Questions tagged [algebraic-groups]

Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

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Hasse principle for $H^2$ of a maximal torus of a connected quasisplit group?

Let $k$ be a number field and let $G$ be a quasisplit reductive algebraic group over $k$. Does there exist a maximal torus in $G$ such that the Hasse principle in dimension $2$ holds, i.e., such that ...
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Centraliser of a maximal $k$-split torus of a reductive $k$-group

Let $G$ be a connected reductive group defined over a field $k$, and let $S$ be a maximal $k$-split $k$-torus of $G$. Then the centraliser $\mathscr Z_{G}(S)$ is defined over $k$. In fact, it is a ...
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When the action of reductive group on algebraic variety is not equidimensional?

I saw the question When is an almost geometric quotient flat? which said "The quotient $\pi$ is flat if and only if $\pi$ is equidimensional and $X$ is smooth". I am curious is there an ...
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Cohomology of compact open subgroups of semisimple groups over local fields

Let $E$ be a local field, $\mathcal{O}$ its ring of integers, $k$ its residue field, and $G$ a split semisimple group over $\mathcal{O}$. Let $K$ be an open subgroup of $G(\mathcal{O})$; more ...
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Conjugates of relative root groups by an element of the Weyl group

Let $G$ be a reductive group (over an algebraically closed field), $T$ a maximal torus, and $\Phi$ the root system of $(G,T)$. Then for each root $\alpha \in \Phi$ there is a unique connected $T$-...
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Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group?

Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of Tits building of a linear algebraic group. I ...
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combinatorial way to count representatives of conjugacy class of elements of ord 5

I am trying to find a representative of each conjugacy class of order 5 elements in PGL$_6$($\mathbb C$). Let $r$ in $\mathbb C$ such that $r^5 = 1$ and [ ] denote modular the center of GL$_6(\mathbb ...
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217 views

Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero

I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
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Dimension of $\text{K}_0(\operatorname{Rep}G)$ for non-connected groups

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\rank{rank}$Remember that if $G$ is a connected reductive group with maximal torus $T$ (whose dimension $r$ is called ...
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The Jacquet module of the Steinberg Representation

I have also posted this question also on Math Stack Exchange, please inform me if the level is too low for this forum. Let $G=GL_2(F)$ where $F$ is a non-Archimedean local field of characteristic $0$, ...
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Can we use formal groups to recover Lie-theoretic representation theory in characteristic p?

In differential geometry, Lie's theorems allow us to integrate any Lie algebra representation to a Lie group representation. The algebraic version of this is more complicated (and I'm not terribly ...
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1 answer
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Algebraic groups acting on affine varieties with finite-dim orbits in the coordinate ring

Let $K$ be an algebraically closed field of characteristic zero, and $X$ be an affine $K$-variety (identify $X$ with its set of $K$-points). Let $G$ be group acting "abstractly" on $X$, by ...
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1 answer
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Parahoric subgroup over a local field

$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...
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Intersection of open orbits in homogeneous space

Let $G$ be a simple complex algebraic group. Let $P(\alpha_i),P(\alpha_k)$ be maximal standard parabolic subgroups of $G$ associated to simple roots $\alpha_i,\alpha_k$ in the root system associated ...
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1 answer
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General centralizer of algebraic group

Perhaps there is a simple answer, but I'm very puzzled by the following question: Question: Does there exist a (smooth, connected) algebraic group $G$ such that the general centralizer (i.e. the ...
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Classification of semisimple algebraic groups which act transitively on a projective space

Let $k$ be an algebraically closed field of characteristic 0, and $V$ be a vector space on $k$ of dimension $>1$. In this situation, is there a classification of connected semisimple groups (up to ...
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1 answer
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Existence of regular semisimple elements in linear group over local field

Let $ L $ be a finite extension of $p$-adic numbers $ \mathbb{Q}_p $. Let $ \text{GL}_{n}(L) $ denote the general linear group $ \text{GL}_{n}(L) $ over $L$ equipped with the topology induced from the ...
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5 votes
1 answer
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Group scheme with an isotrivial maximal torus

