Weyl Groups and the Nil-Hecke Algebra

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ . We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ .

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by simple reflections and Coxeter relations. In a broad sense this answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumoto’s theorem. Therefore the word problem is solved for the groupoid.

We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a Gamma -permutahedron for some finite reflection group Gamma subset {{,mathrm{O},}}(mathbb {R}^d). The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification of finite reflection groups. We prove that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set to be a root system.

In this talk we introduce generalised Calogero–Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H 3, H 4, and the dihedral group I 2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero–Moser models and for any choices of the potentials are linear combinations of the reflection operators. The equivalence of the Lax pair with the equations of motion is proved by decomposing the root system into a sum of two-dimensional sub-root systems, A 2, B 2, G 2, and I 2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators.

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $Ω$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $Ω_+=Ω\cap C_+$ be the positive part of $Ω$. We define a family $\{-Δ_η^+\}$ of self-adjoint extensions of the Laplacian $-Δ_{Ω_+}$, labeled by homomorphisms $η\colon W\to \{1,-1\}$. In the construction of these $η$-Laplacians $η$-symmetrization of functions on $Ω$ is involved. The Neumann Laplacian $-Δ_{N,Ω_+}$ is included and corresponds to $η\equiv1$. If $H^{1}(Ω)=H^{1}_0(Ω)$, then the Dirichlet Laplacian $-Δ_{D,Ω_+}$ is either included and corresponds to $η={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $Ψ(-Δ_{N,Ω})$ and $Ψ(-Δ_η^+)$, or $Ψ(-Δ_{D,Ω})$ and $Ψ(-Δ_{D,Ω_+})$, where $Ψ$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$.

Parabolic subgroup orbits on finite root systems

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

Classification of crystallographic groups associated with Coxeter groups

Associated to any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections.The Hecke algebra of the group can be represented by monodromy action on the space of functions annihilated by each differential-difference operator in the algebra.For each irreducible representation of G the differential-difference equations lead to a linear system of first-order meromorphic differential equations corresponding to an integrable connection over the G-orbits of regular points in the complexification of the Euclidean space.The fundamental group is the generalized Artin braid group belonging to G, and its monodromy representation factors over the Hecke algebra of G. Monodromy has long been of importance in the study of special functions of several variables, for example, the hyperlogarithms of Lappo-Danilevsky are used to express the flat sections and the work of Riemann on the monodromy of the hypergeometric equation is applied to the case of dihedral groups.

Let \(\mathfrak{g}_{0}\) be a real semisimple Lie algebra. Let \(\mathfrak{g}_{0} = \mathfrak{k}_{0} \oplus \mathfrak{p}_{0}\) be the corresponding Cartan decomposition and \(\mathfrak{h}_{0} = \mathfrak{t}_{0} \oplus \mathfrak{a}_{0}\) be a maximally compact Cartan subalgebra of \(\mathfrak{g}_{0}\). Let \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) and \(\mathfrak{h} = \mathfrak{t} \oplus \mathfrak{a}\) be the corresponding complexifications. The set \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) consists of all the linear forms on \(\mathfrak{t}\) which are the restriction to \(\mathfrak{t}\) of the roots in the root system \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\) of \(\mathfrak{g}\) with respect to \(\mathfrak{h}\). The main result of the paper is to prove that \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) is also a (maybe non-reduced) root system and its Weyl group can be identified with a subgroup of the Weyl group of \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\). Let \(Spin\,\nu :\mathfrak{k} \to End\,S\) be the composition of the isotropy representation \(\nu :\mathfrak{k} \to \mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)}\) with the spin representation \(Spin:\mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)} \to End\,S\). Finally as an application, we give a nice description of the \(\mathfrak{k}\)-module structure on S in terms of the restricted root system \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) and its Weyl group.

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.