| A decorated surface S is an oriented surface, with or without boundary, and a finite set {s1, ..., sn} 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 [23]. So the set of its integral tropical points AG,S&parl0;Z t&parr0; is defined. We introduce a rational positive function W on the space AG,S , called the potential. Its tropicalization is a function Wt:AG, S&parl0;Zt &parr0;→Z . The condition Wt ≥ 0 determines a subset A+G,S &parl0;Zt&parr0; . For G = SL2, we recover the set of positive integral A -laminations on S from [23].;We prove that when S is a disc with n special points on the boundary, the set A+G,S &parl0;Zt&parr0; parametrizes top dimensional components of the convolution varieties. Via the geometric Satake correspondence [72], [43], [74], [9], they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group GL: &parl0;Vl1⊗&ldots;⊗ Vln&parr0;GL . 1 When G = GLm, n = 3, there is a special coordinate system on AG,S [23]. We show that it identifies the set A+GLm,S &parl0;Zt &parr0; with Knutson-Tao's hives [65]. Our result generalizes a theorem of Kamnitzer [56], who used hives to parametrize top components of convolution varieties for G = GLm, n = 3. For G = GLm, n > 3, we prove Kamnitzer's conjecture [56]. We show that our parametrization for any G and n = 3 agrees with Berenstein-Zelevinsky's parametrization [14], whose cyclic invariance is obscure.;We define more general positive spaces with potentials ( A,W ), parametrizing mixed configurations of flags. Using them, we define a generalization of Mirkovic-Vilonen cycles [74], and a new canonical basis in Vl1⊗&ldots;⊗V ln , generalizing the MV basis in Vlambda. Our construction comes naturally with a parametrization of generalized MV cycles. For the classical MV cycles it is equivalent to the one discovered by Kamnitzer [55].;We prove that the set A+G,S &parl0;Zt&parr0; parametrizes top dimensional components in a new moduli space, surface affine Grassmannian, generalizing 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 LocGL,S which reduces to the moduli spaces GL-local systems on S if S has no special points. The set A+G,S &parl0;Zt&parr0; parametrizes a basis in the linear space of regular functions on LocGL,S .;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 AG,S . We conjecture that the pair ( AG,S,W ) is the mirror dual to LocGL,S . In a special case, we recover Givental's description of the quantum cohomology connection for flag varieties and its generalization [41], [79]. We formulate equivariant homological mirror symmetry conjectures parallel to our parametrization of canonical bases.;We relate the above dualities to Fock-Goncharov's cluster Duality Conjecture [25]. We investigate the cluster Duality Conjecture for cluster ensembles of Cartan-Killing type A. We prove that the products of elements of the canonical basis for one cluster space are equivalent to the Minkowski sums of integral tropical points of its dual space. We show that the latter are tropical Stasheff polytopes.. |
