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3
votes
1answer
152 views

Is $G \rightarrow G/P$ surjective on $K$-points over a local field?

Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?
1
vote
2answers
227 views

A question from the proof of affine algebraic group is a linear

In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", ...
7
votes
1answer
162 views

On unramified p-adic groups

Let G be a reductive group over a local field F. Let O be the ring of integers of F. The following are equivalent (and groups satisfying these conditions are called unramified): (a) G is quasisplit ...
12
votes
3answers
784 views

(un)decidability in matrix groups

Given a collection of matrices $S=\{M_1, \dots, M_k\}$ in (say) $SL(n, Z), \ n>2$ does $S$ generate $SL(n, Z)?$ Similar are questions are undecidable for $n\geq 4$ (eg, given a set $S$ as above, ...
16
votes
4answers
705 views

Computing the Zariski closure of a subgroup of SL(n,Z)

Suppose $\Gamma$ is a finitely generated subgroup of $SL(n,\mathbb{Z})$, given as a list of generators. We would like to (somewhat efficiently) try to compute the Zariski closure of $\Gamma$, which is ...
2
votes
2answers
241 views

Compact elements in $G(K)$ for a reductive group $G$ over a nonarchimedean local field $K$

Let $K$ be a nonarchimedean local field and $G$ a (connected) reductive group over $K$, so that $G(K)$ carries a natural topology. An element $g \in G(K)$ is compact if it is contained in a compact ...
8
votes
1answer
405 views

Are there noncommutative extensions of $\alpha_p$ by $\mathbb{G}_m$?

Let $k$ be a field of characteristic $p > 0$ (algebraically closed, if you want; that doesn't make a difference). Consider a finite type $k$-group scheme $E$ that is a (central) extension of ...
8
votes
3answers
764 views

How do I describe the GL_n torsor attached to a smooth morphism of relative dimension n?

Edit: It seems I had two different constructions mixed up in my head, namely the frame torsor and the automorphism bundle of a vector bundle. This made the main question a bit confusing. The first ...
1
vote
2answers
700 views

Is there a Levi decomposition for Lie group and algebraic group?

Let $G$ be a Lie group and $R$ be the largest connected solvable normal subgroup of $G$. Question 1 Is there a Lie subgroup $S$ such that: (1) $G=SR$; (2) every real representation of $S$ is ...
3
votes
0answers
275 views

polynomials with roots on the unit circle

Suppose $P(x) \in \mathbb{Z}[x]$ is irreducible, and such that at least one of its roots has modulus $1.$ Is there anything we can say about the reduction of $P(x)$ modulo primes? Do these have some ...
4
votes
1answer
276 views

Smooth and $GL(n)$-equivariant implies algebraic?

Context: Let $B_n$ be the space of symmetric bilinear forms on $\mathbb{R}^n$ and $L_n\subset B_n$ be the subset of non-degenerate forms of Lorentzian signature $(-,+,\ldots,+)$. Let $T$ be a finite ...
3
votes
1answer
87 views

algebraic groups over non-archimedean local fields acting on buildings

I was wondering could anyone tell me a reference for the fact that an absolutely quasi-simple algebraic group over a non-archimedean local field which is centreless and non-compact acts faithfully and ...
9
votes
4answers
894 views

Simply connected algebraic groups and reductive subgroups of maximal rank

Recall that a connected semisimple algebraic group $G$ over an algebraically closed field $K$ of arbitrary characteristic was defined by Chevalley to be simply connected if the character group $X(T)$ ...
0
votes
1answer
105 views

On the reductive group [closed]

I know that the automorphic representation can be defined only for reductive algebraic group. What property of algebriac group makes it hinder to define for all algebraic group and what nice property ...
1
vote
0answers
85 views

Intuition for the structure theorem for connected solvable algebraic groups over an algebraically closed field of positive characteristic

In the theory of algebraic groups one of the fundamental results is the structure theorem for connected solvable groups. I never understood the proof in any of the standard textbooks, so I just moved ...
0
votes
1answer
115 views

Maximal subgroups of indefinite special orthogonal group SO(p,q)

Can someone answer the following question: Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions: ...
2
votes
1answer
146 views

surjective homomorphism with compact kernel (Milne's note on Shimura varieties)

I'm reading Milne's Introduction to Shimura varieties (http://www.jmilne.org/math/xnotes/svi.pdf) and there is something I don't get. Let $G$ be a connected semisimple algebraic group $G$ over ...
0
votes
0answers
67 views

Can any reductive $k$-group be written as a semidirect product of $k$-linear groups?

