For questions about groups which have additional structure as a algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. Consider using with the (group-theory) tag.

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Jacquet-Langlands double coset decomposition

Let $G=\mathrm{GL}(2,F)$ with $F$ a non-archimedean local field. Let $K=\mathrm{GL}(2,\mathcal{O}_F)$ be a maximal compact subgroup. Every element of $G$ lies in one of the double cosets ...
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1answer
26 views

twists of unipotent algebraic groups

Let $U$ be a unipotent linear algebraic group over some field $k$ with char$k$=0. Let $U'$ be a linear algebraic group over $k$ such that $U'_{\bar{k}} = U_{\bar{k}}$ (ie $U'$ is a $\bar{k}/k$-twist ...
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26 views

Albert- Algebras and Traceforms

Im new to the topic so this could be basic nonsense to you. Any Albert-Algebra $A$ has a trace map $T:A \rightarrow k$ and thus one can assign a quadratic form $q_A$ of rank $27$ by setting $q_A(x) = ...
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3answers
63 views

on the condition “$G$ is defined over $\mathbb{Q}$”

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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1answer
62 views

A point in $ PGL(R) $ not in $ GL(R)/R^{\times} $

A bit of notational background first. Let $k$ be a field and define $ PGL_{n} = Spec(k[x_{ij}]_{(det)}) $, where $i,j = 1,...,n$ and where $k[x_{ij}]_{(det)}$ denote the degree $0$ part of the graded ...
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1answer
79 views

Which categories of linear representations are semisimple?

Let $k$ be a field of characteristic $0$. For which smooth algebraic groups $G$ over $k$ does the abelian category of linear representations $\mathsf{Rep}_k(G)$ (not assumed to be finite-dimensional) ...
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0answers
15 views

Representation groups over Dedekind domains

I am interested on groups defined over $O_K$ the ring of integers of a number field $K$. Given a linear representation $T:Gl_N(O_K)\rightarrow Gl(W)$ with $W$ a free $O_K$-module, What are the main ...
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1answer
32 views

action of $GL_3$ on $P^2$

Find the action of $GL_3(K)$ on $\mathbb P_k^2 $, and compute its orbits and also the isotropy groups for all its orbits. ($K$ is an algebraically closed field) I know that $GL_3$ acts on ...
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1answer
77 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
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1answer
17 views

connected linear algebraic group over the algebraic closure of a field

Let $G$ be a connected linear algebraic group over a field $k$ of characteristic 0. A paper I'm reading seems to imply that $\overline{G}:= G \times_k \overline{k}$ will also be connected, but I don't ...
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1answer
21 views

A sufficient condition for irreducibility of a $G$-variety

Let $G$ be an algebraic group over a field $k$ and let $V$ be a variety on which $G$ acts. Suppose $U\subset V$ is a closed, irreducible, $G$-stable subset which intersects every $G$-orbit ...
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1answer
49 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
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1answer
53 views

The identity element of a group

We define the process in Z. Then, is a group. In this group,which is the identity element? The correct answer is the element 10. why ?
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2answers
18 views

Polynomial Ring of Linear Algebraic Group

During lectures, we defined the Linear Algebraic group as the algebraic set $ GL(V):=k^{n^2}-V(Det) $ Where $V(Det)$ are the matrices with $0$ determinant. Then we proceed by identifying the ...
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1answer
41 views

Definition of unipotent linear algebraic groups over non algebraically closed fields

Suppose we have a field $F$ with $\text{char}\ F=0$ and $F$ is not necessarily algebraically closed. What is the definition of a unipotent linear algebraic group over $F$? I'd really appreciate ...
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1answer
44 views

Lemma 5.2.5 in Springer's Linear Algebraic Groups

I'm stuck trying to understand the first paragraph of this proof. Let $X\rightarrow Y$ be a dominant morphism of affine varieties and denote $B=k[X]$,$A=k[Y]$. Assume there exists $b\in B$ such that ...
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0answers
21 views

What do you get when you pull the Bruhat Decomposition back to the Lie algebra via the exponential map?

