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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42 views

Automorphisms of unipotent groups

I start with a hopelessly broad question: what is known about the structure of the automorphism group of a (smooth, connected) unipotent group (over a field), and particularly about the structure of ...
2
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1answer
122 views

Affine open subsets for algebraic group actions

Let $G$ be a reductive algebraic group over an algebraically closed field $K$ of characteristic zero. Assume that $G$ acts on an affine variety $X$. Assume that $X$ contains an open orbit $U$ (so $\...
6
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1answer
358 views

Wonderful compactification

Suppose $G$ is a semi-simple group of adjoint type over an algebraic closed field, and $X$ its wonderful compactification a la De Concini and Procesi. Let $P=MU$ be a parabolic subgroup in $G$, and ...
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2answers
155 views

Maps between symmetric powers of the natural module for $SL_2 (k)$ in prime characteristic

Let $G=SL_2(k)$ considered as a linear algebraic group over an algebraically closed field of prime characteristic. Let $E$ be the natural module for $G$ and denote by $S^r (E)$ its $r-$th symmetric ...
18
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0answers
286 views

Follow-up to Steinberg's problem (12) in his 1966 ICM talk?

Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with ...
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94 views

Splitting the Tits algebras of a anisotropic group

Assume we are given an anisotropic algebraic group $G$ over a field $k$, having non trivial Tits algebras (i am interested in the $E_7$ adjoint cases). Question: Is it possible that there exists a ...
5
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1answer
245 views

Subgroup schemes of $\mathbb{A}^n$

Let $R$ be an integral $\bar{k}$-algebra of finite type. Let $V(I) \subseteq \mathbb{A}_R^n$ be a reduced (closed) subgroup scheme such that $V(I)\backslash \{0\} \neq \emptyset$ and the $\mathbb{G}_m^...
4
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1answer
97 views

A quotient group of a self-normalizing spherical subgroup

Let $G$ be simply connected, simple algebraic group over $\mathbb{C}$. Let $H\subset G$ be a self-normalizing spherical subgroup of $G$, not necessarily connected or reductive. Here "self-normalizing" ...
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66 views

On orders of stabilisers of group actions and stacks

Let $X$ be a finite type irreducible separated DM-stack over $\mathbb C$. Let $x$ be an object of $X(\mathbb C)$ with stabilizer $G_x$. Let $y$ be an object of $X(\mathbb C)$ with stabilizer $G_y$. ...
8
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0answers
128 views

Smooth quotients of algebraic spaces that are varieties away from codimension $\ge 2$ subset

This is a question about when a smooth complex algebraic space that is very close to being an algebraic variety is actually an algebraic variety. General question: Let $X$ be a smooth separated ...
0
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1answer
64 views

Is the toral component of a connected Lie group equal to the toral component of its radical? [closed]

Given a connected Lie group, define its toral component as the maximal connected and compact subgroup of its center. Is the toral component of a connected Lie group equal to the toral component of ...
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160 views

The center of a group is equal to the center of its radical?

Given a linear algebraic group $G$, is the connected component of the identity of the center of $G$ equal to the connected component of the identity of the center of its solvable radical? If not, is ...
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0answers
86 views

Bounding the number of “generalized $\mathbb{F}_q$-rational points” of a variety in terms of dimension and degree

In what follows, for a prime power $q=p^m$, $\phi_q$ denotes the Frobenius endomorphism $x\mapsto x^q$ of a finite-dimensional affine space over the algebraic closure $\overline{\mathbb{F}_p}$ (the ...
8
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1answer
201 views

Is the image of an $S$-arithmetic subgroup under a surjective $k$-morphism $S$-arithmetic?

Let $k$ be a global field and let $S$ be a non-empty set of places containing all archimedean ones. Suppose $f:G\to H$ is a surjective $k$-morphism of $k$-groups and let $\Gamma\leq G(k)$ be an $S$-...
5
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4answers
260 views

Examples of discrete subgroups of $PSL_2(\mathbf{R})$ with finite covolume and which are not co-compact

Is there a natural example of a discrete subgroup $\Gamma\leq PSL_2(\mathbf{R})$ such that (1) $\Gamma$ has finite covolume (2) $\mathfrak{h}/\Gamma$ is not compact ($\mathfrak{h}$ being the upper ...
4
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0answers
152 views

'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$. Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
5
votes
1answer
645 views

Wrong Tits-Index of E7 from Springer 's book

In the his book Linear algebraic groups, by T.A. Springer, there is a list of possible Tits-Indexes. For the $E_7$ case, there is an index shown, such that vertex $1$ and $7$ are circled (Bourbaki ...
6
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1answer
294 views

Nonabelian $H^2$ and Galois descent

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety ...
2
votes
1answer
107 views

If a Weyl element preserves a root, then it has a representative which preserves the root space?

