# 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.

1,745
questions

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votes

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190 views

### Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...

**1**

vote

**1**answer

159 views

### Taking quotient of a variety by the additive group

1. Let $X$ be a smooth irreducible $\Bbb C$-variety,
on which the algebraic $\Bbb C$-group $G={\bf G}_{a,{\Bbb C}}$
(the additive group) acts freely on the right:
$$ X\times _{\Bbb C} G\to X,\quad (x,...

**3**

votes

**1**answer

106 views

### Questions on norms on Adelic group

This might be stupid question to some experts who works in the realm of automorphic form.
Let $K$ be a number field and $\mathbb{A}$ is a adele ring of $K$. Let $G$ be a connected reductive group ...

**2**

votes

**0**answers

49 views

### The inclusion of Siegel sets of the general linear groups

In some paper, it is written that none of Siegel set of $GL_{n}$ is not contained in any Siegel set of $GL_{n+1}$. I don't understand this because I think it should hold.
To explain my question more ...

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votes

**0**answers

75 views

### The coherence property of center of universal enveloping algebra for reductive Lie algebra?

Let $G' \subset G$ be two reductive Lie groups over $\mathbb{R}$ and $\mathfrak{g}_{\mathbb{C}}' \subset \mathfrak{g}_{\mathbb{C}}$ be their complexified reductive Lie algebra over $\mathbb{C}$, ...

**8**

votes

**1**answer

373 views

### Definition of an arithmetic subgroup of an algebraic group

I'm struggling with the definition of an arithmetic subgroup of an algebraic group defined over $\mathbb{Q}$.
In Wikipedia you can read:
If $\mathrm G$ is an algebraic subgroup of $\mathrm{GL}_n(\...

**5**

votes

**1**answer

131 views

### Signs in Chevalley systems for reductive groups

Let $G$ be a pinned split reductive group. There exists a Chevalley system:
For each root $b$ in its root system there are parametrisations $x_b: \mathbb{G}_a \rightarrow U_b$ of the corresponding ...

**6**

votes

**0**answers

131 views

### Generalization of a standard algebraic group theory result for a tensor problem

$\DeclareMathOperator\GL{GL}$Let $X$, $Y$, $Z$ be $\mathbb{C}$-vector spaces, and let $A\subseteq X$ and $B\subseteq Y$ and $C\subseteq Z$ be linear subspaces. Let $V=X \otimes Y \otimes Z$, acted on ...

**3**

votes

**0**answers

92 views

### Geometry of elements with prescribed multiplicity eigenvalues

Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...

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votes

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49 views

### Question on Iwasawa decomposition of unitary groups over adele ring

Let $E/F$ be a quadratic extension of number fields and $V,\langle,\rangle$ is a hermition vector space over $E$. Let $\mathbb{A}$ be the adele ring of $F$.
Assume that there is a hermitian line $e \...

**4**

votes

**1**answer

136 views

### Extending rational maps to semi-abelian varieties

Setup. Let $k$ be an algebraically closed field of characteristic zero, and let $G/k$ be a semi-abelian variety i.e., $G$ is a commutative algebraic group which is an extension of an abelian variety $...

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votes

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139 views

### Of the group of automorphisms of a vector bundle of finite rank

$\DeclareMathOperator\End{End}\DeclareMathOperator\Id{Id}\DeclareMathOperator\AC{AC}\DeclareMathOperator\GL{GL}$This is a question related to a previous question on existence of almost complex ...

**3**

votes

**0**answers

92 views

### The product of $Z(\mathfrak{g})$-finite functions is also $Z(\mathfrak{g})$-finite?

Let $G$ be a classical group defined over $\mathbb{Q}$.
Let $\mathfrak{g}$ be the Lie algebra of $G(\mathbb{R})$ and $U(\mathfrak{g}_{\mathbb{C}})$ its universal enveloping algebra of $\mathfrak{g}_{\...

**3**

votes

**1**answer

310 views

### Is a comultiplication structure unique?

Let $A$ be an $R$-algebra.
Suppose $A$ has a $R$-coalgebra structure compatible with the algebra structure.
(I.e. there is a comultiplication map $\Delta$ and counit map $\epsilon$ compatible with the ...

**2**

votes

**1**answer

106 views

### Properties of stabilizers of adjoint action general linear group

Let $G=GL(n,\mathbb{C})$ and let us consider $x \in GL(n,\mathbb{C})$. I'd like to know whether the following is true: the stabilizer for the conjugation action $C(x)$ is special in the sense that ...

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vote

**0**answers

99 views

### What is the analogue of Leibniz's rule for universal enveloping algebra?

Let $G$ be a reductive group over $\mathbb{R}$ and $\mathfrak{g}$ its complexitied Lie algebra.
Let $U(\mathfrak{g})$ be the universal enveloping algebra and $Z(\mathfrak{g})$ is the center of $U(\...

