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2018 | Book

Functoriality and the Trace Formula Read chapter Read first chapter

Geometric Aspects of the Trace Formula

Editors: Werner Müller, Sug Woo Shin, Nicolas Templier

Publisher: Springer International Publishing

Book Series : Simons Symposia

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

The second of three volumes devoted to the study of the trace formula, these proceedings focus on automorphic representations of higher rank groups. Based on research presented at the 2016 Simons Symposium on Geometric Aspects of the Trace Formula that took place in Schloss Elmau, Germany, the volume contains both original research articles and articles that synthesize current knowledge and future directions in the field. The articles discuss topics such as the classification problem of representations of reductive groups, the structure of Langlands and Arthur packets, interactions with geometric representation theory, and conjectures on the global automorphic spectrum.

Suitable for both graduate students and researchers, this volume presents the latest research in the field. Readers of the first volume Families of Automorphic Forms and the Trace Formula will find this a natural continuation of the study of the trace formula.

Table of Contents

Frontmatter
Functoriality and the Trace Formula
Abstract
We shall summarize two different lectures that were presented on Beyond Endoscopy, the proposal of Langlands to apply the trace formula to the principle of functoriality. We also include an elementary description of functoriality, and in the last section, some general reflections on where the study of Beyond Endoscopy might be leading.
James Arthur
Graded Hecke Algebras for Disconnected Reductive Groups
Abstract
We introduce graded Hecke algebras \(\mathbb H\) based on a (possibly disconnected) complex reductive group G and a cuspidal local system \({\mathcal L}\) on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and investigated by Lusztig for connected G.
We develop the representation theory of the algebras \(\mathbb H\), obtaining complete and canonical parametrizations of the irreducible, the irreducible tempered and the discrete series representations. All the modules are constructed in terms of perverse sheaves and equivariant homology, relying on work of Lusztig. The parameters come directly from the data \((G,M,{\mathcal L})\) and they are closely related to Langlands parameters.
Our main motivation for considering these graded Hecke algebras is that the space of irreducible \(\mathbb H\)-representations is canonically in bijection with a certain set of “logarithms” of enhanced L-parameters. Therefore, we expect these algebras to play a role in the local Langlands program. We will make their relation with the local Langlands correspondence, which goes via affine Hecke algebras, precise in a sequel to this paper.
Anne-Marie Aubert, Ahmed Moussaoui, Maarten Solleveld
Sur une variante des troncatures d’Arthur
Abstract
We show that, for a large class of test functions, the unipotent contributions in the trace formula for GL(n) over a number field, can be obtained from zeta functions and integrals of Eisenstein series. The main innovation is a new truncation borrowed from a work of Schiffmann on Higgs bundles.
Pierre-Henri Chaudouard
Twisted Endoscopy from a Sheaf-Theoretic Perspective
Abstract
The standard theory of endoscopy for real groups has two parallel formulations. The original formulation of Langlands and Shelstad relies on methods in harmonic analysis. The subsequent formulation of Adams, Barbasch and Vogan relies on sheaf-theoretic methods. The original formulation was extended by Kottwitz and Shelstad to twisted endoscopy. We extend the sheaf-theoretic formulation to the context of twisted endoscopy and provide applications for computing Arthur packets.
Aaron Christie, Paul Mezo
The Subregular Unipotent Contribution to the Geometric Side of the Arthur Trace Formula for the Split Exceptional Group G 2
Abstract
In this paper, a zeta integral for the space of binary cubic forms is associated with the subregular unipotent contribution to the geometric side of the Arthur trace formula for the split exceptional group G 2.
Tobias Finis, Werner Hoffmann, Satoshi Wakatsuki
The Shimura–Waldspurger Correspondence for Mp(2n)
Abstract
We describe some recent developments and formulate some conjectures in the genuine representation theory and the study of automorphic forms of the metaplectic group Mp(2n), from the point of view of the theta correspondence as well as from the point of view of the theory of endoscopy and the trace formula.
Wee Teck Gan, Wen-Wei Li
Fourier Coefficients and Cuspidal Spectrum for Symplectic Groups
Abstract
