All Questions
2,543 questions
2
votes
0
answers
332
views
Spherical building of an algebraic group
Let $G$ be an algebraic group and $X$ its spherical building, that is, $X$ is the set of maximal proper parabolic subgroups of $G$ and the simplices of $X$ are the finite subsets of $X$ of the form $S=...
- 933
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
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 ...
- 96.4k
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
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 ...
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 $T$...
- 815
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 \...
- 285
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...
- 96.4k
4
votes
1
answer
265
views
Symmetrization for hyperalgebras in positive characteristic
Let $G$ be an algebraic group over an algebraically closed field $k$ of arbitrary characteristic and let $U$ be the hyperalgebra of $G$. Recall that $U$ is defined as the subspace of the full linear ...
- 3,637
4
votes
1
answer
474
views
how many Q-forms of SL_n(R) are there for a given Q-rank
Let $G$ be a linear algebraic group defined over $\mathbb Q$.
Suppose that $G$ is isomorphic to $SL_n$ over $\mathbb R$.
Suppose the $\mathbb Q$-rank of $G$ is fixed, say $m$.
How many types are ...
- 853
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'...
- 5,562
2
votes
2
answers
875
views
Parabolic subgroups and BN-pairs
We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of $...
- 933
3
votes
0
answers
289
views
Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
- 31
4
votes
1
answer
796
views
Deformation space of non-ordinary abelian varieties
It is a well known result of Serre and Tate that if $A$ is an ordinary abelian variety over a field $k$ of characteristic $p>0$, then the deformation space $\mathcal{M}$ of $A$ to an abelian ...
- 395
5
votes
2
answers
339
views
Decomposition of the ring of functions on the unipotent radical of a Borel
Background
Let $G$ be a semisimple linear algebraic group over an algebraically closed field $k$ of arbitrary characteristic. Let $B \subseteq G$ be a Borel subgroup of $G$ and let $U \subseteq B$ be ...
- 3,637
4
votes
2
answers
356
views
Infinite products of representations of the additive group
Fix a $\mathbb{Q}$-algebra $R$. Let's call an endomorphism $f : M \to M$ of an $R$-module $M$ locally nilpotent if for every $m \in M$ there is some $n \in \mathbb{N}$ such that $f^n(m)=0$. ...
- 63.1k
6
votes
4
answers
921
views
Coproduct on coordinate ring of finite algebraic group
I'm reading Mukai's book "An introduction to invariants and moduli", and I am having trouble understanding one of his examples. It is example 3.49 on page 101.
The setup is as follows. Let $G$ be a ...
- 63
10
votes
6
answers
2k
views
Proofs in the same vein as Ax-Grothendieck
I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
- 23.4k
11
votes
3
answers
554
views
Uniform setting for computing orders of algebraic groups over finite quotients of the integers?
A couple of recent questions on MO have involved the characters or the orders of specific finite groups of the form $G(\mathbb{Z}/n\mathbb{Z})$ for a familiar algebraic group $G$ defined over $\mathbb{...
- 52.9k
2
votes
0
answers
577
views
A question on algebraic torus
Let $T$ be an algebraic torus defined over $\mathbb Q$, $T_\infty$ be its real points and $\pi_0(T_\infty)$ be the group of connected components of $T_\infty$.
Why is the homomorphism $T(\mathbb Q)\...
- 21
7
votes
1
answer
1k
views
An interesting double coset in the theory of automorphic forms
Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , $H=GL_{n-1}$ ...
- 809
1
vote
2
answers
431
views
Reductive groups over non archimedean local fields.
I want to know if connected reductive groups over non archimedean local fields have a dense countable subset. I was thinking that this should be true because if $G(\mathbb{F})$ is such group where $\...
3
votes
0
answers
803
views
Tamagawa number for functional fields
Let $G$ be a split semi-simple simply connected group over a global field $F$ and let
$\omega$ be a top-degree differential form on $G$ without zeroes (defined over $F$). It is well
known that $\omega$...
- 7,037
3
votes
3
answers
478
views
Description of $GL_3/U$
Let $U$ be the set of unipotent upper triangular matrices and $B$ the upper triangular matrices of $GL_3$. How could I describe $GL_3/U$ ? Using coordinates, in a projective or an affine space.
For ...
- 311
0
votes
1
answer
315
views
intersections of $SO_2^n, SL_2^n$ with $SO_{2n}, Sp_{2n}$
Let $\mathbb{R}^{2n}$ be endowed with the canonical symplectic structure $(\omega, J)$, where $\omega$ the usual nondegenerate symplectic form and $J$ the usual almost complex structure (ie. $\omega(\...
- 2,274
13
votes
0
answers
556
views
Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?
Let $R$ be a commutative ring, and, for $n\ge0$,
${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series
$u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which
$a_0\in R^\times$ and $u(x)\equiv x\pmod{x^...
- 4,193
6
votes
0
answers
436
views
Do there exist weak Lefschetz-type statements that were proved for varieties over complex numbers, but were not proved in finite characteristic?
