This paper consists of three interconnected parts. Parts I,III study the relationship between the cohomology of a reductive group and that of a Levi subgroup. For example, we provide a necessary condition, arising from Kazhdan-Lusztig theory, for the natural map on Ext-groups of irreducible modules to be surjective. In cohomological degree 1, the map is always an isomorphism, under our hypothesis. These results were inspired by recent work of Hemmer obtained for general linear groups, and they both extend and improve upon his work when our condition is met. Part II obtains results on Lusztig character formulas for reductive groups, obtaining new necessary and sufficient conditions for such formulas to hold. In the special case of general linear groups, these conditions can be recast in a striking way completely in terms of explicit representation theoretic properties of the symmetric group (and the results improve upon the sufficient cohomological conditions established recently in the authors).
In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(Δ,R)$ of the Chevalley group $G(Φ,R)$ for the three types of pair $(Φ,Δ)$ (the root system and its subsystem) such that the group $G(Δ,R)$ is (up to torus) a Levi subgroup of the parabolic subgroup with abelian unipotent radical. Namely we show that for any such an overgroup $H$ there exists a unique pair of ideals $σ$ of the ring $R$ such that $E(Φ,Δ,R,σ)\le H\le N_{G(Φ,R)}(E(Φ,Δ,R,σ))$.
In this paper we give a complete invariant of the action of GL_n(F)\times GL_m(F) acting on the Euclidean building B[GL_n(F)], where F is a non-archimedian field. We then use this invariant to give a natural metric on the resulting quotient space. In the special case of the torus acting on B[GL_2(F)] this gives a method of calculating of any vertex to any fixed apartment.
We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = RS where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.
We investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate. These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.
We introduce the notion of ideal triangle in the Bruhat-Tits building associated to a split group -- it is analogous to the usual notion of triangle, but one vertex is "at infinity" in a certain direction. We prove that the algebraic variety of based ideal triangles with prescribed side-lengths is naturally isomorphic to a suitable variety of genuine triangles. From theorems pertaining to genuine triangles, we deduce saturation theorems related to branching to Levi subgroups and to the constant term homomorphisms.
For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ ξ^P_λ: Rep^\mathbb{C}_{λ-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{λ-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(λ)$ of G with highest weight $λ$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\leq k\leq n$ (with the choice of $V(λ) $ to be the defining representation $V(ω_1) $ in $\…
This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
We show that, in a weakly regular $p$-adic Lie group $G$, the subgroup $G_u$ spanned by the one-parameter subgroups of $G$ admits a Levi decomposition. As a consequence, there exists a regular open subgroup of $G$ which contains $G_u$.
Associated to every group with a weak spherical Tits system of rank n+1 with an appropriate rank n subgroup, we construct a relative spectral sequence involving group homology of Levi subgroups of both groups. Using the fact that such Levi subgroups frequently split as semidirect products of smaller groups, we prove homological stability results for unitary groups over division rings with infinite centre as well as for special linear and special orthogonal groups over infinite fields.
