The hypoelliptic Laplacian on the cotangent bundle

Introduction 380 1. Generalized metrics and determinants 386 1.1. Determinants 386 1.2. A generalized metric on E· 387 1.3. The Hodge theory of h · 387 1.4. A generalized metric on detE· 389 1.5. Determinant of the generalized Laplacian and generalized metrics 391 1.6. Truncating the spectrum of the generalized Laplacian 392 1.7. Generalized metrics on determinant bundles 393 1.8. Determinants and flat superconnections 394 1.9. The Hodge theorem as an open condition 395 1.10. Generalized metrics and flat superconnections 395 1.11. Analyticity and the Hodge condition 397 1.12. The equivariant determinant 397 2. The adjoint of the de Rham operator on the cotangent bundle 399 2.1. Clifford algebras 400 2.2. Vector spaces and bilinear forms 400 2.3. The adjoint of the de Rham operator with respect to a nondegenerate bilinear form 402 2.4. The symplectic adjoint of the de Rham operator 403 2.5. The de Rham operator on T ∗X and its symplectic adjoint 404 2.6. A bilinear form on T ∗X and the adjoint of d ∗X 407 2.7. A fundamental symmetry 408 2.8. A Hamiltonian function 409 2.9. The symmetry in the case where H is r-invariant 412 2.10. Poincaré duality 412 2.11. A conjugation of the de Rham operator 413 2.12. The scaling of the variable p 415 2.13. The classical Hamiltonians 418