Some invariants associated to symplectic principal bundle

Question

In the  following  question we  have  a  principal bundle M \to X  where the  group  structure is  a  Lie  group G  with Lie  algebra g. The  total  space  is a  symplectic  manifold acted  symplectically by  G. The  goal is  to  associate  some  invariant  to this  structure. To  do that  we pose  some  questions  which are mentioned in the  following  question.  I  would  appreciate  if you  give  comments  on these  questions.

Best regards,

Ali Taghavi

https://mathoverflow.net/questions/451250/invariants-associated-to-a-principal-bundle-whose-total-space-is-a-symplectic-ma