The hypoelliptic Laplacian on a compact Lie group |
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Authors: | Jean-Michel Bismut |
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Institution: | Département de Mathématique, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France |
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Abstract: | Let G be a compact Lie group, and let g be its Lie algebra. In this paper, we produce a hypoelliptic Laplacian on G×g, which interpolates between the classical Laplacian of G and the geodesic flow. This deformation is obtained by producing a suitable deformation of the Dirac operator of Kostant. We show that various Poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory. The paper was motivated by papers by Atiyah and Frenkel, in connection with localization formulas in equivariant cohomology and with Kac's character formulas for affine Lie algebras. In a companion paper, we will use similar methods in the context of Selberg's trace formula. |
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Keywords: | Infinite-dimensional Lie groups and their Lie algebras Hypoelliptic equations Index theory and related fixed point theorems |
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