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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Questions about nilpotent and reductive groups.

I am reading the lecture notes. On Page 37, it is said that $N_{\alpha}$ is a nilpotent $p$-adic group and $M_{\alpha}$ is a reductive group. I am trying to prove these results. By difinition, a group ...
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95 views

Motivation?: Lie algebra and algebraic group Cohomology

This is just an a-priori question to get a motivational heuristic idea: If an algebraic group G (more generally, G an affine group scheme), is connected over an algebraically closed base-field k. ...
3
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1answer
63 views

The identity component of an algebraic group is always parabolic

Essentially I was wondering if the quotient of an algebraic group $G$ by its identity component $G^0$ is necessarily always parabolic. My argument: This seems right since $G^0$ is a closed ...
3
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1answer
76 views

Why is the map: $GL_n(K)\times GL_n(K) \to GL_n(K)$ regular?

Let $K$ be a field and $GL_n(K)$ the set of all invertible $n$ by $n$ matrices over $K$. Let $m: GL_n(K)\times GL_n(K) \to GL_n(K)$ be the usual multiplication of matrices. Why the map $m$ is regular? ...
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1answer
40 views

How to show that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$?

I am reading the lecture notes. On Page 16, Line 1, it is said that the restriction of $\pi$ to the subrepresentation $W$ factor through $G/N$. What does "factor through" mean? How to show that the ...
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1answer
69 views

Why $G/N$ is discrete?

I am reading the lecture notes. On page 15, Line -5, why $G/N$ is discrete? Thank you very much.
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1answer
51 views

How to show that two representations are equivalent?

I am reading the lecture notes. On page 14, example of $C_{c}^{\infty}(G)$. I am trying to show that the map $A$ takes $f$ to $g\mapsto f(g^{-1})$ is an invertible element of ...
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1answer
58 views

How to show that $\pi^*(g)=\chi(\det g)^{-1}$?

I am reading the lecture notes. On page 14, how could we show that $\pi^*(g)=\chi(\det g)^{-1}$? I think that $\langle \pi^*(g)\lambda, v \rangle = \langle \lambda, \chi(\det g)^{-1} v \rangle = ...
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1answer
48 views

Questions about reductive groups.

I am reading the lecture notes. Let $G$ be a reductive group and $(\pi, V)$ a representation of $G$. For $v \in V$, define $\operatorname{Stab}(v)=\{g\in G \mid \pi(g)v=v\}$ and $V^{\infty}=\{v\in V ...
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2answers
161 views

Automorphism Groups of Projective Algebraic Surfaces

When does a projective algebraic surface have an infinite automorphism group? Is there a simple criterion, or at least a sufficient condition?
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27 views

Is there any good books or papers about Coxeter groups with negative weight function?

As to Coxeter group with weight function, most people are concerned with positive weight functions, including Lusztig, writing Hecke algebras with unequal parameters. This book don't deny the negative ...
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34 views

Calculating this quotient?

What is the quotient of the algebraic Groups $GL_n$ by $Sp_n$ equal to? I was conisering a different example and would use the universal property to establish it, but I wasn't certain what it should ...
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0answers
35 views

Complexification of the inclusion $\text{U}_n\subset \text{GL}_n(\mathbb{C})$

What is the map $\text{GL}_n(\mathbb{C}) \to \text{GL}_n(\mathbb{C}) \times \text{GL}_n(\mathbb{C})$ named in the title? I guess it has something to do with the polar decomposition, but I can't manage ...
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1answer
170 views

What are the Borels/parabolics of the orthogonal or symplectic groups?

Does anyone know where I can find info about the Borel subgroups and parabolic subgroups of algebraic groups? I know what they are for the general linear group and for the special linear group you ...
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102 views

Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
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0answers
50 views

The symplectic transvection group is isomorphic to what $A^1$, $A^2$?

The transvections generate the symplectic group, but to demonstrate its connectedness I need to establish their connectedness. I was thinking of doing this through an isomorphism of alg. groups with ...
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1answer
67 views

Representation Theory - Example of a not-G-stable V

I'm studying linear representations for algebraic groups for the moment. And I kind of got stuck on some theorem. The existence of a finite linear representation makes use of the fact that $V$ is ...
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1answer
135 views

Is taking inverse automatically well-defined?

