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On full asymptotics of analytic torsions for compact locally symmetric orbifolds

Bingxiao Liu

Abstract

We consider a certain sequence of flat vector bundles on a compact locally symmetric orbifold, and we evaluate explicitly the associated asymptotic Ray-Singer real analytic torsion. The basic idea is to computing the heat trace via Selberg's trace formula, so that a key point in this paper is to evaluate the orbital integrals associated with nontrivial elliptic elements. For that purpose, we deduce a geometric localization formula, so that we can rewrite an elliptic orbital integral as a sum of certain identity orbital integrals associated with the centralizer of that elliptic element. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations.

On full asymptotics of analytic torsions for compact locally symmetric orbifolds

Abstract

We consider a certain sequence of flat vector bundles on a compact locally symmetric orbifold, and we evaluate explicitly the associated asymptotic Ray-Singer real analytic torsion. The basic idea is to computing the heat trace via Selberg's trace formula, so that a key point in this paper is to evaluate the orbital integrals associated with nontrivial elliptic elements. For that purpose, we deduce a geometric localization formula, so that we can rewrite an elliptic orbital integral as a sum of certain identity orbital integrals associated with the centralizer of that elliptic element. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations.

Paper Structure

This paper contains 29 sections, 38 theorems, 423 equations.

Table of Contents

  1. Introduction
  2. Ray-Singer analytic torsion
  3. Orbifolds and orbifold vector bundles
  4. Flat vector bundles and analytic torsions of orbifolds
  5. Orbital integrals and locally symmetric spaces
  6. Real reductive Lie group
  7. Symmetric space
  8. Bismut's formula for semisimple orbital integrals
  9. Compact locally symmetric spaces
  10. Selberg's trace formula
  11. Analytic torsions for compact locally symmetric spaces
  12. A vanishing result on the analytic torsions
  13. Symmetric spaces of noncompact type with fundamental rank $1$
  14. Cartan subalgebra and root system of $G$ when $\delta(G)=1$
  15. Reductive Lie algebra with fundamental rank $1$
  16. ...and 14 more sections

Key Result

Theorem \oldthetheorem

Assume that $\delta(G)=1$. There exists a (real) polynomial $P(d)$ in $d$, and for each $[\gamma]\in E^{+}[\Gamma]$, there exists a nice exponential polynomial $PE^{[\gamma]}(d)$ in $d$ (i.e., a finite sum of the terms of the form $\alpha d^{j}e^{2\pi\sqrt{-1}\beta d}$ with $\alpha\in\mathbb{C}, Moreover, the degrees of $P(d)$, $PE^{[\gamma]}(d)$ can be determined in terms of $\lambda$, $\lamb

Theorems & Definitions (92)

  • Theorem \oldthetheorem
  • Theorem 1.0.1'
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • ...and 82 more