Simple Algebraic Group Research Articles

On extensions of the Jacobson–Morozov theorem to even characteristic

AbstractLet be a simple algebraic group over an algebraically closed field of characteristic 2. We consider analogues of the Jacobson–Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3‐dimensional Lie overalgebra in and also those with overalgebras isomorphic to the algebras and . This leads us to calculate the dimension of the Lie automiser for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.

Linkage and translation for tensor products of representations of simple algebraic groups and quantum groups

AbstractLet be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient. For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Generic Stabilizers in Actions of Simple Algebraic Groups

In this paper we treat faithful actions of simple algebraic groups on irreducible modules and on the associated Grassmannian varieties. By explicit calculation, we show that in each case, with essentially one exception, there is a dense open subset any point of which has stabilizer conjugate to a fixed subgroup, called the generic stabilizer. We provide tables listing generic stabilizers in the cases where they are non-trivial; in addition we decide whether or not there is a dense orbit, or a regular orbit for the action on the module.

Topological generation of simple algebraic groups

Let G be a simple algebraic group over an algebraically closed field and let X be an irreducible subvariety of G^{r} with r \geqslant 2 . In this paper, we consider the general problem of determining if there exists a tuple (x_{1}, \ldots, x_{r}) \in X such that \langle x_{1}, \ldots, x_{r} \rangle is Zariski dense in G . We are primarily interested in the case where X = C_{1} \times \cdots \times C_{r} and each C_{i} is a conjugacy class of G comprising elements of prime order modulo the center of G . In this setting, our main theorem gives a complete solution to the problem when G is a symplectic or orthogonal group. By combining our results with earlier work on linear and exceptional groups, this gives an almost complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago.

Aut(Fn) actions on representation spaces

J. Wiegold conjectured that if n≥3 and G is a finite simple group, then the action of Aut(Fn) on Epi(Fn,G) is transitive. In this note we consider analogous questions where G is a compact Lie group, a non-compact simple analytic group or a simple algebraic group.

Discrepancy of rational points in simple algebraic groups

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

Galois cohomology and profinitely solitary Chevalley groups

For every number field and every Cartan Killing type, there is an associated split simple algebraic group. We examine whether the corresponding arithmetic subgroups are profinitely solitary so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. Assuming Grothendieck rigidity, we essentially solve the problem by Galois cohomological means.

Multiplicity-free Representations of Algebraic Groups

Let K K be an algebraically closed field of characteristic zero, and let G G be a connected reductive algebraic group over K K . We address the problem of classifying triples ( G , H , V ) (G,H,V) , where H H is a proper connected subgroup of G G , and V V is a finite-dimensional irreducible G G -module such that the restriction of V V to H H is multiplicity-free – that is, each of its composition factors appears with multiplicity 1. A great deal of classical work, going back to Dynkin, Howe, Kac, Stembridge, Weyl and others, and also more recent work of the authors, can be set in this context. In this paper we determine all such triples in the case where H H and G G are both simple algebraic groups of type A A , and H H is embedded irreducibly in G G . While there are a number of interesting familes of such triples ( G , H , V ) (G,H,V) , the possibilities for the highest weights of the representations defining the embeddings H > G H>G and G > G L ( V ) G>GL(V) are very restricted. For example, apart from two exceptional cases, both weights can only have support on at most two fundamental weights; and in many of the examples, one or other of the weights corresponds to the alternating or symmetric square of the natural module for either G G or H H .

Minimum codimension of eigenspaces in irreducible representations of simple classical linear algebraic groups

Let k be an algebraically closed field of characteristic p ≥ 0 , let G be a simple simply connected classical linear algebraic group of rank l and let T be a maximal torus in G with rational character group X ( T ) . For a nonzero p-restricted dominant weight λ ∈ X ( T ) , let V be the associated irreducible kG-module. We define ν G ( V ) as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for ν G ( V ) for G of type A l and dim ( V ) ≤ l 3 2 , and for G of type B l , C l , or D l and dim ( V ) ≤ 4 l 3 . Moreover, we give the exact value of ν G ( V ) for G of type A l with l ≥ 15 ; for G of type B l or C l with l ≥ 14 ; and for G of type D l with l ≥ 16 .

Adjoint Jordan blocks for simple algebraic groups of type $$C_{\ell }$$ in characteristic two

Adjoint Jordan blocks for simple algebraic groups of type $$C_{\ell }$$ in characteristic two

Simple algebraic groups with the same maximal tori, weakly commensurable Zariski-dense subgroups, and good reduction

We provide a new condition for an absolutely almost simple algebraic group to have good reduction with respect to a discrete valuation of the base field which is formulated in terms of the existence of maximal tori with special properties. This characterization, in particular, shows that the Finiteness Conjecture for forms of an absolutely almost simple algebraic group over a finitely generated field that have good reduction at a divisorial set of places of the field (cf. [66]) would imply the finiteness of the genus of the group at hand. It also leads to a new phenomenon that we refer to as “killing the genus by a purely transcendental extension.” Yet another application deals with the investigation of “eigenvalue rigidity” of Zariski-dense subgroups (cf. [64]), which in turn is related to the analysis of length-commensurable Riemann surfaces and general locally symmetric spaces. Finally, we analyze the Finiteness Conjecture and the genus problem for simple algebraic groups of type F4.

