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Jean Michel BISMUT - Fokker-Planck Operators and the Center of the Enveloping Algebra

Institut des Hautes Etudes Scientifiques (IHES)
71.4K subscribers
1K views 5 years ago
The heat equation method in index theory gives an explicit local formula for the index of a Dirac operator. Its Lagrangian counterpart involves supersymmetric path integrals. Similar methods can be developed to give a geometric formula for semi simple orbital integrals associated with the Casimir operator of a reductive group, this computation being related to Selberg's trace formula. The analogue of the heat equation method is now a suitable deformation of the Laplacian by a family of Fokker-Planck operators. In joint work with Shu SHEN, we have also shown how to extend these formulas to orbital integrals involving general elements of the center of the enveloping algebra. ...more
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Intro

0:00

The Euler characteristic

2:30

The heat equation method

3:07

Four questions

5:20

The analogy

6:55

Selberg's trace formula

8:02

Casimir and Kostant

12:23

A formula of Kostant

15:01

The operator

18:08

The hypoelliptic Laplacian as a deformation

23:00

The case of locally symmetric spaces

27:13

A fundamental identity

27:58

Splitting the identity

30:46

Geometric description of the orbital integral

33:47

A second fundamental identity

35:07

Semisimple orbital integrals

39:15

The center of the enveloping algebra

41:39

The Duflo isomorphism

42:18

Center and enveloping algebras

43:12

The formula for general orbital integrals

44:30

Principle of the proof

46:35

The Langevin equation

48:49
Jean Michel BISMUT - Fokker-Planck Operators and the Center of the Enveloping Algebra
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1,086Views
2020Feb 15
The heat equation method in index theory gives an explicit local formula for the index of a Dirac operator. Its Lagrangian counterpart involves supersymmetric path integrals. Similar methods can be developed to give a geometric formula for semi simple orbital integrals associated with the Casimir operator of a reductive group, this computation being related to Selberg's trace formula. The analogue of the heat equation method is now a suitable deformation of the Laplacian by a family of Fokker-Planck operators. In joint work with Shu SHEN, we have also shown how to extend these formulas to orbital integrals involving general elements of the center of the enveloping algebra.

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Institut des Hautes Etudes Scientifiques (IHES)

71.4K subscribers

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