Explicit Kazhdan constants for representations of semisimple and arithmetic groups

Shalom, Yehuda

Annales de l'Institut Fourier, Tome 50 (2000) no. 3, pp. 833-863.

On considère un groupe algébrique simple non compact, défini sur un corps localement compact non discret, satisfaisant la propriété $\left(T\right)$ de Kazhdan. Étant donné un tel groupe $G$, nous décrivons un ensemble de Kazhdan à deux éléments, et nous calculons sa meilleure constante de Kazhdan. Alors, répondant à une question de Serre et de la Harpe et Valette, nous obtenons des constantes de Kazhdan explicites pour tout réseau $\Gamma$ dans $G$, pour un système générateur “géométrique” de la forme $\Gamma \cap B_r$ où $\Gamma \cap B_r$ est une boule de rayon $r$, la dépendance de $r$ en fonction de $\Gamma$ étant décrite de façon explicite. De plus, pour tous les groupes de Lie de rang un, nous en déduisons des constantes de Kazhdan explicites, pour toute famille de représentations admettant une lacune spectrale. Nous discutons également plusieurs applications de nos méthodes, notamment une extension du théorème de Howe-Moore.

Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property $\left(T\right)$. For any such group, $G$, we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice $\Gamma$ in $G$ are obtained, for a “geometric” generating set of the form $\Gamma \cap B_r$, where $B_r\subset G$ is a ball of radius $r$, and the dependence of $r$ on $\Gamma$ is described explicitly. Furthermore, for all rank one Lie groups we derive explicit Kazhdan constants, for any family of representations which admits a spectral gap. Several applications of our methods are discussed as well, among them, an extension of Howe-Moore’s theorem.

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