Asymptotic boundary KZB operators and quantum Calogero-Moser spin chains
Nicolai Reshetikhin, Jasper Stokman
备注：
31 pages. Typos corrected, and a particular central element $z_λ$ introduced in v1 just before Cor. 5.6 is removed from the text since it is zero by the new, stronger version of Cor. 5.6
Asymptotic boundary KZB equations describe the consistency conditions of degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov conformal field theory on a cylinder. In the first part of the paper we define asymptotic boundary KZB operators for connected real semisimple Lie groups G with finite center. We prove their main properties algebraically using coordinate versions of Harish-Chandra's radial component map. We show that their commutativity is governed by a system of equations involving coupled versions of classical dynamical Yang-Baxter equations and reflection equations. We use the coordinate radial components maps to introduce a new class of quantum superintegrable systems, called quantum Calogero-Moser spin chains. A quantum Calogero-Moser spin chain is a mixture of a quantum spin Calogero-Moser system associated to the restricted root system of G and an one-dimensional spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB operators provide explicit expressions for its first order quantum Hamiltonians. We also explicitly describe the Schrödinger operator.
In this paper, we study properties of the algebras of planar quasi-invariants. These algebras are Cohen-Macaulay and Gorenstein in codimension one. Using the technique of matrix problems, we classify all Cohen-Macaulay modules of rank one over them and determine their Picard groups. In terms of this classification, we describe the spectral modules of the planar rational Calogero-Moser systems. Finally, we elaborate the theory of the algebraic inverse scattering method, computing a new unexpected explicit example of a deformed Calogero-Moser system.
Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author have attached to any finite complex reflection group $W$ several notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular characters), completing the notion of Calogero-Moser families defined by Gordon. If moreover $W$ is a Coxeter group, they conjectured that these notions coincide with the analogous notions defined using the Hecke algebra by Kazhdan and Lusztig (or Lusztig in the unequal parameters case). In the present paper, we aim to investigate these conjectures whenever $W$ is a …
Using the representation theory of Cherednik algebras at $t=0$ and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-s…
We study symplectic leaves of Calogero-Moser spaces of type $G(\ell,1,n)$. We prove that the normalization of the closure of each symplectic leaf is isomorphic to some Calogero-Moser space. We also give a nice combinatorial parameterization of the symplectic leaves.
We prove that every irreducible component of the fixed point variety under the action of $d$-th roots of unity in a smooth Caloger-Moser space is isomorphic to a Calogero-Moser space associated with another reflection group.
The goal of this paper is to compute the cuspidal Calogero-Moser families for all infinite families of finite Coxeter groups, at all parameters. We do this by first computing the symplectic leaves of the associated Calogero-Moser space and then by classifying certain "rigid" modules. Numerical evidence suggests that there is a very close relationship between Calogero-Moser families and Lusztig families. Our classification shows that, additionally, the cuspidal Calogero-Moser families equal cuspidal Lusztig families for the infinite families of Coxeter groups.
We continue the study of Calogero-Moser spaces associated with dihedral groups by investigating in more details the equal parameter case: we obtain explicit equations, some informations about the Poisson bracket, the structure of the Lie algebra associated with the cuspidal point and the action of $SL_2(\mathbb{C})$.
We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by an element of finite order of the normalizer of the associated complex reflection group $W$. We give a parametrization {\it à la Harish-Chandra} of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.
We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero-Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. This allows computing the IC stalks of the Uhlenbeck compactification.
We study the spectrum of a family of algebras, the inhomogeneous Gaudin algebras, acting on the $n$-fold tensor representation $\mathbb{C}[x_1, \ldots, x_r]^{\otimes n}$ of the Lie algebra $\mathfrak{gl}_r$. We use the work of Halacheva-Kamnitzer-Rybnikov-Weekes to demonstrate that the Robinson-Schensted-Knuth correspondence describes the behaviour of the spectrum as we move along special paths in the family. We apply the work of Mukhin-Tarasov-Varchenko, which proves that the rational Calogero-Moser phase space can be realised as a part of this spectrum, to relate this to behaviour at $t=0$ o…
