Here trace φ<submath>''\varphi_{j'',''x''}</submath> means the trace of φ<submath>''\varphi_{j'',}</submath> at a fixed point ''x'' of ''f'', and δ<math>\delta(''x'')</math> is the [[determinant]] of the endomorphism I − ''Df'' at ''x'', with ''Df'' the derivative of ''f'' (the non-vanishing of this is a consequence of transversality). The outer summation is over the fixed points ''x'', and the inner summation over the index ''j'' in the elliptic complex.
Specializing the Atiyah–Bott theorem to the de Rham complex of smooth differential forms yields the original Lefschetz fixed-point formula. A famous application of the Atiyah–Bott theorem is a simple proof of the [[Weyl character formula]] in the theory of [[Lie groups]].{{clarify|date=May 2012}}