Let $G$ be a reductive group defined over a field $F$. Let $\Sigma$ be the set of roots of $G$ with respect to a Borel subgroup $B=TU$ with torus $T$. Let $W=N_G(T)/T$ be the Weyl group of $G$. For $\...
6
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1answer
153 views

Does every cocompact lattice admit a homomorphism (with infinite image) into a compact Lie group?

Let $\Gamma$ be a cocompact arithmetic lattice in a semisimple algebraic group. Does it admit a homomorphism $\Gamma \to K$ with infinite image into a compact real Lie group $K$?
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2answers
615 views

Is the wonderful compactification of a spherical homogeneous variety always projective?

Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}_G(H)=H$. Then by results of Brion and Pauer ...
5
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1answer
224 views

Constructing groups of Type E7 with certain Tits Index

In a new survey on $E_8$, namely Skip Garibaldi - E8 the most exceptional group , the author gives an example (Example 8.4., page 15) on how to construct a group of type E8 with a prescribed Tits-...
0
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1answer
71 views

representing base changes of the unit section

Let $S$ be a scheme and $G$ be a sheaf in groups on the big étale site over $S$. Let $e:S\rightarrow G$ be the unit section. Is it true that given an algebraic space in groups $H$, étale over $S$, and ...
5
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1answer
121 views

Centreless semisimple Lie group that is not real algebraic

Let $G$ be a connected semisimple Lie group with trivial centre and $\mathfrak{g}$ its Lie algebra. The adjoint representation of $G$ defines an isomorphism of $G$ onto the connected component of the ...
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0answers
50 views

What's the symplectic form preserved by a rational representation of a semisimple group

Let $\mathbb{H}_1$, $\mathbb{H}_2$ be two quaternion algebras over $\mathbb{Q}$ and $G_1 = SL_1(\mathbb{H}_1)$, $G_2 =SL_1(\mathbb{H}_2)$. Over $\mathbb{C}$, $G_1\sim G_2 \sim SL_2(\mathbb{C})$. I'm ...
4
votes
0answers
126 views

Centralizer action on components of Springer fibers

Let $G$ be a complex adjoint group. Let $u\in G$ be unipotent. The group $A(u):=\pi_0(Z_G(u))$ acts on the set of components of the Springer fiber $\mathcal{B}_u$, the variety of Borel subgroups that ...
12
votes
2answers
380 views

Affine GIT is an open map?

Let $k$ be a field, $X= \text{Spec}\,A$ be an affine scheme, with $A$ a finitely generated $k$-algebra. $G=\text{Spec}\,R$ is a linearly reductive group acting rationally on A, i.e. every element of $...
3
votes
1answer
216 views

Irreducible representations containing simple actions of $\mathrm{SL}(2,\mathbb{C})$

Let $G$ be a complex semisimple Lie group and let $\rho: G \longrightarrow \mathrm{SL}(n,\mathbb{C})$ be a faithful irreducible representation of $G$ with $n \geq 3$. Suppose that $G$ contains a copy ...
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2answers
166 views

How to prove that $A$ is supersingular iff the Picard number $\rho(A)$ is equal to the second $l$-adic Betti number $b_2(A) = 6$?

Let $A$ be an abelian surface over algebraically closed field $k$ of characteristic $p > 2$. How to prove that $A$ is supersingular (in other words, there is an isogeny between $A$ and $E^2$, where ...
2
votes
1answer
192 views

Chow ring of an algebraic group for another equivalence relation than rational

For $G$ a split algebraic group of arbitrary Dynkin typ, the Chow ring with rational equivalence and $\mathbb{Z}/p\mathbb{Z}$, for $p$ some torsion prime of $G$, is well known and will be denoted as ...
2
votes
0answers
63 views

Can we write an element in a super Grassmannian as a pair of matrices?

Super Grassmannians are introduced by Manin, see for example. Elements in a grassmannian can be written as matrices, see for example. Can we write an element in a super Grassmannian as a pair of ...
14
votes
4answers
700 views

Groups of matrices in which all elements have all eigenvalues equal in modulus

I am writing a research article in which I need to use the following fact: if $G$ is a subgroup of $GL_3(\mathbb{R})$ which is irreducible in the sense that no proper nontrivial subspace of $\mathbb{R}...
7
votes
1answer
343 views

Quasi-split tori and algebraic groups

Let $k$ be a perfect field. Recall that an algebraic torus $T$ over $k$ is called quasi-split if there exists some finite étale $k$-algebra $A$ such that $$T \cong \mathrm{R}_{A/k} \mathbb{G}_m.$$ A ...
7
votes
1answer
242 views

Is it known whether every symmetric pair of finite groups of Lie type is a Gelfand pair?

