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Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Book Description

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.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Contents
  6. Acknowledgments
  7. Introduction
    1. 0.1 The trace formula as a Lefschetz formula
    2. 0.2 A short history of the hypoelliptic Laplacian
    3. 0.3 The hypoelliptic Laplacian on a symmetric space
    4. 0.4 The hypoelliptic Laplacian and its heat kernel
    5. 0.5 Elliptic and hypoelliptic orbital integrals
    6. 0.6 The limit as b → 0
    7. 0.7 The limit as b → + ∞: an explicit formula for the orbital integrals
    8. 0.8 The analysis of the hypoelliptic orbital integrals
    9. 0.9 The heat kernel for bounded b and the Malliavin calculus
    10. 0.10 The heat kernel for large b, Toponogov, and local index
    11. 0.11 The hypoelliptic Laplacian and the wave equation
    12. 0.12 The organization of the book
  8. 1. Clifford and Heisenberg algebras
    1. 1.1 The Clifford algebra of a real vector space
    2. 1.2 The Clifford algebra of V ⊕ V*
    3. 1.3 The Heisenberg algebra
    4. 1.4 The Heisenberg algebra of V ⊕ V*
    5. 1.5 The Clifford-Heisenberg algebra of V ⊕ V*
    6. 1.6 The Clifford-Heisenberg algebra of V ⊕ V* when V is Euclidean
  9. 2. The hypoelliptic Laplacian on X = G / K
    1. 2.1 A pair (G, K)
    2. 2.2 The flat connection on TX ⊕ N
    3. 2.3 The Clifford algebras of g
    4. 2.4 The flat connections on Λ (T* X ⊕ N)
    5. 2.5 The Casimir operator
    6. 2.6 The form
    7. 2.7 The Dirac operator of Kostant
    8. 2.8 The Clifford-Heisenberg algebra of
    9. 2.9 The operator
    10. 2.10 The compression of the operator
    11. 2.11 A formula for
    12. 2.12 The action of on quotients by K
    13. 2.13 The operators and
    14. 2.14 The scaling of the form B
    15. 2.15 The Bianchi identity
    16. 2.16 A fundamental identity
    17. 2.17 The canonical vector fields on X
    18. 2.18 Lie derivatives and the operator
  10. 3. The displacement function and the return map
    1. 3.1 Convexity, the displacement function, and its critical set
    2. 3.2 The norm of the canonical vector fields
    3. 3.3 The subset X (γ) as a symmetric space
    4. 3.4 The normal coordinate system on X based at X (γ)
    5. 3.5 The return map along the minimizing geodesics in X (γ)
    6. 3.6 The return map on
    7. 3.7 The connection form in the parallel transport trivialization
    8. 3.8 Distances and pseudodistances on and
    9. 3.9 The pseudodistance and Toponogov’s theorem
    10. 3.10 The flat bundle (TX ⊕ N) (γ)
  11. 4. Elliptic and hypoelliptic orbital integrals
    1. 4.1 An algebra of invariant kernels on X
    2. 4.2 Orbital integrals
    3. 4.3 Infinite dimensional orbital integrals
    4. 4.4 The orbital integrals for the elliptic heat kernel of X
    5. 4.5 The orbital supertraces for the hypoelliptic heat kernel
    6. 4.6 A fundamental equality
    7. 4.7 Another approach to the orbital integrals
    8. 4.8 The locally symmetric space Z
  12. 5. Evaluation of supertraces for a model operator
    1. 5.1 The operator and the function
    2. 5.2 A conjugate operator
    3. 5.3 An evaluation of certain infinite dimensional traces
    4. 5.4 Some formulas of linear algebra
    5. 5.5 A formula for
  13. 6. A formula for semisimple orbital integrals
    1. 6.1 Orbital integrals for the heat kernel
    2. 6.2 A formula for general orbital integrals
    3. 6.3 The orbital integrals for the wave operator
  14. 7. An application to local index theory
    1. 7.1 Characteristic forms on X
    2. 7.2 The vector bundle of spinors on X and the Dirac operator
    3. 7.3 The McKean-Singer formula on Z
    4. 7.4 Orbital integrals and the index theorem
