Related papers: Quantum affine algebras and affine Hecke algebras
We prove a highest weight theorem classifying irerducible finite--dimensional representations of quantum affine algebras and survey what is currently known about the structure of these representations.
In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster…
We classify the irreducible finite-dimensional representations of the twisted quantum affine algebras.
We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of…
The so called quantized algebras of functions on affine Hecke algebras of type A and the corresponding q-Schur algebras are defined and their irreducible unitarizable representations are classified.
We introduce an explicit representation of the double affine Hecke algebra (of type $A_1$) at $q=1$ that gives rise to a periodic counterpart of a well-known Fourier transform associated with the affine Hecke algebra.
We classify the finite dimensional representations of the double affine Hecke algebra of type $C^{\vee}C_1$ in the case when $q$ is not a root of 1.
We classify finite dimensional simple spherical representations of rational double affine Hecke algebras, and we study a remarkable family of finite dimensional simple spherical representations of double affine Hecke algebras.
This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible…
In this paper, we study the relation between the cocenter and the representation theory of affine Hecke algebras. The approach is based on the interaction between the rigid cocenter, an important subspace of the cocenter, and the dual…
We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine)…
To each symmetric algebra we associate a family of algebras that we call quantum affine wreath algebras. These can be viewed both as symmetric algebra deformations of affine Hecke algebras of type $A$ and as quantum deformations of affine…
We classify the finite dimensional irreducible representations of affine Hecke algebras of type B_2 with unequal parameters.
We classify and construct irreducible completely splittable representations of affine and finite Hecke-Clifford algebras over an algebraically closed field of characteristic not equal to 2.
Affine Hecke algebras arise naturally in the study of smooth representations of reductive $p$-adic groups. Finite dimensional complex representations of affine Hecke algebras (under some restriction on the isogeny class and the parameter…
We use one-dimensional double affine Hecke algebras to introduce q-counterparts of the Gauss integrals and new types of Gauss-Selberg sums at roots of unity.
We give a summary of the theory of (weak) quantum vertex $\C((t))$-algebras and the association of quantum affine algebras with (weak) quantum vertex $\C((t))$-algebras.
We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by $\mathcal{H}_q (f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as…
In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations…
Generalizing our earlier work, we introduce the homogeneous quantum $Z$-algebras for all quantum affine algebras $\alg$ of type one. With the new algebras we unite previously scattered realizations of quantum affine algebras in various…