Positive G-laminations and surface affine Grassmannian

This is a joint work with Linhui Shen (Yale).

Let  G be a split reductive group and S a topological surface with a finite set of points on
the boundary  modulo isotopy. Our goal is to define a canonical basis in the space of regular
functions on the space of G^L-local systems on S. Here G^L is the Langlands dual  group.

We define a set of positive G-lamination on S. It is the set of positive integral tropical points
of positive moduli space with potential. We introduce a surface afffine Grassmannian related
to (S, G) and prove that positive G-laminations parametrise its  top components.
The latter give rise to our canonical basis.

When S is a disc, parametrise canonical bases in invariants of tensor products of representations.
For general (S, G) we  prove a part of our duality conjectures  with Vladimir Fock on canonical
bases, known before for SL_2.