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.
2,111
questions
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$\text{PGL}_n(\mathbf{Q}_p)$ and the Congruence Subgroup Property
Suppose $\Gamma$ is a torsion-free lattice of $\text{PGL}_n(\mathbf{Q}_p)$ for $n\geq 3$. Then I know that by the Margulis arithmeticity theorem, $\Gamma$ must be arithmetic. My question is does $\...
user72293
5
votes
0
answers
193
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Endomorphism rings of commutative algebraic groups are Noetherian?
Let $G$ be a smooth connected commutative algebraic group over an algebraic closed field $K$. By an endomorphism of $G$, we mean a homomorphism $G \to G$ in the category of algebraic groups. As $G$ is ...
- 51
13
votes
3
answers
2k
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Naïve definition of parahoric subgroup
Background
Let $F$ be a $p$-adic local field, and let $G$ be a connected reductive group over $F$. Recall that there is a rich theory of compact open subgroups of $G(F)$ which is, essentially, ...
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3
votes
1
answer
249
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Reduction of structure group for stacks
Consider an action of a smooth linear algebraic group $G$ on a variety $X$ over an arbitrary field $k$, and the quotient stack $[X/G]$. Let $p$ be a $k$-point of $X$. If the action is transitive (i.e. ...
- 95
8
votes
1
answer
765
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Tate modules of commutative group schemes over finite field
Let $G$ be a finite type commutative group scheme over a finite field $k=\Bbb F_q$ with $\Gamma=Gal(\bar k/k)$ and $l$ be a prime such that $(l,q)=1$, we can also define the Tate module $T_lG =\...
- 6,158
37
votes
0
answers
5k
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Homology of $\mathrm{PGL}_2(F)$
Update: As mentioned below, the answer to the original question is a strong No. However, the case of $\pi_4$ remains, and actually I think that this one would follow from Suslin's conjecture on ...
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12
votes
0
answers
345
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Is the quotient of two linear group schemes linear?
Let $S$ be an affine scheme.
Call a group scheme $G\to S$ linear if there exists an $S$-group morphism $G\to \mathrm{GL}_{n,S}$ with trivial kernel.
Assuming this, suppose $H\to S$ is a central closed ...
- 2,846
5
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0
answers
125
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What is the minimum number of steps for two elements of a Lie algebra to generate the whole Lie group?
Consider a compact, connected and simply connected Lie Group $G$, and two elements in the corresponding Lie algebra $X$ and $Y$. By successive action of exponential map you can get the following ...
- 101
1
vote
0
answers
123
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Rational points of torsors over a separable closure
I already asked this question on Math Stack few days ago ( torsors over a separable closure ), but did not receive any answer, so I post it here.
Let $G$ be a smooth linear algebraic group defined ...
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6
votes
2
answers
500
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Torsors over complete local fields
Let $G$ be a linear algebraic group scheme, and let $R$ be a complete discrete valuation ring, with quotient field $K$ and residue field $k$.
If $T$ is an $R$-torsor, it yields by base change a $k$-...
- 1,661
2
votes
1
answer
336
views
Open subgroups of finite index of p-adic semisimple groups
My set up is the following: I have an affine algebraic group $G$ over a $p$-adic field $F$, we assume that $G$ is semisimple and simply connected. I have an abstract subgroup $H\leq G(F)$ of the group ...
- 415
10
votes
1
answer
363
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Bounds on Tamagawa numbers of reductive groups
Let $G$ be a reductive algebraic group over a number field $k$. Weil's conjecture on Tamagawa numbers (now a theorem) tells us that the Tamagawa number $\tau(G)$ of $G$ is 1 if $G$ is semisimple and ...
- 3,699
4
votes
0
answers
120
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Prehomogeneous vector spaces for reductive groups
Recall that a prehomogeneous vector space, is a representation $V$ of a linear algebraic group $G$ having an open $G$-orbit. Let $Z$ be the neutral connected component of the stabilizer of a point of ...
- 1,550
2
votes
1
answer
200
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When does F4 split over its homogeneous variety X_4?
Let $k$ be a real closed field.
Let $G$ be a anisotropic algebraic group of type $F_4$ over $k$.
Consider the projective, homogenous $F_4$-variety $X_4$ (Bourbaki enumeration).
To avoid confusion: ...
- 1,419
6
votes
0
answers
142
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Group homomorphisms out of the additive group is formal completion at the unipotent cone
Let $G$ be an affine algebraic group over an algebraically closed field of characteristic zero $k$. Then, the functor $\text{Hom}_{grp}(\mathbb{G}_a, G)$ is representable by a colimit of schemes (...
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votes
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answers
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Demazure’s principle for tower of reductive groups and analogy with Teichmuller tower
Grothendieck in his Esquisse states (page 6/7 of the English translation) that the “principle” that the Teichmuller tower, i.e., the system of profinite fundamental groupoids $\hat{T}_{g,\nu}$ of ...
user122285
6
votes
1
answer
229
views
Are indecomposable representations of a finite group of Lie type absolutely indecomposable?
