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Register a new userCoherent IC-sheaves on type $A_{n}$ affine Grassmannians and dual canonical basis of affine type $A_{1}$
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The convolution ring $K^{GL_n(mathcal{O})rtimesmathbb{C}^times}(mathrm{Gr}_{GL_n})$ was identified with a quantum unipotent cell of the loop group $LSL_2$ in [Cautis-Williams, J. Amer. Math. Soc. 32 (2019), pp. 709-778]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.
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We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory $mathcal{C}_{ell}$ of $U_q(hat{mathfrak{sl}_n})$-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux. Via the isomorphism, we define an element ch(T) in a Grassmannian cluster algebra for every rectangular tableau T. By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T) for some T. Using formulas of Arakawa-Suzuki, we give an explicit expression for ch(T), and also give explicit q-character formulas for finite-dimensional $U_q(hat{mathfrak{sl}_n})$-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.
The article is a contribution to the local theory of geometric Langlands correspondence. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra, thought of as an algebra of Iwahori bi-invariant functions on a semi-simple group over a local non-Archimedian field, and Grothendieck group of equivariant coherent sheaves on Steinberg variety of the Langlands dual group; this isomorphism due to Kazhdan--Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures. The paper is a continuation of an earlier joint work with S. Arkhipov, it relies on technical material developed in a paper with Z. Yun.
We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent.
We prove the existence of a one-parameter family of nondisplaceable Lagrangian tori near a linear chain of Lagrangian 2-spheres in a symplectic 4-manifold. When the symplectic structure is rational we prove that the deformed Floer cohomology groups of these tori are nontrivial. The proof uses the idea of toric degeneration to analyze the full potential functions with bulk deformations of these tori.
Let $Q$ be a finite quiver without loops and $mathcal{Q}_{alpha}$ be the Lusztig category for any dimension vector $alpha$. The purpose of this paper is to prove that all Frobenius eigenvalues of the $i$-th cohomology $mathcal{H}^i(mathcal{L})|_x$ for a simple perverse sheaf $mathcal{L}in mathcal{Q}_{alpha}$ and $xin mathbb{E}_{alpha}^{F^n}=mathbb{E}_{alpha}(mathbb{F}_{q^n})$ are equal to $(sqrt{q^n})^{i}$ as a conjecture given by Schiffmann (cite{Schiffmann2}). As an application, we prove the existence of a class of Hall polynomials.
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