Related papers: Strata of Weyl groups

An element w of a Weyl group W is called elliptic if it has no eigenvalue 1 in the standard reflection representation. We determine the order of any representative g in a semisimple algebraic group G of an elliptic element w in the…

The problem of interpreting a set of ${\cal W}$-algebra constraints constructed in terms of an arbitrarily twisted scalar field as the recursion relations of a topological theory is addressed. In this picture, the conventional models of…

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl group in the same two-sided cell. We use this map to parametrize…

Let $D$ be a connected component of a possibly disconnected reductive group $G$ over an algebraic closed field. We define a partition of $D$ into finitely many Strata each of which is a union of $G^0$-conjugacy classes of fixed dimension.…

Let G be a simple reductive group over the complex numbers. Let W be the Weyl group of G. We propose a description of the Springer representations of W associated to various unipotent classes of G by a purely algebraic method involving the…

Given a grading $\Gamma: A=\oplus_{g\in G}A_g$ on a nonassociative algebra $A$ by an abelian group $G$, we have two subgroups of the group of automorphisms of $A$: the automorphisms that stabilize each homogeneous component $A_g$ (as a…

Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element.…

We show that various invariants of a unipotent conjugacy class in a connected semisimple group can be recovered purely in terms of data involving the Weyl group.

A Weyl structure is usually defined by an equivalence class of pairs $({\bf g}, \boldsymbol{\omega})$ related by Weyl transformations, which preserve the relation $\nabla {\bf g}=\boldsymbol{\omega}\otimes{\bf g}$, where ${\bf g}$ and…

Let W be a Weyl group. We introduce the notion of positive conjugacy class in W. This generalizes the notion of regular elliptic conjugacy class in the sense of Springer.

Given a grading on a nonassociative algebra by an abelian group, we have two subgroups of automorphisms attached to it: the automorphisms that stabilize each homogeneous component (as a subspace) and the automorphisms that permute the…

We consider the finite Weyl groups of classical type -- $W(A_{r})$ for $r \geq 1$, $W(B_{r}) = W(C_{r})$ for $r \geq 2$, and $W(D_{r})$ for $r \geq 4$ -- as supergroups in which the reflections are of odd superdegree. Viewing the…

Let W be a Weyl group, presented as a crystallographic reflection group on a Euclidean vector space V, and C an open Weyl chamber. In a recent paper, Waldspurger proved that the images (id-w)(C), for Weyl group elements w, are all disjoint,…

We define (in two, equivalent ways) the notion of a rigid stratum of a reductive group. This generalizes the notion of rigid unipotent class.

In this note a combinatorial formula related to the symmetric group is generalized to an arbitrary finite Weyl group.

The orbits of Weyl groups W(A(n)) of simple A(n) type Lie algebras are reduced to the union of orbits of the Weyl groups of maximal reductive subalgebras of A(n). Matrices transforming points of the orbits of W(An) into points of subalgebra…

We compute the Weyl group (in the sense of Segal) of the group of holomorphic isometries of a K\"ahler toric manifold with real analytic K\"ahler metric.

It is known that signature of a Weyl group element is defined in terms of the number of its simple Weyl reflections. Actual calculations hence are not always possible especially for Weyl groups with higher order like $E_8$ Weyl group. By…

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair $(X,Y)$ of subsets of the symmetric group $\mathfrak{S}_n$, the…

Let w be an elliptic element of the Weyl group of a connected reductive group G. Let X be the set of pairs (g,B) where g is an element of G, B is a Borel subgroup of G and B,gBg^{-1} are in relative position w. Then G acts naturally on X.…