We show that Hecke algebra of type B and a coideal subalgebra of the type A quantum group satisfy a double centralizer property, generalizing the Schur-Jimbo duality in type A. The quantum group of type A and its coideal subalgebra form a quantum symmetric pair. A new theory of canonical bases arising from quantum symmetric pairs is initiated. It is then applied to formulate and establish for the first time a Kazhdan-Lusztig theory for the BGG category O of the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$. In particular, our approach provides a new formulation of the Kazhdan-Lusztig theory for Lie algebras of type B/C.

We study the resonances of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the non-compact type. The symmetric space is assumed to have rank-one but the irreducible representation τ of K defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that τ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable.

In this paper, we generalize the categorifical construction of a quantum group and its canonical basis by Lusztig (\cite{Lusztig,Lusztig2}) to the generic form of whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in \cite{Green} and the restriction functor in \cite{Lusztig2}. By a geometric way to prove Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.

The first author constructed a q-parameterized spherical category $\sC$ over C(q) in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.

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We show that the FI-homology of an FI-module can be computed via a Koszul complex. As an application, we prove that the Castelnuovo-Mumford regularity of a finitely generated torsion FI-module is equal to its degree.

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The authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichmuller theory by the last two-named authors, they further develop the supergeometry of these surfaces and establish asymptotic growth rate of their length spectra.

We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac–Moodyalgebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Heckealgebras.This yields a proof of a conjecture of Chuang–Miyachi relating the level-rank duality with the Ringel–Koszul duality of cyclotomic rational double affine Hecke algebras.

We study the functorially finite maximal rigid subcategories in 2-CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously functorially finite and maximal rigid; we prove that the converse is true if the 2-CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of Keller and Reiten (2007) [KR], we prove that any functorially finite maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via simple mutations, then their endomorphism algebras have the same representation type.