016

[16] R. Lipsman, Representation theory of almost connected groups, Pacific J. of Math., 42, (1972), 453-467. | MR 48 #6317 | Zbl 0242.22008

[17] J.Y. Charbonnel, La formule de Plancherel pour un groupe de Lie résoluble connexe, Lecture notes, n° 587 (1977), 32-76. | MR 58 #6067 | Zbl 0365.22009