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Riemann-Roch and the trace formula Jean-Michel Bismut Institut de - PowerPoint PPT Presentation

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Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and physics References Riemann-Roch and the trace formula Jean-Michel

  1. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Selberg’s explicit formula as a local formula The left-hand side is global, the right-hand side is ‘local’. The formula in the right-hand side looks like Riemann-Roch. 1 Is Selberg explicit formula a Riemann-Roch formula ? 2 Is there a global-local deformation principle? Jean-Michel Bismut Riemann-Roch and the trace formula 6 / 40

  2. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Geodesics Jean-Michel Bismut Riemann-Roch and the trace formula 7 / 40

  3. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A reductive Lie group Jean-Michel Bismut Riemann-Roch and the trace formula 8 / 40

  4. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A reductive Lie group G reductive Lie group, K maximal compact. Jean-Michel Bismut Riemann-Roch and the trace formula 8 / 40

  5. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A reductive Lie group G reductive Lie group, K maximal compact. g = p ⊕ k Cartan splitting. Jean-Michel Bismut Riemann-Roch and the trace formula 8 / 40

  6. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A reductive Lie group G reductive Lie group, K maximal compact. g = p ⊕ k Cartan splitting. B invariant bilinear form > 0 on p , < 0 on k . Jean-Michel Bismut Riemann-Roch and the trace formula 8 / 40

  7. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A reductive Lie group G reductive Lie group, K maximal compact. g = p ⊕ k Cartan splitting. B invariant bilinear form > 0 on p , < 0 on k . X = G/K symmetric space, Riemannian with curvature ≤ 0. Jean-Michel Bismut Riemann-Roch and the trace formula 8 / 40

  8. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A reductive Lie group G reductive Lie group, K maximal compact. g = p ⊕ k Cartan splitting. B invariant bilinear form > 0 on p , < 0 on k . X = G/K symmetric space, Riemannian with curvature ≤ 0. Example G = SL 2 ( R ), K = S 1 , X upper half-plane. Jean-Michel Bismut Riemann-Roch and the trace formula 8 / 40

  9. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  10. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  11. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Z = Γ \ X compact locally symmetric space. Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  12. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Z = Γ \ X compact locally symmetric space. p Z t , p X t smooth heat kernels on X, Z . Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  13. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Z = Γ \ X compact locally symmetric space. p Z t , p X t smooth heat kernels on X, Z . Selberg: Tr C ∞ ( Z, R ) � e t ∆ Z / 2 � [ γ ] Vol [ γ ] Tr [ γ ] � � = � p X . t Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  14. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Z = Γ \ X compact locally symmetric space. p Z t , p X t smooth heat kernels on X, Z . Selberg: Tr C ∞ ( Z, R ) � e t ∆ Z / 2 � [ γ ] Vol [ γ ] Tr [ γ ] � � = � p X . t Tr [ γ ] � � p X orbital integral. t Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  15. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Z = Γ \ X compact locally symmetric space. p Z t , p X t smooth heat kernels on X, Z . Selberg: Tr C ∞ ( Z, R ) � e t ∆ Z / 2 � [ γ ] Vol [ γ ] Tr [ γ ] � � = � p X . t Tr [ γ ] � � p X orbital integral. t Tr [ γ ] � � � p X Z ( γ ) \ G p X t ( g − 1 γg ) dg . = t Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  16. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Z = Γ \ X compact locally symmetric space. p Z t , p X t smooth heat kernels on X, Z . Selberg: Tr C ∞ ( Z, R ) � e t ∆ Z / 2 � [ γ ] Vol [ γ ] Tr [ γ ] � � = � p X . t Tr [ γ ] � � p X orbital integral. t Tr [ γ ] � � � p X Z ( γ ) \ G p X t ( g − 1 γg ) dg . = t Orbital integrals considered as generalized Euler characteristic. Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  17. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References A locally symmetric space Γ ⊂ G torsion free discrete subgroup. Z = Γ \ X compact locally symmetric space. p Z t , p X t smooth heat kernels on X, Z . Selberg: Tr C ∞ ( Z, R ) � e t ∆ Z / 2 � [ γ ] Vol [ γ ] Tr [ γ ] � � = � p X . t Tr [ γ ] � � p X orbital integral. t Tr [ γ ] � � � p X Z ( γ ) \ G p X t ( g − 1 γg ) dg . = t Orbital integrals considered as generalized Euler characteristic. Will be computed explicitly by Riemann-Roch formula. Jean-Michel Bismut Riemann-Roch and the trace formula 9 / 40

