  A decorated surface S is a surface with a finite set of special points on the
boundary, considered modulo isotopy. Let G be a split reductive group. A pair
(G, S) gives rise to a moduli space A(G, S), closely related to the space of
G-local systems on S. It has a positive structure. So the set of its integral
tropical points is defined. We introduce a rational positive function W on A(G,
S), the potential. The condition that its tropicalisation is non-negative
determines its subset. For SL(2), we recover the set of integral laminations on
S. We prove that when S is a disc with n special points on the boundary, this
set parametrises top dimensional components of the convolution varieties. Thus,
via geometric Satake correspondence, they provide a canonical basis in tensor
product invariants of irreducible modules for the Langlands dual group. When
G=GL(m), n=3, there is a special coordinate system on A(G,S). We show that it
identifies our set with the set of with Knutson-Tao's hives. Our result
generalises a theorem of Kamnitzer, who used hives to parametrise top
components of convolution varieties for GL(m), n=3. For n>3, we prove
Kamnitzer's conjecture. We define canonical bases in tensor products,
generalizing the Mirkovic-Vilonen basis in a single representation. We prove
that for any S, the set of positive integral tropical points of A(G, S)
parametrise top components in a new space, surface affine Grasmannian. We view
W as a potential for Landau-Ginzburg model on A(G,S). We conjecture that the
pair (A(G,S), W) is the mirror dual to the moduli space of local systems on S
for the Langlands dual group. In a special case, we recover Givental's
description of the quantum cohomology connection for flag varieties.
