# 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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### 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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### 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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### 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 $V$ ...
1 vote
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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 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 ...
1 vote
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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 ...
1 vote
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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?
### 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 ...