All Questions
2,543 questions
6
votes
1
answer
644
views
question about equivariant embeddings of riemannian symmetric domains
Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a ...
CommunityBot
- 1
11
votes
2
answers
2k
views
Groups of matrices that preserve several quadratic forms
Given two (or more) quadratic forms (on the same vector space) consider the group of matrices that preserve these forms, i.e. $Q_i=U Q_i U^T$, $i=1,2..,k$ What is known about such groups? (at least ...
- 536
4
votes
1
answer
497
views
Can a non-trivial action of a connected group on a reduced scheme be trivial on a dense open?
It is well-known that if a reduced algebraic group $G$ acts on a separated reduced scheme $X$, and $G$ acts trivially on a dense open subscheme $U\subseteq X$, then the action is trivial.
If $X$ is ...
- 24.1k
3
votes
1
answer
481
views
Turing-Complete Cellular Automata and Sym(Z)
Does there exist a Turing complete, cellular automata with universe and alphabet $\mathbb{Z}$ such that the only allowable configurations are permutations of $\mathbb{Z}$? Formally, consider $\tau : ...
- 237k
7
votes
0
answers
207
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
- 1,199
3
votes
2
answers
675
views
The group G^+ of algebraic groups over local fields
Let $G$ be an algebraic group defined over a char 0
local field $k$. Following Borel and Tits (73) we define
the group $G^+(k)$ or $G^+$ by the subgroup of $G(k)$
generated by the unipotent elements ...
- 63.9k
4
votes
1
answer
624
views
Quick easy question - representation theory
1) What is the proper term for a closed subgroup H of an algebraic group G such that every linear representation of H arises as the restriction of a representation of G?
2) Where can I read about ...
- 52.9k
5
votes
0
answers
504
views
More familiar description of wonderful compactification of SL_n/S(GL_2 \times GL_n-2)
I am trying to learn a bit about spherical geometry and wonderful compactifications. Please correct any misconceptions. If I've understood http://www.springerlink.com/content/x62342v721707828/ ...
- 3,758
5
votes
2
answers
1k
views
Quotient space of algebraic group
Let $H \subset G$ closed subgroup of an algebraic group. We want to prove the existence of the quotient $G/H$ which is a quasi-projective variety and homogeneous G-space.
We can find a vector $0 \ne ...
- 23
4
votes
1
answer
918
views
Heisenberg group in characteristic two
I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
- 45.7k
11
votes
2
answers
2k
views
The anticanonical bundle on a flag variety is ample
Hello,
I would like to get references or answers, for the following. How do I show that the anti-canonical line bundle (i.e. dual to top wedge power of cotangent bundle) on a flag variety (of a ...
- 52.9k
4
votes
2
answers
1k
views
Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?
If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
- 18.7k
1
vote
0
answers
238
views
Classification of fibres in pencils of curves of genus two
For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
- 11
2
votes
1
answer
319
views
Reference request for Cartier Duality of algebraic tori
Hi,
I need a reference for the following result:
Let $S$ be a scheme and let $X$ be an algebraic torus over $S$. Then the functor $F_X :S'\mapsto Hom_{S'}(X\times S',\mathbb{G}_M\times S')$ is ...
- 45.7k
2
votes
1
answer
236
views
Double coset isomorphism
Let $G$ be a connected reductive group (although I don't think this is relevant here), $B$ a Borel subgroup containing a maximal torus $T$ and $U$ the associated unipotent radical ($U^-$ the opposite ...
- 6,614
9
votes
2
answers
519
views
Are algebraic groups defined by their invariants in tensor spaces?
Let $K$ be a field of characteristic zero, and let $G \subseteq \mathrm{GL}_V$ be an algebraic group over $K$, acting faithfully on a finite dimensional vector space $V$. Let $H \subseteq \mathrm{GL}...
- 44.4k
2
votes
2
answers
636
views
Are there other ways to show Pic(G)is trivial when G is a simple-connected semisimple algebraic groups over C?
Indeed,I'm reading the book《representation theory and complex geometry》,there is a proof of the fact that Pic(G)is trivial when G is a simple-connected semisimple algebraic group over C,but the proof ...
- 44.4k
7
votes
2
answers
994
views
commuting elements in a reductive group
Does anyone know if the following holds?
Conjecture: Any two commuting elements in a reductive algebraic group G over C of rank>1 lie in a proper parabolic subgroup of G.
