Descent to double coset

The form does not descend from to ; but we will prove that for some subgroups , it can be modified in a natural way, so that the modified form does descend to

#### 1` `Straightforward descent does not work

We are identifying with the
double coset .
Let be a special
canonical transformation, i.e. . It follows immediately
that is invariant under
the left shift . But
is not horizonthal; for we have:

Therefore:

does not descend from to

#### 2` `Modified PDF

However, we identify a class of subalgebras for which we can construct the base form . Being a base form, it descends to , where is the Lie group generated by the flows of elements of .