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.
2,105
questions
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Relation between a conjecture of Pink and semi-abelian varieties
A conjecture of Pink says that in a mixed Shimura variety, every Hodge-generic point is Galois generic (conjecture 6.8 of "A Combination of the Conjectures of Mordell-Lang and Andr\'e-Oort). One can ...
2
votes
1
answer
426
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Automorphism of the Dynkin diagram
I have a question about a passage from Springer's Linear Algebraic Groups (Birkhauser). The setup is: $F$ is an arbitrary field, $G$ is an $F$-split group, and $B$ is a Borel subgroup of $G$ ...
10
votes
1
answer
342
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Is Zariski closure of finitely generated matrix semigroup computable?
In general, can the Zariski closure of the semigroup of matrices $\langle M_1, \ldots, M_k \rangle$ be algorithmically computed (at least in theory)?
For this purpose I'm happy to assume the ...
4
votes
1
answer
149
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Is the complement of the open $B$-orbit in a spherical variety cut out by one equation?
Let $X$ be an affine spherical variety for some reductive algebraic group $G$. Let $X^0$ be the open orbit in $X$ under a fixed Borel subgroup $B \subseteq G$. Does there exists a function $f$ on $X$ ...
10
votes
2
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696
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Square root in complex reductive groups
Let $G$ be a connected complex reductive linear algebraic group. Does every $g\in G$ have a square root? (That is, some $a\in G$ such that $a^2=g$.)
2
votes
0
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422
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Center of a split unipotent group
Let $N$ be a unipotent algebraic group over a field $k$ of characteristic $p>0$.
Assume that $N$ is split (i.e. it admits a filtration whose quotients are isomorphic to the additive group).
In ...
4
votes
1
answer
345
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A generalization of Siegel property
In reduction theory of arithmetic groups, one has the following finiteness property.
Proposition 1 (Siegel property). Let $G$ be a reductive group over $\mathbb{Q}$ and let $\Gamma\subset G(\mathbb{...
4
votes
1
answer
284
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Automorphisms of products of $GL_n(\mathbb{Z})$ 's
It is a Theorem of Hua y Reiner (1951) that the group or outer automorphisms $Out(GL_n(\mathbb{Z}))$ is either isomorphic to $\mathbb{Z}/2$, if $n$ odd or $n=2$, or to $\mathbb{Z}/2 \times \mathbb{Z}/...
10
votes
1
answer
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Number of connected components of the intersection of two maximal tori
Let $G$ be a connected complex semisimple Lie group and $S$, $T$ two maximal tori in $G$. Is there a known upper bound on the number of connected components of $S\cap T$? For example, is it bounded by ...
4
votes
1
answer
308
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Understanding full set of sections as in Katz-Mazur
I was reading this question, specifically Brian's answer. In particular I am having a bit of trouble digesting the following sentence:
Being a "full set of sections" of $Z/S$ is something which is ...
5
votes
1
answer
231
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Is the set of points with smallest stabilizer open?
Let $G$ be a complex reductive group acting linearly on a complex affine variety $X$, and let $K$ be the kernel of the action, i.e.
$$K:=\{g\in G:g\cdot x=x\text{ for all }x\in X\}.$$
Is
$$X_K:=\{x\...
4
votes
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answers
200
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Schur functors for representations of the borel
Let $k$ be a field of characteristic $0$. Let's denote by $V_{std} = k^n$ the standard representation of $GL_n$. We know that every irreducible representation of $GL_n$ can be obtained by applying a ...
3
votes
0
answers
204
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A criterion for a $G$-variety to be isomorphic to $G/H$
Let $k$ be an algebraically closed field of characteristic 0.
Let $G$ be a connected linear algebraic group over $k$.
Let $H\subset G$ be an algebraic $k$-subgroup.
Let $Y$ be an algebraic variety ...
5
votes
2
answers
677
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What condition makes unitary reductive group unramified?
I am a little bit confused with the definition of an unramified unitary group.
Let $F$ be a local field of characteristic zero whose residue field is finite field of characteristic $p$.
Then for a ...
3
votes
1
answer
377
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variations of finite stabilizer in the action of an algebraic group on an affine variety
Assume that $G$ is an affine reductive algebraic group (I am mostly interested in the case $GL_n$) over an algebraically closed field $K$ of characteristic zero. Assume also that $G$ acts on an affine ...
3
votes
1
answer
168
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Is any spherical subgroup conjugate to a subgroup defined over a smaller algebraically closed field?
Let $G_0$ be a connected semisimple algebraic group defined over an algebraically closed field $k_0$. Let $k\supset k_0$ be a larger algebraically closed field.
We write $G=G_0\times_{k_0} k$ for the ...
12
votes
2
answers
572
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Bounding weight multiplicities by number of certain Coxeter elements
This question concerns lower bounds of certain weight multiplicities in finite dimensional representations of algebraic groups (or Lie groups, Lie algebras).
