equivariant differential form

In differential geometry, an equivariant differential form on a manifold M acted upon by a Lie group G is a polynomial map

:$\backslash alpha:\; \backslash mathfrak\{g\}\; \backslash to\; \backslash Omega^*(M)$

from the Lie algebra $\backslash mathfrak\{g\}\; =\; \backslash operatorname\{Lie\}(G)$ to the space of differential forms on M that are equivariant; i.e.,

:$\backslash alpha(\backslash operatorname\{Ad\}(g)X)\; =\; g\backslash alpha(X).$

In other words, an equivariant differential form is an invariant element ofProof: with $V\; =\; \backslash Omega^*(M)$, we have: $\backslash operatorname\{Mor\}\_G(\backslash mathfrak\{g\},\; V)\; =\; \backslash operatorname\{Mor\}(\backslash mathfrak\{g\},\; V)^G\; =\; (\backslash operatorname\{Mor\}(\backslash mathfrak\{g\},\; \backslash mathbb\{C\})\backslash otimes\; V)^G.$ Note $\backslash mathbb\{C\}[\backslash mathfrak\{g\}]$ is the ring of polynomials in linear functionals of $\backslash mathfrak\{g\}$; see ring of polynomial functions. See also https://math.stackexchange.com/q/101453 for M. Emerton's comment.

:$\backslash mathbb\{C\}[\backslash mathfrak\{g\}]\; \backslash otimes\; \backslash Omega^*(M)\; =\; \backslash operatorname\{Sym\}(\backslash mathfrak\{g\}^*)\; \backslash otimes\; \backslash Omega^*(M).$

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

:$(d\_\backslash mathfrak\{g\}\; \backslash alpha)(X)\; =\; d(\backslash alpha(X))\; -\; i\_\{X^\backslash \#\}(\backslash alpha(X))$

where d is the usual exterior derivative and $i\_\{X^\backslash \#\}$ is the interior product by the fundamental vector field generated by X.

It is easy to see $d\_\backslash mathfrak\{g\}\; \backslash circ\; d\_\backslash mathfrak\{g\}\; =\; 0$ (use the fact the Lie derivative of $\backslash alpha(X)$ along $X^\backslash \#$ is zero) and one then puts

:$H^*\_G(X)\; =\; \backslash ker\; d\_\backslash mathfrak\{g\}/\backslash operatorname\{im\}\; d\_\backslash mathfrak\{g\}\; ,$

which is called the equivariant cohomology of M (which 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.

$d\_\backslash mathfrak\{g\}$-closed or $d\_\backslash mathfrak\{g\}$-exact forms are called equivariantly closed or equivariantly exact.

The integral of an equivariantly closed form may be evaluated from its restriction to the fixed point by means of the localization formula.