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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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 ...
Sebastian Eterovic's user avatar
2 votes
1 answer
426 views

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 $G$ ...
Not a grad student's user avatar
10 votes
1 answer
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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 ...
Joël Ouaknine's user avatar
4 votes
1 answer
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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 $X$ ...
Anonymous's user avatar
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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 $a^2=g$.)
Pete's user avatar
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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 ...
Arkandias's user avatar
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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{...
Golden Wave 's user avatar
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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}/...
Luis Jorge's user avatar
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 ...
Thomas's user avatar
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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 ...
aytio's user avatar
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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\...
user113147's user avatar
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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 ...
bob's user avatar
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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 ...
Mikhail Borovoi's user avatar
5 votes
2 answers
677 views

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 ...
Monty's user avatar
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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 ...
Ehud Meir's user avatar
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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 ...
Mikhail Borovoi's user avatar
12 votes
2 answers
572 views

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 ...
Jingren Chi's user avatar
5 votes
2 answers
507 views

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 ...
sabrebooth's user avatar
3 votes
0 answers
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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 ...
Anonymous's user avatar
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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-...
Evan Wilson's user avatar
2 votes
1 answer
150 views

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 ...
Golden Wave 's user avatar
3 votes
1 answer
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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. ...
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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 ...
Z.A.Z.Z's user avatar
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3 votes
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217 views

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(...
Z.A.Z.Z's user avatar
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13 votes
1 answer
641 views

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 ...
Cristian D. Gonzalez-Aviles's user avatar
4 votes
1 answer
176 views

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 ...
Ariel Weiss's user avatar
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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 ...
C.Niculescu's user avatar
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2 answers
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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 ...
Cristian D. Gonzalez-Aviles's user avatar
2 votes
0 answers
854 views

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 $Z(G)$, ...
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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 $\...
Hang's user avatar
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7 votes
0 answers
408 views

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 $M(\rho, x)$...
Feng Hao's user avatar
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7 votes
0 answers
313 views

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 $k$-...
Mikhail Borovoi's user avatar
5 votes
2 answers
392 views

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 $\Delta$ ...
D_S's user avatar
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3 votes
0 answers
138 views

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\...
Tatyana's user avatar
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0 answers
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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 $\...
Alexander's user avatar
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4 votes
1 answer
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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 ...
Mikhail Borovoi's user avatar
2 votes
0 answers
82 views

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 $\Psi : K^{2n}\times K^{2n}\rightarrow K$ ...
Z.A.Z.Z's user avatar
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6 votes
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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 ...
Not a grad student's user avatar
4 votes
0 answers
299 views

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 ...
Desiderius Severus's user avatar
2 votes
0 answers
803 views

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 ...
Not a grad student's user avatar
6 votes
0 answers
268 views

$G$ 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 ...
D_S's user avatar
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1 vote
0 answers
222 views

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 $G$-...
A Rock and a Hard Place's user avatar
1 vote
0 answers
279 views

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 \...
Golden Wave 's user avatar
5 votes
0 answers
162 views

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 ...
D_S's user avatar
  • 6,100
9 votes
1 answer
536 views

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 ...
LSpice's user avatar
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2 votes
1 answer
209 views

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 ...
user148212's user avatar
  • 1,534
8 votes
3 answers
598 views

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 ...
Tippy Tipper's user avatar
6 votes
0 answers
188 views

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 ...
Not a grad student's user avatar
10 votes
1 answer
277 views

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 ...
Mark's user avatar
  • 185
10 votes
2 answers
2k views

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{...
D_S's user avatar
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