A decorated surface S is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over Q. A pair (G,S) gives rise to a moduli space AG,S, closely related to the moduli space of G-local systems on S. It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). So a set AG,S(ℤt) of its integral tropical points is defined. We introduce a rational positive function W on the space AG,S, called the potential. Its tropicalisation is a function (Formula presented.). The condition (Formula presented.) defines a subset of positive integral tropical points (Formula presented.). For (Formula presented.), we recover the set of positive integral A-laminations on S from Fock and Goncharov (Publ Math IHES 103:1–212, 2006). We prove that when S is a disc with n special points on the boundary, the set (Formula presented.) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Astérisque 101–102:208–229, 1983; Ginzburg,1995; Mirkovic and Vilonen, Ann Math (2) 166(1):95–143, 2007; Beilinson and Drinfeld, Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, 2004) they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group (Formula presented.).When (Formula presented.), n=3, there is a special coordinate system on (Formula presented.) (Fock and Goncharov, Publ Math IHES 103:1–212, 2006). We show that it identifies the set (Formula presented.) with Knutson–Tao’s hives (Knutson and Tao, The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture, 1998). Our result generalises a theorem of Kamnitzer (Hives and the fibres of the convolution morphism, 2007), who used hives to parametrise top components of convolution varieties for G=GLm, n=3. For G=GLm, n>3, we prove Kamnitzer’s conjecture (Kamnitzer, Hives and the fibres of the convolution morphism, 2012). Our parametrisation is naturally cyclic invariant. We show that for any G and n=3 it agrees with Berenstein–Zelevinsky’s parametrisation (Berenstein and Zelevinsky, Invent Math 143(1):77–128, 2001), whose cyclic invariance is obscure. We define more general positive spaces with potentials (A,W), parametrising mixed configurations of flags. Using them, we define a generalization of Mirković–Vilonen cycles (Mirkovic and Vilonen, Ann Math (2) 166(1):95–143, 2007), and a canonical basis in (Formula presented.), generalizing the Mirković–Vilonen basis in Vλ. Our construction comes naturally with a parametrisation of the generalised MV cycles. For the classical MV cycles it is equivalent to the one discovered by Kamnitzer (Mirkovich–Vilonen cycles and polytopes, 2005). We prove that the set (Formula presented.) parametrises top dimensional components of a new moduli space, surface affine Grasmannian, generalising the fibers of the convolution maps. These components are usually infinite dimensional. We define their dimension being an element of a Z-torsor, rather then an integer. We define a new moduli space (Formula presented.), which reduces to the moduli spaces of (Formula presented.)-local systems on S if S has no special points. The set (Formula presented.) parametrises a basis in the linear space of regular functions on (Formula presented.). We suggest that the potential W itself, not only its tropicalization, is important—it should be viewed as the potential for a Landau–Ginzburg model on (Formula presented.). We conjecture that the pair (Formula presented.) is the mirror dual to (Formula presented.). In a special case, we recover Givental’s description of the quantum cohomology connection for flag varieties and its generalisation (Gerasimov et al., New integral representations of Whittaker functions for classical Lie groups, 2012; Rietsch, A mirror symmetric solution to the quantum Toda lattice, 2012). We formulate equivariant homological mirror symmetry conjectures parallel to our parametrisations of canonical bases.
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