As far as I understood the situation (reading section 3.5B of Lazarsfeld's "Positivity in algebraic geometry" and also some of the references), some of weak Lefschetz-type statements known rely on '...
- 16.9k
1
vote
0
answers
189
views
Exotic Chains for Group Homology of a Complex Lie Group
Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
4
votes
0
answers
184
views
Exotic Chains for Group Cohomology of a Complex Lie Group
Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
- 18.7k
5
votes
1
answer
1k
views
Minimal Model Program for surfaces over algebraically closed fields of characteristic p
Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...
- 2,215
4
votes
0
answers
1k
views
Cartan decomposition for upper triangular matrices
Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...
- 11.2k
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.
- 314
10
votes
0
answers
424
views
Polynomial function from $S^3$ to $S^3$ and quaternions
I am searching the polynomial functions from $S^3$ to $S^3$.
($S^3$ is the set of vectors $x$ in $\mathbb{R}^4$ such that $\|x\|=1$)
We say $g$ is a polynomial function from $S^3$ to $S^3$, if there ...
- 663
3
votes
1
answer
559
views
Springer isomorphisms and parabolics
Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $...
- 3,637
2
votes
0
answers
464
views
understanding Milne's article "Duality in the flat cohomology of a surface"
I have trouble understanding a point of Milne's article "Duality in the flat cohomology of a surface" http://jmilne.org/math/articles/1976a.pdf
see the "Alternatively" on p. 177, paragraph before ...
user19475
7
votes
2
answers
571
views
abelian centralizers in almost simple groups
Hallo!
I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.
Let $S$ be a non-abelian finite ...
- 73
2
votes
2
answers
571
views
What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building?
$F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$?
This can be phrased also as question about ...
- 11.2k
3
votes
0
answers
308
views
Invertible Hasse-Witt for non-ordinary curves
Assume that $S$ is a smooth curve of over a field $k$ of characteristic $p>0$ and $f\colon X\to S$ is a relative curve over $S$ (i.e., the fibers are curves). It is well-known that when all the ...
- 395
3
votes
1
answer
805
views
Finite connected groups over a perfect field of characteristic p
In 14.4 of "Introduction to Affine Group Schemes" it is proved (!) that if $A$ represents a finite connected group scheme over a perfect field $k$ of characteristic $p$ then $A$ has the form $k[X_{1}, ...
- 163
2
votes
2
answers
765
views
Can the intersection of a maximal parabolic with a closed sub-group contain more than one maximal parabolic?
Suppose that we have a closed embedding $G_1\hookrightarrow G_2$ of reductive groups (say over $\mathbb{Q}$), and suppose that we have a maximal parabolic sub-group $P_2\subset G_2$, and a minimal ...
- 3,032
2
votes
1
answer
302
views
finiteness of class number: a bound for semi-simple groups?
Let $F$ be a number field, and $G$ a connected semi-simple linear algebraic $F$-group, which does not contain anisotropic (simple) $F$-factors. Write $\hat{F}$ for the ring of finite adeles $F\otimes\...
- 1,305
11
votes
2
answers
959
views
Spherical building of an exceptional group of Lie type
I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
- 805
2
votes
2
answers
544
views
Reference request: The geometry of $GL_2(\mathbb{R})$ and related questions
Can anyone please recommend some good reading on the geometry of linear groups and their actions?
An example of the kind of question I am interested in: Explicitly describe a fundamental domain for ...
3
votes
1
answer
464
views
Action of Non-Split Torus in Deligne-Lustzig induction
Recently I have been trying to understand Deigne-Lustzig induction in the case of $G = \text{Sl}(2,\mathbb{F}_p).$
In this case the appropriate Deligne Lustzig variety is given by $X:xy^q-y^qx = 1,$ ...
- 565
10
votes
2
answers
812
views
Cohomology vanishing for tensor powers of tangent bundle on the flag variety
Let $X$ denote the flag variety of a semi-simple group $G$ (in characteristic 0)
and let $T_X$ denote its tangent bundle. I would like to ask the following question(s):
1) Is it true that for any $n\...
- 7,037
1
vote
0
answers
157
views
On closed abelian reductive subgroups of Real reductive groups
Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions
Let $\mathrm{G}=\mathrm{K} \exp(\...
- 11
3
votes
1
answer
459
views
Frobenius functor and length of local cohomology
Let $(R,\mathfrak{m})$ be a Noetherian local ring of positive prime characteristic $p$ and let $F$ be the Frobenius functor. Write $d$ for dimension of $R$. Assume that for some $0\leq i< d $ the ...
- 3,101
24
votes
4
answers
4k
views
Is strong approximation difficult?
Recently a colleague and I needed to use the fact that the natural map $SL_2(\mathbb{Z}) \rightarrow SL_2(\mathbb{Z}/N\mathbb{Z})$ is surjective for each $N$. I happily chugged my way through an ...
- 7,347
2
votes
2
answers
839
views
Possible Borel subgroups of GL_n?
I am trying to understand the interaction between Borel subgroups of $GL_n$ and its roots. Is it correct to say that for any choice of roots among each pair of reciprocal roots
there is a Borel ...
- 2,842