**4**

votes

**1**answer

128 views

### Maximal torus of linear algebraic group over a ring

Let $G$ be a linear algebraic group over a $k$-algebra $A$, where $k$ is an algebraically closed field.
Consider the structure morphism $G\rightarrow U={\rm Spec}(A)$.
Assume that for every $k$-point ...

**0**

votes

**0**answers

36 views

### Product of subsets of a unipotent group

Let $N$ be a unipotent algebraic group and $X,Y$ be two algebraic subsets of $N$.
It is known that if $X,Y$ are algebraic subgroups of $N$, then the product
$X\cdot Y$ is closed (algebraic subset).
My ...

**2**

votes

**0**answers

41 views

### Canonical parabolics vs Levi subgroups

Let $G$ be a reductive group over a field $k$ of characteristic zero. The Jacobson-Morozov theorem gives a method of embedding any unipotent element into an $\mathfrak{sl}_2$ triple, which in turn ...

**2**

votes

**1**answer

108 views

### Cartan subspace of graded Lie algebras

Suppose $\mathfrak{g}$ is a complex reductive Lie algebra and $\theta$ is an automorphism of order $2$. Let $\mathfrak{g} = \mathfrak{g_0} \oplus \mathfrak{g}_1$ be the corresponding $\mathbb Z_2$-...

**3**

votes

**0**answers

117 views

### Functoriality for compactifications of locally symmetric spaces

Let $X$ be a symmetric space associated to an algebraic group $G$ defined over $\mathbb{Q}$ and $G(\mathbb{R})$ acts on $X$ from the left. Let $\Gamma \subset G(\mathbb{Q})$ be an arithmetic subgroup ...

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votes

**0**answers

27 views

### Centralizers of completely reducible subgroups

Let $k$ be a field of characteristic $p \geq 0$. Let $G$ be a connected reductive group defined over $k$ with $p$ good for $G$.
In what follows, I will cite results from the following two papers: ...

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votes

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75 views

### Commutators and brackets in nilpotent Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over an algebraically closed field $k$ of characteristic zero. Throughout, let $x,y$ and $z$ be elements of $\mathfrak{g}$. The Baker-...

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votes

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126 views

### 2-fold linear cover of reductive group of type A

Let $F$ be a nonarchimedean local field of characteristic zero. Let $G=\operatorname{Res}_{E/F}\operatorname{GL}_n$ or $\operatorname{Res}_{E/F}\operatorname{U}_n$, where $\operatorname{U}_n$ is any ...

**1**

vote

**1**answer

60 views

### Character constructed from Kummer local system lifts to representation of algebraic torus

I'm currently reading Mar's and Springer's Character Sheaves. In Chapter 2 (Kummer local systems on tori), they provide a construction of Kummer local systems on a torus $T$ by way of the $m^{th}$ ...

**8**

votes

**2**answers

227 views

### Parabolics and simple roots for a special unitary group: reference request

I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed.
...

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votes

**0**answers

76 views

### Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$

Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...

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votes

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36 views

### Equivalence classes of Hermitian elements in a central simple algebra with an involution of second kind

Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension,
and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$.
Then $^\...

**3**

votes

**1**answer

69 views

### Image of square map on reductive group

$\DeclareMathOperator\ss{ss}$Let $G$ be a reductive group over a field $F$ of characteristic 0. (Here not necessarily $F=\overline{F}$.) Consider the square map
$$
G(F)\longrightarrow G(F), \quad g\...

**2**

votes

**1**answer

192 views

### Stabilizer in $G(\mathbb{Z})$ of point in fundamental domain $G(\mathbb{Z}) \backslash G(\mathbb{R}) / K$

Let $G$ be a semisimple group (the cases of primary interest to me are where $G$ is a special linear group or a special orthogonal group), let $K$ be a maximal compact subgroup of $G(\mathbb{R})$, and ...

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vote

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33 views

### Distributivity property for smooth parabolic induction

Let $G$ be a reductive group over a local field $k$ of characteristic zero with maximal split torus $T$, Weyl group $W$, Borel $B$ and a parabolic subgroup $P$ such that $P\supset B \supset T$. Denote ...

**4**

votes

**1**answer

149 views

### $\mathbb{Q}$-Zariski Closure not equal to smallest Q-subgroup

I wonder if there is a simple instance of the following phenomena : an abstract subgroup S $\subset GL_n(\mathbb{C})$ whose $\mathbb{Q}$-Zariski closure isn't a group ?
Is there some criteria to ...

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vote

**0**answers

75 views

### $H(\mathbb A)^0/H(k)$ is homeomorphic to a closed set in $G(\mathbb A)/G(k)$

I'm reading Godement's paper Domaines fondamentaux des groupes arithmetiques and am confused about a part in a proof. The setting is $k$ is a number field, $\mathbb A$ is the ring of adeles of $k$, $...