If $G$ is a connected, reductive, complex group with Borel subgroup $B < G$ and Weyl group $W$, we can write $$G = \bigsqcup_{w \in W} B w B$$ If $\mathfrak{g}$ is the Lie algebra of $G$, we have ...
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0answers
40 views

Is a connected unipotent subgroup always contained in a Borel subgroup?

As the question says, is a connected unipotent subgroup $U$ of a linear algebraic group scheme $G$ always contained in a Borel subgroup of $G$? I have an argument for why the answer is yes, and I ...
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1answer
40 views

The structure of maximal tori in finite simple groups

Let $\mathbf{G}$ be a linear algebraic group over an algebraic closed field of characteristic $p$ and $F$ a proper frobenius map on it with fixed point group $\mathbf{G}^f=G$ such that ...
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0answers
24 views

Ideas for seminar talk on Algebraic Groups related to Number Theory

In a few weeks I have to give a seminar talk in an algebraic groups seminar, and the topic is number theory (possibly elliptic curves). I am not very knowledgeable in the subject, so I was hoping I ...
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0answers
37 views

diagonalizable group

The following is an exercise from Humphrey's Linear Algebraic Groups (page 108): Let $G$ be an algebraic group, $ \displaystyle H= \cap_{ \chi \in X (G)} \ker ( \chi) $. Prove that: (a) $H$ ...
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0answers
30 views

Computing cohomology of finite groups of Lie type

Let $G_{/\mathbf{Z}}$ be a Chevalley-Demazure group scheme, i.e. a split reductive group scheme over $\mathbf{Z}$. Let $\rho:G\to \operatorname{GL}(V_{/\mathbf{Z}})$ be a representation. If $k$ is a ...
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69 views

A Question about Zariski topology

Zariski topology which is used in the definition of an algebraic group is only defined for affine and projective varieties, isn't it?
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162 views

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

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
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0answers
38 views

Quotients of linear algebraic groups in different categories?

I have a question on quotients of linear algebraic groups. Let $G$ be a linear algebraic group and $H$ a linear algebraic group acting on $G$ as an algebraic group. I would like to know what the ...
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0answers
23 views

Is a form of a linear algebraic group over $k$ a linear algebraic group over $k$?

As the title says. $k$ : field with char($k$) = $0$ form: if $G$ is a l.a.g. over $k$ a form of $G$ is an algebraic group $G'$ over $k$ such that $G_{\bar{k}} \cong G'_{\bar{k}}$.
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1answer
52 views

Does a projective variety have a torus fixed point?

Let $X$ be a projective variety over $\mathbb{C}$ and let $T=(\mathbb{C^*})^k$ act on it. Is it true that there is a fixed point of this action on every irreducible component of $X$ just because $X$ ...
2
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2answers
42 views

Projective Special Linear Group is an Linear Algebraic Group

So I was wondering why the group $PSL(2,K)$ is a linear algebraic group, in the case that the characteristic of $K$ is not equal to $2$. Actually there is a description of $PSL(2,K)$, namely: ...
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1answer
45 views

Variety of Connected Components

In Milne's text http://www.jmilne.org/math/CourseNotes/iAG.pdf (A71), he introduces the "variety of connected components" of a finite type scheme $X$ over $k$ as the universal example of a zero ...
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1answer
23 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
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58 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
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0answers
18 views

QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
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1answer
39 views

Relative center of relative group scheme

This might be an easy question. Let $p: X \rightarrow S$ be a relative group scheme. In particular the fibers are group schemes. I want to know if there are constructions like the ``relative ...
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1answer
25 views

Question on subgroups of reductive groups

A linear algebraic group $G$ over some field $k$, which I assume being of characteristic 0, is reductive if $R_u(G^0_{\overline{k}})$ is trivial, where $R_u$ denotes the unipotent radical, $G^0$ is ...
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0answers
24 views

The simply connected form of a semisimple algebraic group

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$, so that $G$ is an almost-direct product of its minimal closed connected normal subgroups of positive dimension, ...
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2answers
169 views

Is there a group between $SO(2,\mathbb{R})$ and $SL(2,\mathbb{R})$?

