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