Questions tagged [algebraic-groups]

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

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On axiomatic definition of affine root systems

In Macdonald's book Affine Hecke Algebras and Orthogonal Polynomials, chapter 1 introduces affine root systems. I will recall the definition here: Let $E$ be a non-zero real Euclidean space (finite ...
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How to compute that the volume of $Z(\mathbb{A})\backslash \frak{S}$ for a Siegel set is finite?

I found it quite hard to find a complete reference for the (adelic) reduction theory of algebraic groups used in automorphic representations. I want to show that for $G=GL(n)$ over a number field $F$, ...
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Why are Iwahori subgroups compact?

I read the definition of Iwahori subgroup but I don't know why it is compact. Does anyone know how to prove it?
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Epimorphism of algebras restricted to ring of invariants

Suppose $G$ is an affine algebraic group that acts rationally on $K$-algebras $R$ and $R'$, ie every element is contained in a rational representation: a finite-dimensional $G$-stable vector subspace $...
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Reference request: multiplicative group of a central simple algebra, their reductivity and parabolic subgroups

During my study of the theory of automorphic forms and $L$-functions, I never found any literature dealing with the following: Suppose $D$ is a central division algebra over a local or global field $F$...
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Why is the multiplicative group of a central division algebra anisotropic?

Let $k$ be any field. Suppose $D$ is a central division algebra over $k$ of degree $n^2$, then we can understand its multiplicative group $D^{\times}$ as an algebraic group (defined over $k$). I ...
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Explicit equations describing complex spin groups as affine algebraic groups

Although there are tons of questions about spin groups here on math.SE, I could not find there what I want. What I want is this. Take the complex spin group $\operatorname{Spin}(2n,\mathbb C)$. It has ...
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Coordinate algebra of image of homomorphism between linear algebraic groups

Let $f: G\to H$ be a homomorphism of linear algebraic groups. Let $f^*: k[H]\to k[G]$ be the corresponding Hopf algebra morphism. Then $f^*$ factors as $k[H] \twoheadrightarrow k[H]/I\hookrightarrow ...
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Correspondence closed subgroup of $GL(n,k)$ and closed subgroup scheme of $GL_n$

Sorry for my bad English. Let $k$ be a field (if necessary ch $k=0$). We can think a general linear group $GL(n,k)$ is a topology group by Zariski topology, and hence think closed subgroups $H\subset ...
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Orbits closed under scalar multiplication

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic $0$. Let $V$ be a rational irreducible $KG$-module. Suppose that $V$ is a prehomogeneus space, i.e. $G$ ...
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Are there any references to study Proalgebraic group defined by Serre?

I am trying to read the paper 'Groupes proalgébriques' by Serre and is there any other reference which covers the topics in this paper? Any help is appreciated.
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component B in the commuting algebra

Theorem $3.1$ from Robert Griess' "Elementary abelian p-subgroups of algebraic groups". Let p be a prime, $\mathbb{K}$ an algebraically closed field, G = SL($n+1$, $\mathbb{K}$), Z $\...
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Lattice in noncompact simple group is Ad-irreducible

Is every lattice in a Lie group Ad-irreducible? No. This is false for $ G $ compact because any closed subgroup is a lattice. And it is certainly false if $ G $ is not simple since a group can only ...
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Adjoint-irreducible subgroup is either finite or Zariski dense

$ G $ is a linear algebraic group whose real points are connected. $ H $ is a subgroup. The adjoint action of $ H $ on the Lie algebra of $ G $ is irreducible. Is it true that $ H $ is either finite ...
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Computing $\chi(1)$ and $\chi(s)$ for $\chi\in\widehat{\mathrm{GL}_2(\mathbb{F}_q)}$ and semisimple non-regular $s$ using formulas of Deligne-Lusztig

Let $G=\mathrm{GL}_2$ and $s=\left(\begin{smallmatrix} a & \\ & b \end{smallmatrix}\right)$ be semisimple and non-regular in $G(\mathbb{F}_q)=\mathrm{GL}_2(\mathbb{F}_q)$ (i.e. $a\neq b$ and $...
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Torus and dual torus

In the setting of algebraic/reductive group, there is a notion of dual group (defined from the dual root system). Is there an explicit way to see it, or to describe it? For instance if $T \simeq k^\...
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Relation between characters and coordinate ring

I am reading about algebraic groups. I don't fully understand the purpose of the coordinate ring, but I feel this is a way of "parametrizing" characters on the group. Here is an example to ...
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The complement of a Zariski open dense subset in a real linear algebraic group has zero Haar measure?

