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Abstract:
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Let $G$ be a real Lie group acting on a manifold $M$,
let $\mathfrak g$ be the Lie algebra of $G$, and let
$\alpha(X)$ be an equivariantly closed form on $M$ depending
on $X \in \mathfrak g$.
For $X \in \mathfrak g$, we denote by $\operatorname{VF}_X$ the vector
field on $M$ induced by the infinitesimal action of $X$.
Then the Berline-Vergne localization formula says that the integral of
$\alpha(X)$ can be expressed as a sum over the set of zeroes of
$\operatorname{VF}_X$ of certain {\em local} quantities of $M$ and $\alpha$:

$$
\int_M \alpha(X) =
\sum_{{p \in M,} \atop
{\scriptstyle p {\rm \:\, is \: a\: zero \: of} \: \operatorname{VF}_X}}
\mbox{local invariant of $M$ and $\alpha$ at $p$}.
$$
This formula plays a central role in symplectic geometry and many other
branches of mathematics.

For compact groups it was proved by N.Berline and M.Vergne
and independently by M.Atiyah and R.Bott more than twenty years ago.
In this talk I will explain a version of this localization formula for
non-compact group actions.


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