  We generalise the Atiyah-Segal-Singer fixed point theorem to noncompact
manifolds. Using $KK$-theory, we extend the equivariant index to the noncompact
setting, and obtain a fixed point formula for it. The fixed point formula is
the explicit cohomological expression from Atiyah-Segal-Singer's result. In the
noncompact case, however, we show in examples that this expression yields
characters of infinite-dimensional representations. In one example, we realise
characters of discrete series representations on the regular elements of a
maximal torus, in terms of the index we define. Further results are a fixed
point formula for the index pairing between equivariant $K$-theory and
$K$-homology, and a non-localised expression for the index we use, in terms of
deformations of principal symbols. The latter result is one of several links we
find to indices of deformed symbols and operators studied by various authors.