Let's say $G$ is a simple algebraic group ...
5
votes
2
answers
507
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Connected algebraic subgroup of $PGL_3$ and $PGL_2 \times PGL_2$
I am looking for a reference regarding the maximal proper connected algebraic subgroups of $PGL_3$ and $PGL_2 \times PGL_2$ respectively when the base field is any algebraically closed field (of ...
3
votes
0
answers
118
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Finite generation of the module of invariant vector fields
Let $G$ be a linear algebraic group (not necessarily reductive) and let
$X$ be an affine variety with a regular $G$-action (everything defined over the field of complex numbers $\mathbb{C}$). Denote ...
6
votes
1
answer
394
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Leray-Serre spectral sequence for algebraic groups
Let $G$ be a semisimple, simply-connected, complex algebraic group. Fix a Borel subgroup $B$ and let $P$ be a parabolic subgroup properly containing $B$. If $M$ is a $B$-module, then we have the Leray-...
2
votes
1
answer
150
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About the conjugation of reductive subgroups
I am considering the following question.
Question 1. Let $G$ be a reductive algebraic group over $\mathbb{R}$, can we find finitely many reductive $\mathbb{R}$-subgroups $H_1,...,H_m$ such that for ...
3
votes
1
answer
199
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Borel subgroups of centralisers of Lie algebra elements in bad characteristic
Let $G$ be a simple linear algebraic group over an algebraically
closed field $k$ of characteristic $p>0$, and let $\mathfrak{g}=\mathrm{Lie}(G)(k)$
denote (the $k$-points of) the Lie algebra.
...
3
votes
0
answers
132
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Simply connected quasi-split groups are generated by unipotent subgroups
I am particularly interested in the special unitary groups defined over the field $\mathbb C((X))$ with the conjugation $X\rightarrow -X$.
Are these groups generated by some unipotent subgroups?
Does ...
3
votes
0
answers
217
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Is this scheme reduced?
Let $O=\mathbb C[X]$ and denote by $SL_r(O)_1\subset SL_r(O)$ the set of matrices of this form $$I_r+XM(X), \text{ for some matrix } M(X)$$
Consider the involution $\sigma$ on $SL_r(O)_1$ given by $M(...
13
votes
1
answer
641
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The Picard group of a semisimple algebraic group in positive characteristic
Let $k$ be a field of positive characteristic and let $G$ be a connected semisimple algebraic group over $k$ with fundamental group $\mu$. Note that $\mu$ can be non-smooth. It is stated in Sansuc's ...
4
votes
1
answer
176
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Distribution of regular elements in a disconnected algebraic group
Let $k$ be an arbitrary field (in my case $k = \mathbb Q_p$), and $G\subset \mathrm{GL}(n)_{/k}$ a reductive group. Let $G^0$ be its identity connected component.
Suppose that $G^0(k)$ contains an ...
3
votes
0
answers
114
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Algebraic (g,B) modules
Let $\mathfrak{g}$ be a semisimple lie algebra over a field of characteristic $0$, $G$ its adjoint group, with Borel group $B$. I am trying to understand the theory of algebraic $(g,B)$ modules as ...
6
votes
2
answers
395
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A connected reductive algebraic group over a separably closed field is a rational variety
I need either a proof or a reference in modern (scheme-theoretic) language. According to Sansuc, this result can be gleaned from Borel's book on linear algebraic groups, but the old-style algebraic ...
2
votes
0
answers
854
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Description of the center of a reductive group using absolute and relative roots
Let $G$ be a connected, reductive group over a field $k$. Let $T \subseteq B$ be a maximal torus and Borel subgroup of $G$ with corresponding base $\Delta \subseteq X(T)$. Then $T$ contains $Z(G)$, ...
1
vote
0
answers
146
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Generators of the same degree in a graded ring and GIT quotient
Let $S$ be a graded ring i.e. $S_d\cdot S_e \subset S_{d+e}$ where $S_d$ is the degree $d$ part of $S$. Assume $S_{<0}$ vanishes. For simplicity, you may think of $S$ as a graded subring of $\...
7
votes
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408
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Monodromy group from semisimple local system is reductive
Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $\rho: \pi_1(X,x)\rightarrow Gl(n,\mathbb{C})$ be a semisimple representation of fundamental group of $X$. The monodromy group $M(\rho, x)$...
7
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313
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A basic question on a base change of a homogeneous space of a linear algebraic group
I asked this basic question in MSE and got a comment "This belongs to Mathoverflow", so I ask my question here.
Let $G$ be a linear algebraic group over a field $k$, and $H\subset G$ be a $k$-...
5
votes
2
answers
392
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Reference Request: Derived group of $\mathscr R_u(B)$
Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ ...
3
votes
0
answers
138
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Cartan decomposition for $G[z]$
Let $G$ be a reductive group over complex numbers. Fix some maximal torus $T$. Let $\Lambda^{+}$ be the monoid of dominant coweights. It is known that one has a Cartan decomposition $$G((z))=\coprod\...