Let $G$ be a Lie group over a totally disconnected local field and $α$ be an analytic endomorphism of $G$. The contraction group of $α$ ist the set of all $x\in G$ such that $α^n(x)\to e$ as $n\to\infty$. Call sequence $(x_{-n})_{n\geq 0}$ in $G$ an $α$-regressive trajectory for $x\in G$ if $α(x_{-n})=x_{-n+1}$ for all $n\geq 1$ and $x_0=x$. The anti-contraction group of $α$ is the set of all $x\in G$ admitting an $α$-regressive trajectory $(x_{-n})_{n\geq 0}$ such that $x_{-n}\to e$ as $n\to\infty$. The Levi subgroup is the set of all $x\in G$ whose $α$-orbit is relatively compact, and such t…
Weizmann Institute of Science（魏茨曼科学研究所）；Paderborn University, Institute of Mathematics（帕德博恩大学）；University of California at San Diego（加州大学圣地亚哥分校）；Bielefeld University（比勒费尔德大学）
Let $G$ be a simple algebraic group over an algebraically closed field. A closed subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) if, whenever $H$ is contained in a parabolic subgroup $P$ of $G$, it is contained in a Levi factor of $P$. In this paper we complete the classification of connected $G$-cr subgroups when $G$ has exceptional type, by determining the $L_{0}$-irreducible connected reductive subgroups for each simple classical factor $L_{0}$ of a Levi subgroup of $G$. As an illustration, we determine all reducible, $G$-cr semisimple subgroups when $G$ has type $F_4$ and v…
Let $F$ be a local complete field with discrete valuation, and let $G$ be a quasi-split group over $F$ which splits over some unramified extension of $F$. Let $P$ be a parahoric subgroup of the group $G(F)$ of $F$-points of $G$; t he open normal pro-nilpotent subgroups of $P$ can be classified using the standa rd normal filtration subgroups of Prasad and Raghanathan. More precisely, we sho w that if $G$ is quasi-simple and satisfies some additional conditions, $H$ is, modulo a subgroup of some maximal torus of $G$, either one of these filtration s ubgroups or the product of one of them by a st…
A general method for finding subgroups of a totally disconnected, locally compact groups having flat-rank greater than 1 is described. This method uses the internal structure of the group, notably the Levi subgroup of a given flat group, in order to produce a new flat group that, under an additional hypothesis, has flat-rank one greater than that of the given group.
Let $G$ be a simple algebraic group of classical type over an algebraically closed field $k$. Let $P$ be a parabolic subgroup of $G$ and let $\p = \Lie P$ be the Lie algebra of $P$ with Levi decomposition $\p = ł\oplus \u$, where $\u$ is the Lie algebra of the unipotent radical of $P$ and $ł$ is a Levi complement. Thanks to a fundamental theorem of R. W. Richardson, $P$ acts on $\u$ with an open dense orbit; this orbit is called the {\em Richardson orbit} and its elements are called {\em Richardson elements}. Recently, the first author gave constructions of Richardson elements in the case $k =…
A semisimple element $s$ of a connected reductive group $G$ is said {\it quasi-isolated} (respectively {\it isolated}) if $C_G(s)$ (respectively $C_G^0(s)$) is not contained in a Levi subgroup of a proper parabolic subgroup of $G$. We study properties of quasi-isolated semisimple elements and give a classification in terms of the affine Dynkin diagram of $G$. Tables are provided for adjoint simple groups.
Let $G$ be a connected reductive algebraic group defined over an algebraic closure of a finite field and let $F : G \to G$ be an endomorphism such that $F^d$ is a Frobenius endomorphism for some $d \geq 1$. Let $P$ be a parabolic subgroup of $G$ admitting an $F$-stable Levi subgroup. We prove that the Deligne-Lusztig variety $\{gP | g^{-1}F(g)\in P\cdot F(P)\}$ is irreducible if and only if $P$ is not contained in a proper $F$-stable parabolic subgroup of $G$.
We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe higher generating families of parabolic subgroups in Aut(F_n).
Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We fin…
Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a parabolic subgroup P of G, then H is in a Levi subgroup of that parabolic. Building on work of Liebeck and Seitz, we find all triples (X,G,p) such that there exists a closed, connected, simple non-G-cr subgroup H<G with root system X.
