La Formule des Traces Tordue d’après le Friday Morning Seminar

Abstract

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...

Endoscopic Classification of representations of Quasi-Split Unitary Groups

“…The invariant trace formula, in both its standard and twisted version, is established in [A2,A3,LW]. The proof of the main theorems of this paper, in particular the theorems stated in section 2.5, is based on the comparison of the two trace formulas via their stabilized version.…”

Arthur’s multiplicity formula for GSp 4 and restriction to Sp 4

“…We now state the stabilisation of the twisted trace formula proved by Moeglin and Waldspurger in [MW16a], [MW16b] following the case of ordinary (i.e. nontwisted) endoscopy proved by Arthur in [Art02], [Art01], [Art03] (also following [Lan83], [Kot86], [Lab99], and of course [LW13]). We recall some of the definitions needed to state the stabilisation, and mention some simplifications occurring in the cases at hand.…”

A local trace formula for the generalized Shalika model

“…And we set (see [26] §1.9). If the (G, M)-family Y is positive, then v Q M (Y) is just the volume of the convex hull of the projection of (Y P ) P ∈P(M ),P ⊂Q onto A L Q M .…”

“…If the (G, M)-family Y is positive, then v Q M (Y) is just the volume of the convex hull of the projection of (Y P ) P ∈P(M ),P ⊂Q onto A L Q M . Also to a (G, M)-orthogonal set Y we can associate a certain function Γ G M (., Y) on A M (see [26] §1.8) which is just the characteristic function of the sum of the convex hull of Y with A G if Y is positive.…”