The reduced norm-one group
of a central simple algebra is an inner form of the special linear
group, and an involution on the algebra induces an automorphism of
. We
study the action of such automorphisms in the cohomology of arithmetic subgroups
of
.
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.
