Quasi-isolated elements in reductive groups

Generic stabilisers for actions of reductive groups

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 $v\in V$ 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.

准连通半单群的阿贝尔伽罗群上同调

Abelian Galois cohomology of quasi-connected reductive groups

本文研究了准连通半单群的阿贝尔伽罗群上同调，推广了Labesse的构造和结果至任意半单群，并探讨了其函子性性质。

The Reductive Subgroups of G_2

Let $G:=G_2(K)$ be a simple algebraic group of type $G_2$ defined over an algebraically closed field $K$ of characteristic $p>0$. Let $σ$ denote a standard Frobenius automorphism of $G$ such that $G_σ\cong G_2(q)$ with $q\geq 4$. In this paper we find all reductive subgroups of $G$ and quasi-simple subgroups of $G_σ$ in the defining characteristic.

Quasiflats with holes in reductive groups

We give a new proof of a theorem of Kleiner-Leeb: that any quasi-isometrically embedded Euclidean space in a product of symmetric spaces and Euclidean buildings is contained in a metric neighborhood of finitely many flats, as long as the rank of the Euclidean space is not less than the rank of the target. A bound on the size of the neighborhood and on the number of flats is determined by the size of the quasi-isometry constants. Without using asymptotic cones, our proof focuses on the intrinsic geometry of symmetric spaces and Euclidean buildings by extending the proof of Eskin-Farb's quasifla…

Polyhedral compactifications of Bruhat-Tits buildings of quasi-reductive groups

Given a quasi-reductive group $G$ over a local field $k$, using Berkovich geometry, we exhibit a family of $G(k)$-equivariant compactifications of the Bruhat-Tits building $\mathcal B(G, k)$, constructed and investigated by Solleveld and Lourenço. The compactification procedure consists in mapping the building in the analytification $G^{\mathrm{an}}$ of $G$, then composing this map with the projections from $G^{\mathrm{an}}$ to its (in general non-compact) pseudo-flag varieties $(G/P)^{\mathrm{an}}$, for $P$ ranging among the pseudo-parabolic subgroups of $G$. This generalises previous constru…

Adjoint quotients of reductive groups

Let $\rG$ be a reductive group over a commutative ring $k$. In this article, we prove that the adjoint quotient $\adqG$ is stable under base change. Moreover, if $\rG$ has a maximal torus $\rT$, then the adjoint quotient of the torus $\rT$ by its Weyl group will be isomorphic to $\adqG$. Then we focus on the semisimple simply connected group $\rG$ of the constant type. In this case, $\adqG$ is isomorphic to the Weil restriction $\underset{\rD/\spec k}{\prod}\aff^{1}_\rD$, where $\rD$ is the Dynkin scheme of $\rG$. Then we prove that for such $\rG$, the Steinberg's cross-section can be defined …

Semilinear automorphisms of reductive algebraic groups

Let $ G $ be a connected reductive algebraic group over a field $ k $. We study the group of semilinear automorphisms Aut($ G\to $Spec $k$) consisting of algebraic automorphisms of $ G $ over automorphisms of $ k $. We focus on the exact sequence $ 1\to $Aut $G\to $Aut ($ G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1 $. When $G$ is quasi-split, we show that Aut$_{G}(k)$ is isomorphic to Aut$_{\mathcal{R}(G)}(k)$, where $\mathcal{R}(G)$ denotes the scheme of based root datum of $G$. Furthermore, the exact sequence $ 1\to $Aut $G\to $Aut ($ G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1 $ splits if and only if t…

Characters of positive height in blocks of finite quasi-simple groups

Eaton and Moretó proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We show that their conjecture holds for principal blocks of quasi-simple groups, for all blocks of finite reductive groups in their defining characteristic, as well as for all covering groups of symmetric and alternating groups. For the proof, we determine the minimal non-trivial character degrees of Sylow $p$-subgroups of finite reductive groups in characteristic~…

函数域上拟分裂群的玉河数

Tamagawa numbers of quasi-split groups over function fields

本文利用Morris的Eisenstein级数理论，将Harder关于玉河数的计算扩展至全局函数域上的拟分裂群。

Théorie de Bruhat-Tits pour les groupes quasi-réductifs

We extend (scheme-theoretic) Bruhat-Tits theory to quasi-reductive groups i.e. with trivial split unipotent radical over discretely valued henselian non-archimedean fields $K$, whose ring of integers is excellent and residue field is perfect, building on Bruhat-Tits I et II and the classification of pseudo-reductive groups by Conrad-Gabber-Prasad. The buildings had been constructed previously by Solleveld in a different manner, but they ultimately coincide with ours. We were motivated by the geometry of affine Grassmannians and give some applications at the end (compare also with previously su…

Quasi-isolated blocks and the Alperin-McKay conjecture

The Alperin-McKay conjecture is a longstanding open conjecture in the representation theory of finite groups. Späth showed that the Alperin-McKay conjecture holds if the so-called inductive Alperin-McKay (iAM) condition holds for all finite simple groups. In a previous paper, the author has proved that it is enough to verify the inductive condition for quasi-isolated blocks of groups of Lie type. In this paper we show that the verification of the iAM-condition can be further reduced in many cases to isolated blocks. As a consequence of this we obtain a proof of the Alperin-McKay conjecture for…

On the coadjoint orbits of maximal unipotent subgroups of reductive groups

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which h…

Quasi-isolated blocks and Brauer's height zero conjecture

This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpoten…

Distributions invariantes sur les groupes reductifs quasi-deployes

Let $F$ be a nonarchimedean local field, and $G$ the group of $F$-points of a c onnected quasisplit reductive group defined on $F$; in this paper, we will study the distributions on $G$ which are invariant by conjugation, and the vector spa ce of their restrictions to the Hecke algebra $\mathcal{H}$ of the functions on $G$ with compact support and biinvariant by a given Iwahori subgroup $I$. It is first shown that such a distribution on $\mathcal{H}$ is entirely determined by its restriction to the finite-diimensional subspace of $\mathcal{H}$ containing the elements with support in the union …

On Serre's injectivity question and norm principle

Let $k$ be a field of characteristic not $2$. We give a positive answer to Serre's injectivity question for any smooth connected reductive $k$-group whose Dynkin diagram contains connected components only of type $A_n$, $B_n$ or $C_n$. We do this by relating Serre's question to the norm principles proved by Barquero and Merkurjev. We give a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for the non-trialitarian $D_{n}$ case and yield a positive answer to Serre's question for connected reductive $k$-groups whose Dynkin diagrams contain components of …

Geometric invariants of spaces with isolated flats

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as geometrically finite Kleinian groups and numerous 2-dimensional CAT(0) groups. For such a group we show that there is an intrinsic notion of a quasiconvex subgroup which is equivalent to the inclusion being a quasi-isometric embedding. We also show that the visual boundary of the CAT(0) space is actually an invariant of the group. More generally we show tha…