In classical paper by Atiyah-Singer on page 16 (or 560) stated formula $(3.1)$. It should give classical Lefschetz fixed-point formula if the operator is $d + d^* : \Omega^{even} \rightarrow \Omega^{odd}$. But classical Lefschetz formula involves some signs. I wonder how that signs appear from formula $(3.1)$.
By the way, I am not sure, shall I call formula $(3.1)$ as Atyah-Bott formula.
Let $g$ be an element of compact Lie group, acting on our manifold. $ dg_x $ is the differential of g at fixed point $x$. Sign of point $x$ is sign of $ \det (1 - dg)$ . Note that if this determinant is equal to 0 then intersection of graph with diagonal is not transversal and we are not allowed to use fixed point theorem.
If $g$ is an element of compact group then it preserves a positive definite metric (which can be obtained by averaging over group). Finally it is linear algebra exercise to show that $ \det (1 - dg) > 0$ for isometry $g$ without eigenvalues 1 (there is no 1 eigenvalues on fiber of normal bundle to fixed locus).
$$det \begin{pmatrix} 1 - cos{ \theta } & - sin ( \theta )\\ sin \theta & 1 - cos \theta \end{pmatrix} = 2(1 - cos( \theta ))$$
$$det(1 - (-1)) = 2$$
So if $g$ is an element of compact group then we are in special case: each point has sign $+1$.