On the spectral side of Arthur's trace formula --- absolute convergence

Finis, Tobias; Lapid, Erez; Müller, Werner

doi:10.4007/annals.2011.174.1.5

Cited by 43 publications

(61 citation statements)

References 22 publications

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“…To understand the behavior of the distributions J G M ([u] S , f S ) for our test functions, we want to apply our asymptotic expansion for the archimedean weighted integral. To that end we need to separate ∞ from the other places in S which we will do by using Arthur's splitting formula [Ar05, (18.7)]: Suppose that S = S 1 ∪ S 2 definition of the spectral side, and in particular the refinement of the spectral expansion obtained in [FLM11].…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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Analytic torsion and $L^2$-torsion of compact locally symmetric manifolds

Müller¹,

Pfaff²

2013

J. Differential Geom.

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Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Recall the (purely combinatorial) map X L : B P,L → F 1 (M) m with the property that X L (β) ∈ Ξ L (β) for all β ∈ B P,L as defined in [FLM11,. 2…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Analytic torsion and $L^2$-torsion of compact locally symmetric manifolds

Müller¹,

Pfaff²

2013

J. Differential Geom.

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“…This was mentioned by X. Li [Li] (see also [FLM,Remark 7]). Indeed, integrating (1.2) with respect to the normalized Lebesgue measure m, we obtain (1.3).…”

mentioning

confidence: 92%

A model space approach to some classical inequalities for rational functions

Baranov¹,

Zarouf²

2014

Journal of Mathematical Analysis and Applications

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Abstract. We consider the set R n of rational functions of degree at most n ≥ 1 with no poles on the unit circle T and its subclass R n, r consisting of rational functions without poles in the annulus ξ : r ≤ |ξ| ≤ 1 r . We discuss an approach based on the model space theory which brings some integral representations for functions in R n and their derivatives. Using this approach we obtain L p -analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdélyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as n tends to infinity and the poles of the rational functions approach the unit circle T.

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“…Cela fait l'objet du corollaire 2.10.3. Le théorème de convergence dominée montre alors que, si B(0) = 1 cette expression est la limite pour → 0 de De plus, grâce aux travaux récents de Finis, Lapid et Müller [22,23], on sait maintenant que le développement spectral est absolument convergent. Leurs travaux ne concernent que le cas classique (non tordu) mais ils s'étendent sans modification au cas général.…”

Section: Calcul De a T (B)unclassified

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

Labesse

Waldspurger

2013

CRM Monograph Series

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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...

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On the spectral side of Arthur's trace formula --- absolute convergence

Abstract: We derive a refinement of the spectral expansion of Arthur's trace formula. The expression is absolutely convergent with respect to the trace norm.

Cited by 43 publications

References 22 publications

Analytic torsion and $L^2$-torsion of compact locally symmetric manifolds

Analytic torsion and $L^2$-torsion of compact locally symmetric manifolds

A model space approach to some classical inequalities for rational functions

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

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