# zbMATH — the first resource for mathematics

A short proof of the Selberg principle for a $$p$$-adic group. (Une preuve courte du principe de Selberg pour un groupe $$p$$-adique.) (French) Zbl 0978.22018
Let $$F$$ be a local field of characteristic zero, $$G$$ be the group of $$F$$-points of a reductive algebraic group defined over $$F$$. The “abstract Selberg principle” proved by P. Blanc and J.-L. Brylinski [J. Funct. Anal. 109, 289-330 (1992; Zbl 0783.55004)] states that the orbital integrals of the Hattori rank of a finitely generated projective smooth representation of $$G$$ over the conjugacy classes of the non-compact elements vanish.
The author gives a new proof based on Clozel’s integration formula [L. Clozel, Ann. Math. (2) 129, 237-251 (1989; Zbl 0675.22007)] and some $$K$$-theoretic arguments.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 22E35 Analysis on $$p$$-adic Lie groups 19A49 $$K_0$$ of other rings
##### Keywords:
orbital integral; Hattori rank; Selberg principle
Full Text:
##### References:
 [1] Hyman Bass, Algebraic \?-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0174.30302 [2] Représentations des groupes réductifs sur un corps local, Travaux en Cours. [Works in Progress], Hermann, Paris, 1984 (French). · Zbl 0544.00007 [3] Philippe Blanc and Jean-Luc Brylinski, Cyclic homology and the Selberg principle, J. Funct. Anal. 109 (1992), no. 2, 289 – 330. · Zbl 0783.55004 · doi:10.1016/0022-1236(92)90020-J · doi.org [4] Laurent Clozel, Orbital integrals on \?-adic groups: a proof of the Howe conjecture, Ann. of Math. (2) 129 (1989), no. 2, 237 – 251. · Zbl 0675.22007 · doi:10.2307/1971447 · doi.org [5] J.-F. Dat. On the $${K}_0$$ of a $$p$$-adic group. Inv. Math., 140:171-226, 2000. · Zbl 0981.22004 [6] Pierre Deligne, Le support du caractère d’une représentation supercuspidale, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 4, Aii, A155 – A157 (French, with English summary). · Zbl 0336.22009 [7] Harish-Chandra, Admissible invariant distributions on reductive \?-adic groups, Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977) Queen’s Univ., Kingston, Ont., 1978, pp. 281 – 347. Queen’s Papers in Pure Appl. Math., No. 48. [8] Nigel Higson and Victor Nistor, Cyclic homology of totally disconnected groups acting on buildings, J. Funct. Anal. 141 (1996), no. 2, 466 – 495. · Zbl 0921.46083 · doi:10.1006/jfan.1996.0138 · doi.org [9] David Kazhdan, Cuspidal geometry of \?-adic groups, J. Analyse Math. 47 (1986), 1 – 36. · Zbl 0634.22009 · doi:10.1007/BF02792530 · doi.org [10] D. Kazhdan, Representations of groups over close local fields, J. Analyse Math. 47 (1986), 175 – 179. · Zbl 0634.22010 · doi:10.1007/BF02792537 · doi.org [11] Peter Schneider, The cyclic homology of \?-adic reductive groups, J. Reine Angew. Math. 475 (1996), 39 – 54. · Zbl 0849.22007 · doi:10.1515/crll.1996.475.39 · doi.org [12] G. van Dijk, Computation of certain induced characters of \?-adic groups, Math. Ann. 199 (1972), 229 – 240. · Zbl 0231.22018 · doi:10.1007/BF01429876 · doi.org [13] Marie-France Vignéras, On formal dimensions for reductive \?-adic groups, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 225 – 266. · Zbl 0732.22007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.