Classification of connected unimodular Lie groups with discrete series

Anh Nguyen Huu

Annales de l'Institut Fourier, Tome 30 (1980) no. 1, pp. 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 $G/\mathrm{Rad}G$ 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$\left(S\right)/Z$ est compact; d) on a $H\cap S=Z^0$.

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 $\ell$ in the Lie algebra $h$ of $H$, the symplectic for $B_{\ell }$ on $h/z$ given by $\ell \left(\right[x,y\left]\right)$ 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 $G/\mathrm{Rad}G$ is finite, has discrete series if and only if $G$ may be written as $G=H{S}^{\prime }$, $H\cap S=Z^0$, 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$\left(S\right)/Z$ is compact.

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