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# 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.

1,400 questions
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### Definition of automorphic form on non-reductive group $N \rtimes G$

Let me ask the definition of automorphic form. For a connective reductive group $G$, it is well defined the automorphic form on $G$. But sometimes it does often appears the concept of automorphic ...
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### Real points of reductive groups and connected components

Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...
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### Relative position on flag variety

Let $G$ be a semisimple algebraic group over $\mathbb{C}$. Consider the $G$ diagonal action on $G/B \times G/B$, the orbit is indexed by $W$, the Weyl group of $G$ by Bruhat decomposition. There is a ...
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### Orbits of action of the split group of type $F_4$

Let the split group of type $F_4$ act as the automorphism group of the split Albert algebra $A$. Consider the action of $F_4\times \mathbb{G}_m$ on $A$, given by letting $\mathbb{G}_m$ act by scalar ...
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### Variety of conjugacy classes

Consider a reductive group $G$ over an algebraically closed field $K$ of characteristic $0$. I would like to consider the space $X$ of all $G$-conjugacy classes in $G$. Does the space $X$ have some ...
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### Element in finite number of Borel subgroups

Let G is a linear algebraic group over algebraic closed field, B is an Borel subgroup of G. Does there exist g$\in$G which is only in a finite numbers of conjugates of B (they are also Borel ...
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### Levi subgroup of Siegel parabolic of GSpin

I consider the group $G=\mathrm{GSpin(V)}$ as in this question. We have the so called Siegel parabolic $P$ (after fixing a cocharacter) and the associated Levi $M$ (these can also be obtained using ...
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### Criterion for vector bundle to descend to GIT quotient in positive characteristic

Let $k$ be an algebraically closed field of positive characteristic. Let $G$ be an affine reductive group acting on a smooth projective variety $X$. Let $E$ be a vector bundle on $X$ such that the ...
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### Counting points on connected reductive monoids over a finite field

Let $M$ be a connected reductive monoid over a finite field $\mathbb F_q$, is there an explicit formula of $\#M(\mathbb F_q)$? There is something similar to Bruhat decomposition with the Weyl group ...
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### Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?

Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
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### Component Groups of Reductive Groups

Suppose $G$ is a reductive group that is not necessarily connected and $Z \subset G$ is a central subgroup. Suppose $G^0$ is the identity component of $G$. Is it true that $G/G^0Z= \pi_0(G/Z)$? I can ...
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### Reference Request: Structure constants for G2

Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
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Let $S$ be affine. A bit of background: Let us think of $S$-group schemes as abelian sheaves over a given site (etale, Zariski, fppf, etc). When we take a cokernel of a morphism $\phi$ this category: $... 0answers 83 views ### Example of a spherical homogeneous space$G/H$with a pairs of colors and with the center of$G$not contained in$H$? Let$G$be a simply connected simple algebraic group over$\mathbb C$,$B\subset G$a Borel subgroup, and$T\subset B$a maximal torus. Let$\mathcal{S}=\mathcal{S}(G,T,B)$denote the set of simple ... 1answer 83 views ### Splitting of regular semisimple conjugacy classes in$SL_{n}(q)$I have the following question: Consider the following two finite groups:$GL_{n}(q)$and$SL_{n}(q)$. What I am trying to understand is the regular semisimple conjugacy classes of$SL_{n}(q)\$. Now, ...

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