In this paper characters of the normaliser of $d$-split Levi subgroups in $\mathrm {SL}_n(q)$ and $\mathrm {SU}_n(q)$ are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer.These results are applied to verify the Alperin-McKay conjecture for primes $\ell$ with $\ell\nmid 6(q^2-1)$ and the Alperin weight conjecture for $\ell$-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of $\mathrm {SL}_n(q)$ and $\mat…
In this paper we have examined hyperbolicity and relative hyperbolicity of $Γ_n := \mathsf{Out}(G_n)$ , where $G_n = A_1*...*A_n$, is a finite free product and each $A_i$ is a finite group. We have used the $\mathsf{Out}(G_n)$ action on the Guirardel-Levitt deformation space, to find a virtual generating set and prove quasi isometric embedding of a large class of subgroups. We have used ideas from works of Mosher-Handel and Alibegović to prove non-distortion. We have used these subgroups to prove that $Γ_n$ is thick for higher complexities. Thickness was developed by Behrstock-Druţu-Mosher and…
Given a universal elementary Chevalley-Demazure group $E_Φ^{sc}(R)$ for which its (standard) parabolic subgroups are finitely generated, we consider the problem of classifying which parabolics $P(R) \subset E_Φ^{sc}(R)$ are finitely presented. We show that, under mild assumptions, this is equivalent to the finite presentability of a suitable retract of $P$ which contains the Levi factor. If the base ring $R$ is a Dedekind domain of arithmetic type, we combine our results with well-known theorems due to Borel-Serre, Abels, Behr and Bux to give a partial classification of finitely presentable $S…
The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric…
Let $G\subset\GL(V)$ be a complex reductive group where $\dim V<\infty$, and let $π\colon V\to\quot VG$ be the categorical quotient. Let $\NN:=π\invπ(0)$ be the null cone of $V$, let $H_0$ be the subgroup of $\GL(V)$ which preserves the ideal $\I$ of $\NN$ and let $H$ be a Levi subgroup of $H_0$ containing $G$. We determine the identity component of $H$. In many cases we show that $H=H_0$. For adjoint representations we have $H=H_0$ and we determine $H$ completely. We also investigate the subgroup $G_F$ of $\GL(V)$ preserving a fiber $F$ of $π$ when $V$ is an irreducible cofree $G$-module.
We obtain the symplectic group $\SP(V)$ as the universal completion of an amalgam of low rank subgroups akin to Levi components. We let $\SP(V)$ act flag-transitively on the geometry of maximal rank subspaces of $V$. We show that this geometry and its rank $\ge 3$ residues are simply connected with few exceptions. The main exceptional residue is described in some detail. The amalgamation result is then obtained by applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups.
We check that the connected centralisers of nilpotent elements in the orthogonal and symplectic groups have Levi decompositions in even characteristic. This provides a justification for the identification of the isomorphism classes of the reductive quotients as stated in [Liebeck, Seitz; Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras].
Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a natural correspondence of characters, both for $GL_n(q)$ and $SL_n(q)$. A similar result is established for certain finite groups with self-normalizing Sylow $p$-subgroups. We also construct a canonical bijection between the odd-degree irreducible characters of $S_n$ and those of $M$, where $M$ is any maximal subgroup of $S_n$ of odd index; as well as between the …
We obtain the symplectic group as an amalgam of low rank subgroups akin to Levi components. We do this by having the group act flag-transitively on a new type of geometry and applying Tits' lemma. This provides a new way of recognizing the symplectic groups from a small collection of small subgroups. The geometry consists of all subspaces of maximal rank in a vector space of maximal rank with respect to a symplectic form. The main result holds for fields of size at least 3. We analyze the geometry over the field of size 2 and describe its simply connected cover if different from the geometry.
If G is a connected linear algebraic group over the field k, a Levi factor of G is a reductive complement to the unipotent radical of G. If k has positive characteristic, G may have no Levi factor, or G may have Levi factors which are not geometrically conjugate. We give in this paper some sufficient conditions for the existence and the conjugacy of Levi factors of G. Let A be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p>0. Let G be a connected and reductive algebraic group over K. Bruhat and Tits have associated to G certain smooth …
Let $G=G(K)$ be a simple algebraic group defined over an algebraically closed field $K$ of characteristic $p>0$. A subgroup $X$ of $G$ is said to be $G$-completely reducible if, whenever it is contained in a parabolic subgroup of $G$, it is contained in a Levi subgroup of that parabolic. A subgroup $X$ of $G$ is said to be $G$-irreducible if $X$ is in no parabolic subgroup of $G$; and $G$-reducible if it is in some parabolic of $G$. In this thesis, we consider the case that $G$ is of exceptional type. When $G$ is of type $G_2$ we find all conjugacy classes of closed, connected, reductive subgr…
