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Register a new userOvergroups of regular unipotent elements in reductive groups
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We study reductive subgroups $H$ of a reductive linear algebraic group $G$ -- possibly non-connected -- such that $H$ contains a regular unipotent element of $G$. We show that under suitable hypotheses, such subgroups are $G$-irreducible in the sense of Serre. This generalizes results of Malle, Testerman and Zalesski. We obtain analogous results for Lie algebras and for finite groups of Lie type. Our proofs are short, conceptual and uniform.
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We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k be a purely inseparable field extension of k of degree p^e and let G denote the Weil restriction of scalars R_{k/k}(G) of a reductive k-group G. When G= R_{k/k}(G) we also provide some results on the orders of elements of the unipotent radical RR_u(G_{bar k}) of the extension of scalars of G to the algebraic closure bar k of k.
Let $k/k$ be a finite purely inseparable field extension and let $G$ be a reductive $k$-group. We denote by $G=R_{k/k}(G)$ the Weil restriction of $G$ across $k/k$, a pseudo-reductive group. This article gives bounds for the exponent of the geometric unipotent radical $mathscr{R}_{u}(G_{bar{k}})$, focusing on the case $G=GL_n$.
Fixing an arithmetic lattice $Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Delta$ with $[Gamma : Gamma cap Delta] [Delta: Gamma cap Delta] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smiths Subgroups of finite index in nilpotent groups apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $mathbb{Z}$-group with arithmetic lattice $Gamma$, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in $p^{-s}$, where the degrees of the numerator and denominator are independent of $p$. This gives regularity results for the set of arithmetic lattices in $G$.
Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H$ of $G$, unique up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL_n$ and with Serres $G$-analogue of semisimplification for subgroups of $G(k)$). We also show that under some extra hypotheses, one can pick $H$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf and Rousseau.
Let $G$ be a reductive algebraic group over an algebraically closed field and let $V$ be a quasi-projective $G$-variety. We prove that the set of points $vin V$ such that ${rm dim}(G_v)$ is minimal and $G_v$ is reductive is open. We also prove some results on the existence of principal stabilisers in an appropriate sense.
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