Related papers: Connection problems for quantum affine KZ equation…

Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of…

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or $A_{1}^{(1)}$ model, Izergin-Korepin or $A_{2}^{(2)}$ model, sl(2|1) model and osp(2|1) model. We find…

We define categories $\mathcal{O}^w$ of representations of Borel subalgebras $\mathcal{U}_q\mathfrak{b}$ of quantum affine algebras $\mathcal{U}_q\hat{\mathfrak{g}}$, which come from the category $\mathcal{O}$ twisted by Weyl group elements…

For W a finite (2-)reflection group and B its (generalized) braid group, we determine the Zariski closure of the image of B inside the corresponding Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and generated in the…

We define the degenerate two boundary affine Hecke-Clifford algebra $\mathcal{H}_d$, and show it admits a well-defined $\mathfrak{q}(n)$-linear action on the tensor space $M\otimes N\otimes V^{\otimes d}$, where $V$ is the natural module…

A key tool for the study of an affine Hecke algebra $\mathcal{H}$ is provided by Springer theory of the Langlands dual group via the realization of $\mathcal{H}$ as equivariant $K$-theory of the Steinberg variety. We prove a similar…

We use the structural similarity of certain Coxeter Artin Systems to the Yang--Baxter and Reflection Equations to convert representations of these systems into new solutions of the Reflection Equation. We construct certain Bethe ansatz…

We prove that irreducible representations of the elliptic affine Hecke algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence with certain nilpotent Higgs bundles on the elliptic curve. The main tool we use is the…

We apply the theory of $\phi$-coordinated modules, developed by H.-S. Li, to the Etingof--Kazhdan quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$. We prove, for a certain associate $\phi$ of the…

In this paper, we completely classify the rational weights $k$ for which the Kaneko-Zagier (KZ) differential equation admits a fundamental system of solutions consisting of modular forms for a principal congruence subgroup $\Gamma(N)$. By…

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a…

We prove that the category of finitely generated graded modules over the quiver Hecke algebra of arbitrary type admits numerous stratifications in the sense of Kleshchev. A direct consequence is that the full subcategory corresponding to…

Given a Hecke symmetry $R$, one can define a matrix bialgebra $E_R$ and a matrix Hopf algebra $H_R$, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to $R$. We show that for an even…

The Hecke group algebra $HW_0$ of a finite Coxeter group $W_0$, as introduced by the first and last author, is obtained from $W_0$ by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent…

Aiming for a revival of the theory of crystallographic complex reflection groups, we compute (minimal) Coxeter-like reflection presentations for the infinite families of those non-genuine groups which satisfy Steinberg's fixed point…

For an admissible affine vertex algebra $V_k(\mathfrak{g})$ of type $A$, we describe a new family of relaxed highest weight representations of $V_k(\mathfrak{g})$. They are simple quotients of representations of the affine Kac-Moody algebra…

We define and study cyclotomic quotients of affine Hecke algebras of type B. We establish an isomorphism between direct sums of blocks of these algebras and a generalisation, for type B, of cyclotomic quiver Hecke algebras which are a…

We study the algebraic properties of the five-parameter family $H(t_1,t_2,t_3,t_4;q)$ of double affine Hecke algebras of type $C^\vee C_1$. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by…

Motivated by a study of the crossing symmetry of the `gemini' representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to…

We introduce and begin to study Lie theoretical analogs of symplectic reflection algebras for a finite cyclic group, which we call "cyclic double affine Lie algebra". We focus on type A : in the finite (resp. affine, double affine) case, we…