# 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.

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### Hasse principle for $H^2$ of a maximal torus of a connected quasisplit group?

Let $k$ be a number field and let $G$ be a quasisplit reductive algebraic group over $k$. Does there exist a maximal torus in $G$ such that the Hasse principle in dimension $2$ holds, i.e., such that ...
80 views

### Centraliser of a maximal $k$-split torus of a reductive $k$-group

Let $G$ be a connected reductive group defined over a field $k$, and let $S$ be a maximal $k$-split $k$-torus of $G$. Then the centraliser $\mathscr Z_{G}(S)$ is defined over $k$. In fact, it is a ...
1 vote
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### When the action of reductive group on algebraic variety is not equidimensional?

I saw the question When is an almost geometric quotient flat? which said "The quotient $\pi$ is flat if and only if $\pi$ is equidimensional and $X$ is smooth". I am curious is there an ...
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### Cohomology of compact open subgroups of semisimple groups over local fields

Let $E$ be a local field, $\mathcal{O}$ its ring of integers, $k$ its residue field, and $G$ a split semisimple group over $\mathcal{O}$. Let $K$ be an open subgroup of $G(\mathcal{O})$; more ...
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### Conjugates of relative root groups by an element of the Weyl group

Let $G$ be a reductive group (over an algebraically closed field), $T$ a maximal torus, and $\Phi$ the root system of $(G,T)$. Then for each root $\alpha \in \Phi$ there is a unique connected $T$-...
142 views

### Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group?

Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of Tits building of a linear algebraic group. I ...
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### Finite commutative group schemes whose exponent coincides with its rank

In group theory, a finite commutative group $G$ contains an element whose order is the exponent of $G$. Thus, If the exponent of $G$ is the same as the order of $G$, it must be that $G$ is cyclic. ...
1 vote
105 views

### Koszul complex of equations defining a stabilizer

Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0. Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we ...
253 views

### Abstract-group isomorphisms between linear algebraic groups

Let $G$ and $H$ be linear algebraic groups over the rationals. Suppose that we know that $G(\mathbb Q)$ and $H(\mathbb Q)$ are isomorphic as abstract groups. Does it under any circumstances follow ...
1 vote
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### Why do such a birational map exists? And why it is unique?

Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra. Suppose that $H \subset G$ is a 1-dimensional torus such that the action of $H$ ...
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### Is the image of the exponential map of a complex semisimple group Zariski open?

Let $G$ be a semisimple complex algebraic group. Is the image of the exponential map $$\exp : \mathfrak{g} \to G$$ Zariski open in $G$?
1 vote
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### Cohomology with coefficient in sheaf of morphisms of an algebraic group

Let $G$ be an affine algebraic group over ${\mathbb C}$. We denote the sheaf of morphisms from ${\mathbb A}^1$ to $G$ by $\bf G$. Then $H^1({\mathbb A}^1,\bf G)=0$ (Cech cohomlogy). Is this fact true? ...
1 vote
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### Properness of Hom-schemes for finite group schemes

In SGA 3 XI proposition 3.12 (b) (https://webusers.imj-prg.fr/~patrick.polo/SGA3/Expo11.pdf) It is shown that: If $G$ is an affine group scheme over a base $S$ and $H$ is a finite group scheme over $S$...
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### How to check the non-emptyness of the A-packet of non-split $\operatorname{SO}(2n+1)$?

Let $F$ be a number field and $G_n=\operatorname{SO}(2n+1)$ be the split group over $F$. Let $G_n^{\times}$ be a non-split group over $F$. Let $\tau$ be an irreducible cuspidal automorphic ...
1 vote
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### Is the manifold of complex points of a quotient of compact groups just the tangent bundle?

In great generality a Lie group mod its maximal compact subgroup is contractible (for example this is true for all connected Lie groups). Whenever this is true then the Lie group $D$ is ...
238 views

### Universal covering groups of simple linear algebraic group schemes

Let $R$ be a Dedekind domain with fraction field $K$, and let $G$ be a smooth affine group scheme over $S = \text{Spec }R$ whose geometric fibers are connected and simple linear algebraic groups (i.e.,...
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### Flatness of the closure of a closed subgroup of the generic fiber of an algebraic group inside an integral model of the ambient group

$\DeclareMathOperator\GL{GL}$Let $K$ be a number field and $R$ its ring of integers. Let $G$ be a connected reductive closed subgroup of $\GL_{n,K}$. On p55 of Brian Conrad's notes Reductive group ...
Let $G$ be a unipotent complex algebraic group and $K\subseteq G$ a closed subgroup. Let $V$ be a finite-dimensional $K$-module, and consider the induced module $\operatorname{Ind}_{K}^{G}V$, the $G$-...