To make things easier, you ...
- 52.9k
5
votes
0
answers
460
views
Has anyone used this theorem of P. Cartier?
In "Groupes Algebriques et Groupes Formels", Conf. au coll. sur la theorie des groupes algebriques, Bruxelles 1962, P. Cartier proves the following in Section 9, Theoreme 1:
(What follows is my ...
- 13.3k
2
votes
2
answers
757
views
Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
- 8,493
14
votes
1
answer
1k
views
Lie groups vs. algebraic groups and deformations
I am interested in deformations of (discrete subgroups of) Lie groups. But, as I understand it, deformation theory, as a theory, prefers to speak schemes.
At least the classical Lie groups can be ...
- 31.2k
11
votes
1
answer
1k
views
Fontaine's classification of p-divisible groups
Let k be a finite extension of $\mathbb{F}_p$, and W its ring of Witt vectors. Write W[F,V] for the Dieudonn\'e ring.
Let G be a connected p-divisible group which is finite-dimensional over k, and ...
- 4,807
1
vote
0
answers
617
views
levi subgroup generated by maximal tori?
In the levi decomposition of an connected algebraic group $G$, is the levi subgroup generated by maximal tori of $G$?.
- 11
1
vote
0
answers
116
views
Reference request: a verification of a nonstandard subgroup being a Tits subgroup.
I have a particular infinite-index subgroup $H$ of the genus 2 symplectic group $Sp(2, \mathbb{R})$. This subgroup is self-normalizing (ie. $gHg^{-1}=H$ only if $g\in H$). I am looking to determine ...
- 2,274
2
votes
0
answers
279
views
deRham cohomoloy of CM liftings of Jacobians
Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
- 637
3
votes
1
answer
168
views
homogenous bundles
Let $G$ be a reductive algebraic group over $\mathbb{C}$ and $H$ be an algebraic subgroup of $G$. We suppose that $H$ acts on some scheme $S$, where $S$ is of finite type over $\mathbb{C}$. Then I ...
- 3,637
3
votes
0
answers
162
views
Does the following characterization of subgroups of $GL_2(\mathbb{F}_p)$ generalise?
Let $p$ be a prime number. By a Cartan subgroup of $GL_n(\mathbb{F}_p)$ I mean an absolutely semisimple maximal abelian subgroup.
When $n=2$, it is well-known* that, for $G \subset GL_n(\mathbb{F}_p)$...
- 2,827
3
votes
0
answers
742
views
Explicit description of O^{cris}_n in Fontaine/Messing
Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
0
votes
2
answers
212
views
A kind of orthogonal subtorus
Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup
$S = \{x \in \...
CommunityBot
- 1
4
votes
2
answers
579
views
Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
- 108k
1
vote
1
answer
282
views
Spectral decomposition of parabolic induced for GL2(Zp)
Let $F$ be a number field and let $o$ be its ring of integers. Let $o_p$ resp. $F_p$ be the completion at a prime ideal $p$ in $o$. Let $B$ be the group of upper triangular matrices in $GL_2$. Let $\...
- 11.2k
7
votes
1
answer
901
views
Bruhat decomposition for G(R), R local ring or R=Z/p^r
Is there an invariant, which encodes the failure of the Bruhat decomposition to hold for a reductive group with coefficients in a local ring like the p-adic integers or the ring $\mathbb{Z}/\mathfrak{...
- 11.2k
7
votes
1
answer
561
views
How does the right regular of GL(n, R) and GL(n,Qp) decompose?
The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right ...
- 11.2k
1
vote
1
answer
185
views
Let G be an affine connected algebraic group. When a subvariety of G with codimension one is a subgroup.
Let G be an affine connected algebraic group, and K[G] be its coordinate ring. Let Y be a subvariety of G defined as a zero set for some f in K[G]. For which f, Y is a closed subgroup of G
- 6,614
2
votes
1
answer
510
views
hyperalgebras (positive characteristic)
The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$.
How to ...
- 829
8
votes
5
answers
927
views
Why is p=2 special, if we want to classify cplx. representation of GL2(Zp)?
I am currently reading Shalika's article "Representation of the two by two unimodular group over local fields" and various other related articles, which deal with the classification of complex ...
