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The reduced norm-one group G of a central simple algebra is an inner form ofthe special linear group, and an involution on the algebra induces anautomorphism of G. We study the action of such automorphisms in the cohomologyof arithmetic subgroups of G. The main result is a precise formula forLefschetz numbers of automorphisms induced by involutions of symplectic type.Our approach is based on a careful study of the smoothness properties of groupschemes associated with orders in central simple algebras. Along the way wealso derive an adelic reformulation of Harder's Gauss-Bonnet Theorem.

Lefschetz numbers of symplectic involutions on arithmetic groups