Let $G$ be a real linear algebraic group (so it is locally compact and Hausdorff), equipped with a left-invariant Haar measure. Let $U$ be a Zariski open dense subset of $G$. I wonder how to show that ...
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$N^-D N^+$ is Zariski open dense in $\text{GL}(n, \mathbb R)$

Let $N^-$ (and resp. $N^+$) denote the upper (resp. lower) triangular unipotent subgroup of $\text{GL}(n,\mathbb R)$. Let $D$ denote the full diagonal subgroup of $\text{GL}(n,\mathbb R)$ I wonder how ...
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Inequality involving degrees of field extensions

Suppose we have fields $F \subset E \subset K$ and $\alpha \in K$. Suppose also that $K$ is algebraic over $E$. Let $\alpha \in K$. Then we have $p(\alpha)=0$ for some $p(x) = a_{0} + a_{1}x+ \dots+a_{...
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Intuitive understanding of Tamagawa measure and its relationship between local measures?

Weil's book Basic Number Theory, (1973, second edition) pp. 113 mentioned the Tamagawa measure on $k_\mathbb{A}$, where $k$ is a global field and $k_\mathbb{A}$ its ring of adeles (an old fashioned ...
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Lie algebra of linear algebraic group $\operatorname{GL}_n$

Let $k$ be a field. Then we can define the algebra of dual numbers $k[\varepsilon]:= k[x]/(x^2)=k\oplus \varepsilon k$, with $\varepsilon^2=0$. We can then also consider the projection $\pi: k[\...
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Generating the algebra $\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]$ (regular functions)

I recently asked a question about the regular functions in $\mathbf{GL}(n,\mathbb{C})$ but now that I have read the appendix I am again confused; here is my past question: Understanding regular ...
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Can closure of an orbit under a reductive action contain infinitely many orbits?

To be more specific, let $G\subseteq GL(V)$ be a reductive linear algebraic group over an algebraically closed field of characteristic zero. For $v\in V$, can the (Zariski) closure of $Gv$ in $V$ ...
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1 answer
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Question on isogenies of degree d.

I am trying to understand the following question (Proposition 5.12. in ABELIAN VARIETIES, Bas Edixhoven, Gerard van der Geer, and Ben Moonen) If $f: X \to Y$ is an isogeny of degree $d$ between ...
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Understanding regular functions of $\mathbf{GL}(n,\mathbb{C})$

I am following the book Symmetry, Representations and Invariants and I have a confusion with the definition of a regular function on $\mathbf{GL}(n,\mathbb{C})$: For the group $\mathbf{GL}(n,\mathbb{...
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Inverse limit of affine algebraic groups

Let $G_i$ be an inverse system of affine Algebraic groups (ie. Group objects in category of affine varieties). Is the inverse limit an affine algebraic group scheme? More precisely, does the inverse ...
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5 votes
1 answer
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Decomposition of symmetric powers of the standard representation of $SO(n)$

Let $V$ be the n-dimensional standard-representation of $SO(n)$. Since $SO(n)$ preserves a bilinear form on $V$ there is a trivial 1-dimensional subrepresentation in $S^2V$. So, in general, $S^k V$ ...
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Real structure on algberaic group

Let $G$ be an algberaic group defined over $\mathbb R$. We have the following different notions. The real points of $G$, ie all $x\in G$ such that $O_x/m_x = \mathbb R$ A real form on $G$, ie. the ...
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Parabolic induction of p-adic groups independent of the choice of parabolic.

I noticed many papers concerning the theory of smooth representations of connected reductive p-adic groups, omit the mention of the specific parabolic subgroup $P\subseteq G$ used in defining the ...
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Classification of Reductive groups.

Let $K$ be a field with a non trivial discrete valuation and with $K$ complete and $\bar{K}$ perfect. What role do the Buildings play in the classification of reductive groups over $K$ ? Also there ...
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Given a prime number why cant a cyclic group have more than $p-1$ elements of order $p$

The other day I stumbled across this question: Let $p$ be a prime number. If a group has more than $p-1$ elements of order $p$. Can it be cyclic? I've tried to solve this by trying to prove that ...
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If an arithmetic group is map to another one, is the same true for the corresponding $\mathbb{Q}$-forms?