By the usual definition, Lie group is a manifold $G$ with a group structure on it such that the multiplication $m\colon G\times G\to G$ and taking inverse $i\colon G\to G$ are both smooth maps. But it ...
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1answer
70 views

Is Bruhat cell dense in p-adic topology?

I've seen in literature a statement like 'there exists an open and dense Bruhat cell'. In $GL(2,F)$ for example, where $F$ is a p-adic field, let $\omega=\begin{pmatrix} & 1 \\ 1 & ...
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0answers
50 views

The set of regular points in an algebraic group

I saw the following fact in a paper I was reading and I was wondering if someone could provide a reference. Let $K$ be a non-archimedean local field (say of characteristic $0$), and let $G$ be a ...
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0answers
91 views

“Invariant integral” for linearly reductive groups, and the Reynolds operator

Let $G$ be an affine algebraic group. Let $$\lambda\colon G\to GL(k[G]),\quad \lambda(g)(f)=(h\mapsto f(g^{-1}h)),$$ be the left-translation, which is a rational representation of $G$ on its ...
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1answer
138 views

Abelian subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that $$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$ so ...
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0answers
67 views

What is the Weyl group of this group?

Let $G$ be the group $GL_{n_1}(q^{l_1})\times GL_{n_2}(q^{l_2})\times GL_{n_3}(q^{l_3})$. Here $GL_{n_1}(q^{l_1})$ is the rational points of $GL(n,\bar {\mathbf{F}}_q)$. My question iis what is the ...
5
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1answer
111 views

The regular representation for affine group schemes

I want to understand the regular representation of an affine algebraic group. An affine algebraic group as I know it, is a functor from the category of $k $ -algebras to groups that is representable ...
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30 views

If Lie(H) preserves a subspace, must H also preserve that subspace?

Assume $H \subset G$ is a closed connected subgroup of a linear algebraic group over an arbitrary field (both assumed to be smooth). Assume $G$ acts linearly on the (finite dimensional) vector space ...
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0answers
124 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
4
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1answer
91 views

Isogeny of algebraic groups

Let $f:G\to H$ be an isogeny between connected linear algebraic groups. What invariants (rank, semisimple rank, reductive rank, being semisimple...) do these groups share? Are there any properties ...
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1answer
65 views

Groups of transformations

I tried to find literature and articles about groups of transformations, but mostly what I found is either about groups or about transformations. Can you suggest me literature where groups of ...
3
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0answers
38 views

Subvariety of an Algebraic Group.

Given an algebraic group $G$ over an algebraically closed field $K$, if $H$ is a subvariety of $G$, then is $H$ a subgroup of $G$? This seems rather strong. If it is indeed false, is there a geometric ...
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0answers
119 views

In a solvable algebraic group all maximal tori are conjugate to each other

I want to prove the following statement: All maximal tori in a solvable algebraic group are conjugate to each other. I use the following facts: Theorem. Let $G$ be an irreducible solvable ...
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0answers
37 views

“adding” numbers $\in\mathbb{U\left(2\right)}\times \mathbb{R}^+_0$

Let a unitary number be one that corresponds to a matrix of the form: $$\left( \begin{array}{cc} w+i x & y+i z \\ -e^{i p} (y-i z) & e^{i p} (w-i x) \end{array} \right)$$ This is analogous to ...
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0answers
82 views

generators for the multiplicative characters of a linear algebraic group

Let $G\leq \mathrm{GL}_n(\mathbb{C})$ be a linear algebraic group. Assume we know its defining ideal $$\mathcal{I}(G)=\langle f_\alpha \mid \alpha \in S\rangle \subseteq \mathbb{C}[x_{i,j}, ...
4
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1answer
75 views

Algebraic group homomorphism

Let $\phi$ be an algebraic group homomorphism from $G_{m}$ to $GL_{n}(\mathbb C)$,where $G_{m}= \mathbb C^{*} $. Then image of $\phi$ lies in the $D(n,\mathbb C) \cap GL_{n}(\mathbb C)$. Moreover each ...
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0answers
40 views

Borel subgroup which is a maximal torus is nilpotent

I am struggling with this simple fact. Let $B$ be a Borel subgroup of a connected algebraic group $G$. Let $T$ be a maximal torus. Then, if $B = T$, we have $B$ is nilpotent. I don't see why this is ...
4
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1answer
103 views