On minimal epimorphic subgroups in simple algebraic groups of rank 2

The category of linear algebraic groups admits non-surjective epimorphisms. For simple algebraic groups of rank 2 defined over algebraically closed fields, we show that the minimal dimension of a closed epimorphic subgroup is 3.

Conformal blocks for Galois covers of algebraic curves

We study the spaces of twisted conformal blocks attached to a $\Gamma$-curve $\Sigma$ with marked $\Gamma$-orbits and an action of $\Gamma$ on a simple Lie algebra $\mathfrak {g}$, where $\Gamma$ is a finite group. We prove that if $\Gamma$ stabilizes a Borel subalgebra of $\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed $\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let $\mathscr {G}$ be the parahoric Bruhat–Tits group scheme on the quotient curve $\Sigma /\Gamma$ obtained via the $\Gamma$-invariance of Weil restriction associated to $\Sigma$ and the simply connected simple algebraic group $G$ with Lie algebra $\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic $\mathscr {G}$-torsors on $\Sigma /\Gamma$ when the level $c$ is divisible by $|\Gamma |$ (establishing a conjecture due to Pappas and Rapoport).

Upper bounds for eigenvalue multiplicities of almost cyclic elements in irreducible representations of simple algebraic groups

Upper bounds for eigenvalue multiplicities of almost cyclic elements in irreducible representations of simple algebraic groups

Matrices of simple spectrum in irreducible representations of cyclic extensions of simple algebraic groups

Matrices of simple spectrum in irreducible representations of cyclic extensions of simple algebraic groups

Closures of certain matrix varieties and applications

We prove some results about closures of certain matrix varieties consisting of elements with the same centralizer dimension. This generalizes a result of Dixmier and has applications to topological generation of simple algebraic groups.

Hochschild cohomology of generalised Grassmannians

We compute the Hochschild–Kostant–Rosenberg decomposition of the Hochschild cohomology of generalised Grassmannians, i.e., partial flag varieties associated to maximal parabolic subgroups in a simple algebraic group, in terms of representation-theoretic data.We explain how the decomposition is concentrated in global sections for the (co)minuscule and (co)adjoint generalised Grassmannians, and conjecture that for (almost) all other cases the same vanishing of the higher cohomology does not hold. Our methods give an explicit partial description of the Gerstenhaber algebra structure for the Hochschild cohomology of cominuscule and adjoint generalised Grassmannians. We also consider the case of adjoint partial flag varieties in type A, which are associated to certain submaximal parabolic subgroups.

On the Topological Generation of Exceptional Groups by Unipotent Elements

AbstractLet G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $$p \geqslant 0$$ p ⩾ 0 which is not algebraic over a finite field. Let $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t be non-central conjugacy classes in G. In earlier work with Gerhardt and Guralnick, we proved that if $$t \geqslant 5$$ t ⩾ 5 (or $$t \geqslant 4$$ t ⩾ 4 if $$G = G_2$$ G = G 2 ), then there exist elements $$x_i \in \mathcal {C}_i$$ x i ∈ C i such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , x t ⟩ is Zariski dense in G. Moreover, this bound on t is best possible. Here we establish a more refined version of this result in the special case where $$p>0$$ p > 0 and the $$\mathcal {C}_i$$ C i are unipotent classes containing elements of order p. Indeed, in this setting we completely determine the classes $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t for $$t \geqslant 2$$ t ⩾ 2 such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , x t ⟩ is Zariski dense for some $$x_i \in \mathcal {C}_i$$ x i ∈ C i .

Local types of (Γ,G)$(\Gamma ,G)$‐bundles and parahoric group schemes

AbstractLet be a simple algebraic group over an algebraically closed field . Let be a finite group acting on . We classify and compute the local types of ‐bundles on a smooth projective ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in . When , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings.

Rational Elements in Representations of Simple Algebraic Groups, I

An element g of a group G is called rational if g is conjugate to $$g^i$$ for every integer i coprime to |g|. We determine all triples $$(G,g,\phi )$$ , where G is a simple algebraic group of type $$A_n$$ , $$B_n$$ or $$C_n$$ over an algebraically closed field of characteristic $$p\ge 0$$ , $$g\in G$$ is a rational odd order semisimple element and $$\phi $$ is an irreducible representation of G such that $$\phi (g)$$ has eigenvalue 1.