Let $\mathrm{G}$ and $\mathrm{H}$ be $\mathbb{Q}$-linear simple algebraic groups. Let $\phi:\mathrm{G}\to\mathrm{H}$ be a $\mathbb{R}$-morphism. Let $\Gamma$ be an arithmetic subgroup of $\mathrm{G}(\...
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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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Algebraic group vs Lie group

I know the definitions of algebraic and Lie groups. I know the difference between them in terms of definition; loosely, the first is a variety plus group, while the second is a smooth manifold plus ...
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Show that $(\operatorname{SL}_2\times\operatorname{SL}_2)/N$ is an almost direct product

A linear algebraic group $G$is an almost direct product of its subgroups $G_1,\dots,G_r$ if the product map $$ G_1\times\dots\times G_r\to G:(g_1,\dots,g_r)\mapsto g_1\cdots g_r$$ is a surjective ...
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The Transporter is a submonoid

I am reading Humprheys' Algebraic Groups, stuck at an apparently simple point In section 8.2 Actions of Algebraic groups(line 6, paragraph 1), The transporter is defined: Let $Y ,Z$ be subsets of $X$ (...
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Homomorphisms from a linear algebraic group to a finite group

Is there any example of a linear algebraic group (over an algebraically closed field) $G$, a finite group $H$ (with discrete topology) and a group homomorphism from $G$ to $H$ which is not continuous, ...
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Does Repf($G$) have enough injectives?

Let $G$ be a group scheme over a field $k$. My question is Does Repf(G), the category of finite-dimensional linear representations of $G$, have enough injectives? It is well-known that Rep(G), the ...
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Obstruction of existing a 2-cocycle in Galois cohomology with given local components

Let $K$ be a global field and $T$ be a torus which satisfy Hesse principle defined over $K$ and spilt over $L$. By tate-nakayama theorem we now that at every palace $l$,a cocharcter $\mu_l$ defined ...
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Is every finite subgroup the integer points of a linear algebraic group?

Let $ K $ be a compact Lie group. For every finite subgroup $ \Gamma $ of $ K $ does there exist a linear algebraic group $ G $ such that the integer points are $$ G_\mathbb{Z} \cong \Gamma $$ and ...
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1 vote
1 answer
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Relating the $F$-rational points of a torus to the character group

If $T$ is an algebraic torus over a field $F$, then I keep reading that $T(F) \cong X_*(T) \otimes F^\times$, where $X_*(T)$ is the cocharacter lattice. What is the isomorphism between them?
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4 votes
1 answer
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Character group of non-split torus in $GL_2$

Let $E=\mathbb Q(\sqrt{-d})$ be an imaginary quadratic field and let $R_{E/\mathbb Q}(\mathbb G_m)$ be the restriction of scalars of the multiplicative group, i.e. $R_{E/\mathbb Q}(\mathbb G_m)(X) = \...
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2 votes
1 answer
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Algebraic group scheme is a torsor under kernel of a group homomorphism.

In the book by 'Algebraic Groups' by Milne, a right $G$- torsor over $S_0$ is a scheme $S$ faithfully flat over $S_0$ together with an action $S\times_{S_0}G\to S$ of $G$ on $S$ such that the map $$(s,...
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Is it possible to realize the Moebius strip as a linear group orbit?

Is the Moebius strip a linear group orbit? In other words: Does there exists a Lie group $ G $, a representation $ \pi: G \to GL(V) $, and a vector $ v \in V $ such that the orbit $$ \mathcal{O}_v=\{ \...
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3 votes
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Cohomology of $G/T$ for complex reductive group $G$

Okay, I will try this again, last time people didn't like how I asked it :) So let's say $G$ is a reductive complex algebraic group; it could be $GL_n(\mathbb{C})$ if that makes you happy (and in ...
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Finite simple group which is maximal closed subgroup of lie group

The icosahedral subgroup of $ G=PSU_2 \cong SO_3(\mathbb{R}) $ is a finite simple group which is also maximal closed in $ G $. Do other Lie groups admit maximal closed subgroups which are also finite ...
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1 vote
1 answer
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Dimension (as an algebraic variety) of the symplectic group over a finite field

The dimension of the symplectic group Sp(2n, F) is $2n^2+n$ if F is $\mathbb{R}$ or $\mathbb{C}$. I would like to know if the same is true if F is the algebraic closure of a finite field. To elaborate:...
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1 vote
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Non-split algebraic group can be split at some local places?

Let me take an example for special orthogonal subgroup $SO(3)$. Let $F$ be a number field and $\alpha,\beta \in F^{\times}$ such that $x^2+\alpha y^2+ \beta z^2$ does not represent $0$. Let $J$ be a $...
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What is $\dim(M_1wP_2)$ for parabolics $P_i = M_iN_i$ of a reductive group?

Let $B$ be a Borel subgroup of a connected reductive algebraic group $G$ with maximal torus $T$. Let $\Phi = \Phi(G,T)$ be the root system, $\Delta$ be the basis of $\Phi$ defined by $B$, and $W$ be ...
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