Geometry of canonical bases and mirror symmetry

• 成果类型: Article
• 署名作者: Goncharov, Alexander; Shen, Linhui
• 署名单位: Yale University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0568-2
• 发表日期: 2015
• 页码: 487-633
• 关键词: langlands duality; flag manifolds; quantum; REPRESENTATIONS; VARIETIES; calculus; crystals; analog

摘要

A decorated surface is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let be a split reductive group over . A pair gives rise to a moduli space , closely related to the moduli space of -local systems on . It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1-212, 2006). So a set of its integral tropical points is defined. We introduce a rational positive function on the space , called the potential. Its tropicalisation is a function . The condition defines a subset of positive integral tropical points . For , we recover the set of positive integral -laminations on from Fock and Goncharov (Publ Math IHES 103:1-212, 2006). We prove that when is a disc with special points on the boundary, the set parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Ast,risque 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 G(L): (V-lambda 1 circle times ... circle times V-lambda n)(GL) (1) When , , there is a special coordinate system on (Fock and Goncharov, Publ Math IHES 103:1-212, 2006). We show that it identifies the set 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 , . For , , 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 and 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 , parametrising mixed configurations of flags. Using them, we define a generalization of MirkoviA double dagger-Vilonen cycles (Mirkovic and Vilonen, Ann Math (2) 166(1):95-143, 2007), and a canonical basis in , generalizing the MirkoviA double dagger-Vilonen basis in . 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 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 -torsor, rather then an integer. We define a new moduli space , which reduces to the moduli spaces of -local systems on if has no special points. The set parametrises a basis in the linear space of regular functions on . We suggest that the potential itself, not only its tropicalization, is important-it should be viewed as the potential for a Landau-Ginzburg model on . We conjecture that the pair is the mirror dual to . 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.