    5. 7.5 A proof of (7.4.4)
    6. 7.6 The case of complex symmetric spaces
    7. 7.7 The case of an elliptic element
    8. 7.8 The de Rham-Hodge operator
    9. 7.9 The integrand of de Rham torsion
  15. 8. The case where
    1. 8.1 The case where G = K
    2. 8.2 The case
    3. 8.3 The case where G = SL2(R)
  16. 9. A proof of the main identity
    1. 9.1 Estimates on the heat kernel away from
    2. 9.2 A rescaling on the coordinates (f, Y)
    3. 9.3 A conjugation of the Clifford variables
    4. 9.4 The norm of α
    5. 9.5 A conjugation of the hypoelliptic Laplacian
    6. 9.6 The limit of the rescaled heat kernel
    7. 9.7 A proof of Theorem 6.1.1
    8. 9.8 A translation on the variable YTX
    9. 9.9 A coordinate system and a trivialization of the vector bundles
    10. 9.10 The asymptotics of the operator
    11. 9.11 A proof of Theorem 9.6.1
  17. 10. The action functional and the harmonic oscillator
    1. 10.1 A variational problem
    2. 10.2 The Pontryagin maximum principle
    3. 10.3 The variational problem on an Euclidean vector space
    4. 10.4 Mehler’s formula
    5. 10.5 The hypoelliptic heat kernel on an Euclidean vector space
    6. 10.6 Orbital integrals on an Euclidean vector space
    7. 10.7 Some computations involving Mehler’s formula
    8. 10.8 The probabilistic interpretation of the harmonic oscillator
  18. 11. The analysis of the hypoelliptic Laplacian
    1. 11.1 The scalar operators on
    2. 11.2 The Littlewood-Paley decomposition along the fibres TX
    3. 11.3 The Littlewood-Paley decomposition on X
    4. 11.4 The Littlewood Paley decomposition on
    5. 11.5 The heat kernels for
    6. 11.6 The scalar hypoelliptic operators on
    7. 11.7 The scalar hypoelliptic operator on with a quartic term
    8. 11.8 The heat kernel associated with the operator
  19. 12. Rough estimates on the scalar heat kernel
    1. 12.1 The Malliavin calculus for the Brownian motion on X
    2. 12.2 The probabilistic construction of exp over
    3. 12.3 The operator and the wave equation
    4. 12.4 The Malliavin calculus for the operator
    5. 12.5 The tangent variational problem and integration by parts
    6. 12.6 A uniform control of the integration by parts formula as b → 0
    7. 12.7 Uniform rough estimates on for bounded b
    8. 12.8 The limit as b → 0
    9. 12.9 The rough estimates as b → +∞
    10. 12.10 The heat kernel on
    11. 12.11 The heat kernel on
  20. 13. Refined estimates on the scalar heat kernel for bounded b
    1. 13.1 The Hessian of the distance function
    2. 13.2 Bounds on the scalar heat kernel on for bounded b
    3. 13.3 Bounds on the scalar heat kernel on for bounded b
  21. 14. The heat kernel for bounded b
    1. 14.1 A probabilistic construction of exp
    2. 14.2 The operator and the wave equation
    3. 14.3 Changing Y into −Y
    4. 14.4 A probabilistic construction of exp
    5. 14.5 Estimating V.
    6. 14.6 Estimating W.
    7. 14.7 A proof of (4.5.3) when E is trivial
    8. 14.8 A proof of the estimate (4.5.3) in the general case
    9. 14.9 Rough estimates on the derivatives of for bounded b
    10. 14.10 The behavior of V. as b → 0
    11. 14.11 The limit of as b → 0
  22. 15. The heat kernel for b large
    1. 15.1 Uniform estimates on the kernel over
    2. 15.2 The deviation from the geodesic flow for large b
    3. 15.3 The scalar heat kernel on away from
    4. 15.4 Gaussian estimates for near iaX (γ)
    5. 15.5 The scalar heat kernel on away from
    6. 15.6 Estimates on the scalar heat kernel on near
    7. 15.7 A proof of Theorem 9.1.1
    8. 15.8 A proof of Theorem 9.1.3
    9. 15.9 A proof of Theorem 9.5.6
    10. 15.10 A proof of Theorem 9.11.1
  23. Bibliography
  24. Subject Index
  25. Index of Notation