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Formal degree and existence of stable arithmetic lattices of cuspidal representations of p-adic reductive groups. (English) Zbl 0715.22021

Let \(\underline G\) be a connected reductive group over a local non- archimedean field F. An irreducible complex representation \(\pi\) of \(G=\underline G(F)\) is called cuspidal (quasicuspidal) if the support of any matrix coefficient of \(\pi\) is compact (compact-mod-center). Let \(\pi\) be a cuspidal representation and let \(dg_{\pi}=d_{\pi}\cdot dg\) be the formal degree of \(\pi\). A. Borel and G. Harder [J. Reine Angew. Math. 298, 53-64 (1978; Zbl 0385.14014)] showed that \(vol(H\setminus G,dg_{\pi})\in {\mathbb{N}}\) for H a discrete cocompact torsion free subgroup of G. The author pushes the method to prove the existence of a G-stable arithmetic lattice of \(\pi\), i.e., a G-stable B- lattice in the space of \(\pi\), where B is the ring of integers in a number field. This implies in characteristic zero that \(vol(\Gamma,dg_{\pi})^{-1}\in {\mathbb{N}}_{(q)}=\{nq^ r|\) \(n\in {\mathbb{N}}\), \(r\in {\mathbb{Z}}\}\), where \(\Gamma\) is any open compact pro-p- subgroup of G, and the order q of the residue field of F is a power of the prime p. Thus, modulo a power of p, \(dg_{\pi}\) divides \(vol(\Gamma,dg)^{-1}dg\). A theorem of D. Kazhdan [J. Anal. Math. 47, 175-179 (1986; Zbl 0634.22010)] allows the extension of the results to arbitrary characteristic. The results also extend to quasicuspidal \(\pi\), and can be used to study modular representations and properties of idempotents in the Bernstein center.
Reviewer: C.D.Keys

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22E35 Analysis on \(p\)-adic Lie groups
22E40 Discrete subgroups of Lie groups
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References:

[1] Bernstein, J.-N.: Redigé par Deligne, P., Le centre de Bernstein. In: Représentations des groupes réductifs sur un corps local. Travaux en cours. Paris: Hermann 1984
[2] Bernstein, J.-N., Zelevinski, A.V.: Representations of the groupGL(n, F) whereF is a nonarchimedean local field. Russ. Math. Surv.31, 1-68 (1976); Usp. Mat. Nauk31, 5-70 (1976) · Zbl 0348.43007 · doi:10.1070/RM1976v031n03ABEH001532
[3] Borel, A.: Admissible representations of a semi-simple group over a local field. Invent. Math.35, 233-259 (1976) · Zbl 0334.22012 · doi:10.1007/BF01390139
[4] Borel, A., Harder, G.: Existence of discrete cocompact subgroups of reductive groups over local fields. J. Reine Angew. Math.298, 53-64 (1978) · Zbl 0385.14014
[5] Bourbaki, N.: Groupes et Algèbres de Lie, Chapitres 4, 5 et 6. Paris: Hermann 1968 · Zbl 0186.33001
[6] Corwin, L., Moy, A., Sally, P.: Degrees and formal degrees for division algebras andGL n over ap-adic field. Preprint (1987) · Zbl 0689.22009
[7] Ellison, W.J., Mendès-France, M.: Les nombres premiers. Paris: Hermann 1970
[8] Harish-Chandra.: Harmonic analysis on reductivep-adic groups. (Lect. Notes Math., Vol. 162). (Notes by G, van Dyk). Berlin-Heidelberg-New York: Springer 1970
[9] Harish-Chandra: Admissible invariant distributions on reductivep-adic groups. Lie theories and applications. Queen’s Pap. Pure Appl. Math.48, 281-347 (1978)
[10] Kazhdan, D.: Representations of groups over close local fields. J. Anal. Math.47, 175-179 (1986) · Zbl 0634.22010 · doi:10.1007/BF02792537
[11] Kottwitz, R.E.: Tamagawa measures. Ann. Math.127, 629-646 (1988) · Zbl 0678.22012 · doi:10.2307/2007007
[12] Moy, A., Sally, P.: Supercuspidal representations ofSL n over ap-adic field: the tame case. Duke Math. J.1, 149-161 (1984) · Zbl 0539.22014 · doi:10.1215/S0012-7094-84-05108-1
[13] Rogawski, J. D.: An application of the building to orbital integrals. Compos. Math.32, 417-423 (1981) · Zbl 0471.22020
[14] Rogawski, J.D.: Representations ofGL n and divisions algebras over ap-adic field. Duke Math. J.50, 161-196 (1983) · Zbl 0523.22015 · doi:10.1215/S0012-7094-83-05006-8
[15] Serre, J.-P.: Oeuvres II, III. Berlin-Heidelberg-New York: Springer 1986
[16] Vignéras, M.-F.: Représentations modulaires deGL(2,F) en caractéristiquel, F corpsp-adique. Compos. Math. (to appear)
[17] Waldspurger, J.-L.: Algèbres de Hecke et induites de représentations cuspidales pourGL(N). J. Reine Angew. Math.370, 127-191 (1986) · Zbl 0586.20020 · doi:10.1515/crll.1986.370.127
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