Holomorphic torsion for Hermitian locally symmetric spaces
The holomorphic torsion of a compact locally symmetric manifold is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.
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arXiv subjects
Anton Deitmar
Publications and source records attributed to Anton Deitmar.
Torus actions on compact quotients
We consider the action of a noncompact torus H on the compact quotient G/L, where G is a Lie group containing H and L is a uniform lattice in G. Using harmonic analysis on G we prove a formula relating the compact orbits of H to the action of H on the (infinite dimensional) tangential cohomology. The formula may be viewed as a Lefschetz formula or an equivariant index theorem for noncompact group actions.
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Regularized and $L^2$-Determinants
It is shown that in a tower of coverings the regularized determinant of a generalized Laplacian converges to the $L^2$-determinant. This shows generic nontriviality of analytic torsion or regularized determinants since the $L^2$-counterparts are easier to compute. We further have an "Euler product expansion" for regularized determinants in terms of equivariant $L^2$-determinants.
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Equivariant torsion of locally symmetric spaces
The equivariant holomorphic torsion of a compact locally symmetric manifold and an automorphism is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.
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On the index of Dirac operators on arithmetic quotients
Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient over a totally real field with at least two real embeddings as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of rank one spaces. In this case the index theorem looks exactly as in the compact case.
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A Lefschetz formula for flows
For the geodesic flow of an odd dimensional hyperbolic manifold we prove a Lefschetz type formula. The local terms are Fuller indices of the closed orbits. The global "Frobenius operator" is the generator of the flow and its action on tangential cohomology.
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Geometric Zeta Functions, $L^2$-Theory, and Compact Shimura Manifolds
We define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray and Singer and the holomorphic $L^2$-torsion, where the latter is defined via the $L^2$-theory of Atiyah. For higher fundamental rank twisted torsion numbers appear.
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