This might be an easy question for the experts, so I apologise in advance. By a reductive group over a field $k$, I mean a linear algebraic group (not necessarily connected) such that the unipotent ...
0
votes
0answers
107 views

Irreducible action of an algebraic group

Is the following claim true?: Let $G$ be an algebraic group such that $G^\circ$ is reductive. Suppose $G$ acts irreducibly on $V$. Is it true that $V$ is decomposed (written as direct sum) into ...
1
vote
0answers
74 views

Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers. Can one classify all reductive group schemes over $C$? Certainly, you have the trivial ones (coming from pulling-back ...
2
votes
1answer
106 views

Characteristic polynomials of reductive subgroup over C

Can any one provide a hint to prove the following statement? : Let $H$ be a complex reductive subgroup (not necessarily connected) contained in $SO(n,\mathbb{C)})$. Consider the map $H \rightarrow ...
4
votes
1answer
288 views

how many Q-forms of SL_n(R) are there for a given Q-rank

Let $G$ be a linear algebraic group defined over $\mathbb Q$. Suppose that $G$ is isomorphic to $SL_n$ over $\mathbb R$. Suppose the $\mathbb Q$-rank of $G$ is fixed, say $m$. How many types are ...
-1
votes
0answers
86 views

Compact elements of $\mathrm{GL}_n(F)$, where $F$ is a nonarchimedean local field

Let $F$ be a nonarchimedean local field and let $\mathcal{O}$ be its ring of integers. An element $g$ of $\mathrm{GL}_n(F)$ is called compact if the cyclic subgroup that it generates has compact ...
0
votes
1answer
48 views

Reference for finite number of Weyl groups of reductive groups of rank $r$

I'm posting this question on behalf of someone without access to mathoverflow: Can anybody give me a reliable reference (not a proof) to the following statement? Up to isomorphism, there are only ...
4
votes
0answers
163 views

Tannaka categories and reductive groups

The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple. Pro-reductive groups make sense over any scheme. Is there an extension of the theory ...
9
votes
3answers
298 views

Maximal compact subgroup of p-adic lie groups

Let $k$ be a number field and $S$ be a finite set of places of $k$. Let $G$ be a connected semisimple algebraic group over $k$. Let $k_S=\prod_{v\in S}k_v$ where $k_v$ is the completion of $k$ at $v$. ...
4
votes
1answer
154 views

Pulling back quasi-coherent sheaves from a quotient stack

In a problem I am trying to solve, the following situation occurs. $X$ is a smooth variety and $G$ is a reductive group acting transitively on $X$. We have the stack $X/G$ and a morphism $\pi : X \to ...
6
votes
1answer
243 views

An inequality on representations and subgroups of general linear groups over finite field

Let $q$ be a power of $p$, let $l$ be a prime different from $p$, and let $H_1$ and $H_2$ be two subgroups of $GL_n(\mathbb F_q)$ that are $l$-groups. If for all characteristic $0$ representations ...
10
votes
2answers
835 views

Why are $S$-arithmetic groups interesting?

Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$. Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion ...
2
votes
2answers
185 views

Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$

Let $K$ be a global field and set $O := \prod_{v\nmid \infty} O_v$ where $v$ runs over the finite places of $K$. Equip $\mathrm{GL}_n(O) = \prod_v \mathrm{GL}_n(O_v)$ with the product of the $v$-adic ...
3
votes
1answer
124 views

Compactness of adelic quotients for unipotent groups over global fields

Let $K$ be a global field, $\mathbb{A}_K$ the ring of adeles, and $U$ a unipotent algebraic group over $K$. Why is $U(\mathbb{A}_K)/U(K)$, when endowed with the quotient topology, compact?
7
votes
1answer
936 views

An interesting double coset in the theory of automorphic forms

Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , $H=GL_{n-1}$ ...
4
votes
2answers
380 views

Are linear algebraic groups rigid?