A pair of groups $(G,H)$ is called a symmetric pair if $H$ is the group of fixed points of an involutive automorphism of $G$, for example $(GL(2n,\mathbb{F}_q),Sp(2n,\mathbb{F_q}))$ is a symmetric ...
21
votes
4answers
1k views

Hasse principle for rational times square

Does a Hasse principle hold for the property of being a rational times a square ? Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...
1
vote
0answers
64 views

Representation equivalent lattices

Suppose $G$ is a absolutely almost simple algebraic groups over a number field $K$. Let $\Gamma_1$ and $\Gamma_2$ be two lattices in $G(K)$. Then $\Gamma_1$ and $\Gamma_2$ are said to be ...
0
votes
0answers
106 views

Normalizer of non-split tori

Let $\mathbb{G}$ be a connected reductive group over $\mathbb{C}$. Let $G:=\mathbb{G}(\mathbb{C}(\!(t)\!))$. Let $T$ be a maximal torus in $G$. Question: What do we know about the normalizer $N_G(T)$...
2
votes
1answer
152 views

Product of Bruhat Cells

Fix a $(B,N)$ pair (Tits system) of a semisimple Lie group $G$. Let $u$ and $v$ be two Weyl group elements such that $l(uv)=l(u)+l(v)$. It is known that $BuvB=(BuB)(BvB)$ (see for example Humphreys's ...
1
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0answers
83 views

representability of a certain extension of group algebraic spaces

Let S be a scheme. Suppose we have sheaves in abelian groups $A,B,C$ over the big étale site of $S$. Suppose that $A$ and $C$ are representable by algebraic spaces in groups locally of finite type ...
5
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0answers
305 views

étale cohomology of rings of integers of number fields and Shafarevich-Tate groups

Let $K$ be a number field, $A$ an abelian variety over $K$. Let $\mathcal{O}$ be the ring of integers of $K$, $\mathcal{A}$ the Néron model abelian scheme of $A$ over $\text{Spec}(\mathcal{O})$. For ...
14
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1answer
369 views

Does the ring $R = \mathbb{Z}[X^{\pm1}]$ of Laurent polynomials over $\mathbb{Z}$ satisfy $SL_2(R) = E_2(R)$?

Let $R = \mathbb{Z}[X^{\pm1}]$ be the ring of Laurent polynomials on one indeterminate over $\mathbb{Z}$. Let $E_2(R)$ be the subgroup of $GL_2(R)$ generated by the matrices that differ from the ...
4
votes
0answers
169 views

Is the class of commutative generalized Euclidean rings stable under quotient and localization?

Let $R$ be an associative ring with indentity and let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by matrices obtained from the identity matrix by replacing an off-diagonal entry by some $r∈R$. ...
12
votes
2answers
526 views

1-st cohomology of multiplicative group in a vector space

Let $\mathbb k$ be a field of characteristic $p$ and let $\mathbb k_n$ be a 1-dimensional representation of $\mathbb k^\times$, where the action is given by $t\circ v= t^n v$. Is it known what are the ...
3
votes
0answers
77 views

Correspondence between dual center and linear characters of finite reductive group

Let $(G,F)$ be a connected reductive group defined over $\mathbb{F}_q$ via the Frobenius $F$ and let $(G^*,F^*)$ be a group in duality with $(G,F)$ with respect to rational maximal tori $T \subseteq G$...
3
votes
2answers
184 views

Do all reductive group schemes over semilocal rings admit finite-dimensional free faithful representations?

The definition of a reductive group scheme is as in SGA III. Frankly, I only know that they exist for the adjoint group (the adjoint representation). In SGA III, I could only find a result for general ...
8
votes
1answer
200 views

Algebraic points of uniformly bounded degree on an algebraic variety

Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$. Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$. Does there exist a natural number $d=d(\...
7
votes
1answer
197 views

Characters of simply connected semsimple algebraic groups over local fields

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$. However, it is quite possible ...
3
votes
1answer
123 views

A more precise description of conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...
4
votes
1answer
245 views

About the conjugation of semi-simple subgroups

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $...
4
votes
2answers
238 views

Adjoint semi-simple algebraic groups over non-algebraically closed fields

Let $k$ be a field of characteristic zero and let $G$ be an adjoint semi-simple algebraic group over $k$. On p34 of the paper "Sansuc - Groupe de Brauer et arithmétique des groupes algébriques ...