Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property . For any such group, , 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 in are obtained, for a “geometric” generating set of the form , where is a ball of radius , and the dependence of on 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.
On considère un groupe algébrique simple non compact, défini sur un corps localement compact non discret, satisfaisant la propriété de Kazhdan. Étant donné un tel groupe , 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 dans , pour un système générateur “géométrique” de la forme où est une boule de rayon , la dépendance de en fonction de 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.
@article{AIF_2000__50_3_833_0, author = {Shalom, Yehuda}, title = {Explicit {Kazhdan} constants for representations of semisimple and arithmetic groups}, journal = {Annales de l'Institut Fourier}, pages = {833--863}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {50}, number = {3}, year = {2000}, doi = {10.5802/aif.1775}, zbl = {0966.22004}, mrnumber = {2001i:22019}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1775/} }
TY - JOUR AU - Shalom, Yehuda TI - Explicit Kazhdan constants for representations of semisimple and arithmetic groups JO - Annales de l'Institut Fourier PY - 2000 SP - 833 EP - 863 VL - 50 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1775/ DO - 10.5802/aif.1775 LA - en ID - AIF_2000__50_3_833_0 ER -
%0 Journal Article %A Shalom, Yehuda %T Explicit Kazhdan constants for representations of semisimple and arithmetic groups %J Annales de l'Institut Fourier %D 2000 %P 833-863 %V 50 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1775/ %R 10.5802/aif.1775 %G en %F AIF_2000__50_3_833_0
Shalom, Yehuda. Explicit Kazhdan constants for representations of semisimple and arithmetic groups. Annales de l'Institut Fourier, Volume 50 (2000) no. 3, pp. 833-863. doi : 10.5802/aif.1775. https://aif.centre-mersenne.org/articles/10.5802/aif.1775/
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