Let $G$ be a reductive group scheme over a normal ring $A$. Then, we know that Zariski locally it admits a maximal torus. Let us assume that it admits a maximal torus after a finite surjective (resp. ...
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8 votes
1 answer
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Abelianization of $\mathrm{GL}_2(R)$

$\DeclareMathOperator\GL{GL}$Let $R$ be a number ring. Are there known lower bounds for $H_1(\GL_2(R);\mathbb Q)$ or $H_1(\GL_2(R),\GL_1(R);\mathbb Q)$ in terms of properties of $R$ (class number, ...
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1 answer
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Compactifications of group varieties

Let $V$ be a nonempty, irreducible, smooth projective variety over $\mathbf{C}$. Is there a smooth projective variety $X$ over $\mathbf{C}$, a surjective map $X\to V$ of varieties over $\mathbf{C}$, ...
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4 votes
0 answers
215 views

Number of homomorphisms from a group to $\mathrm{GL}_n(\mathbb{F}_q)$

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms ...
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Which Lie groups are a central extension of an algebraic group?

Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
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1 answer
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An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre

Let $\mathfrak{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathfrak{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(...
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6 votes
1 answer
365 views

Is every finite subgroup the integer points of a linear algebraic group?

Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link Is every finite subgroup the integer points of a linear algebraic group? Let $ ...
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2 votes
0 answers
147 views

Understanding the proof of a theorem by Van Den Bergh

I'm trying to understand the proof of a theorem by Van Den Bergh, which is Proposition 6 in the paper Bessenrodt, Christine and Lieven Le Bruyn. “Stable rationality of certain PGLn-quotients.” ...
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3 votes
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262 views

Analogies to the chromatic layers of the sphere spectrum

Is there an analogy between the chromatic layers the sphere spectrum $\mathbb{S}$ and the ramification groups of the absolute Galois group $G(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})$?
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1 answer
264 views

$ S_4 $ subgroups and $ \operatorname{SO}_3(\mathbb{R}) $

$\DeclareMathOperator\SO{SO}$I posted this on MSE 10 days ago and it got 3 upvotes but no answers or comments, so I'm cross-posting to MO. Background: The group of rotations $ \SO_3(\mathbb{R}) $ has ...
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5 votes
1 answer
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Is there a non-split algebraic torus (over a finite field) satisfying the following properties?

Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties? $T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
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3 votes
1 answer
129 views

Question regarding semistability of a point of GIT quotient

$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\...
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Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group

Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
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5 votes
2 answers
319 views

Quotient of a quotient stack: interesting examples?

Let $X$ be a scheme acted on by an algebraic group $G$. Also, let $H$ be an algebraic group acting on the quotient stack $X/G$, for the definition of "act", see Romagny - Group Actions on ...
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3 votes
2 answers
192 views

Closed subgroups of $\operatorname{GL}_n(\mathbb C)$ with Lie algebra $\mathfrak{so}_n(\mathbb C)$

What is the classification of (Zariski) closed subgroups in $\operatorname{GL}_n(\mathbb C)$ (viewed as a linear algebraic group) with Lie algebra $\mathfrak{so}_n(\mathbb C)$? Is it true that every ...
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22 votes
2 answers
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Is it possible to realize the Moebius strip as a linear group orbit?

On MSE this got 5 upvotes but no answers not even a comment so I figured it was time to cross-post it on MO: Is the Moebius strip a linear group orbit? In other words: Does there exists a Lie group $ ...
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3 votes
0 answers
122 views

Finite commutative group schemes whose exponent coincides with its rank

In group theory, a finite commutative group $G$ contains an element whose order is the exponent of $G$. Thus, If the exponent of $G$ is the same as the order of $G$, it must be that $G$ is cyclic. ...
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1 vote
1 answer
105 views

Koszul complex of equations defining a stabilizer

Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0. Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we ...
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4 votes
0 answers
253 views

Abstract-group isomorphisms between linear algebraic groups

Let $G$ and $H$ be linear algebraic groups over the rationals. Suppose that we know that $G(\mathbb Q)$ and $H(\mathbb Q)$ are isomorphic as abstract groups. Does it under any circumstances follow ...
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Why do such a birational map exists? And why it is unique?

Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra. Suppose that $H \subset G$ is a 1-dimensional torus such that the action of $H$ ...
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2 votes
0 answers
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Is the image of the exponential map of a complex semisimple group Zariski open?

Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map $$\exp : \mathfrak{g} \to G$$ Zariski open in $G$?
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Cohomology with coefficient in sheaf of morphisms of an algebraic group

Let $G$ be an affine algebraic group over ${\mathbb C}$. We denote the sheaf of morphisms from ${\mathbb A}^1$ to $G$ by $\bf G$. Then $H^1({\mathbb A}^1,\bf G)=0$ (Cech cohomlogy). Is this fact true? ...
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1 vote
0 answers
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Properness of Hom-schemes for finite group schemes

In SGA 3 XI proposition 3.12 (b) (https://webusers.imj-prg.fr/~patrick.polo/SGA3/Expo11.pdf) It is shown that: If $G$ is an affine group scheme over a base $S$ and $H$ is a finite group scheme over $S$...
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2 votes
1 answer
104 views

An extension of algebraic torus

Let $T_1$ and $T_2$ be algebraic tori over a field of characteristic 0. Let $T$ be an extension of $T_1$ by $T_2$, namely $$ 1\longrightarrow T_1\longrightarrow T\longrightarrow T_2\longrightarrow 1. $...
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  • 551
4 votes
1 answer
203 views

Criteria for Zariski density of subgroups of reductive groups

Let $G$ be a reductive group over a number field $K$. Let $\Gamma\subset G(K)$ be a subgroup. My extremely naive question is - When can you deduce that $\Gamma$ is Zariski-dense? I'm looking for ...
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5 votes
0 answers
133 views

Sections of $\mathcal{L}_{\lambda}$ on intersections of open cover on a flag variety

Let $G$ be a reductive complex algebraic group, $P$ a parabolic subgroup, $\mathbb{C}_{\lambda}$ a one-dimensional representation of $P$ and $\mathcal{L}_{\lambda}$ the corresponding line bundle on $G/...
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1 vote
0 answers
98 views

How to check the non-emptyness of the A-packet of non-split $\operatorname{SO}(2n+1)$?

Let $F$ be a number field and $G_n=\operatorname{SO}(2n+1)$ be the split group over $F$. Let $G_n^{\times}$ be a non-split group over $F$. Let $\tau$ be an irreducible cuspidal automorphic ...
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1 vote
1 answer
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Is the manifold of complex points of a quotient of compact groups just the tangent bundle?

In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $ D $ is ...
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4 votes
1 answer
238 views

Universal covering groups of simple linear algebraic group schemes

Let $R$ be a Dedekind domain with fraction field $K$, and let $G$ be a smooth affine group scheme over $S = \text{Spec }R$ whose geometric fibers are connected and simple linear algebraic groups (i.e.,...
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6 votes
1 answer
189 views

Flatness of the closure of a closed subgroup of the generic fiber of an algebraic group inside an integral model of the ambient group

$\DeclareMathOperator\GL{GL}$Let $K$ be a number field and $R$ its ring of integers. Let $G$ be a connected reductive closed subgroup of $\GL_{n,K}$. On p55 of Brian Conrad's notes Reductive group ...
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3 votes
0 answers
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Splitting of induced module

Let $G$ be a unipotent complex algebraic group and $K\subseteq G$ a closed subgroup. Let $V$ be a finite-dimensional $K$-module, and consider the induced module $\operatorname{Ind}_{K}^{G}V$, the $G$-...
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2 votes
0 answers
149 views

Reference request - obtaining finite simple groups from algebraic groups

I'm looking for references for the following statements, which I believe are true: Let $G$ be a simply connected simple linear algebraic group over a finite field $k$ of cardinality $q\ge 4$. Let $Z\...
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2 votes
1 answer
111 views

Finite, normal subgroups of reductive groups in positive characteristic

Consider the following statement about a connected, reductive group $G$ over a field $k$: Every finite, normal subgroup $N$ of $G$ is central. In characteristic $0$, this is true, and the proof is ...
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