  18. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References More general orbital integrals Jean-Michel Bismut Riemann-Roch and the trace formula 10 / 40

  19. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References More general orbital integrals C g Casimir operator on G generalized Laplacian. Jean-Michel Bismut Riemann-Roch and the trace formula 10 / 40

  20. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References More general orbital integrals C g Casimir operator on G generalized Laplacian. ρ : K → Aut ( E ) representation, descends to vector bundle F on X . Jean-Michel Bismut Riemann-Roch and the trace formula 10 / 40

  21. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References More general orbital integrals C g Casimir operator on G generalized Laplacian. ρ : K → Aut ( E ) representation, descends to vector bundle F on X . C g acts as C g ,X on C ∞ ( X, F ). Jean-Michel Bismut Riemann-Roch and the trace formula 10 / 40

  22. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References More general orbital integrals C g Casimir operator on G generalized Laplacian. ρ : K → Aut ( E ) representation, descends to vector bundle F on X . C g acts as C g ,X on C ∞ ( X, F ). For t > 0, Tr [ γ ] � � �� − tC g ,X / 2 exp orbital integral for heat kernel on C ∞ ( X, F ). Jean-Michel Bismut Riemann-Roch and the trace formula 10 / 40

  23. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The centralizer of γ Jean-Michel Bismut Riemann-Roch and the trace formula 11 / 40

  24. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The centralizer of γ γ ∈ G semisimple, γ = e a k − 1 , a ∈ p , k ∈ K , Ad ( k ) a = a . Jean-Michel Bismut Riemann-Roch and the trace formula 11 / 40

  25. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The centralizer of γ γ ∈ G semisimple, γ = e a k − 1 , a ∈ p , k ∈ K , Ad ( k ) a = a . Z ( γ ) ⊂ G centralizer of γ . Jean-Michel Bismut Riemann-Roch and the trace formula 11 / 40

  26. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The centralizer of γ γ ∈ G semisimple, γ = e a k − 1 , a ∈ p , k ∈ K , Ad ( k ) a = a . Z ( γ ) ⊂ G centralizer of γ . Z ( γ ) reductive group, z ( γ ) = p ( γ ) ⊕ k ( γ ) Cartan splitting. Jean-Michel Bismut Riemann-Roch and the trace formula 11 / 40

  27. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Semisimple orbital integrals Jean-Michel Bismut Riemann-Roch and the trace formula 12 / 40

  28. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Semisimple orbital integrals Theorem (B. 2011) � � Y k , Y k There is an explicit function J γ 0 ∈ i k ( γ ), such that 0 Jean-Michel Bismut Riemann-Roch and the trace formula 12 / 40

  29. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Semisimple orbital integrals Theorem (B. 2011) � � Y k , Y k There is an explicit function J γ 0 ∈ i k ( γ ), such that 0 � � − | a | 2 / 2 t Tr [ γ ] � � � � �� = exp C g ,X − c exp − t / 2 (2 πt ) p/ 2 � � ρ E � �� � � k − 1 e − Y k Y k J γ Tr 0 0 i k ( γ ) � � � � dY k � 2 / 2 t � Y k 0 exp − (2 πt ) q/ 2 . 0 Jean-Michel Bismut Riemann-Roch and the trace formula 12 / 40

  30. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Semisimple orbital integrals Theorem (B. 2011) � � Y k , Y k There is an explicit function J γ 0 ∈ i k ( γ ), such that 0 � � − | a | 2 / 2 t Tr [ γ ] � � � � �� = exp C g ,X − c exp − t / 2 (2 πt ) p/ 2 � � ρ E � �� � � k − 1 e − Y k Y k J γ Tr 0 0 i k ( γ ) � � � � dY k � 2 / 2 t � Y k 0 exp − (2 πt ) q/ 2 . 0 Note the integral on i k ( γ ). . . Jean-Michel Bismut Riemann-Roch and the trace formula 12 / 40

  31. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References � � Y k , Y k The function J γ 0 ∈ i k ( γ ) 0 Jean-Michel Bismut Riemann-Roch and the trace formula 13 / 40

  32. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References � � Y k , Y k The function J γ 0 ∈ i k ( γ ) 0 Definition � � � � � � � Y k A ad | p ( γ ) 1 0 Y k � � J γ = � � � � 0 1 / 2 � � � A ad Y k � det (1 − Ad ( γ )) | z ⊥ � 0 k ( γ ) 0 � 1 det (1 − Ad ( k − 1 )) | z ⊥ 0 ( γ ) � � �� � 1 / 2 k − 1 e − Y k det 1 − Ad | k ⊥ 0 0 ( γ ) � � 0 �� . k − 1 e − Y k det 1 − Ad | p ⊥ 0 ( γ ) Jean-Michel Bismut Riemann-Roch and the trace formula 13 / 40

