Classification of some Global Integrals related to groups of type $A_n$

In this paper we start a classification of certain global integrals. First, we use the language of unipotent orbits to write down a family of global integrals. We then classify all those integrals which satisfy the dimension equation we set. After doing so, we check which of these integrals are global unipotent integrals. We do all this for groups of type $A_n$, and using all this we derive a certain interesting conjecture about the length of these integrals.