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A Lefschetz fixed point formula for Hecke operators. (Eine Lefschetzsche Fixpunktformel für Hecke-Operatoren.) (German) Zbl 0589.12013
Mathematisch-Naturwissenschaftliche Fakultät der Rheinischen Friedrich- Wilhelm-Universität zu Bonn. Bonn. Math. Schr. 164, 124 S. (1985).
In this paper, coincidence formulas for Lefschetz numbers of twisted Hecke operators acting on the cohomology of arithmetic groups are calculated by using topological methods. These methods are of the same type as those used commonly in order to prove Lefschetz fixed point formulas.
First the author constructs the Hecke operation by defining it on cochain level. Thus he obtains the first coincidence formula in terms of adeles as follows: Let \(G\) be a reductive group over \(\mathbb Q\) with arbitrary \(\mathbb Q\)-rank and trivial character group over \(\mathbb Q\), \(\tau\) a \(G\)-automorphism of finite order and \(\alpha \in G(\mathbb A_f)\). For every system \(\tilde M\) of local coefficients one obtains the following formula:
\[ \begin{split} L(T_{\alpha}\tau^{*-1},H^*(G(\mathbb Q)\setminus G(\mathbb A)/K_{\infty}K_f,\tilde M)) = \\ = \text{vol}(K_f)^{-1}\sum_{\xi \in G(\mathbb Q)/\sim} J(\xi)\cdot \text{tr}(\xi^{-1}\tau,M) \int_{G_\xi\;{}^\tau(\mathbb A_f)\setminus G(\mathbb A_f)}1_{K_f\alpha K_f}(^{\tau}g_f^{-1}\xi g_f) \,dg_f. \end{split} \] The numbers \(J(\xi)\) depend only on the \(\tau\)-conjugation class of \(\xi\) and do not disappear save in a finite number of cases, and in these cases the orbital integral converges.
Afterwards the author proves explicit formulas for \(\mathbb Q\)-rank one by decomposing \(L(T_{\alpha}\tau^{*-1})\) in an elliptic part \(L_{\text{e}}\) (being a sum of Euler characteristics) and a boundary part \(L_{\text{\partial}}\) induced by a correspondence on the boundary of the Borel-Serre compactification. As a consequence of these formulas one obtains some well known class number relations.
Reviewer: Maria Heep (Bonn)

MSC:
11F75 Cohomology of arithmetic groups
11R56 Adèle rings and groups
20G10 Cohomology theory for linear algebraic groups
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
22E40 Discrete subgroups of Lie groups
55M20 Fixed points and coincidences in algebraic topology
55N25 Homology with local coefficients, equivariant cohomology