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vote

**0**answers

60 views

### The splitting pattern of the Killing form of an algebraic group and the Tits index

Let us assume that $G$ is an anisotropic semisimple, connected algebraic group over a field $k$ of characteristic zero.
Let $K_G$ denote the class of its Killing form in the Witt ring of $k$.
Let $X$ ...

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votes

**0**answers

62 views

### Relation between root subgroups and the root system in unitary groups

Consider a 4-dimensional non-degenerate unitary space over a field of order 4. It can be shown that there are 45 isotropic lines. For each such a line one can associate a unitary transvection and each ...

**0**

votes

**1**answer

177 views

### Definition of group scheme [closed]

Consider the definition of group scheme in Stack Project [022R]. In the paragraph following definition 39.4.1, it is said that
We have morphisms of schemes over $S$: (identity) $e:S\rightarrow G$ and ...

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votes

**0**answers

325 views

### Does this subset of $\mathbb{Z}$ form a subgroup?

Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ with basis $e_1,\ldots,e_n$. Consider the abelian group $N = \mathbb{Z}e_1\oplus\ldots\oplus\mathbb{Z}e_n$.
Assume that $A$ is any ...

**8**

votes

**1**answer

237 views

### Does $G_2(\mathbb{Z})$ depend on the choice of an integral model?

I am trying to understand constructions of exceptional groups of type $G_2$ (over rings). In this post, by a model (of type $G_2$) I mean an affine smooth group scheme over $\mathbb{Z}$ such that the ...

**8**

votes

**2**answers

389 views

### Action of symmetric matrices under $\mathrm{O}(n)$

$\DeclareMathOperator\Sym{Sym}\DeclareMathOperator\O{O}\DeclareMathOperator\GL{GL}$Let $k$ be an algebraically closed field of characteristic 0 (it can even be $\mathbb{C}$ if you like), and let $n\in\...

**3**

votes

**1**answer

169 views

### Schubert cells in G/P for reductive G

All literature on the Schubert cells of the generalized flag varieties $G/P$ ("generalized" here means that $P$ is an arbitrary parabolic) assumes that $G$ is a semisimple complex group. I ...

**6**

votes

**2**answers

169 views

### Relation between finite dimensional representations of an affine group scheme and quasicoherent sheaves on the classifying stack

Let $G$ be an affine group scheme over a field $k$ of characteristic zero.
I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) ...

**3**

votes

**1**answer

169 views

### When an action on open dense subvariety by an algebraic group extends to variety

A toric variety $X$ over $k$ is a variety which contains an algebraic torus
($T= \mathbb{G}_k^s$)
as a dense open subset such that the action of the torus on itself extends to the whole of
$X$. Slogan:...

**1**

vote

**0**answers

161 views

### Blow up of a flag variety at a point

What is the description of the blow-up space of $G/B$ at $B.$ Specifically:
(1) Can we describe it explicitly using an embedding of $G/B$ into some projective space associated to a dominant character ...

**2**

votes

**1**answer

134 views

### Picard groups of determinantal varieties

Consider a general $4\times 4$ matrix:
$$
X:=\left(
\begin{array}{cccc}
X_0 & X_1 & X_2 & X_3 \\
X_4 & X_5 & X_6 & X_7 \\
X_8 & X_9 & X_{10} & X_{11} \\
X_{12} &...

**4**

votes

**1**answer

120 views

### Picard group of $(SL(n)\times SL(m))$-orbits

Let $\mathbb{P}^N$ be the projective space of $n\times m$ matrices with complex entries modulo scalar. Consider the $(SL(n)\times SL(m))$-action on $\mathbb{P}^N$ given by $((A,B),Z)\mapsto AZB^{T}$. ...

**16**

votes

**4**answers

748 views

### Is $O_n({\bf Q})$ dense in $O_n({\bf R})$?

I am wondering if the orthogonal group $O_n({\bf Q})$ is dense in $O_n({\bf R})$?
It is easily checked for $n = 2$ but I think that there is a general principle concerning compact algebraic groups ...

**4**

votes

**1**answer

260 views

### p-torsion in the Picard group of a regular projective curve

Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$.
If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...

**3**

votes

**0**answers

125 views

### Harmonic analysis on reductive groups over $\mathbb{R}$

A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...

**3**

votes

**0**answers

130 views

### Trivial morphism between local cohomology groups

I have two questions concerning morphism between local cohomology groups which I think are related.
Let $G$ be a reductive group with Weyl group $W$ and $B \subset G$ a Borel. Let $X=G/B$ be the flag ...

**12**

votes

**1**answer

319 views

### Non-conjugate subgroups that are conjugate in complexification

In trying to come up with a counter-example in my line of research, I would like to find an example as follows:
$G$ is a semisimple Lie group with complexification $G^{\mathbb{C}}$. $H_1, H_2 \...