Transfert d’intégrales orbitales pour le groupe métaplectique

AbstractWe set up a formalism of endoscopy for metaplectic groups. By defining a suitable transfer factor, we prove an analogue of the Langlands–Shelstad transfer conjecture for orbital integrals over any local field of characteristic zero, as well as the fundamental lemma for units of the Hecke algebra in the unramified case. This generalizes prior work of Adams and Renard in the real case and serves as a first step in studying the Arthur–Selberg trace formula for metaplectic groups.

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SPECTRAL TRANSFER FOR METAPLECTIC GROUPS. I. LOCAL CHARACTER RELATIONS

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748016000384 ◽

2016 ◽

Vol 18 (1) ◽

pp. 25-123 ◽

Cited By ~ 1

Author(s):

Wen-Wei Li

Keyword(s):

Internal Structure ◽

Local Field ◽

Characteristic Zero ◽

Test Functions ◽

Metaplectic Groups ◽

Transfer Factors ◽

Orbital Integrals ◽

The Core ◽

Local Character ◽

Stable Trace Formula

Let $\widetilde{\text{Sp}}(2n)$ be the metaplectic covering of $\text{Sp}(2n)$ over a local field of characteristic zero. The core of the theory of endoscopy for $\widetilde{\text{Sp}}(2n)$ is the geometric transfer of orbital integrals to its elliptic endoscopic groups. The dual of this map, called the spectral transfer, is expected to yield endoscopic character relations which should reveal the internal structure of $L$-packets. As a first step, we characterize the image of the collective geometric transfer in the non-archimedean case, then reduce the spectral transfer to the case of cuspidal test functions by using a simple stable trace formula. In the archimedean case, we establish the character relations and determine the spectral transfer factors by rephrasing the works by Adams and Renard.

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On the Fundamental Lemma for Standard Endoscopy: Reduction to Unit Elements

Canadian Journal of Mathematics ◽

10.4153/cjm-1995-051-5 ◽

1995 ◽

Vol 47 (5) ◽

pp. 974-994 ◽

Cited By ~ 12

Author(s):

Thomas C. Hales

Keyword(s):

Trace Formula ◽

Simple Form ◽

Hecke Algebras ◽

Orbital Integrals ◽

Fundamental Lemma ◽

Stable Trace Formula

AbstractThe fundamental lemma for standard endoscopy follows from the matching of unit elements in Hecke algebras. A simple form of the stable trace formula, based on the matching of unit elements, shows the fundamental lemma to be equivalent to a collection of character identities. These character identities are established by comparing them to a compact-character expansion of orbital integrals.

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Stable Bi-Period Summation Formula and Transfer Factors

Canadian Journal of Mathematics ◽

10.4153/cjm-1999-033-9 ◽

1999 ◽

Vol 51 (4) ◽

pp. 771-791

Author(s):

Yuval Z. Flicker

Keyword(s):

Transfer Factor ◽

Automorphic Forms ◽

Summation Formula ◽

Number Fields ◽

Cusp Forms ◽

Weighted Sums ◽

Standard Case ◽

Transfer Factors ◽

Orbital Integrals ◽

Fundamental Lemma

AbstractThis paper starts by introducing a bi-periodic summation formula for automorphic forms on a group G(E), with periods by a subgroup G(F), where E/F is a quadratic extension of number fields. The split case, where E = F ⊕ F, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups H which occur in the case of standard conjugacy.The spectral side of the bi-period summation formula involves periods, namely integrals over the group of F-adele points of G, of cusp forms on the group of E-adele points on the group G. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces H(E)/H(F). This offers a sharpening of the theory of liftings, where periods play a key role.The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on G and H have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.Finally, the fundamental lemma is verified for SL(2).

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Le lemme fondamental pondéré. I. Constructions géométriques

Compositio Mathematica ◽

10.1112/s0010437x10004756 ◽

2010 ◽

Vol 146 (6) ◽

pp. 1416-1506 ◽

Cited By ~ 13

Author(s):

Pierre-Henri Chaudouard ◽

Gérard Laumon

Keyword(s):

