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.

Filter by
Sorted by
Tagged with
3
votes
2answers
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
1answer
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
1answer
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
0answers
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 ...
4
votes
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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 ...
4
votes
0answers
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
1answer
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 $...
-1
votes
0answers
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
0answers
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
1answer
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
1answer
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 ...
1
vote
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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 ...
2
votes
0answers
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: ...
2
votes
0answers
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-...
3
votes
0answers
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
1answer
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
2answers
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. ...
5
votes
0answers
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 $K$ ...
2
votes
0answers
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
1answer
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
1answer
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 ...
1
vote
0answers
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
1answer
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 ...
1
vote
0answers
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 $k$, $...
1
vote
0answers
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 $X$ ...
3
votes
0answers
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
1answer
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 ...
5
votes
0answers
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
4answers
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
1answer
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
0answers
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
0answers
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
1answer
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 \...

1
2 3 4 5
35