Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau, and Shen)

Astérisque ◽  
2019 ◽  
Vol 407 ◽  
pp. 333-389
Author(s):  
Xiaonan MA
Keyword(s):  
Orbital Integrals

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

scholarly journals On the vanishing of some non-semisimple orbital integrals

2010 ◽  
Vol 28 (3) ◽  
pp. 276-289 ◽  
Author(s):  
Gaëtan Chenevier ◽  
David Renard
Keyword(s):  
Orbital Integrals

scholarly journals CUSP FORMS ON GSp(4) WITH SO(4)-PERIODS

2011 ◽  
Vol 07 (04) ◽  
pp. 855-919
Author(s):  
YUVAL Z. FLICKER
Keyword(s):  
Quadratic Extension ◽  
Summation Formula ◽  
Unipotent Radical ◽  
Automorphic Representations ◽  
Orbital Integrals ◽  
Relative Trace Formula ◽  
Almost Everywhere ◽  
Genus 2 ◽  
Technical Advances ◽  
Ramanujan Conjecture

The Saito–Kurokawa lifting of automorphic representations from PGL(2) to the projective symplectic group of similitudes PGSp(4) of genus 2 is studied using the Fourier summation formula (an instance of the "relative trace formula"), thus characterizing the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F) associated to a quadratic extension E of the global base field F, and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The image is nongeneric and almost everywhere nontempered, violating a naive generalization of the Ramanujan conjecture. Technical advances here concern the development of the summation formula and matching of relative orbital integrals.

A formula for semisimple orbital integrals

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Heat Kernel ◽  
Explicit Formula ◽  
Wave Operator ◽  
Orbital Integrals ◽  
Gaussian Integral ◽  
Wave Kernel

This chapter states without proof the main result of this volume, which expresses the elliptic orbital integrals associated with the heat kernel as a Gaussian integral on t(γ‎). In addition, the chapter shows how to derive from this formula a corresponding formula for other kernels, which include the wave kernel. First, the chapter gives an explicit formula for the orbital integrals associated with the heat kernel of ℒAX in terms of a Gaussian integral on t(γ‎). From the formula for the heat kernel, this chapter derives a corresponding formula for the semisimple orbital integrals associated with the wave operator of ℒAX.

On Connected Components of Shimura Varieties

2002 ◽  
Vol 54 (2) ◽  
pp. 352-395 ◽  
Author(s):  
Thomas J. Haines
Keyword(s):  
Trace Formula ◽  
High Power ◽  
Weighting Factor ◽  
Connected Components ◽  
Shimura Varieties ◽  
Orbital Integrals ◽  
Crucial Step ◽  
Hecke Operator ◽  
Twisted Trace Formula ◽  
Twisted Orbital Integrals

AbstractWe study the cohomology of connected components of Shimura varieties coming from the group GSp2g, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character ϖ on the group of connected components of we define an operator L(ω) on the cohomology groups with compact supports Hic(, ), and then we prove that the virtual trace of the composition of L(ω) with a Hecke operator f away from p and a sufficiently high power of a geometric Frobenius , can be expressed as a sum of ω-weighted (twisted) orbital integrals (where ω-weighted means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character ϖ). As the crucial step, we define and study a new invariant α1(γ0; γ, δ) which is a refinement of the invariant α(γ0; γ, δ) defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

SLn, Orthogonality Relations and Transfer

2007 ◽  
Vol 59 (3) ◽  
pp. 449-464 ◽  
Author(s):  
Alexandru Ioan Badulescu
Keyword(s):  
Division Algebra ◽  
Local Field ◽  
Finite Dimension ◽  
Complex Conjugate ◽  
Orbital Integrals ◽  
Central Division ◽  
Zero Characteristic ◽  
Central Division Algebra ◽  
Orthogonality Relations ◽  
Square Integrable Representation

AbstractLet π be a square integrable representation of G′ = SLn(D), with D a central division algebra of finite dimension over a local field F of non-zero characteristic. We prove that, on the elliptic set, the character of π equals the complex conjugate of the orbital integral of one of the pseudocoefficients of π. We prove also the orthogonality relations for characters of square integrable representations of G′. We prove the stable transfer of orbital integrals between SLn(F) and its inner forms.

scholarly journals Orbital integrals and K-theory classes

2019 ◽  
Vol 4 (2) ◽  
pp. 185-209 ◽  
Author(s):  
Peter Hochs ◽  
Hang Wang
Keyword(s):  
Orbital Integrals ◽  
K Theory

Orbital Integrals in Reductive Groups

1972 ◽  
Vol 96 (3) ◽  
pp. 505 ◽  
Author(s):  
R. Ranga Rao
Keyword(s):  
Reductive Groups ◽  
Orbital Integrals
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