Geometry of canonical bases and mirror symmetry

Abstract

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 spaceAG,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(Z ) 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 W t : AG,S(Z ) → Z. The condition W t ≥ 0 defines a subset of positive integral tropical points AG,S(Zt ). For G = SL2, we recover the set of positive integralA-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 AG,S(Zt ) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Asterisque 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 GL : (Vλ1 ⊗ . . .⊗ Vλn )G L . (1) A. Goncharov (B) · L. Shen Mathematics Department, Yale University, New Haven, CT 06520, USA e-mail: alexander.goncharov@yale.edu