Let $G = G(K)$ be a Chevalley group over an algebraically closed field $K$ of characteristic $p > 0$. Consider the finite group $G(q) = G(\mathbb{F}_q)$. (For example, if $G = \operatorname{SL}_n(K)...
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0
answers
84
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How can I compute minimal distance of the AG-code on the Hirzebruch surface $\mathbb{F}_3$?
Let $\mathbb{F}_3$ be the Hirzebruch surface (with index $3$) over a finite field $\mathbb{F}_q$ and $\pi\!: \mathbb{F}_3 \to \mathbb{P}^1$ be the unique $\mathbb{P}^1$-fibration on $\mathbb{F}_3$. ...
- 1,801
8
votes
0
answers
145
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Semisimple Lie groups admitting a free algebra of invariants
Assume we work over an algebraically closed field of characteristic zero.
I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ...
- 189
2
votes
1
answer
273
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Decomposition into Weyl modules
Let $G$ be a split reductive group over an arbitrary field $k$. By definition, see Jantzen (*), an ascending chain $$0 = V_0 \subset V_1 \subset V_2 \subset \dots$$ of submodules of a $G$-module $V$ ...
3
votes
1
answer
298
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Is there a "big open cell" analogue for parabolic subgroups?
Let $G$ be a reductive group over a $p$-adic field. Let $P$ be a parabolic subgroup of $G$ containing a minimal parabolic $P_0$. Let $S$ be a maximal split torus of $P_0$, and let $\Delta$ be a set ...
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vote
1
answer
255
views
Is the Borel subgroup the only closed double coset?
Let $G$ be a quasisplit connected reductive group over a $p$-adic field $k$. Identify $G$ with its rational points. Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$, both defined ...
- 6,110
3
votes
1
answer
478
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Orbits of unipotent groups over local fields are closed?
Let $H$ be a connected, unipotent linear algebraic group defined over a local field $k$. Let $H \times_k X \rightarrow X$ be an action of $H$ on an irreducible, affine $k$-variety $X$ which is ...
- 6,110
9
votes
1
answer
783
views
Interesting examples of pro-algebraic completions of groups
Fix a field $k$. Given a discrete group $G$, the pro-algebraic completion (or Hochschild-Mostow completion) is the pro-algebraic group $A_{k}(G)$ with is universal with respect to finite dimensional ...
- 1,635
5
votes
1
answer
333
views
Forms of automorphism groups of algebraic varieties
Let $k$ be an arbitrary field and $X$ be a proper scheme over $k$. Mastumura and Oort proved that the functor $S \longmapsto Aut_{S-sch}(X \times S)$ is representable by a group scheme, locally of ...
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0
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130
views
What do I know about a group if its representations are filtered?
Let $G$ be an affine group scheme over a field.
Say that, for every finite-dimensional representation of $G$, I have a $\mathbb{Z}$-grading on the underlying vector space, compatible with tensor ...
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1
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0
answers
77
views
Rationality of the representation of $(SL_2)^r$ arising from a $\mathbb{Q}$-VHS
In the multivariable $SL_2$-orbit theorem (which describes the asymptotic behavior of the period map associated to a polarized $\mathbb{Q}$-VHS, can we choose a suitable base point such that the ...
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votes
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answers
754
views
When is GIT quotient of a projective variety smooth?
It might be very basic, but my google search could not find an answer for this, nor the search in previous related questions.
Let $X$ be a smooth projective variety, and let $G$ be a reductive group ...
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4
votes
1
answer
326
views
For tori $S \subseteq T$, every character of $S(k)$ extends to a character of $T(k)$?
Let $k$ be a $p$-adic field, $T$ a torus over $k$, and $S$ an $k$-subtorus of $T$. If $\chi: S(k) \rightarrow \mathbb{C}^{\ast}$ is a smooth (resp. continuous) homomorphism, then does $\chi$ ...
- 6,110
8
votes
1
answer
780
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Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
- 6,110
1
vote
1
answer
138
views
does $Aut^0$ act trivially on the Neron-Severi group?
Let $X$ be a projective integral scheme over an algebraically closed field $k$. Does $\mathrm{Aut}^0_{X/k}(k)$ act trivially on $NS(X)$?
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votes
0
answers
149
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Connected unipotent groups acting on an affine variety (re: stabilizers)
Let $U$ be a connected unipotent algebraic group over a field of characteristic $p>0$. Assume $U$ acts on an affine variety $X$ by regular maps.
Is it true that the stabilizers of rational points ...
- 993
0
votes
2
answers
283
views
quasi-affine-ness [closed]
Let $G$ be a group. Let $H$ be a subgroup of $R_u(G)$. Then $G/H\rightarrow G/R_u(G)$ is a $R_u(G)/H$ fibration. It is well known that
$R_u(G)/H=\mathbb{A}^n$. Is $G/H$ a quasi-affine variety?
user111251
9
votes
2
answers
1k
views
Hecke algebra of GL(2,F)
I was studying about the Hecke algebra from Bernstein's notes on p-adic representation theory and various other sources. First a disclaimer: everything below is fairly new to me so please feel free to ...