Properties of Algebraic Groups

Let $G(X_1, \cdots, X_n) \subseteq GL(V)$ denote the smallest algebraic group containing $\{X_i\}_{i=1}^{n} \subseteq GL(V)$, where $V$ is a finite-dimensional vector space over $\mathbb{C}$. Let $S$ ...
3
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1answer
161 views

How to find root subgroups

$\newcommand{\GL}{\text{GL}}\newcommand{\diag}{\text{diag}}$For $G = \GL_n(k)$ let $B$ be the upper triangular matrices and $T$ be the diagonal matrices in $G$. In this case I understand that the ...
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2answers
100 views

The Radical of $SL(n,k)$

For an algebraically closed field $k$, I'd like to show that the algebraic group $G=SL(n,k)$ is semisimple. Since $G$ is connected and nontrivial, this amounts to showing that the radical of $G$, ...
4
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1answer
126 views

Tangent bundle of an algebraic group

Let $G$ be a linear algebraic group over a field $k$. I think that the tangent bundle should be the sheaf $Der(\mathcal{O}_G,\mathcal{O}_G)$, which is isomorphic to the dual of the differentials. ...
5
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2answers
150 views

Definition of Unipotent in Positive Characteristic

Let $G$ be an affine algebraic group over an algebraically closed field $k$ whose characteristic is $p>0$. Can $\mathcal{U}(G)$, the set of unipotent elements of $G$, be characterized as all ...
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1answer
81 views

Eigenvectors of algebraic group representation

In a paper of Kollar and Szabo there is a lemma (Lemma $1$) in which the following terminology is used: "Every representation $H\rightarrow GL(n,K)$ has an $H$-eigenvector" ($H$ is an algebraic ...
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2answers
145 views

Definitions in a Theorem of Lang

I'm trying to understand the following theorem due to Serge Lang (Algebraic Groups over Finite Fields, 1956, Theorem 2): Let $p$ be a prime, and let $k$ be a finite field of $q=p^n$ elements. Let ...
5
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2answers
224 views

How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?

the answer should be $$|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|=N^{3}\prod_{p|N}(1-{1 /p^2})$$ But first how to prove $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
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34 views

How to show $G_{n}^{\perp}=nX$?

Let $G$ be a diagonalizable algebraic group, and $G_{n}$ denotes the subgroup of elements of order dividing $n$. Assuming $k$ is algebraically closed and has characteristic $p$ and $p$ does not divide ...
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0answers
72 views

Reduction of closed orbit lemma by Galois descent

I am reading a proof of the closed orbit lemma: If $G$ is a smooth $k$-group of finite type acting on a finite-type $k$-scheme $X$, and $x$ is a $k$-point of $X$, then the orbit map of $x$ ...
4
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1answer
88 views

Why is this group action a morphism of varieties?

I am examining a proof that any affine connected algebraic group is a closed subgroup of some $GL_n$, and I am stuck on a fine point. Let $G$ be an affine algebraic group in the sense that it is a ...
2
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1answer
65 views

Dense Torsion in Torus

I am reading Onishchik and Vinberg's "Lie Groups and Algebraic Groups". Upon introducing the torus $T=\mathbb{K}^{\times} \times \cdots \times \mathbb{K}^{\times} = (\mathbb{K}^{\times})^n$ (where ...
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1answer
40 views

Why the projection map to the semisimple part is a morphism for algebraic groups?

From Springer, Linear Algebraic Groups, first page of Chapter 3. Let $G$ be a commuative algebraic group. If we regard $G$ as a closed subgroup of $GL_{n}$, then we can identify its semisimple part ...
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1answer
126 views

If a group scheme $G$ operates on another scheme $X$, how do you define orbits?

In my specific case, $G=\mathrm{Spec}(k[M])$ is an algebraic torus acting on a toric variety $X_\Sigma$ corresponding to a fan $\Sigma$ when $k$ is not necessarily algebraically closed (or maybe even ...
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48 views

Do “multiples” of open dense sets of an algebraic group union to the whole group?

Let $G$ be an algebraic group. From algebraic geometry we know there exists an open dense subset $U$ of $G$ such that $U$ is nonsingular. Since left multiplication of $U$ by elements of $G$ is an ...
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2answers
41 views

Co-algebra structure on $k[G]$

Let $k$ be a field. Given an affine algebraic group $G$ (defined as a functor from the category of $k$-algebras to the category of sets) then we have the coordinate ring (or the $k$-algebra ...