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Geometry of canonical bases and mirror symmetry. (English) Zbl 1355.14030
The paper under review studies the set \(\mathcal{A}_{G,S}^+(\mathbb{Z}^t)\) of positive integral tropical points of a certain moduli space of \(G\)-local systems on a decorated surface \(S\), where \(G\) is a split reductive group over \(\mathbb{Q}\). When \(S\) is a disc with \(n\) special points on the boundary, it is shown that the set \(\mathcal{A}_{G,S}^+(\mathbb{Z}^t)\) parameterizes top dimensional components of fibers of the convolution maps. Via the geometric Satake correspondence one thus obtains a canonical basis in a certain invariant space of a tensor product of irreducible modules for the Langland dual group. When \(G=\mathrm{GL}_m\), it is shown that \(\mathcal{A}_{G,S}^+(\mathbb{Z}^t)\) can be identified with Knutson-Tao’s hives which implies, for \(m>3\), that hives parameterize top components of convolution varieties, thus proving a conjecture of Kamnitzer.
The authors also define more general spaces with potentials which parameterize mixed configurations of flags. This is used to define a generalization of Mirkovic-Vilonen cycles and Mirkovic-Vilonen basis in irreducible modules. Again, the set \(\mathcal{A}_{G,S}^+(\mathbb{Z}^t)\) parameterizes top dimensional components of a new moduli space, which the authors call the surface affine Grassmannian. The authors also define yet another moduli space and conjecture that it is the mirror dual to \((\mathcal{A}_{G,S},\mathcal{W})\), where \(\mathcal{W}\) is the potential. In a special case, the authors recover Givental’s description of the quantum cohomology connection for flag varieties. Finally, the authors formulate equivariant homological mirror symmetry conjectures parallel to their parameterization of canonical bases.

MSC:
14J33 Mirror symmetry (algebro-geometric aspects)
14D20 Algebraic moduli problems, moduli of vector bundles
14L24 Geometric invariant theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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