Finite Field ◽

Affine Space ◽

Base Space ◽

Characteristic Polynomials ◽

Selberg Trace Formula ◽

Orbital Integrals ◽

Fundamental Lemma ◽

Open Set ◽

Simple Set ◽

Hitchin Fibration

AbstractThis work is the geometric part of our proof of the weighted fundamental lemma, which is an extension of Ngô Bao Châu’s proof of the Langlands–Shelstad fundamental lemma. Ngô’s approach is based on a study of the elliptic part of the Hichin fibration. The total space of this fibration is the algebraic stack of Hitchin bundles and its base space is the affine space of ‘characteristic polynomials’. Over the elliptic set, the Hitchin fibration is proper and the number of points of its fibers over a finite field can be expressed in terms of orbital integrals. In this paper, we study the Hitchin fibration over an open set larger than the elliptic set, namely the ‘generically regular semi-simple set’. The fibers are in general neither of finite type nor separated. By analogy with Arthur’s truncation, we introduce the substack of ξ-stable Hitchin bundles. We show that it is a Deligne–Mumford stack, smooth over the base field and proper over the base space of ‘characteristic polynomials’. Moreover, the number of points of the ξ-stable fibers over a finite field can be expressed as a sum of weighted orbital integrals, which appear in the Arthur–Selberg trace formula.

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A stable trace formula for Igusa varieties

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748010000046 ◽

2010 ◽

Vol 9 (4) ◽

pp. 847-895 ◽

Cited By ~ 3

Author(s):

Sug Woo Shin

Keyword(s):

Trace Formula ◽

Positive Characteristic ◽

Galois Representations ◽

Type A ◽

Shimura Varieties ◽

Explicit Method ◽

Good Reduction ◽

Orbital Integrals ◽

Fundamental Lemma ◽

Stable Trace Formula

AbstractIgusa varieties are smooth varieties in positive characteristic p which are closely related to Shimura varieties and Rapoport–Zink spaces. One motivation for studying Igusa varieties is to analyse the representations in the cohomology of Shimura varieties which may be ramified at p. The main purpose of this work is to stabilize the trace formula for the cohomology of Igusa varieties arising from a PEL datum of type (A) or (C). Our proof is unconditional thanks to the recent proof of the fundamental lemma by Ngô, Waldspurger and many others.An earlier work of Kottwitz, which inspired our work and proves the stable trace formula for the special fibres of PEL Shimura varieties with good reduction, provides an explicit way to stabilize terms at ∞. Stabilization away from p and ∞ is carried out by the usual Langlands–Shelstad transfer as in work of Kottwitz. The key point of our work is to develop an explicit method to handle the orbital integrals at p. Our approach has the technical advantage that we do not need to deal with twisted orbital integrals or the twisted fundamental lemma.One application of our formula, among others, is the computation of the arithmetic cohomology of some compact PEL-type Shimura varieties of type (A) with non-trivial endoscopy. This is worked out in a preprint of the author's entitled ‘Galois representations arising from some compact Shimura varieties’.

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Some Orbital Integrals and a Technique for Counting Representations of GL 2(F)

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-037-3 ◽

1978 ◽

Vol 30 (02) ◽

pp. 431-448 ◽

Cited By ~ 1

Author(s):

T. Callahan

Keyword(s):

Haar Measure ◽

Local Field ◽

Maximal Ideal ◽

Characteristic Zero ◽

Maximal Compact Subgroup ◽

Residue Field ◽

Compact Subgroup ◽

Orbital Integrals ◽

Ring Of Integers

Let F be a local field of characteristic zero, with q elements in its residue field, ring of integers uniformizer ωF and maximal ideal . Let GF = GL2(F). We fix Haar measures dg and dz on GF and ZF, the centre of GF, so that meas(K) = meas where K = GL2() is a maximal compact subgroup of GF. If T is a torus in GF we take dt to be the Haar measure on T such that means(TM)=1 where TM denotes the maximal compact subgroup of T.

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Local intertwining relation for metaplectic groups

Compositio Mathematica ◽

10.1112/s0010437x20007253 ◽

2020 ◽

Vol 156 (8) ◽

pp. 1560-1594

Author(s):

Hiroshi Ishimoto

Keyword(s):

Local Field ◽

Characteristic Zero ◽

Orthogonal Groups ◽

Metaplectic Groups ◽

Langlands Correspondence ◽

Intertwining Relation ◽

Local Langlands Correspondence

AbstractIn an earlier paper of Wee Teck Gan and Gordan Savin, the local Langlands correspondence for metaplectic groups over a nonarchimedean local field of characteristic zero was established. In this paper, we formulate and prove a local intertwining relation for metaplectic groups assuming the local intertwining relation for non-quasi-split odd special orthogonal groups.

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