LEADER 07751nam 2201741 a 450 001 9910827349603321 005 20240516081923.0 010 $a1-283-16387-X 010 $a9786613163875 010 $a1-4008-4057-0 024 7 $a10.1515/9781400840571 035 $a(Au-PeEL)EBL729954 035 $a(CaPaEBR)ebr10481984 035 $a(CaONFJC)MIL316387 035 $a(OCoLC)747411206 035 $a(DE-B1597)446690 035 $a(OCoLC)979593679 035 $a(DE-B1597)9781400840571 035 $z(PPN)199244537 035 $a(PPN)187958386 035 $a(CKB)2550000000040205 035 $a(FR-PaCSA)88838025 035 $a(MiAaPQ)EBC729954 035 $a(EXLCZ)992550000000040205 100 $a20110803d2011 uy 0 101 0 $aeng 135 $aurznu---uuuuu 181 $ctxt 182 $cc 183 $acr 200 10$aHypoelliptic Laplacian and orbital integrals /$fJean-Michel Bismut 205 $aCourse Book 210 $aPrinceton $cPrinceton University Press$d2011 215 $a1 online resource (320 p.) 225 1 $aAnnals of mathematics studies ;$vno. 177 300 $aDescription based upon print version of record. 311 $a0-691-15129-6 311 $a0-691-15130-X 320 $aIncludes bibliographical references and indexes. 327 $tFrontmatter --$tContents --$tAcknowledgments --$tIntroduction --$tChapter One. Clifford and Heisenberg algebras --$tChapter Two. The hypoelliptic Laplacian on X = G/K --$tChapter Three. The displacement function and the return map --$tChapter Four. Elliptic and hypoelliptic orbital integrals --$tChapter Five. Evaluation of supertraces for a model operator --$tChapter Six. A formula for semisimple orbital integrals --$tChapter Seven. An application to local index theory --$tChapter Eight. The case where [k (?) ; p0] = 0 --$tChapter Nine. A proof of the main identity --$tChapter Ten. The action functional and the harmonic oscillator --$tChapter Eleven. The analysis of the hypoelliptic Laplacian --$tChapter Twelve. Rough estimates on the scalar heat kernel --$tChapter Thirteen. Refined estimates on the scalar heat kernel for bounded b --$tChapter Fourteen. The heat kernel qXb;t for bounded b --$tChapter Fifteen. The heat kernel qXb;t for b large --$tBibliography --$tSubject Index --$tIndex of Notation 330 $aThis book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof. 410 0$aAnnals of mathematics studies ;$vno. 177. 606 $aDifferential equations, Hypoelliptic 606 $aLaplacian operator 606 $aDefinite integrals 606 $aOrbit method 610 $aBianchi identity. 610 $aBrownian motion. 610 $aCasimir operator. 610 $aClifford algebras. 610 $aClifford variables. 610 $aDirac operator. 610 $aEuclidean vector space. 610 $aFeynman-Kac formula. 610 $aGaussian integral. 610 $aGaussian type estimates. 610 $aHeisenberg algebras. 610 $aKostant. 610 $aLeftschetz formula. 610 $aLittlewood-Paley decomposition. 610 $aMalliavin calculus. 610 $aPontryagin maximum principle. 610 $aSelberg's trace formula. 610 $aSobolev spaces. 610 $aToponogov's theorem. 610 $aWitten complex. 610 $aaction functional. 610 $acomplexification. 610 $aconjugations. 610 $aconvergence. 610 $aconvexity. 610 $ade Rham complex. 610 $adisplacement function. 610 $adistance function. 610 $aelliptic Laplacian. 610 $aelliptic orbital integrals. 610 $afixed point formulas. 610 $aflat bundle. 610 $ageneral kernels. 610 $ageneral orbital integrals. 610 $ageodesic flow. 610 $ageodesics. 610 $aharmonic oscillator. 610 $aheat kernel. 610 $aheat kernels. 610 $aheat operators. 610 $ahypoelliptic Laplacian. 610 $ahypoelliptic deformation. 610 $ahypoelliptic heat kernel. 610 $ahypoelliptic heat kernels. 610 $ahypoelliptic operators. 610 $ahypoelliptic orbital integrals. 610 $aindex formulas. 610 $aindex theory. 610 $ainfinite dimensional orbital integrals. 610 $akeat kernels. 610 $alocal index theory. 610 $alocally symmetric space. 610 $amatrix part. 610 $amodel operator. 610 $anondegeneracy. 610 $aorbifolds. 610 $aorbital integrals. 610 $aparallel transport trivialization. 610 $aprobabilistic construction. 610 $apseudodistances. 610 $aquantitative estimates. 610 $aquartic term. 610 $areal vector space. 610 $arefined estimates. 610 $arescaled heat kernel. 610 $aresolvents. 610 $areturn map. 610 $arough estimates. 610 $ascalar heat kernel. 610 $ascalar heat kernels. 610 $ascalar hypoelliptic Laplacian. 610 $ascalar hypoelliptic heat kernels. 610 $ascalar hypoelliptic operator. 610 $ascalar part. 610 $asemisimple orbital integrals. 610 $asmooth kernels. 610 $astandard elliptic heat kernel. 610 $asupertraces. 610 $asymmetric space. 610 $asymplectic vector space. 610 $atrace formula. 610 $aunbounded operators. 610 $auniform bounds. 610 $auniform estimates. 610 $avariational problems. 610 $avector bundles. 610 $awave equation. 610 $awave kernel. 610 $awave operator. 615 0$aDifferential equations, Hypoelliptic. 615 0$aLaplacian operator. 615 0$aDefinite integrals. 615 0$aOrbit method. 676 $a515.7242 700 $aBismut$b Jean-Michel$044924 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910827349603321 996 $aHypoelliptic Laplacian and orbital integrals$9241722 997 $aUNINA