Is there a non-trivial subgroup $H \subset SL(2,\mathbb{R})$ such that $H \supset SO(2,\mathbb{R})$ ? My intuition is that, since $\dim SO(2)=1$ and $\dim SL(2)=3$, there should be some group ...
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2answers
18 views

How to show that $GL_n/U$ is birationally isomorphic to $B^-$?

It is said that $GL_n/U$ is birationally isomorphic to $B^-$. Here $U$ acts by right multiplication on $GL_n$. I think that $GL_n/U$ consisting of cosets. Two matrices in the same coset if any two ...
2
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1answer
64 views

quasi-split algebraic group

While reading papers, there usually an assumption "quasi-split" for reductive algebraic groups. To use their results I need to know which groups are quasi-split. For the case I am interested in ...
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1answer
28 views

Weyl group of $B_n$ and $D_n$

Is it true that the Weyl group $W(D_n)$ is also a quotient of the Weyl group $W(B_n)$? One can see that $W(D_n)$ is a normal subgroup of $W(B_n)$ irrespective of $n$ even or odd.
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35 views

Is there anywhere some explicit Bruhat decompositions are written down?

Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given. Also, I calculated the following regarding the Bruhat decomposition of ...
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1answer
64 views

Two equivalent definitions of GIT semistable points

Let $X$ be a projective variety, acted on by a reductive algebraic group $G$. We fix a linearization given by the $G$-equivariant ample line bundle $L\to X$. I am aware of two definitions of the ...
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2answers
72 views

T//W for adjoint type group PGL3

Let $G$ be a reductive algebraic group and $T$ a maximal torus (over $\mathbb{C}$). It is well known that if $G$ is simply connected type then $T//W = \mathbb{A}^r$. I want to verify that the ...
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1answer
46 views

Complex conjugation of positive roots

I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a ...
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51 views

Examples of inner forms

Let $G$ and $G′$ be two linear algebraic groups over a field $F$. From what I understand, $G$ is called an inner form of $G′$ if $G$ and $G′$ are isomorphic over a the (or an?) algebraic closure of ...
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38 views

$p$-divisible group of tori

I am looking for a reference of the following question which should be well known. Let $k$ be any field and $T$ an algebraic torus over $k$ which is not necessarily split. Let $T(l)$ be the ...
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35 views

Orbits of $Sp(n,R)$ under action of $Gl(2n,R)$ by conjugation

These questions arose from a question related to K-theory, I am hoping for (big) results from the theory of linear algebraic groups to be helpful. Maybe somebody with a better background there can ...
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1answer
27 views

Maximal tori in $SO(n,\mathbb{C})$

What are maximal tori in $SO(n,\mathbb{C})$? (not $SO(n,\mathbb{R})$) Can a maximal torus in $SO(n,\mathbb{C})$ be written as $T\cap SO(n,\mathbb{C})$ for some maximal torus $T$ in ...
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0answers
17 views

Is this a compact group?

Consider $$x(t)=e^{-iHt}x(0)$$ and define $$G=\{ e^{-iHt}\mid t\in \mathbb{R}_{\geq 0}\}$$ Also write $\bar{G}$ to the closure of $G$ wrt the Euclidean topology. Q: is $\bar{G}$ a compact group? ...
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1answer
63 views

Algebraic groups of multiplicative type in char 0

For a number field $k$ (so of char 0), are algebraic $k$-groups of multiplicative type always linear?
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1answer
32 views

Order of a group of matrices

I have to calculate the order of the group $G=\operatorname{GL}(n,\mathbb{Z}_m)$ with $m=p^k$ for $p$ prime and $k\in \mathbb{N}$. So I was thinking on the homomorphism $\det: G \rightarrow ...