  33. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Applications Jean-Michel Bismut Riemann-Roch and the trace formula 14 / 40

  34. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Applications In work with Shu SHEN, we extended our formula to arbitrary elements of the center of the enveloping algebra. Jean-Michel Bismut Riemann-Roch and the trace formula 14 / 40

  35. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Applications In work with Shu SHEN, we extended our formula to arbitrary elements of the center of the enveloping algebra. Harish-Chandra had obtained non-explicit formulas in terms of Cartan subalgebras. Jean-Michel Bismut Riemann-Roch and the trace formula 14 / 40

  36. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Applications In work with Shu SHEN, we extended our formula to arbitrary elements of the center of the enveloping algebra. Harish-Chandra had obtained non-explicit formulas in terms of Cartan subalgebras. On complex locally symmetric spaces, Riemann-Roch and “automorphic Riemann-Roch”. Jean-Michel Bismut Riemann-Roch and the trace formula 14 / 40

  37. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Applications In work with Shu SHEN, we extended our formula to arbitrary elements of the center of the enveloping algebra. Harish-Chandra had obtained non-explicit formulas in terms of Cartan subalgebras. On complex locally symmetric spaces, Riemann-Roch and “automorphic Riemann-Roch”. Applications to eta invariants and analytic torsion. Jean-Michel Bismut Riemann-Roch and the trace formula 14 / 40

  38. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The method Jean-Michel Bismut Riemann-Roch and the trace formula 15 / 40

  39. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The method We will use cohomological methods. Jean-Michel Bismut Riemann-Roch and the trace formula 15 / 40

  40. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The method We will use cohomological methods. Global-local interpolation. Jean-Michel Bismut Riemann-Roch and the trace formula 15 / 40

  41. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The method We will use cohomological methods. Global-local interpolation. We proceed formally as in the heat equation method for RR-Hirzebruch and Lefschetz RR. Jean-Michel Bismut Riemann-Roch and the trace formula 15 / 40

  42. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat equation method for Riemann-Roch Jean-Michel Bismut Riemann-Roch and the trace formula 16 / 40

  43. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat equation method for Riemann-Roch X compact complex, F holomorphic vector bundle. Jean-Michel Bismut Riemann-Roch and the trace formula 16 / 40

  44. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat equation method for Riemann-Roch X compact complex, F holomorphic vector bundle. � X � Ω 0 , • ( X, F ) , ∂ Dolbeault complex, cohomology H • ( X, F ). Jean-Michel Bismut Riemann-Roch and the trace formula 16 / 40

  45. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat equation method for Riemann-Roch X compact complex, F holomorphic vector bundle. � X � Ω 0 , • ( X, F ) , ∂ Dolbeault complex, cohomology H • ( X, F ). X ∗ adjoint of ∂ X + ∂ X , D X = ∂ X ∗ g TX K¨ ahler metric, ∂ Dirac operator. Jean-Michel Bismut Riemann-Roch and the trace formula 16 / 40

  46. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat equation method for Riemann-Roch X compact complex, F holomorphic vector bundle. � X � Ω 0 , • ( X, F ) , ∂ Dolbeault complex, cohomology H • ( X, F ). X ∗ adjoint of ∂ X + ∂ X , D X = ∂ X ∗ g TX K¨ ahler metric, ∂ Dirac operator. � X ∗ � D X, 2 = X , ∂ ∂ Hodge Laplacian. Jean-Michel Bismut Riemann-Roch and the trace formula 16 / 40

  47. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat equation method for Riemann-Roch X compact complex, F holomorphic vector bundle. � X � Ω 0 , • ( X, F ) , ∂ Dolbeault complex, cohomology H • ( X, F ). X ∗ adjoint of ∂ X + ∂ X , D X = ∂ X ∗ g TX K¨ ahler metric, ∂ Dirac operator. � X ∗ � D X, 2 = X , ∂ ∂ Hodge Laplacian. McKean-Singer: For any s > 0, � � − sD X, 2 �� L ( g ) = Tr s g exp . Jean-Michel Bismut Riemann-Roch and the trace formula 16 / 40

