Related papers: Complete Reducibility in Good Characteristic

A $G$-grading on a complex semisimple Lie algebra $L$, where $G$ is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems,…

For $G$ a connected, reductive group over an algebraically closed field $k$ of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent variety of $G$…

Parahoric group schemes are certain possibly non-reductive, smooth, affine integral models of reductive group schemes defined over a henselian discretely valued field $K$ whose residue field is perfect. We show that any such group scheme…

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of…

Let G be a reductive algebraic group over a field of positive characteristic and denote by C(G) the category of rational G-modules. In this note we investigate the subcategory of C(G) consisting of those modules whose composition factors…

A subgroup $H$ of a finite group $G$ is said to be an NC-subgroup of $G$, if $ H^G N_G (H) =G$, where $H^G$ denotes the normal closure of $H$ in $G$. A finite group $G$ is called a PNC-group, if any subgroup of $G$ is an NC-subgroup of $G$,…

Let G be a finite non-abelian simple group and let p be a prime. We classify all pairs (G,p) such that the sum of the complex irreducible character degrees of G is greater than the index of a Sylow p-subgroup of G. Our classification…

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…

For a finite group $G$, we denote by $c(G)$, the minimal degree of faithful representation of $G$ by quasi-permutation matrices over the complex field $\mathbb{C}$. For an irreducible character $\chi$ of $G$, the codegree of $\chi$ is…

The (reduced) characteristic group of a locally conformally product manifold is obtained by restricting the action of its fundamental group to the non-flat factor of the universal cover, and taking the connected component of the identity in…

Let $(\mathfrak{g},[p])$ be a finite-dimensional restricted Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $p>0$, and $G$ be the adjoint group of $\mathfrak{g}$. We say that $\mathfrak{g}$ satisfying the {\sl…

The reduced C*-algebra of a countable linear group G is shown to be simple if and only if G has no nontrivial normal amenable subgroups. Moreover, these conditions are shown to be equivalent to the uniqueness of tracial state on the…

Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a Local Langlands Correspondence…

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental…

Let G be a connected reductive group (over $\mathbb{C}$) and H a connected semisimple subgroup. The dimension data of H (realative to its given embedding in G) is the collection of the numbers $\{{\rm dim} V^{H}\}$, where V runs over all…

A group $G$ is said to have restricted centralizers if for each $g \in G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. We take…

Let F be a local henselian nonarchimedean field of residual field k, and let G be the group of F-points of a connected reductive group defined over F. It is well-known that the quotient of any parahoric subgroup of G by its first congruence…

Let G be a finite group and let p be a prime. A module for G over a field of characteristic p is called algebraic if it satisfies a polynomial, with addition and multiplication given by direct sum and tensor product. In some sense, having…

Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\mathcal{F}$ associates to $\mathbf{G}$ a linear algebraic group…

The reduced power graph $\mathcal{RP}(G)$ of a group $G$ is the graph with vertex set $G$ and two vertices $u$ and $v$ are adjacent if and only if $\left\langle v\right\rangle \subset \left\langle u \right\rangle $ or $\left\langle…