By Jean-Michel Bismut
This publication makes use of the hypoelliptic Laplacian to judge semisimple orbital integrals in a formalism that unifies index concept and the hint formulation. The hypoelliptic Laplacian is a family members of operators that's speculated to interpolate among the standard Laplacian and the geodesic circulate. it truly is basically the weighted sum of a harmonic oscillator alongside the fiber of the tangent package, and of the generator of the geodesic movement. during this publication, semisimple orbital integrals linked to the warmth kernel of the Casimir operator are proven to be invariant less than an appropriate hypoelliptic deformation, that is built utilizing the Dirac operator of Kostant. Their specific assessment is received by way of localization on geodesics within the symmetric house, in a formulation heavily regarding the Atiyah-Bott mounted aspect formulation. Orbital integrals linked to the wave kernel also are computed.
Estimates at the hypoelliptic warmth kernel play a key function within the proofs, and are acquired through combining analytic, geometric, and probabilistic thoughts. Analytic ideas emphasize the wavelike features of the hypoelliptic warmth kernel, whereas geometrical issues are had to receive right keep watch over of the hypoelliptic warmth kernel, specifically within the localization procedure close to the geodesics. Probabilistic strategies are particularly correct, simply because underlying the hypoelliptic deformation is a deformation of dynamical structures at the symmetric area, which interpolates among Brownian movement and the geodesic circulate. The Malliavin calculus is used at serious levels of the proof.
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1) Let g F be the natural Hermitian metric on F , let ∇F be the canonical Hermitian connection of F , and let RF be its curvature. Then K acts unitarily on Λ· (g∗ )⊗S · (g∗ )⊗E. Equivalently K acts unitarily on Λ· (g∗ )⊗L2 (g)⊗E. If k ∈ K, and s ∈ Λ· (g∗ ) ⊗ L2 (g) ⊗ E, ks is given by · ∗ ks (Y ) = ρΛ (g )⊗E (k) s Ad k −1 Y . 3). If e ∈ k, then [e, Y ] is a Killing vector field on g = p ⊕ k, and the corresponding Lie derivative LV[e,Y ] acting on C ∞ (g, Λ· (g∗ )) is given by ek i[e,ek ] . 3) Traceglobfin June 3, 2011 36 Chapter 2 Equivalently, LV[e,Y ] = ∇V[e,Y ] − (c + c) (ad (e)) .
Then c (ad (e)) ∈ c (p). 12), we get m+n c (−κg ) = −2 c (ei ) c (ad (ei ) |p ) + c −κk . 5) 1 c (ei ) (ei + c (ad (ei ) |p )) + c −κk . 5), we get g D g = DH + DVg . 6) Now we recall a fundamental result of Kostant [Ko76, Ko97]. 2. The following identities hold: 1 g DH , DVg = 0, Dg,2 = −C g − B ∗ (κg , κg ) . 7) Traceglobfin June 3, 2011 The hypoelliptic Laplacian on X = G/K 31 Proof. 7). We take e1 , . . 5). In particular for 1 ≤ i ≤ m + n, c (ei ) c (e∗i ) = 1. 2), we get 1 Dg,2 = B (e∗i , e∗i ) e2i + c (e∗i ) c e∗j [ei , ej ] 2 1 1 2 + [c (e∗i ) , c (−κg )] ei + c (−κg ) .
2), we get 1 Dg,2 = B (e∗i , e∗i ) e2i + c (e∗i ) c e∗j [ei , ej ] 2 1 1 2 + [c (e∗i ) , c (−κg )] ei + c (−κg ) . 9) 2 4 Moreover, B (e∗i , e∗i ) e2i = −C g . Also one verifies easily that 1 1 c (e∗i ) c e∗j [ei , ej ] + [c (e∗i ) , c (−κg )] ei = 0. 2 2 We will now show that 2 c (−κg ) = −B ∗ (κg , κg ) . 13), 2 c (−κg ) = 1 g κ (ei , ej , ek ) κg e∗i , e∗j , e∗k 36 c (e∗i ) c e∗j c (e∗k ) c (ei ) c (ej ) c (ek ) . 13), the indices (i, j, k) and (i , j , k ) are all distinct. By antisymmetry the sum over disjoint triples of (i, j, k) , (i , j , k ) vanishes.