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L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ I.-ORBITAL INTEGRALS i. Notations.-Let r ^ 2, n ^ i be integers, and F a local or global field of characteristic o which contains the group ^ of n-th roots of i. If n ^ 3 and F is global then F is totally imaginary. If F is global and v is a place of F, we write F^ for the completion ofF in the valuation | |y, normalized as usual so that the product formula holds. Ifv is non-archimedean, we put p for the residual characteristic of F^; R = Rf or the ring of integers; n for a uniformizer; q for the cardinality of the field R/wR.

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A theorem of Grothendieck asserts that over a perfect field k k of cohomological dimension one, all non-abelian H 2 H^{2} -cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization — to the context of perfect fields of virtual cohomological dimension one — takes the form of a local-global principle for the H 2 H^{2} -sets with respect to the orderings of the field. This principle asserts in particular that an element in H 2 H^{2} is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of k k . Our techniques provide a new proof of Grothendieck’s original theorem. An application to homogeneous spaces over k k is also given.

Let E=F be a quadratic separable extension of global elds, and A E , A the corresponding rings of adeles. Fix a character ! 0 on the group A E =E of E-idele classes which is trivial on the F-idele classes, and an irreducible, automorphic discrete-series representation of GL(2; A E ) with central character ! 0 realized (as a closed invariant subspace) in the space of automorphic forms. Then is said to be GL(2; A )-distinguished (or cyclic) if there exists a form in the space of such that its integral (or period) R (x)dx over the space (or cycle) PGL(2; F)nPGL(2; A ) is non-zero.One purpose of this paper is to compare the notion of being GL(2; A )-distinguished with the notion (de ned below) of being distinguished with respect to another subgroup of GL(2; A E ). Using a \relative trace formula", Jacquet and Lai JL] carried out such comparisons in certain cases. To extend their results, one could either develop an extensive theory of orbital integrals for the relative trace formula, as is done in H3], or give a relative version of the Deligne-Kazhdan \simple trace formula," in which this theory simpli es. We adopt the latter approach. Another objective of this work is to consider such a comparison and a \relative trace" (or \bi-period summation") formula in the higher rank case.Distinguished representations were introduced in a similar context by Waldspurger Wa], and in our context by Harder, Langlands and Rapoport HLR] to study Tate's conjectures T] on algebraic cycles in the case of Hilbert modular surfaces. Then Lai L] ? using the comparisons of distinguished representations in JL]