Let M be a compact manifold provided with an action of a lie group G.
What is the meaning when we say that M is G-oriented and if possible what is the intuition behind this notion?
Actually I'm reading an article on equivariant cohomology which say: "if M is G-oriented, integration over M gives us a map from $H^{-\infty}_G(M)$ (G-equivariant de Rham cohomology of M with generalized coefficients)to $C^{-\infty}(\mathfrak{g})^G$ (the subspace of generalized functions on $\mathfrak{g}$ which are invariant under the action of G endowed from the adjoint action)".
Thanks!
We say that $M$ is $G$-oriented if $M$ is oriented and the $G$-action preserves the orientation of $M$.
If $M$ (not necessarily compact) is $G$-oriented, then we can talk about integrating (compactly supported) smooth equivariant form $\alpha\in\mathcal{A}^\infty_G(\mathfrak{g},M)=C^\infty(\mathfrak{g},\mathcal{A}(M))^G$ $$ \int_M\alpha\colon X\in\mathfrak{g}\mapsto\int_M\alpha(X) $$ so get a map $$ \int_M\colon \mathcal{H}^\infty_{G,c}(\mathfrak{g},M)\to C^\infty(M)^G $$ where the differential defining the $G$-equivariant (compactly supported) cohomology group $\mathcal{H}^\infty_{G,c}$ is $(d_\mathfrak{g}\alpha)(X)=(d_M\alpha)(X)-i(X_M)(\alpha(X))$. Now your article generalises this to distributions and do some more exotic things.