J. Arthur (The endoscopic classification of representations: orthogonal and Symplectic groups. Colloquium Publication, vol 61. American Mathematical Society, 2013) classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their implication to the structure of the cuspidal spectrum for symplectic groups (Jiang, Automorphic integral transforms for classical groups I: endoscopy correspondences. In: Automorphic forms and related geometry: assessing the legacy of I.I. Piatetski-Shapiro. Contemp. Math., vol 614, pp 179–242. AMS, 2014; Jiang and Liu, Fourier coefficients for automorphic forms on quasisplit classical groups. In: Advances in the theory of automorphic forms and their L-functions. Contemp. Math., vol 664, pp 187–208. AMS, 2016). As a result, we obtain certain characterization and construction of small cuspidal automorphic representations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak (Notes on the generalized Ramanujan conjectures. In: Harmonic analysis, the trace formula, and Shimura varieties. Clay Math. Proc., vol 4, pp 659–685. Amer. Math. Soc., Providence, RI, 2005).
Dihua Jiang, Baiying Liu
Symmetry Breaking for Orthogonal Groups and a Conjecture by B. Gross and D. Prasad
Abstract
We consider irreducible unitary representations A i of G = SO(n + 1, 1) with the same infinitesimal character as the trivial representation and representations B j of H = SO(n, 1) with the same properties and discuss H-equivariant homomorphisms \(\operatorname {Hom}_H(A_i,B_j)\). For tempered representations our results confirm the predictions of conjectures by B. Gross and D. Prasad.
Toshiyuki Kobayashi, Birgit Speh
Conjectures About Certain Parabolic Kazhdan–Lusztig Polynomials
Abstract
Irreducibility results for parabolic induction of representations of the general linear group over a local non-Archimedean field can be formulated in terms of Kazhdan–Lusztig polynomials of type A. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan–Lusztig polynomials known as parabolic Kazhdan–Lusztig polynomials satisfy properties analogous to those of the ordinary ones.
Erez Lapid
Sur les paquets d’Arthur aux places réelles, translation
Résumé.
This article is part of a project which aims to describe as explicitly as possible the Arthur packets of classical real groups and to prove a multiplicity one result for them. Let G be a symplectic or special orthogonal real group, and \(\psi : W_{\mathbb R}\times \mathbf {SL}_2(\mathbb C)\rightarrow { }^{LG}\) be an Arthur parameter for G. Let A(ψ) the component group of the centralizer of ψ in \(\widehat G\). Attached to ψ is a finite length unitary representation π A(ψ) of G × A(ψ), which is characterized by the endoscopic identities (ordinary and twisted) it satisfies.
In (Moeglin et Renard, Sur les paquets d’Arthur des groupes classiques réels, arXiv :1703.07226) we gave a description of the irreducible components of π A(ψ) when the parameter ψ is “very regular, with good parity”. In the present paper, we use translation of infinitesimal character to describe π A(ψ) in the general good parity case from the representation π A(ψ +) attached to a very regular, with good parity, parameter ψ + obtained from ψ by a simple shift.
Colette Moeglin, David Renard
Inverse Satake Transforms
Abstract
Let H be a split reductive group over a local non-Archimedean field, and let \(\check {H}\) denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight representation of the dual group, considered as a series of elements in the Hecke algebra of H. This offers an alternative approach to a solution of the same problem by Wen-Wei Li. Moreover, we generalize the notion of “Satake transform” and perform the analogous calculation for a large class of spherical varieties.
Yiannis Sakellaridis
On Generalized Fourier Transforms for Standard L-Functions
Abstract
Any generalization of the method of Godement–Jacquet on principal L-functions for GL(n) to other groups as perceived by Braverman–Kazhdan/Ngo requires a Fourier transform on a space of Schwartz functions. In the case of standard L-functions for classical groups, a theory of this nature was developed by Piatetski-Shapiro and Rallis, called the doubling method. It was later that Braverman and Kazhdan, using an algebro-geometric approach, different from doubling method, introduced a space of Schwartz functions and a Fourier transform, which projected onto those from doubling method. In both methods a normalized intertwining operator played the role of the Fourier transform. The purpose of this paper is to show that the Fourier transform of Braverman–Kazhdan projects onto that of doubling method. In particular, we show that they preserve their corresponding basic functions. The normalizations involved are not the standard ones suggested by Langlands, but rather a singular version of local coefficients of Langlands–Shahidi method. The basic function will require a shift by 1/2 as dictated by doubling construction, reflecting the global theory, and begs explanation when compared with the work of Bouthier–Ngo–Sakellaridis. This matter is further discussed in an appendix by Wen-Wei Li.
Freydoon Shahidi
On Unitarizability in the Case of Classical p-Adic Groups
Abstract
In the introduction of this paper we discuss a possible approach to the unitarizability problem for classical p-adic groups. In this paper we give some very limited support that such approach is not without chance. In a forthcoming paper we shall give additional evidence in generalized cuspidal rank (up to) three.
Marko Tadić
Metadata
Title
Geometric Aspects of the Trace Formula
Editors
Werner Müller
Sug Woo Shin
Nicolas Templier
Copyright Year
2018
Electronic ISBN
978-3-319-94833-1
Print ISBN
978-3-319-94832-4
DOI
https://doi.org/10.1007/978-3-319-94833-1

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