2
votes
0
answers
169
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Absolute and Relative Coroots
$G$ is a connected reductive group over a field $k$. $T$ is a maximal torus and $S \subset T$ is a maximal $k$-split torus. We have an embedding $X_*(S) \hookrightarrow X_*(T)$. Is it true that if $\...
4
votes
1
answer
215
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Action of $N(H)/H$ on the colors of a spherical homogeneous space $G/H$
Let $G$ be a semisimple group over $\mathbb C$, and $X=G/H$ be a spherical homogeneous space of $G$.
Let $T\subset B\subset G$ be a maximal torus and a Borel subgroup.
Let $S=S(G,T,B)$ denote the ...
2
votes
0
answers
82
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What is the classification of this group?
Let $K=\mathbb C((t))$ and $O=\mathbb C[[t]]$, and $n\geq 1$. Consider the matrix $$J_{2n}=\begin{pmatrix} 0& I_n \\ -I_n & 0\end{pmatrix},$$ And let $\Psi : K^{2n}\times K^{2n}\rightarrow K$ ...
6
votes
0
answers
282
views
Is there a list of the inner forms of the quasisplit groups over local and global fields of characteristic 0?
From what I've gathered from trying to learn the classification of reductive groups, the classification of semisimple groups over a local or global field $F$ of characteristic 0 proceeds in several ...
4
votes
0
answers
299
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Using the Bruhat-Tits tree for unitary groups
For now I always worked in the setting of the Bruhat-Tits tree in the $SL(2)$ setting (like in the book of Serre), without any further background about Bruhat-Tits buildings. I would like to adapt ...
2
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0
answers
803
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Why is the radical of a reductive group equal to the connected component of the center?
If $G$ is a connected reductive group over a perfect field $k$ (The definition given in Milne's "Algebraic Groups": $G$ is a connected group variety containing no non-trivial connected unipotent ...
6
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268
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$G$ is quasisplit at almost all places
Let $G$ be a connected, reductive group over a global field $k$. I am trying to understand why $G_v = G \times_k k_v$ is quasisplit for almost all places $v$ of $k$. There are several equivalent ...
1
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0
answers
222
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Open/closed immersion and quotient stacks
I'm quite new to stacks, so this might be very easy. In particular, if there is a canonical reference I can consult for these things, please feel free to point it out.
Let $f:X\to Y$ be a $G$-...
1
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0
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279
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A question about Mostow's theorem for self-adjoint groups
In Mostow's paper Self-adjoint groups, one can find the following property.
Theorem. Let $G\subset \mathrm{GL}_{n,\mathbb{R}}$ be a reductive real algebraic subgroup. Then there exists $a\in \...
5
votes
0
answers
162
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Definition of the homomorphism $\textrm{Gal}(k_s/k) \rightarrow \textrm{Aut}(\psi_0(G))$
Let $G$ be a connected, reductive group over a field $k$. Identify $G$ with its $\overline{k}$-points. Let $\Gamma = \textrm{Gal}(k_s/k) = \textrm{Aut}(\overline{k}/k)$. Let $B$ be a Borel subgroup ...
9
votes
1
answer
536
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Endoscopic group that is not a subgroup
The question is a very little more than what's in the title. It is easy (for some values of ‘easy’) to produce examples of endoscopic groups that are not subgroups. When I asked a colleague, he ...
2
votes
1
answer
209
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On cuspidal maximal tori of a connected reductive group
Let $G$ be the base change of a connected reductive group over ${\mathbb{F}}_q$ to $\overline{\mathbb{F}}_q$, and let $F$ be the associated geometric Frobenius on $G$. Let $W$ be the Weyl group of a ...
8
votes
3
answers
598
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Centralizers of subtori in reductive groups, derived subgroups
Let $G$ be a split, almost-simple connected reductive group over a field $F$ with split maximal torus $T$. I am trying to understand precisely the groups $[G_{\alpha}, G_{\alpha}]$, where $\alpha$ is ...
6
votes
0
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188
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How can I get access to Hijikata's mimeographed notes?
In several papers on buildings, Hijikata's mimeographed notes from Yale ("Maximal compact subgroups of some p-adic classical groups") are often cited as a precursor to the work of Bruhat-Tits. While ...
10
votes
1
answer
277
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Lagrangian Grassmannian from an Involution
I don't know if this is already answered somewhere in MO. The dynkin involution of $SL_{2n}$ that is $\alpha_i \mapsto \alpha_{2n-i}$ gives an outer automorphism of $SL_{2n}$ and then the maximal ...
10
votes
2
answers
2k
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A reductive group has a quasi-split inner form
Let $G$ be a connected, reductive group over a field $k$. Let $\Gamma = \textrm{Gal}(k_s/k)$. I think my question is better suited using the classical language: think of $G$ as an affine $\overline{...