The algebraic-groups tag has no wiki summary.

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### 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:
...

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### 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 ...

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### Lifting one parameter subgroups of algebraic groups

Let $G$ be a linear algebraic group over an algebraically closed field $\mathbb C$ of characteristic zero and $U$ its unipotent radical, then $H:=G/U$ is a reductive group. Assume that I have a one ...

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### what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...

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### The definition of $SK_1$ for an arbitrary ring

Let $R$ be a unitary associative ring. If $R$ is commutative, then one defines $SK_1(R)$ as the quotient $$SK_1(R)=SL(R)/E(R)$$ (Definition 2.8 of ...

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### An extension of group schemes admitting Neron models

Let $R$ be a discrete valuation ring, $K$ its field of fractions, and
$$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$
a short exact sequence of smooth $K$-group schemes of ...

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### Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...

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### Geometrically connected components of an algebraic group

Suppose that $G$ is an algebraic group over a field $k$. Let $G^o$be the connected component of the identity. Since $G^0$ contains a $k$-rational point (the identity) therefore it is geometrically ...

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### Comparison of two Chevalley basis

Let $G$ be a connected reductive group over an algebraically closed field and $T$ a maximal torus.
Let $H$ be a pseudo-Levi subgroup, say the neutral component of a centralizer of a semisimple element ...

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### Torsors and twists of algebraic groups

Let $G/S$ be an affine group scheme. Then the automorphism group of every $G$-torsor over $S$ is a twist of $G$, but it this functor isn't essentially surjective in general (It may be not full nor ...

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### Decomposing quasi-finite separated group schemes

Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps)
Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...

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### orders of maximal abelian subgroups

What are the orders of maximal abelian subgroups of the simple groups $F_4(q)$ and $C_4(q)$, where $F_4(q)$ is an exceptional group and $C_4(q)$ is a symplectic group?

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### on the condition “$G$ is defined over $\mathbb{Q}$” [migrated]

This might be a stupid question, but I cannot understand the "technical condition" when studying some basics of arithmetic groups, that is an algebraic group is defined over $\mathbb{Q}$. ...

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### Quotient of a (non-linear) algebraic group by a closed subgroup

Let G be a (not necessarily linear) algebraic group and H a closed algebraic subgroup. Does the, say categorical, quotient G/H exist?
If yes, where do I find a proof?
If no, do you have a ...

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### Representation of GL(n, F_p) over F_p, for n small

The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...

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### Analogies, Riemann surfaces and Algebraic groups

Let G be a complex simple Lie group of adjoint type. Then, it is well known that every such $G$ contains, unique up to conjugacy, an irreducibly embedded copy of $PSL(2,\mathbb{C}).$ This fact seems ...

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### Picard functor of an algebraic group

Let $K$ be a field, $G$ a $K$-group scheme of finite type, and $X$ a $G$-torsor. Is the Picard functor $\mathrm{Pic}_{X/K}$ representable? I recall that $\mathrm{Pic}_{X/K}$ is the fppf sheafification ...

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### Is there a scheme parametrizing the closed subgroups of an algebraic group?

In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...

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### $\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$

I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?

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### Connected unipotent algebraic groups

Let $G$ be a connected unipotent algebraic (affine) group defined over a perfect field $k$. Then $G$ is $k$-isomorphic as an algebraic $k$-variety to the affine space $\mathbb A^n_k$.
Is there any ...

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### conjugacy classes of proper semisimple subgroups in parabolic groups

Is it true that there are only finitely many conjugacy classes of proper semisimple subgroups in a parabolic subgroup which in turn is contained in an semisimple algebraic group?
Let $G$ be a ...

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### Meaning of fibration in Kazhdan and Lusztig's paper on affine flag manifolds

Kazhdan and Lusztig's paper "Fixed point varieties on affine flag manifolds" has the following definition on p.143: define inductively a variety $Z$ to be an "almost affine space" if $Z$ is affine or ...

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### On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned:
Is the map
$g_3 ...

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### Reducible reductive Lie subalgebras of so(p,q)

Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...

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### Reductive subgroup and its derived subgroup with an irreducible represenation

Could you please answer the following question: Let $V$ be a faithful irreducible representation of a connected reductive group $H$ defined over $\mathbb{R}$ Is it true that the derived group of $H$, ...

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### Representation theory of the general linear group over a finite prime field

I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained.
The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...

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### exponential and anisotropic torus

Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...

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### Optimal definition of “paving by affine spaces”?

Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine ...

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### on lifting extensions

Let $G$ be a connected reductive group with $G_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$.
We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus ...

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### Is a “central” extension of $\mathbb{Z}/m\mathbb{Z}$ by $\mathrm{GL}_n$ necessarily split?

Let $m \ge 1$ be an integer, let $k$ be a field of characteristic $0$, and let
$$
1 \rightarrow \mathrm{GL}_n \rightarrow E \rightarrow \mathbb{Z}/m\mathbb{Z} \rightarrow 1
$$
be an extension of ...

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### endoscopy and simply connectedness

Let $G$ be a connected reductive group with $G_{der}$ simply connected over a local field $F$.
Let $\hat{G}$ be its Langlands dual, $s$ a semisimple element in $\hat{G}$ and $\hat{H}=\hat{G}_{s}$.
...

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### Generators of the algebra of invariant polynomials on a Lie algebra and the root-space decomposition

Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ ...

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### When is fiber dimension upper semi-continuous?

Suppose $f\colon X \to Y $ is a morphism of schemes. We can define a function on the topological space $Y$ by sending $y\in Y$ to the dimension of the fiber of $f$ over $y$.
When is this function ...

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### 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?

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### 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", ...

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### 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 ...

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### (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, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $X(T)$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...