Stable local trace formula

Langlands program is the central concern of modern number theory. In general, we think that the main tool for studying this program is the trace formula. Arthur-Selberg’s trace formula is successful in solving the endoscopic case of Langlands program. Arthur stabilized the geometric side of the local trace formula in the general case, but did not stabilize the general spectral. This paper aims to solve the general case by different method in the Archimedean case, by directly stabilizing the spectral side of the local trace formula, and obtains the stable local trace formula.     

On an open problem of Sankaranarayanan

Let λ(n) be the n-th normalized Fourier coeficient of a holomorphic Hecke eigenform f(z) ∈ Sk(Γ) for the full modular group.In one of his papers,Sankaranarayanan mentioned that it is an open problem to give a non-trivial estimate for the sum of λ(n) over cubes,i.e.n x λ(n3).In this paper,we are able to use the analytic properties of symmetric power L-functions to solve his problem.More precisely,we prove that Σn≤zλ(n3) x(3/4 +ε),Σn≤zλ(n4) x(97+ε).     

On the Action of the Dual Group on the Cohomology of Perverse Sheaves on the Affine Grassmannian

It was proved by Ginzburg, Mirkovic and Vilonen that the G(O)-equivariant perverse sheaves on the affine Grassmannian of a connected reductive group G form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group G . In this paper we construct explicitly the action of G on the global cohomology of a perverse sheaf.     

THE COMPLETE LANGLANDS PARAMETERS CORRESPONDING TO THE REPRESENTATIONS OF THE CLASSICAL GROUPS WITH INTEGRAL REGULAR INFINITESIMAL CHARACTERS

1.CompleteLanglandsParametersandL-PacketsInthissectionwereviewsomebasicfactsfrom[1].Wedenotethesetsofrealnumbers,complexnumbersandintegersbyR,CandZ,respectively.SupposethatGisaconnectedcomplexreductivealgebraicgroup.WehaveanextendedgrouPGrforG,andvGrisanE-groupforGr.WedenotethesetofLanglandsparametersandthesetofgeometricparametersforvGrbYp(vG')andX(VG'),respectively.ThereisanaturalmapfromP(VG')toX(vG').ItinducesabijectionfromequivalenceclassesofLanglandsparameterstogeometricparamet…     

关于自守型式的Langlands函子性

本文系该著者有关自守型式的Langlands函子性的演讲稿的细述及扩充文稿.特别强调了与该著者的近期工作相关的课题.     

The Uniqueness of AF Diagonals in Regular Limit Algebras

Necessary and sufficient conditions are obtained for the uniqueness of standard regular AF maximal abelian self-adjoint algebras in regular limit algebras up to approximate inner unitary equivalence.     

论连通和单连通可裂G_2的A_q(λ)的酉性

设G为一连通,单连通的G2型可裂实李群,K为G的最大紧致子群,g为G的李代数的复化,Aq（λ）为 Knapp在[7]中的定义（g,K）-模.（MAN,σ,υ）表示由[7]中猜测方法确定的Aq（λ）的自然不可约分量的Langlands参数.本文确定了当λ位于弱好区域外时,哪些Langlands商J（MAN,σ,υ）为酉的.     

The essentially tame local Langlands correspondence, I

Let be a non-Archimedean local field (of characteristic or ) with finite residue field of characteristic . An irreducible smooth representation of the Weil group of is called essentially tame if its restriction to wild inertia is a sum of characters. The set of isomorphism classes of irreducible, essentially tame representations of dimension is denoted . The Langlands correspondence induces a bijection of with a certain set of irreducible supercuspidal representations of . We consider the set of isomorphism classes of certain pairs , called ``admissible', consisting of a tamely ramified field extension of degree and a quasicharacter of . There is an obvious bijection of with . Using the classification of supercuspidal representations and tame lifting, we construct directly a canonical bijection of with , generalizing and simplifying a construction of Howe (1977). Together, these maps give a canonical bijection of with . We show that one obtains the Langlands correspondence by composing the map with a permutation of of the form , where is a tamely ramified character of depending on . This answers a question of Moy (1986). We calculate the character in the case where is totally ramified of odd degree.

     

Affine A3^（1） N = 2 Monopole as the D Module and AiTine ADHMN Sheaf

A Higgs-Yang-Mills monopole scattering spherical symmetrically along light cones is given. The left incoming anti-self-dual α plane fields are holomorphic, but the right outgoing SD β plane fields are antiholomorphic, meanwhile the diffeomorphism symmetry is preserved with mutual inverse afiine rapidity parameters μ and μ^-1. The Dirac wave function scattering in this background also factorized respectively into the （anti）holomorphic amplitudes. The holomorphic anomaly is realized by the center term of a quasi Hopf algebra corresponding to an integrable conformal affine massive field. We find explicit Nahm transformation matrix （Fourier Mukai transformation） between the Higgs YM BPS （fiat） bundles （1） modules） and the affinized blow up ADHMN twistors （perverse sheafs）. Thus we establish the algebra for the ＇t Hooft Hecke operators in the Hecke correspondence of the geometric Langlands program.     

Functoriality of automorphic L-functions through their zeros

Let E be a Galois extension of Q of degree , not necessarily solvable. In this paper we first prove that the L-function L(s,π) attached to an automorphic cuspidal representation π of GLm(EA) cannot be factored nontrivially into a product of L-functions over E. Next, we compare the n-level correlation of normalized nontrivial zeros of L(s,π1)···L(s,πk), where πj, j = 1,...,k, are automorphic cuspidal representations of GLmj(QA), with that of L(s,π). We prove a necessary condition for L(s,π) having a factoriz...