4
votes
1
answer
155
views
Group algebraic spaces that are locally of finite type and have only finitely many points
Let $k$ be an algebraically closed field of characteristic zero. Let $G$ be a group algebraic space over $k$ such that $G\to $ Spec $k$ is locally of finite type.
Suppose that $G(k)$ is finite.
...
- 181
3
votes
0
answers
158
views
Levi decomposition and minimal parabolics
Let G be reductive group over a field $k$. Let $P_0$ be a minimal k-parabolic subgroup of G and fix a Levi decomposition $P_0=M_0 N_0$. If P is a parabolic containing $P_0$, then why is there only ...
- 91
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answers
473
views
Sheaf-theoretic Grothendieck groups
Let $S$ be a scheme, $M\to S$ a commutative monoid object in algebraic $S$-spaces, ie. an algebraic $S$-space such that, functorially on $S$-schemes $T$, $M(T)$ is a commutative monoid with neutral ...
user113393
3
votes
1
answer
161
views
Connected subgroup of $\mathbb G^n_m$ of Zariski dimension 1
The following Question has a yes answer when $K$ is algebraically closed. I am looking for an elementary proof of it, as well as an answer for arbitrary $K$ of characteristic $0$.
Question. ...
- 1,555
9
votes
3
answers
687
views
Reduction mod $n$ of symplectic group
Let $g,n$ be positive integers, is there a reference that $\mathrm{Sp}(2g,\mathbb{Z})\to\mathrm{Sp}(2g,\mathbb{Z}/n\mathbb{Z})$ is surjection?
The only reference I could find is lemma 5.16 in Deligne–...
user39380
5
votes
1
answer
509
views
Uniqueness of the wonderful compactification of a semi-simple group
Let $G$ be a semi-simple group over an algebraically closed field of characteristic zero. In which cases there is a unique wonderful compactification of $G$ (modulo isomorphism)?
For instance, is the ...
user117617
1
vote
0
answers
159
views
Isogenies of type A_n, basis of cocharacter lattice
The isogenies of type $A_n$ are indexed by the subgroups of $\mathbb{Z} / (n+1) \mathbb{Z}$, i.e. by the positive divisors of $n+1$. If $a$ is a positive integer and $a \mid (n+1)$, then the ...
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4
votes
1
answer
382
views
Is the subgroup generated by a conjugacy class of semisimple elements Zariski closed?
Let $k$ be an arbitrary field with $\operatorname{char}(k) \neq 2$. Let $G$ be a linear algebraic group over $k$. Let $X$ be the conjugacy class of a semisimple element $s \in G(k)$ of order 2 (or a ...
5
votes
0
answers
127
views
subgroups of $\mathrm{Sp}_{2g}(\mathbb{Z}_2)$ whose mod-2 image is the symmetric group
Let $G \subseteq \mathrm{Sp}_{2g}(\mathbb{Z}_2)$ be a closed subgroup of the symplectic group over the $2$-adic integers whose image under the mod-$2$ homomorphism $\pi : \mathrm{Sp}_{2g}(\mathbb{Z}_2)...
- 1,308
6
votes
2
answers
337
views
The Tits classes of simply connected simple real groups
Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$).
Then $G$ is an inner form of a quasi-split $k$-group $G_{\rm qs}$:
...
- 13.6k
14
votes
2
answers
827
views
Split rank of inner forms
Let $G$ be a (connected) reductive group over some ground field $F$ and $G^*$ its unique quasi-split inner form. Denote by $\operatorname{rank}_F G$ the split rank of $G$, i.e. the dimension of a ...
- 1,744
1
vote
0
answers
220
views
Geometric quotient becomes a quotient manifold when passing to rational points
Let $k$ be a local field of characteristc zero. I'm interesting in understanding how morphisms of schemes of finite type over $k$ become morphisms of analytic manifolds on passing to rational points. ...
- 6,110
7
votes
1
answer
949
views
Rosenlicht's theorem and rationality questions
Let $G$ be a connected algebraic group over an algebraically closed field $\overline{k}$ acting on an irreducible variety $X$. A geometric quotient is a morphism of varieties $\pi: X \rightarrow X/\...
- 6,110
7
votes
2
answers
423
views
"skyscraper group scheme"
Is there a skyscraper group scheme?
Let $S$ be a DVR. Is there a group scheme $\mathcal{G}$ over $S$ which is generically {1} trivial i.e, identity group, but at the closed point some nontrivial ...
user111251
6
votes
1
answer
542
views
Commutative group algebraic spaces
Is the category of commutative group algebraic spaces (commutative group objects in algebraic spaces) locally of finite type over a field, an abelian category?
I would benefit from a reference
user119470
3
votes
0
answers
82
views
Particular decomposition of $SU(n)$
Given $a,b \in \mathfrak{su(n)}$ which generate the full algebra, it is possible to write and $G \in SU(n)$ as:
$G = \exp(\alpha_1 a)\exp(\beta_1 b) \ldots \exp(\alpha_m a)\exp(\beta_m b)$
for some ...
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