  48. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat equation method for Riemann-Roch X compact complex, F holomorphic vector bundle. � X � Ω 0 , • ( X, F ) , ∂ Dolbeault complex, cohomology H • ( X, F ). X ∗ adjoint of ∂ X + ∂ X , D X = ∂ X ∗ g TX K¨ ahler metric, ∂ Dirac operator. � X ∗ � D X, 2 = X , ∂ ∂ Hodge Laplacian. McKean-Singer: For any s > 0, � � − sD X, 2 �� L ( g ) = Tr s g exp . Tr s [ g exp ( − sD X, 2 )] | s> 0 • L ( g ) | s =+ ∞ − − − − − − − − − − − − → Fixed point formula | s =0 . � �� � � �� � local global Jean-Michel Bismut Riemann-Roch and the trace formula 16 / 40

  49. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat operator of X Jean-Michel Bismut Riemann-Roch and the trace formula 17 / 40

  50. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat operator of X X compact Riemannian manifold. Jean-Michel Bismut Riemann-Roch and the trace formula 17 / 40

  51. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat operator of X X compact Riemannian manifold. ∆ X Laplacian on X . Jean-Michel Bismut Riemann-Roch and the trace formula 17 / 40

  52. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The heat operator of X X compact Riemannian manifold. ∆ X Laplacian on X . � � t ∆ X / 2 For t > 0, g = exp heat operator acting on C ∞ ( X, R ). Jean-Michel Bismut Riemann-Roch and the trace formula 17 / 40

  53. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  54. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? 1 Can I find a resolution C ∞ ( X, R ) by a complex ( R, d )? Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  55. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? 1 Can I find a resolution C ∞ ( X, R ) by a complex ( R, d )? 2 Does ( R, d ) have a Hodge theory? Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  56. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? 1 Can I find a resolution C ∞ ( X, R ) by a complex ( R, d )? 2 Does ( R, d ) have a Hodge theory? 3 Does the heat kernel g lift to ( R, d )? Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  57. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? 1 Can I find a resolution C ∞ ( X, R ) by a complex ( R, d )? 2 Does ( R, d ) have a Hodge theory? 3 Does the heat kernel g lift to ( R, d )? 4 Can I write a formula of the type Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  58. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? 1 Can I find a resolution C ∞ ( X, R ) by a complex ( R, d )? 2 Does ( R, d ) have a Hodge theory? 3 Does the heat kernel g lift to ( R, d )? 4 Can I write a formula of the type R � � �� Tr C ∞ ( X, R ) [ g ] = Tr s − D 2 g exp R,b / 2 . Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  59. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? 1 Can I find a resolution C ∞ ( X, R ) by a complex ( R, d )? 2 Does ( R, d ) have a Hodge theory? 3 Does the heat kernel g lift to ( R, d )? 4 Can I write a formula of the type R � � �� Tr C ∞ ( X, R ) [ g ] = Tr s − D 2 g exp R,b / 2 . 5 By making b → + ∞ , do we obtain Selberg’s trace formula ? Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  60. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References Is Tr C ∞ ( X, R ) [ g ] an Euler characteristic? 1 Can I find a resolution C ∞ ( X, R ) by a complex ( R, d )? 2 Does ( R, d ) have a Hodge theory? 3 Does the heat kernel g lift to ( R, d )? 4 Can I write a formula of the type R � � �� Tr C ∞ ( X, R ) [ g ] = Tr s − D 2 g exp R,b / 2 . 5 By making b → + ∞ , do we obtain Selberg’s trace formula ? 6 Is Selberg’s trace formula a Lefschetz formula? Jean-Michel Bismut Riemann-Roch and the trace formula 18 / 40

  61. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The analogy Jean-Michel Bismut Riemann-Roch and the trace formula 19 / 40

  62. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The analogy Tr s [ g exp ( − sD X, 2 )] | s> 0 L ( g ) | s =+ ∞ − − − − − − − − − − − − → Fixed point formula | s =0 . � �� � � �� � local global Jean-Michel Bismut Riemann-Roch and the trace formula 19 / 40

  63. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References The analogy Tr s [ g exp ( − sD X, 2 )] | s> 0 L ( g ) | s =+ ∞ − − − − − − − − − − − − → Fixed point formula | s =0 . � �� � � �� � local global Tr s [ g exp ( − D R, 2 b )] | b> 0 Tr C ∞ ( X, R ) [ g ] b =0 − − − − − − − − − − − − → Selberg t . f . | b =+ ∞ . � �� � � �� � global local via closed geodesics Jean-Michel Bismut Riemann-Roch and the trace formula 19 / 40

  64. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 1 . Finding a resolution of C ∞ ( X, R ) Jean-Michel Bismut Riemann-Roch and the trace formula 20 / 40

