Equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted by a compact Lie group G is a polynomial map from the Lie algebra ${\mathfrak {g}}=\operatorname {Lie} (G)$ to the space of differential forms on M that is equivariant; i.e., $\alpha (gX)=g\alpha (X)$. Since

$\operatorname {Hom} _{G}(\mathbb {C} [{\mathfrak {g}}],\Omega ^{*}(M))\to \operatorname {Hom} (\mathbb {C} [{\mathfrak {g}}],\Omega ^{*}(M))^{G}=(\mathbb {C} [{\mathfrak {g}}]\otimes \Omega ^{*}(M))^{G},$

one can view an equivariant differential form as an invariant element of $\mathbb {C} [{\mathfrak {g}}]\otimes \Omega ^{*}(M)$ as well.

For an equivariant differential form $\alpha$, the equivariant exterior derivative $d_{\mathfrak {g}}\alpha$ of $\alpha$ is defined by

$(d_{\mathfrak {g}}\alpha )(X)=d(\alpha (X))-i_{X^{\#}}(\alpha (X))$

where d is the usual exterior derivative and $i_{X^{\#}}$ is the interior product by the fundamental vector field generated by X. It is easy to see $d_{\mathfrak {g}}\circ d_{\mathfrak {g}}=0$ (use the fact the Lie derivative of $\alpha (X)$ along $X^{\#}$ is zero) and this makes the space of equivariant differential forms a complex. One can then put

$H_{G}^{*}(X)=\operatorname {ker} d_{\mathfrak {g}}/\operatorname {im} d_{\mathfrak {g}}$,

which is called the equivariant cohomology of M (This coincides with the ordinary equivariant cohomology defined in terms of Borel construction.) The definition is due to H. Cartan. The notion has an application to the equivariant index theory.

$d_{\mathfrak {g}}$-closed or $d_{\mathfrak {g}}$-exact forms are often called equivariantly closed or equivariantly exact.

References

• Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag