Center of Infinitesimal Hecke Algebra of $sl_2$

On maximal primitive quotients of infinitesimal Cherednik algebras of gl_n

We prove analogues of some of Kostant's theorems for infinitesimal Cherednik algebras of gl_n. As a consequence, it follows that in positive characteristic the Azumaya and smooth loci of the center of these algebras coincide.

Frobenius map for the centers of Hecke algebras

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.

On infinitesimal Cherednik algebras of gl_2

We prove that the center of an infinitesimal Cherednik algebra of $\mf{gl}_2$ is the polynomial algebra of two variables over the field of characteristic 0. In positive characteristic we show that any infinitesimal Cherednik algebra is a finitely generated module over its center.

Double affine Hecke algebra in logarithmic conformal field theory

We construct the representation of Double Affine Hecke Algebra whose symmetrization gives the center of the quantum group U_q(sl(2)) and by Kazhdan--Lusztig duality the Verlinde algebra of (1,p) models of logarithmic conformal field theory.

渐近Hecke范畴的中心与单位特征sheaves

The center of the asymptotic Hecke category and unipotent character sheaves

本文证明了渐近Hecke范畴的中心为模ularity张量范畴，并研究了其在二面体群及H3、H4部分细胞中的应用。

Axiomatic framework for the BGG Category O

The main goal of this paper is to show that a wide variety of infinite-dimensional algebras all share a common structure, including a triangular decomposition and a theory of weights. This structure allows us to define and study the BGG Category O, generalizing previous definitions of it. Having presented our axiomatic framework, we present sufficient conditions that guarantee finite length, enough projectives, and a block decomposition into highest weight categories. The framework is strictly more general than the usual theory of O; this is needed to accommodate (quantized or higher rank) inf…

Hecke-Clifford algebras and spin Hecke algebras I: the classical affine type

Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.

On Iwahori-Hecke Algebras for p-adic Loop Groups: Double Coset Basis and Bruhat Order

We study the $p$-adic loop group Iwahori-Hecke algebra $\mathcal{H}(G^+,I)$ constructed by Braverman, Kazhdan, and Patnaik and give positive answers to two of their conjectures. First, we algebraically develop the "double coset basis" of $\mathcal{H}(G^+,I)$ given by indicator functions of double cosets. We prove a generalization of the Iwahori-Matsumoto formula, and as a consequence, we prove that the structure coefficients of the double coset basis are polynomials in the order of the residue field. The basis is naturally indexed by a semi-group $\mathcal{W}_\mathcal{T}$ on which Braverman, K…

Scattering matrices and affine Hecke algebras

We construct the scattering matrices for an arbitrary Weyl group in terms of elementary operators which obey the generalised Yang-Baxter equation. We use this construction to obtain the affine Hecke algebras. The center of the affine Hecke algebras coincides with commuting Hamiltonians. These Hamiltonians have q-deformed affine Lie algebras as symmetry algebra.

Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra

This paper builds on the previous paper arXiv:math/0612730 by the author, where a relationship between Zhedanov's algebra AW(3) and the double affine Hecke algebra (DAHA) corresponding to the Askey-Wilson polynomials was established. It is shown here that the spherical subalgebra of this DAHA is isomorphic to AW(3) with an additional relation that the Casimir operator equals an explicit constant. A similar result with q-shifted parameters holds for the antispherical subalgebra. Some theorems on centralizers and centers for the algebras under consideration will finally be proved as corollaries …

The Whittaker model of the center of the quantum group and Hecke algebras

In 1978 Kostant suggested the Whittaker model of the center of the universal enveloping algebra U(g) of a complex simple Lie algebra g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b), where b is a Borel subalgebra in g. This observation is used in the theory of principal series representations of the corresponding Lie group G and in the proof of complete integrability of the quantum Toda lattice. In this thesis we generalize the Kostant's construction to quantum groups. In our construction we use quantum analogues of regular nilpotent elements defi…

Centers in Generalized Reflection Equation algebras

In the Reflection Equation (RE) algebra associated with an involutive or Hecke symmetry $R$ the center is generated by elements ${\rm Tr}_R L^k$ (called the quantum power sums), where $L$ is the generating matrix of this algebra and ${\rm Tr}_R$ is the $R$-trace associated with $R$. We consider the problem: whether it is so in certain RE-like algebras depending on spectral parameters. Mainly, we deal with algebras similar to those considered in \cite{RS} (we call them algebras of RS type). These algebras are defined by means of some current (i.e. depending on parameters) $R$-matrices arising f…

A categorical approach to classical and quantum Schur-Weyl duality

We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct act…

Center of the universal Askey--Wilson algebra at roots of unity

Inspired by a profound observation on the Racah--Wigner coefficients of $U_q(\mathfrak{sl}_2)$, the Askey--Wilson algebras were introduced in the early 1990s. A universal analog $\triangle_q$ of the Askey--Wilson algebras was recently studied. For $q$ not a root of unity, it is known that $Z(\triangle_q)$ is isomorphic to the polynomial ring of four variables. A presentation for $Z(\triangle_q)$ at $q$ a root of unity is displayed in this paper. As an application, a presentation for the center of the double affine Hecke algebra of type $(C_1^\vee,C_1)$ at roots of unity is obtained.

Algebres de Hecke affines generiques

Let $H$ be a generic affine Hecke algebra (Iwahori-Matsumoto definition) over a polynomial algebra with a finite number of indeterminates over the ring of integers. We prove the existence of an integral Bernstein-Lusztig basis related to the Iwahori-Matsumoto basis by a strictly upper triangular matrix, from which we deduce that the center $Z$ of $H$ is finitely generated and that $H$ is a finite type $Z$-module (this was proved after inversion of the parameters by Bernstein-Lusztig), and we give some applications to the theory of $H$-modules where the parameters act by 0. These results are re…

Double affine Hecke algebras of rank 1 and affine cubic surfaces

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 Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if $q=1$, then the spectrum of the center of $H$ is an affine cubic surface $C$, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of $H$ for s…

The Character Theory of a Complex Group

We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. Our focus is the Hecke category of Borel-equivariant D-modules on the flag variety of a complex reductive group G (equivalently, the category of Harish Chandra bimodules of trivial central character) and its monodromic variant. The Hecke category is a categorified analogue of the finite Hecke algebra, which is a finite-dimensional semi-simple symmetric Frobenius algebra. We establish parallel properties of the Hecke category, showing it is a two-dualizable Calabi-Yau …

Generalized double affine Hecke algebras of rank 1 and quantized Del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D4,E6,E7,E8). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z_l\ltimes Z^2, where l is 2,3,4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their def…

Cohomology of finite tensor categories: duality and Drinfeld centers

We consider the finite generation property for cohomology of a finite tensor category C, which requires that the self-extension algebra of the unit Ext*_C(1,1) is a finitely generated algebra and that, for each object V in C, the graded extension group Ext*_C(1,V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C. For example, the stated result holds when C is a braided tensor category of odd Fro…