Equivariant integration (localization formula) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:34:46Z http://mathoverflow.net/feeds/question/120473 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120473/equivariant-integration-localization-formula Equivariant integration (localization formula) Trakrendal 2013-01-31T23:11:19Z 2013-02-01T16:17:44Z <p>We consider the action of $S^{1}$ on $S^{2}$ by rotation respect the vertical axes. We want to integrate the $2-$ equivariant form </p> <p>$$\alpha(X)= -X\cos(\phi) +\sin(\phi)\,d\phi\,d\theta.$$ </p> <p>We have the localization formula: $$\int_{S^{2}} \alpha(X)= (-2\pi)^{\ell} \sum_{p \in M_{0}(X)} \frac{\alpha(X)(p)}{\det(L_{p})^{1/2}}$$</p> <p>(where $\ell=\dim(M)/2$, $\alpha(X)(p)$ the value of the function <code>$\alpha(X)_{[0]}$</code> in the point $p$, $M_{0}(X)$ is the set of zeros of $X$ and for $p \in M_{0}(X)$ $L_{p}$ is a linear automorphism of $T_{p}(M)$ induced by the Lie action $L(X)\xi = [X, \xi]$ ). So we consider $X \in Lie(S^{1})= \mathbb{R}$. How can I explicity integrate this form ($\int_{S^{2}}\alpha(X)$)? (I know that the result is $2 \pi$)</p>