A note on the eta function for quotients of \(PSL_ 2({\mathbb{R}})\) by co- compact Fuchsian groups
The authors compute the value at 0 of the Atiyah-Patodi-Singer eta invariant associated to the Dirac operator on \(\Gamma \setminus PSL_ 2({\mathbb{R}})\), where \(\Gamma\) is a cocompact Fuchsian group. They do this by considering a family of metrics parameterized by \(t\in (0,\infty)\) and developing explicit formulae for the eigenvalues of the parameterized Dirac operator. Their approach makes use of a great deal of information concerning representations of \(SL_ 2({\mathbb{R}})\). Their main results are the following: Theorem 1: \(\Gamma \setminus PSL_ 2({\mathbb{R}})\) has no harmonic spinors for any cocompact Fuchsian group \(\Gamma\). Theorem 2: If \(\Gamma \subset PSL_ 2({\mathbb{R}})\) is a cocompact Fuchsian group of signature \(\{\) g; \(\alpha_ 1,\alpha_ 2,...,\alpha_ n\}\) then \[ \eta (0)=\frac{1}{48\pi}vol(F_{\Gamma})+\frac{1}{6}(\sum^{n}_{i=1}(\alpha _ i-1/\alpha_ i))=\frac{1}{24}\{2g-2+n+\sum^{n}_{i=1}(4\alpha_ i-5/\alpha_ i)\}, \] where \(F_{\Gamma}\) is a fundamental domain for the action of \(\Gamma\) on \({\mathbb{H}}\).
representations of \(SL_ 2({\mathbb{R}})\)
scientific article; zbMATH DE number 4086605