Unitary representations of the Cherednik algebra: $V^*$-homology

从切列德尼克代数到纽结同调：经由尖点特征D-模

From Cherednik algebras to knot homology via cuspidal D-modules

本文证明环面纽结的三重分级 Khovanov-Rozansky 同调可由有理切列德尼克代数的有限维表示及其尖点特征 D-模的 Hodge 滤过恢复，并通过 shuffle 代数与 Hilbert 概型等变 K 理论的关系证明了有理主猜想。

Affine Springer Fibers, Procesi bundles, and Cherednik algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{t}$ its Cartan subalgebra and $W$ the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus \mathfrak{t}^*)/W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimo…

从有理Cherednik代数的表示到抛物Hilbert概形：经由Dunkl-Opdam子代数

From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra

本文通过Dunkl-Opdam子代数研究有理Cherednik代数的组合表示论，构造其在平面曲线奇点抛物Hilbert概形等变同调上的作用，并推广到量子化Gieseker代数及非既约曲线情形。

Homology of torus knots

Using the method of Elias-Hogancamp and combinatorics of toric braids we give an explicit formula for the triply graded Khovanov-Rozansky homology of an arbitrary torus knot, thereby proving some of the conjectures of Aganagic-Shakirov, Cherednik, Gorsky-Negut and Oblomkov-Rasmussen-Shende.

The Dunkl-Cherednik Deformation of a Howe duality

We consider the deformed versions of the classical Howe dual pairs $(O(r),\mathfrak{s}\mathfrak{l}(2))$ and $(O(r),\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2))$ in the context of a rational Cherednik algebra $H_c=H_c(W,\mathfrak{h})$ associated to a finite Coxeter group $W$ at the parameters $c$ and $t=1$. For the first pair, we compute the centraliser of the well-known copy of $\mathfrak{s}\cong\mathfrak{s}\mathfrak{l}(2)$ inside $H_c$. For the second pair, we show that the classical copy of $\mathfrak{g}\cong\mathfrak{s}\mathfrak{p}\mathfrak{o}(2|2)$ inside the Weyl-Clifford algebra $\mathcal{…

Homology representations of unitary reflection groups

This paper continues the study of the poset of eigenspaces of elements of a unitary reflection group (for a fixed eigenvalue), which was commenced in [6] and [5]. The emphasis in this paper is on the representation theory of unitary reflection groups. The main tool is the theory of poset extensions due to Segev and Webb ([16]). The new results place the well-known representations of unitary reflection groups on the top homology of the lattice of intersections of hyperplanes into a natural family, parameterised by eigenvalue.

有理 Cherednik 代数与环面纽结不变量

Rational Cherednik Algebras and Torus Knot Invariants

本文研究有理 Cherednik 代数表示上的滤过与环面纽结 Khovanov-Rozansky 同调之间的猜想联系，证明代数滤过和归纳滤过这两个候选滤过相等。

Cyclotomic double affine Hecke algebras (with an appendix by Hiraku Nakajima and Daisuke Yamakawa)

We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, we define a new $q$-deformation of this algebra, which we call cyclotomic DAHA. Namel…

Unitary representations of Hecke algebras of complex reflection groups

We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the unitary representations in the category $\OO$ for the rational Cherednik algebra of type A studied in \cite{ESG} correspond to the unitary representations of the corresponding Hecke algebra via the KZ functor defined in \cite{GGOR}.

