Haar measure and the Artin conductor. (English) Zbl 0991.20033

Gross, Benedict H.; Gan, Wee Teck

Trans. Am. Math. Soc. 351, No. 4, 1691-1704 (1999).

Summary: Let \(G\) be a connected reductive group, defined over a local, non-Archimedean field \(k\). The group \(G(k)\) is locally compact and unimodular. B. H. Gross [Invent. Math. 130, No. 2, 287-313 (1997; Zbl 0904.11014)] defined a Haar measure \(|\omega_G|\) on \(G(k)\), using the theory of Bruhat and Tits. In this note, we give another construction of the measure \(|\omega_G|\), using the Artin conductor of the motive \(M\) of \(G\) over \(k\). The equivalence of the two constructions is deduced from a result of G. Prasad.

MSC:

20G25	Linear algebraic groups over local fields and their integers
11G09	Drinfel’d modules; higher-dimensional motives, etc.
43A05	Measures on groups and semigroups, etc.

Keywords:

connected reductive groups; local fields; Tamagawa numbers

Citations:

Zbl 0904.11014

References:

[1]	E. Hlawka, Interpolation analytischer Funktionen auf dem Einheitskreis, Number Theory and Analysis (Papers in Honor of Edmund Landau), Plenum, New York, 1969, pp. 97 – 118 (German). F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197 – 376 (French).
[2]	C. Chevalley, Invariants of Finite Groups Generated by Reflections, Amer. J. Math. 77(1955), Pg 778-782. · Zbl 0065.26103
[3]	B.H. Gross, On the Motive of a Reductive Group, Invent. Math. 130 (1997), 287-313. CMP 98:02
[4]	Robert E. Kottwitz, Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc. 278 (1983), no. 1, 289 – 297. · Zbl 0538.22010
[5]	Gérard Laumon, Cohomology of Drinfeld modular varieties. Part I, Cambridge Studies in Advanced Mathematics, vol. 41, Cambridge University Press, Cambridge, 1996. Geometry, counting of points and local harmonic analysis. Gérard Laumon, Cohomology of Drinfeld modular varieties. Part II, Cambridge Studies in Advanced Mathematics, vol. 56, Cambridge University Press, Cambridge, 1997. Automorphic forms, trace formulas and Langlands correspondence; With an appendix by Jean-Loup Waldspurger. · Zbl 0837.14018
[6]	John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. · Zbl 0292.10016
[7]	Takashi Ono, Arithmetic of algebraic tori, Ann. of Math. (2) 74 (1961), 101 – 139. · Zbl 0119.27801 · doi:10.2307/1970307
[8]	Gopal Prasad, Volumes of \?-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91 – 117. With an appendix by Moshe Jarden and the author. · Zbl 0695.22005
[9]	Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. · Zbl 0355.20006
[10]	Jean-Pierre Serre, Conducteurs d’Artin des caractères réels, Invent. Math. 14 (1971), 173 – 183 (French). · Zbl 0229.13006 · doi:10.1007/BF01418887
[11]	Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. · Zbl 0423.12016
[12]	T. A. Springer, Reductive groups, Automorphic forms, representations and \?-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3 – 27.
[13]	Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. · Zbl 0164.02902
[14]	John Tate, Les conjectures de Stark sur les fonctions \? d’Artin en \?=0, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. · Zbl 0545.12009