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3
votes
1answer
155 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
164 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 ...
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 ...
3
votes
0answers
276 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 ...
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 ...
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 ...
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 ...
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
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 ...
-1
votes
0answers
87 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
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: ...
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 ...
4
votes
1answer
157 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 ...
9
votes
3answers
299 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 $v$. ...
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 ...
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?
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 ...
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 ...
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
77 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
350 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 ...
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
308 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 ...
2
votes
0answers
58 views

Springer Isomorphisms for Adjoint Simple Exceptional Groups

I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint ...
1
vote
0answers
79 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space ...
6
votes
3answers
413 views

Does a lisse $\ell$-adic sheaf give rise to an affine group scheme?

Let $k$ be a finitely generated field, $\ell$ a prime different from the characteristic of $k$, $S$ a $k$-variety, and $\mathcal{V}$ a lisse $\ell$-adic sheaf on $S$. Fix an algebraic closure ...
5
votes
1answer
195 views

Over a finite field, does a torsor under the component group of G lift to a torsor under G?

Let $k$ be a finite field and $G$ a finite type smooth $k$-group scheme. Let $G^0$ and $\Gamma$ be the connected component of identity and the component group of $G$, so there is an exact sequence $1 ...
0
votes
0answers
120 views

Base extension of the Weil restriction

It is a problem in Waterhouse's book, "Affine group schemes, p 61": (a) Let $B$ be finite dimensional (commutative) $k$-algebra. Let $G$ be an affine group scheme over $B$. Define the Weil ...
3
votes
1answer
198 views

The cardinality of first non-abelian Galois cohomology

Let $G$ be a linear algebraic group over a non-archimedean local field $F$. Let $H^1(F,G)$ be the first non-abelian Galois cohomology. It is known that when $F$ is of characteristic 0, i.e. finite ...
8
votes
1answer
260 views

Known norm varieties and the Bloch-Kato conjecture

The Bloch-Kato conjecture states that $K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$. A important part in the proof of the Bloch-Kato conjecture is to ...
6
votes
1answer
125 views

About G-modules with good filtrations

Let $k$ be an algebraically closed field of positive characteristic, and let $G$ be a reductive algebraic group over $k$ (for instance a classical group). Let $V$ be a (rational) $G$-module. We say ...
6
votes
1answer
94 views

Number of Richardson orbits in simple Lie algebras of types $E_n$?

This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and ...
7
votes
2answers
221 views

$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$

Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has ...
1
vote
2answers
121 views

Explicit description of algebraic hull

I wonder for an explicit description of the "algebraic hull" (and it's associated Hopf algebra) of a given (discrete) group. I know the answer only for finite groups $G$ which is equal to Spec of the ...