高维Heegaard Floer同调与双重仿射Hecke代数的多项式表示

Higher-dimensional Heegaard Floer homology and the polynomial representation of double affine Hecke algebras

本文证明高维Heegaard Floer同调可恢复A型双重仿射Hecke代数的多项式表示，并给出Cherednik内积的拓扑解释。

Deformations of unitary Howe dual pairs

We study deformations of the Howe dual pairs $(\mathrm{U}(n),\mathfrak{u}(1,1))$ and $(\mathrm{U}(n),\mathfrak{u}(2|1))$ to the context of a rational Cherednik algebra $H_{1,c}(G,E)$ associated with a real reflection group $G$ acting on a real vector space $E$ of even dimension. For each pair, we show that the Lie (super)algebra structure of one partner is preserved under the deformation, which leads to a multiplicity-free decomposition of the standard module or its tensor product with a spinor space. For the case where $E$ is two-dimensional and $G$ is a dihedral group, we provide complete de…

Representations of Rational Cherednik algebras with minimal support and torus knots

We obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type A_{n-1} for c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Et…

Twisted homology for the mirabolic nilradical

The notion of derivatives for smooth representations of $GL(n,\mathbb{Q}_p)$ was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the "adduced" representation. In [AGS] derivatives of all orders were defined for smooth admissible Frechet representations (of moderate growth). A key ingredient of this definition is the functor of twisted coinvariants with respect to the nilradical of the mirabolic subgroup. In this paper we prove exactness of this functor and compute it on a certain class of …

Generalized affine Springer theory and Hilbert schemes on planar curves

We show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky-Kottwitz-MacPherson. Using a generalization of affine Springer theory for Braverman-Finkelberg-Nakajima's Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes, and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn-Kamnitzer-Weekes, which may be of independent interest. In the spherical cas…

The product formula in unitary deformation K-theory

We prove that "unitary deformation K-theory" takes products of finitely generated groups to coproducts of algebra spectra over ku, the connective K-theory spectrum. Additionally, we give spectral sequences for computing the homotopy groups of the unitary deformation K-theory of a group G and the cofiber of the Bott map in terms of PU(n)-equivariant K-theory and homology of spaces of G-representations.

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator

The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a real…

半单群的酉上同调

On the unitary cohomology of semisimple groups

本文研究半单李群在任意酉表示下的连续（上）同调，将可约表示情形分解为Hausdorff部分与挠部分，分别由不可约上同调表示的内含性与弱内含性刻画，并给出秩1群及SO°(n,1)的完整结果，同时证明挠上同调首阶在性质(T)下受对称空间维数平方根下界限制。

Homological realization of the restricted Kostka polynomials

In this paper we give two realizations of the restricted Kostka polynomials for $\sl_2$. Firstly we identify the restricted Kostka polynomials with a characters of the zero homology of the current algebra with a coefficients in a certain modules. As a corollary we reobtain the alternating sum formula. Secondly we show that the restricted Kostka polynomials are a $q$-multiplicities of the decomposition of the certain integrable $\hat{\sl}_2$-modules to the irreducible components. This allows to write a kind of fermionic formula for the Virasoro unitary models.

Topology of eigenspace posets for imprimitive reflection groups

This paper studies the poset of eigenspaces of elements of an imprimitive unitary reflection group, for a fixed eigenvalue, ordered by the reverse of inclusion. The study of this poset is suggested by the eigenspace theory of Springer and Lehrer. The posets are shown to be isomorphic to certain subposets of Dowling lattices (the `d-divisible, k-evenly coloured Dowling lattices'). This enables us to prove that these posets are Cohen-Macaulay, and to determine the dimension of their top homology.

约化 $p$-进群的 Hochschild 同调

Hochschild homology of reductive $p$-adic groups

本文研究约化 $p$-进群的 Hecke 与 Harish-Chandra–Schwartz 代数，利用光滑表示族和 Bernstein 分解证明其 Hochschild、循环同调与交叉积代数对应，并联系 K-理论。

紧致双曲轨道曲面上的扭曲 Ruelle zeta 函数与 Reidemeister-Turaev 挠率

The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion

该文研究紧致双曲轨道曲面上扭曲 Ruelle zeta 函数在零点取值与 Reidemeister-Turaev 挠率的关系，推广 Fried 猜想至非酉表示并解决相位符号问题，同时计算消失阶与首项系数。