Configurations of decorated flags and finite dimensional representations

We consider the space of configurations of decorated flags. It has a positive structure introduced by Fock and Goncharov. One can thus consider the tropicalization of this space. We introduce a canonical potential on such a space. It provides a subset of ${\it positive integral tropical points}$. We show that it has a natural connection with finite dimensional representations of Langlands dual group. This is part of co-project with Prof. Goncharov.