Construction of base form

Let denote the equivariant form, given
by Eq. (30). We will denote the
corresponding base form :

We will work under the assumption that:

the action of on does not have fixed points

Therefore can be considered
a principal -bundle.
In order to construct the base form from the Cartan’s , we first choose on
this principal bundle
some connection . (We understand the connection as a -valued 1-form
on computing the “vertical component” of a vector.) Then we apply
the horizonthal projection
i.e. replace

Finally, we replace with
the curvature of the connection ; we get:

To conclude:

Generally speaking, it is not true that descends from to
. But we found a class of subalgebras of , of the form ,
such that for any subalgebra from this class we can construct an equivariant form
. Then the standard procedure can be used to construct
the corresponding base form which descends to
.
The result, generally speaking, does depend on the choice of .