The underlying variety of a linear elgebraic group (say, over an algebraically closed field) is affine, so doesn't have nontrivial (infinitesimal) deformations. I'm curious to know whether it's ...
4
votes
2answers
121 views

Invariant planes of a nilpotent matrix with two Jordan blocks of size two

Describe all the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map $$ N = \begin{pmatrix} 0 & 1 & & \\ 0 & 0 & & \\ & & 0 ...
3
votes
0answers
107 views

Non-linearly isomorphic non-equivalent $G-$representations?

Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow ...
2
votes
0answers
122 views

Can the commuting condition in Jordan-Chevalley decomposition be replaced with this global criterion?

Let $G$ be a reductive linear algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra. The Jordan-Chevalley ...
1
vote
0answers
84 views

Symmetric spaces which are compact modulo the unipotent radical are compact

Is the following true? Let $X = G/H$ be a symmetric space of a reductive group over a p-adic field $F$. Let $X^0$ be an open orbit w.r.t. the action of the minimal parabolic $B$ of $G$. Let $U$ be ...
0
votes
0answers
96 views

Parahoric group schemes over curves

Let $X$ be a smooth projective curve over $\mathbb{C}$ and Let $G$ be a complex reductive group. By a parahoric group scheme $\mathcal{G}$ over $X$, I mean a smooth group scheme over $X$ whose ...
1
vote
1answer
76 views

Global centralizers in Jordan-Chevalley decomposition in bad characteristic

Let $G$ be an affine algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra. Given $X\in\mathfrak{g}$, it has ...
10
votes
1answer
349 views

The Mordell and Bogomolov problems in linear groups

Many things in the arithmetic of abelian varieties have counterparts not only in linear tori, but also for semisimple linear groups. Two examples are the Tamagawa number and the conjectured finiteness ...
3
votes
1answer
76 views

Density of $\Gamma(N)$ in $\mathrm{Sp}_{2g}(\mathbb{Z}_{\ell})$ where $\ell \not | N$

Let $\mathrm{Sp}_{2g}$ denote the symplectic group of $2g \times 2g$ matrices for some $g \geq 1$, and let $\Gamma(N)$ be the level-$N$ principal congruence subgroup of $\mathrm{Sp}_{2g}(\mathbb{Z})$. ...
3
votes
2answers
227 views

Specialisations of flag varieties

Recall that a flag variety over a field $k$ is a smooth projective variety over $k$, which is a homogeneous space for some linear algebraic group. My question concerns specialisations of flag ...
7
votes
1answer
445 views

Uniform proof of dimension formula for minimal special nilpotent orbit?

Given a simple Lie algebra over an algebraically closed field of good characteristic such as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...
14
votes
1answer
874 views

What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense: Conjecture ...
4
votes
0answers
135 views

Is a semiabelian algebraic space a scheme?

Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is ...
2
votes
0answers
92 views

$X$-points of reductive group schemes, if $X$ is a proper smooth curve over a finite field

Let $X$ be a connected proper smooth curve over a finite field (so the generic point of $X$ is the spectrum of a global field $K$), and let $G \rightarrow X$ be an affine $X$-group scheme of finite ...
6
votes
1answer
224 views

Generalization of Frobenius groups

Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. In other words, if in a ...
4
votes
1answer
174 views

Automorphisms of SO_n(k,f)

Let $k$ be a field, $n\in\mathbb{N}$ and $f:k^n\times k^n\to k$ a non-degenerate symmetric bilinear form. Let $$O_n(k,f):=\{ g\in GL_n(k) \mid \forall x,y\in k^n : f(x,y)=f(g.x,g.y) \}$$ and ...
0
votes
3answers
130 views

How do I show that a separable isogeny is central?

I've been trying to prove this (probably very simple) result that is stated in a paper that I'm reading: Let $G$ and $H$ be connected semisimple algebraic groups defined over a field $F$, and let $f: ...
1
vote
0answers
307 views

Rational structures on the flag variety over a finite field

Some Notions A variety over a field is defined to be a scheme of finite type over this field. An $\mathbb{F}_q$-rational structure of an $\bar{\mathbb{F}}_q$-variety $V$ is a $\mathbb{F}_q$-variety ...