- 11.2k
2
votes
1
answer
265
views
Any local algebraic group is birationally equivalent to an algebraic group
In this paper, page $6$ the authors state the following:
By Weil’s theorem $[17]$, any local algebraic group is birationally
equivalent to an algebraic group.
Where
$[17]$ A.Weil. On ...
CommunityBot
- 1
3
votes
0
answers
234
views
Generators and relations for the enveloping algebra of a unipotent radical
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0 and let $B \subseteq G$ be a Borel subgroup with unipotent radical $U$. Let $P \supseteq B$ be a ...
- 3,637
4
votes
1
answer
283
views
Minimal relative Schubert modules
I am trying to better understand the definition of certain objects called minimal relative Schubert modules. My primary reference is Chapters 1 and 2 of Wilberd van der Kallen's Lectures on Frobenius ...
- 3,637
4
votes
1
answer
570
views
Soft(?) algebraic groups question
Suppose $G$ is a linear algebraic group over $\mathbb{C}$, defined over $\mathbb{Z}$ (for example, $SL(n, \mathbb{C})$ is defined by $\det x = 1,$ which visibly has integer coefficients). Let $H$ be ...
- 52.9k
2
votes
1
answer
740
views
finite non-commutative local group schemes
Can I have some examples of finite non-commutative connected group schemes over a field $k$?
I would like also to see some non-trivial torsors over a $k$-scheme $X$ under such group schemes. Thanks.
- 19.6k
1
vote
0
answers
457
views
Why do twists of an algebraic group over k correspond to k-torsors over G
Let $G$ be an algebraic group over a field $k$. Let $k^s$ be the separable closure of $k$.
I can't seem to figure out why isomorphism classes of twists of $G$ correspond to $k$-torsors over $G$.
It'...
- 1,213
4
votes
0
answers
360
views
On a resolution of sections of line bundles on the cotangent bundle of a flag variety
Background
Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic 0. Let $B \subseteq G$ be a Borel subgroup and let $U \subseteq B$ be its unipotent radical....
- 3,637
12
votes
3
answers
1k
views
Is the Gelfand-Graev character isomorphic to a cohomology group for some sheaf on a Deligne-Lusztig variety?
Deligne-Lusztig theory
is awesome. You take a maximal torus $T$, you take a character $\theta$, construct a variety $X_T$$^*$, take etale cohomology, get a virtual character $R_T^\theta$, maybe it's ...
- 686
8
votes
1
answer
982
views
Is there a really big ring of differential operators in characteristic p?
$k$ is a field of characteristic $p$.
$k[t]$ has canonical first-order differential operator $\partial$
As an endomorphism of $k[t]$, $\partial^p=0$.
First way to fix it:
Use the divided power ...
CommunityBot
- 1
0
votes
0
answers
129
views
A kind of orthogonal subgroup
Let $n$ a positive integer and $k \in \mathbb{Z}^n$ such that for all integer $a \geq 2$ and $h \in \mathbb{Z}^n$ we have $k \neq ah$. Here $\cdot$ is the scalar product.
Is it true that $\{x \in \...
- 285
4
votes
2
answers
740
views
Weyl group of a singular torus
Let $G$ be a semisimple algebraic group over an algebraically closed field, and let $T$ be a torus in $G$.
If $T$ is a maximal torus, then $N_G(T)/Z_G(T)=N_G(T)/T$ is the Weyl group $W$ of $G$. If $T$...
- 52.9k
5
votes
3
answers
2k
views
Rationality of algebraic groups
The Cayley parametrization of $O(n),$ as in my answer to this question makes one wonder: which algebraic groups are actually rational? I am sure this is very well understood, just not by me...
CommunityBot
- 1
4
votes
3
answers
677
views
About $G$-modules versus $Lie(G)$-modules for algebraic groups
Hello,
I would like to know clear references about the following facts:
Let $G$ be a connected algebraic group (over alg. closed field in char. 0), $Lie(G)$ its Lie algebra, $M$ a $G$-module. I don'...
- 52.9k
3
votes
2
answers
506
views
Parabolic induction for GL(2,Z/pn)
Fix a finite extension $F$ of $\mathbb{Q}_p$. Consider its ring of integers $\mathfrak{o}$ with maximal ideal $\mathfrak{p}$. Set $R_n = \mathfrak{o}/\mathfrak{p}^n$. Let $\mathrm{B}$ be the upper ...
- 11.2k