Classification of connected unimodular Lie groups with discrete series
Annales de l'Institut Fourier, Tome 30 (1980) no. 1, p. 159-192
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/ 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 HS=Z 0 .
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 in the Lie algebra h of H, the symplectic for B on h/z given by ([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=HS , HS=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.
@article{AIF_1980__30_1_159_0,
     author = {Anh Nguyen Huu},
     title = {Classification of connected unimodular Lie groups with discrete series},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {30},
     number = {1},
     year = {1980},
     pages = {159-192},
     doi = {10.5802/aif.779},
     zbl = {0418.22010},
     mrnumber = {82a:22016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1980__30_1_159_0}
}
Anh Nguyen Huu. Classification of connected unimodular Lie groups with discrete series. Annales de l'Institut Fourier, Tome 30 (1980) no. 1, pp. 159-192. doi : 10.5802/aif.779. http://www.numdam.org/item/AIF_1980__30_1_159_0/

[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