The notion of L-indistinguishability, like many others current in the study of L-functions, has yet to be completely defined, but it is in our opinion important for the study of automorphic forms and of representations of algebraic groups. In this paper we study it for the simplest class of groups, basically forms of SL(2). Although the definition we use is applicable to very few groups, there is every reason to believe that the results will have general analogues [12].The phenomena which the notion is intended to express have been met–and exploited–by others (Hecke [5] § 13, Shimura [17]). Their source seems to lie in the distinction between conjugacy and stable conjugacy. If F is a field, G a reductive algebraic group over F, and the algebraic closure of F then two elements of G(F) may be conjugate in without being conjugate in G(F).

Classification mathématique 2010. primaire 11F72 ; secondaire 11R56, 20G35 La formule des traces tordue : Avant-proposThe trace formula as an algebraic idea is as old as representation theory itself and can be regarded as a form of the Frobenius reciprocity theorem. Suppose that G is a finite group and Γ a subgroup. Consider the representation ρ of G on the functions on Γ\G, g : φ → ρ(g)φ, where ρ(g)φ(h) = φ(hg). It is the representation induced from the trivial representation of Γ and the action of a function on G on the space of ρ isThe trace of ρ(f ) is readily calculated as 1 |G|where the measure on Γ\G is implicitly normalized to 1 by a factor |Γ\G| and g 1 runs over Γ\G. The integral on the right can be expressed in various ways, as a sum over conjugacy classes in Γ,where Γ γ is the centralizer of γ in Γ, or asWe apply these simple formulas to the function f (g) = χ π (g) given by the character χ π of an irreducible representation π of G. By the orthogonality relations for matrix coefficients tr ρ(f ) is the number of times π is contained in ρ. The present function f is a class function, so that the formula (1) for this trace reduces towhich is the number of times the trivial representation is contained in the restriction of π to Γ. In other words, we arrive at Frobenius reciprocity for the trivial representation of Γ.One of the achievements of Selberg, in some regards perhaps the major achievement of his career, was to recognize that there was not only a formula similar to (1) for discrete groups Γ with compact quotient but also one, the trace formula, for discrete subgroups of rank one. He, himself, was primarily concerned with subgroups of SL(2, R), but the principles are similar for all groups of rank one for which the usual reduction theory is valid. The spectral theory for the invariant differential operators is similar to the classical spectral theory for a second-order differential equation on a half-line: a one-dimensional continuous spectrum, empty if Γ\G is compact, together with a discrete spectrum. This is a classical theory with which Selberg was more than familiar. If the quotient is not compact, the eigenfunctions for the continuous spectrum are constructed by the analytic continuation of Eisenstein series, a topic initiated by Maaß and Roelcke, the central difficulty being resolved by Selberg.The rank-one theory together with the reduction theory for general arithmetic groups suggested a more general theory, but there were difficulties, often misunderstood, even underestimated, by commentators with limited familiarity with the methods used for their solution. Even the general reduction theory developed in the nineteenth century by Eisenstein, H. J. S. Smith, Minkowski, Hermite and others and rescued, I am tempted to suggest, from oblivion by C. L. Siegel in the twentieth, with the last proofs being provided by Borel and Harish-Chandra, is sorely in need of a competent historical description. It is not surprising that, in the fifties, when Siegel was still alive, still at the Institute fo...

w 0. IntroductionLet Fc G(E) be an arithmetic subgroup of a semi simple connected algebraic group G defined over an algebraic number field E, and let V be a finite dimensional complex representation of the group G~ = G(N| of real points of G. Then the cusp cohomology it/cusp(/", V) is a subspace of the cohomology of F with coefficients in V defined in analytical terms [3]; using the identification of the cohomology H*(F, V) of F with the relative Lie algebra cohomology it is given as Hc*usp(F, V) = H*(g, Ko~; L20(F\G 0o) | V) (1) and, in particular, it isolates a finite set (depending on V) of representations n of G o occurring in the cuspidal spectrum L20(F\G~) with finite multiplicities re(n, F). The study of the cusp cohomology of F is therefore the study of special types of cuspidal automorphic forms.For example, if G=SL2/~, FcSL2(~. ) and V= V k is of dimension k, then (1) is the Eichler-Shimura isomorphism 1 H~usp(r, Vk) ~-S~+ 1 (r) | s~+ , (r)where S~+I(F ) is the space of holomorphic or anti-holomorphic cusp forms with respect to F of weight k+ 1; and the dimension of S~+ I(F) is the multiplicity m(D~,F) of the holomorphic (antiholomorphic) discrete series representation D~ of SL2(~ ). More generally, if G~o has discrete series representations the Selberg trace formula allows one to get hold of cusp cohomology classes by computing Euler-Poincar6 characteristics (cf. w But if G~ has no discrete series (e.g. in case G =SL, with n > 2 or G~--H(~) is a complex Lie group) the Euler-Poincar6 characteristics of the representations occurring in (1) vanish and the ordinary trace formula does not seem to be of any direct help.On the other hand, there are geometrical methods as additional tools. For example, if F~SL2((DE), (~s a ring of imaginary quadratic integers, or F

Let G be a reductive algebraic group defined over a number field k. A k-automorphism of G is said to be of Cartan-type if, at achimedean places, it differs from a Cartan involution by an inner automorphism. In [1] the following result regarding the existence of non-trivial cuspidal cohomology classes for S-arithmetic subgroups of G is proved:

This paper deals with the stabilization of the contribution of elliptic elements to the geometric side of the general twisted trace formula. We extend the results of Langlands, Kottwitz and Shelstad to all elliptic elements for the general twisted trace formula.