Related papers: The Dunkl Weight Function for Rational Cherednik A…

We define a family of Dirac operators for the Dunkl angular momentum algebra depending on certain central elements of the group algebra of the Pin cover of the Weyl group inherent to the rational Cherednik algebra. We prove an analogue of…

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group $G(\ell,p,n)$. In particular, we: (1) show a relationship to the Coulomb branch construction of…

Given dominant integral weights $\lambda, \mu, \nu$ of a finite-dimensional simple Lie algebra $\mathfrak{g}$ and an element $w$ of its Weyl group, the refined tensor product multiplicity $c_{\lambda \mu}^\nu(w)$ is the multiplicity of the…

In the 1950s L. Schwartz proved his famous impossibility result: for every k in N there does not exist a differential algebra (A,+,*,D) in which the distributions can be embedded, where D is a linear operator that extends the distributional…

Begin with the Hasse-Weil zeta-function of a smooth projective variety over the rational numbers. Replace the variety with a finite CW-complex, replace etale cohomology with complex K-theory $KU^*$, and replace the $p$-Frobenius operator…

Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of…

Let $j\geq 2$ be a given integer. Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Denote by $\lambda_{\text{sym}^{j}f}(n)$ the $n$th normalized…

Associated to every complex reflection group, we construct a lattice of quotients of its braid monoid-algebra, which we term nil-Hecke algebras, and which are obtained by killing all braid words that are "sufficiently long", as well as some…

Given a finite subgroup $W \subset \GL(\fh)$ of the linear group of a finite-dimensional complex vector field $\fh$, it is a well-studied problem to describe the structure of the symmetric algebra $B= \sym(\fh^*)$ as a representation of…

We define the algebraic Dirac induction map $\Ind_D$ for graded affine Hecke algebras. The map $\Ind_D$ is a Hecke algebra analog of the explicit realization of the Baum-Connes assembly map in the $K$-theory of the reduced $C^*$-algebra of…

In this paper we consider the Hecke algebra $\mathcal {H}$ associated to an extended affine Weyl group of type $\widetilde{B_2}$. We give some interesting formulas on $C_{rt}S_{\lambda}$, which imply some relations between the…

We study the $C^*$-algebra $\mathcal{T}/\mathcal{K}$ where $\mathcal{T}$ is the $C^*$-algebra generated by $d$ weighted shifts on the Fock space of $\mathbb{C}^d$, $\mathcal{F}(\mathbb{C}^d)$, ( where the weights are given by a sequence…

For a finite Lie algebra $G_N$ of rank N, the Weyl orbits $W(\Lambda^{++})$ of strictly dominant weights $\Lambda^{++}$ contain $dimW(G_N)$ number of weights where $dimW(G_N)$ is the dimension of its Weyl group $W(G_N)$. For any…

We consider irreducible lowest-weight representations of Cherednik algebras associated to certain classes of complex reflection groups in characteristic p. In particular, we study maximal graded submodules of Verma modules associated to…

We provide evidence for the existence of a family of generalized Kac-Moody(GKM) superalgebras, G_N, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Delta_{k/2}(Z), for (N,k)=(1,10), (2,6),…

It is shown that the rich algebraic structure of the standard $d$-dimensional Coulomb problem can be extended to its Dunkl counterpart. Replacing standard derivatives by Dunkl ones in the so($d+1$,2) dynamical algebra generators of the…

We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions…

We prove that, over an arbitrary CM field, every symmetric formal Fourier-Jacobi series converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular form. As an application, we show that the Chow-valued Kudla…

In this article, we describe all two sided ideals of a cyclotomic rational Cherednik algebra $H_\mathbf{c}$ and its spherical subalgebra $eH_\mathbf{c} e$ with a Weil generic aspherical parameter $\mathbf{c}$, and further describe the…

A W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of these algebras. Under some conditions on a nilpotent element…