Related papers: From conjugacy classes in the Weyl group to semisi…

Let G be a connected reductive group. We define a map from the set of unipotent classes in G to the set of conjugacy classes in the Weyl group (assuming that the characteristic is not bad). This map is a one sided inverse of a map in the…

Let G be a connected reductive algebraic group over an algebraic closed field. We define a (surjective) map from the set of conjugacy classes in the Weyl group to the set of unipotent classes of G.

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.…

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on…

We define a map from the set of conjugacy classes of a Weyl group W to the representation ring of W tensored with the ring of polynomials in one variable.

We define and study a correspondence between the set of distinguished G^0-conjugacy classes in a fixed connected component of a reductive group G (with G^0 almost simple) and the set of (twisted) elliptic conjugacy classes in the Weyl…

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.

Let G be an almost simple reductive group with Weyl group W. Let B be a Borel subgroup of G. Let C be an elliptic conjugacy class in W and let w be an element of minimal length of C. We investigate the existence of a semisimple class of G…

Random walk on the chambers of hyperplanes arrangements is used to define a family of card shuffling measures $H_{W,x}$ for a finite Coxeter group W and real $x \neq 0$. By algebraic group theory, there is a map from the semisimple orbits…

Suppose $\mathfrak{g}$ is a semisimple complex Lie algebra and $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$. To the pair $(\mathfrak{g},\mathfrak{h})$ one can associate both a Weyl group and a set of Kac diagrams. There is a…

Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set…

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.

In this (partly expository) paper we show, using ideas from the theory of total positivity, how a number of properties of a semisimple group over the complex numbers can be presented purely in terms of the Weyl group. We also describe some…

Let G be a connected reductive group over an algebraically closed field of characteristic p. In an earlier paper we defined a surjective map \Phi_p from the set \underline{W} of conjugacy classes in the Weyl group W to the set of unipotent…

In an earlier paper by Kazhdan and the author, a map from the set of unipotent classes in a reductive connected group over C to the conjugacy classes in the Weyl group was defined. Here we present some experimental evidence for a possibly…

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$.…

In a 2015 paper we have defined a map from the set of conjugacy classes in a Weyl group W to the set of irreducible representations of W (its image parametrizes the strata of a reductive group with Weyl group W). In this paper we provide…

We determine the conjugacy classes of semisimple elements in the symplectic groups ${\rm Sp}(2m,F)$, where $F$ is an arbitrary field of characteristic not $2$. This note was originally a letter dated 23 March, 2006, from G.E. Wall to Cheryl…

Suppose $G$ is a reductive algebraic group, $T$ is a Cartan subgroup, $N=\text{Norm}(T)$, and $W=N/T$ is the Weyl group. If $w\in W$ has order $d$, it is natural to ask about the orders lifts of $w$ to $N$. It is straightforward to see that…

Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for…