A pleasant exercise: Langlands’ Boulder Lemma
[Un plaisant exercice : Lemme combinatoire de Langlands]
Labesse, Jean-Pierre1
1 Institut Mathématique de Luminy UMR 7373 Aix Marseille Univ, CNRS, I2M, Marseille (France)
Annales de l'Institut Fourier, Online first, 13 p.

Dans les Proceedings de la conférence de l’AMS à Boulder en 1965 Langlands énonce un lemme combinatoire qui utilise des familles de fonctions caractéristiques attachées à des partitions ordonnées d’une base obtuse dans un espace vectoriel euclidien de dimension finie. Langlands ne donne aucune indication sur la preuve de ce lemme qu’il déclare être un «  plaisant exercice » . Comme il ne semble pas exister de preuve dans la littérature nous avons décidé d’en donner une. Nous pensons qu’elle peut être intéressante au moins du point de vue de l’histoire du domaine.

In the Proceedings of the AMS Boulder conference in 1965 Langlands states a combinatorial lemma involving families of characteristic functions attached to ordered partitions of an obtuse basis in a finite dimensional euclidean vector space. Langlands does not give any indication about the proof of the lemma which is said to be a “pleasant exercise”. Since we did not find a proof in the literature we decided to give one. We believe it of some interest for the history of the subject.

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DOI : 10.5802/aif.3699
Classification : 05A19, 11F70, 11F72, 22E55
Keywords: Langlands combinatorial lemma
Mots-clés : Lemme combinatoire de Langlands

Labesse, Jean-Pierre 1

1 Institut Mathématique de Luminy UMR 7373 Aix Marseille Univ, CNRS, I2M, Marseille (France) 
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Labesse, Jean-Pierre. A pleasant exercise: Langlands’ Boulder Lemma. Annales de l'Institut Fourier, Online first, 13 p.

[1] Arthur, James The characters of discrete series as orbital integrals, Invent. Math., Volume 32 (1976) no. 3, pp. 205-261 | DOI | MR | Zbl

[2] Arthur, James A trace formula for reductive groups. I. Terms associated to classes in G(Q), Duke Math. J., Volume 45 (1978) no. 4, pp. 911-952 | DOI | MR | Zbl

[3] Arthur, James A trace formula for reductive groups. II. Applications of a truncation operator, Compos. Math., Volume 40 (1980) no. 1, pp. 87-121 | Numdam | MR | Zbl

[4] Borel, Armand; Wallach, Nolan Continuous Cohomology, Discrete Subgroups and Representations of Reductive groups, Annals of Mathematics Studies, 94, Princeton University Press, 1980 | MR | Zbl

[5] Labesse, Jean-Pierre; Waldspurger, Jean-Loup La formule des traces tordue d’après le Friday Morning Seminar. With a foreword by Robert Langlands, CRM Monograph Series, 31, American Mathematical Society, 2013 | MR | Zbl

[6] Langlands, Robert P. Eisenstein series, Algebraic Groups and Discontinuous Subgroups (Proceedings of Symposia in Pure Mathematics), Volume IX, American Mathematical Society, 1966, pp. 235-252 | DOI | Zbl

[7] Langlands, Robert P. On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, 544, Springer, 1976 | DOI | MR | Zbl

[8] Langlands, Robert P. On the Classification of Irreducible Representations of Real Algebraic Groups, Representation theory and harmonic analysis on semisimple Lie groups (Mathematical Surveys and Monographs), Volume 31, American Mathematical Society, 1989, pp. 101-170 | DOI | Zbl

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