Erratum to: A higher rank Lefschetz formula

Abstract

For Proposition 1.4.1 to be correct, one has to insist that the open set C be regular in the sense that C is the open interior of its closure. In this way the boundary of C is also the boundary of the complement. Furthermore, in order to apply Proposition 1.4.1 to the construction of the test function f in Section 4.3, the condition ∂F (C) ⊂ F (∂C) needs to be satisfied. As it stands (page 35), this condition is not satisfied. One can, however, replace the set N in the definition of the set C on page 35 with an open subset U of N which has compact support. This suffices for the application and then the proof of the property ∂F (C) ⊂ F (∂C) runs as follows: Let (kj , nj ,mj , aj) be a sequence in C such that F (kj , nj ,mj , aj) = kjnjajmjn −1 j k −1 j converges to some g ∈ G. As K is compact, we can assume that kj converges to some k ∈ K. Replacing g with k−1gk we reduce to k = 1, so the sequence njajmjn −1 j converges to g. As nj varies in a relatively compact set, we can assume that nj converges to some n ∈ N . We have njajmjn −1 j = aj }{{} ∈A mj }{{} ∈M n ajmj j n −1 j } {{ } ∈N .