  65. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 1 . Finding a resolution of C ∞ ( X, R ) Is C ∞ ( X, R ) the cohomology of ‘some complex’ ? Jean-Michel Bismut Riemann-Roch and the trace formula 20 / 40

  66. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 1 . Finding a resolution of C ∞ ( X, R ) Is C ∞ ( X, R ) the cohomology of ‘some complex’ ? E real vector bundle on X . Jean-Michel Bismut Riemann-Roch and the trace formula 20 / 40

  67. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 1 . Finding a resolution of C ∞ ( X, R ) Is C ∞ ( X, R ) the cohomology of ‘some complex’ ? E real vector bundle on X . � Ω • ( E ) , d E � R = fibrewise de Rham complex. Jean-Michel Bismut Riemann-Roch and the trace formula 20 / 40

  68. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 1 . Finding a resolution of C ∞ ( X, R ) Is C ∞ ( X, R ) the cohomology of ‘some complex’ ? E real vector bundle on X . � Ω • ( E ) , d E � R = fibrewise de Rham complex. By Poincar´ e lemma, cohomology is equal to C ∞ ( X, R ). Jean-Michel Bismut Riemann-Roch and the trace formula 20 / 40

  69. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 2 . Does the resolution have a Hodge theory? Jean-Michel Bismut Riemann-Roch and the trace formula 21 / 40

  70. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 2 . Does the resolution have a Hodge theory? � E, g E � real Euclidean vector bundle. Jean-Michel Bismut Riemann-Roch and the trace formula 21 / 40

  71. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 2 . Does the resolution have a Hodge theory? � E, g E � real Euclidean vector bundle. Standard Laplacian on fibers of E has continuous spectrum. Jean-Michel Bismut Riemann-Roch and the trace formula 21 / 40

  72. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 2 . Does the resolution have a Hodge theory? � E, g E � real Euclidean vector bundle. Standard Laplacian on fibers of E has continuous spectrum. If Y tautological section of E on E , use instead the � − | Y | 2 � volume exp dY . Jean-Michel Bismut Riemann-Roch and the trace formula 21 / 40

  73. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 2 . Does the resolution have a Hodge theory? � E, g E � real Euclidean vector bundle. Standard Laplacian on fibers of E has continuous spectrum. If Y tautological section of E on E , use instead the � − | Y | 2 � volume exp dY . The corresponding fiberwise Laplacian is a harmonic oscillator, has discrete spectrum, and Hodge theory holds. Jean-Michel Bismut Riemann-Roch and the trace formula 21 / 40

  74. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 2 . Does the resolution have a Hodge theory? � E, g E � real Euclidean vector bundle. Standard Laplacian on fibers of E has continuous spectrum. If Y tautological section of E on E , use instead the � − | Y | 2 � volume exp dY . The corresponding fiberwise Laplacian is a harmonic oscillator, has discrete spectrum, and Hodge theory holds. The function 1 on E is L 2 and fiberwise harmonic. Jean-Michel Bismut Riemann-Roch and the trace formula 21 / 40

  75. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 3 . Does g lift to a morphism of complexes? Jean-Michel Bismut Riemann-Roch and the trace formula 22 / 40

  76. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 3 . Does g lift to a morphism of complexes? � � � Ω • ( E, R ) , d E � t ∆ X / 2 exp morphism of ? Jean-Michel Bismut Riemann-Roch and the trace formula 22 / 40

  77. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 3 . Does g lift to a morphism of complexes? � � � Ω • ( E, R ) , d E � t ∆ X / 2 exp morphism of ? ∆ X should lift and commute with d E . Jean-Michel Bismut Riemann-Roch and the trace formula 22 / 40

  78. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 3 . Does g lift to a morphism of complexes? � � � Ω • ( E, R ) , d E � t ∆ X / 2 exp morphism of ? ∆ X should lift and commute with d E . In general, the answer is no! Jean-Michel Bismut Riemann-Roch and the trace formula 22 / 40

  79. Euler characteristic and heat equation Explicit formulas for semisimple orbital integrals Hypoelliptic Laplacian and orbital integrals Hypoelliptic Laplacian, math, and ‘physics’ References 3 . Does g lift to a morphism of complexes? � � � Ω • ( E, R ) , d E � t ∆ X / 2 exp morphism of ? ∆ X should lift and commute with d E . In general, the answer is no! On locally symmetric spaces, the Casimir restricts to ∆ X , lifts to everything, and commutes with everything. Jean-Michel Bismut Riemann-Roch and the trace formula 22 / 40

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