Related papers: Multiple rational normal forms in Lie theory

Let $G$ be a simple algebraic group and $\ggg=\Lie(G)$ over $k=\bar\bbf_q$ where $q$ is a power of the prime characteristic of $k$, and $F$ a Frobenius morphism on $G$ which can be defined naturally on $\ggg$. In this paper, we investigate…

Let $G$ be a reductive complex Lie group with Lie algebra $\mathfrak{g}$ and suppose that $V$ is a polar $G$-representation. We prove the existence of a radial parts map $\mathrm{rad}: \mathcal{D}(V)^G\to A_{\kappa}$ from the $G$-invariant…

We address the Noncommutative Noether's Problem on the invariants of Weyl fields for linear actions of finite groups. We prove that if the variety An(k)/G is rational then the Noncommutative Noether's Problem is positively solved for G and…

We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth…

An element $g$ of a Lie group is called stably elliptic if it is contained in the interior of the set $G^e$ of elliptic elements, characterized by the property that $\mathrm{Ad}(g)$ generates a relatively compact subgroup. Stably elliptic…

Given a reductive group $G$, we give a description of the abelian category of $G$-equivariant $D$-modules on $\mathfrak{g}=\mathrm{Lie}(G)$, which specializes to Lusztig's generalized Springer correspondence upon restriction to the…

The main result describes the Brauer-Nesbitt reduction of unipotent representations of a finite group of Lie type, expressing it as an explicit linear combination of the restriction of Weyl modules from the algebraic group to the group of…

We continue the study of separable elements in finite Weyl groups. These elements generalize the well-studied class of separable permutations. We show that the multiplication map $W/U \times U \to W$ is a length-additive bijection, or…

A differential system $[A] : \; Y'=AY$, with $A\in \mathrm{Mat}(n, \bar{k})$ is said to be in reduced form if $A\in \mathfrak{g}(\bar{k})$ where $\mathfrak{g}$ is the Lie algebra of the differential Galois group $G$ of $[A]$. In this…

Let U be the unipotent radical of a Borel subgroup of a connected reductive algebraic group G, which is defined over an algebraically closed field k. In this paper, we extend work by Goodwin-R\"ohrle concerning the commuting variety of…

Any deformation of a Weyl or Clifford algebra A can be realized through a `deforming map', i.e. a formal change of generators in A. This is true in particular if A is covariant under a Lie algebra g and its deformation is induced by some…

Let $\mathfrak{g}$ be a Lie algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Define the universal grading group $\mathcal{C}(\mathfrak{g})$ as having one generator $g_{\rho}$ for each irreducible…

We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a…

Let G be a reductive algebraic group and V a G-module. We consider the question of when (GL(V), rho(G)) is a reductive pair of algebraic groups, where rho is the representation afforded by V. We first make some observations about general G…

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of complex…

Let $k$ be a nonperfect separably closed field. Let $G$ be a (possibly non-connected) reductive group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In our previous work, we…

In this article, we present a combinatorial formula for computing the Wedderburn decomposition of the rational group algebra associated with an ordinary metacyclic $p$-group $G$, where $p$ is any prime. We also provide a formula for…

In this paper we discuss some of Springer's work on unipotent elements in a reductive groups and on representations of Weyl groups. Among the topics considered are Springer's bijection from the unipotent variety to the nilpotent variety,…

We study the parametrizations of simple modules provided by the theory of basic sets for all finite Weyl groups. In the case of type B, we show the existence of basic sets for the matrices of constructible representations. Then we study…

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x, y) in g\timesg such that x and y are nilpotents and are in the same Borel subalgebra. The main result of this paper is that N is a closed and…