Lefschetz numbers and twisted stabilized orbital integrals
Consider a connected simply connected semisimple algebraic group \(G\) defined over \(\mathbb{Q}\) such that \(G(R)\) is non-compact. Let \(\sigma\) be a \(\mathbb{Q}\)-rational automorphism of \(G\) of finite order, and \(\Gamma\) an arithmetic, \(\sigma\)-stable and torsion-free subgroup of \(G(\mathbb{Q})\). For a complex finite-dimensional representation on a space \(V\) with compatible \(\sigma\)-action we can define an action \(\sigma^ i\) of \(\sigma\) on the cohomology group \(H^ i (\Gamma,V)\) and then define a Lefschetz number \(L(\sigma, \Gamma, V)=\sum (-1)^ i \text{tr } \sigma^ i\), where the sum is taken from \(i=0\) to infinity. The main result of the paper is an explicit formula for the Lefschetz number. As claimed by the authors this formula has the following applications: Computationally, it provides a method to compute the actual Lefschetz number. Topologically, it can be viewed as a \(\sigma\)-twisted variant of the Hirzebruch proportionality principle; the untwisted version of this principle is in this case Harder's Gauss-Bonnet formula. Analytically, it fits into the picture of the conjectural twisted stabilization of the Arthur-Selberg trace formula.
scientific article; zbMATH DE number 537294
scientific article