All Questions
2,543 questions
13
votes
1
answer
2k
views
What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?
Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
- 6,069
1
vote
1
answer
265
views
How we characterize a subgroup of finite group of Lie type with unipotent elements.
Let $G$ be a finite group of Lie type. Let $H$ be a subgroup of $G$ which contains unipotent elements. I want to find a 'nice' subgroup of $G$ that contains $H$, for example a Levi subgroup of $G$ ...
- 35
0
votes
1
answer
160
views
subgroups of a $p$-solvable group and complete reducibility
1.
Let $G$ be a $p$-solvable group and $V$ be a finite dimensional
faithful $kG$-module, where the characteristic of $k$ is $p$. But
$V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...
- 491
1
vote
0
answers
82
views
decomposition lemma in adelic groups II
Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$.
Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$.
On each point, we have an évaluation morphisme $ev_{x}:G(k[[t_{x}...
- 3,472
3
votes
2
answers
477
views
Maximal soluble subgroups in a parabolic subgroup of finite classical simple group
Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ stabilizing an isotropic subspace. Is the Borel subgroup of $G$ maximal soluble in $P$ and is there ...
- 767
1
vote
0
answers
140
views
on a decomposition lemma in adelic groups
Let X a curve over an algebraically closed k.
Fix $x$ and $y$ two distinct closed points of X.
Let G be a connected reductive group over k.
We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...
- 3,472
5
votes
2
answers
708
views
Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers
Consider $2n$ coordinates $x_1,\ldots,x_n,y_1,\ldots,y_n$ and the quadratic form $q = \sum_{i=1}^n x_i y_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with ...
- 32.5k
14
votes
4
answers
3k
views
Is the normalizer of a reductive subgroup reductive?
Let $G$ be a reductive algebraic group over an algebraically closed field (of characteristic zero if it matters) and $H \subset G$ a subgroup, also reductive. Is the identity component of the ...
- 1,500
0
votes
1
answer
678
views
For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ?
Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C_{G}(s)$ is Levi ...
- 35
11
votes
2
answers
684
views
Differential/difference algebraic groups as "group schemes"
While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by ...
- 3,186
0
votes
0
answers
440
views
Foliations in positive characteristic
Ekedahl wrote about foliations in positive characteristic, over the field $\mathbb{Z}/p\mathbb{Z}$ as a subsheaf of the tangent sheaf, that are closed with respect to involution and $p$-power.
My ...
- 1
6
votes
1
answer
666
views
Groups becoming algebraic groups
Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that:
(1) the set of $k$-points has the structure of a group.
(2) for any $g\in G$ the right-...
- 7,722
7
votes
1
answer
840
views
Confusing Point in Proof: Semisimple Automorphism Fixes Torus
I am reading a proof on p.51 of Robert Steinberg in his book "Endomorphisms of Algebraic Groups" and I am having a bit of difficulty understanding one point in the proof.
The setting is as follows. ...
user1437
5
votes
0
answers
783
views
finite etale group scheme over a field
Could I have an example of a finite etale group scheme over a field k which is not a constant group scheme?
I just know that the category of etale group schemes over a k is equivalent to the category ...
- 147
1
vote
1
answer
809
views
On the Steinberg section
Let $\chi:G\rightarrow T/W$ the Steinberg map. I assume that G is simply connected. Then $T/W=\mathbb{A}^{r}$ and Steinberg constructed a section to this map given by
$\epsilon(a_{1},...,a_{r})=x_{\...
- 3,472
21
votes
3
answers
3k
views
Simple Tamagawa number calculations
As is well known, Euler proved the Basel identity $\displaystyle\sum\limits_{i=0}^{\infty} \frac{1}{n^2} = \frac{{\pi}^2}{6}$. By far the most illuminating explanation of this fact that I've seen is ...
- 7,072
12
votes
1
answer
1k
views
Recovering classical Tannaka duality from Lurie's version for geometric stacks
In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects
$$ f \colon X \to Y$$
is equivalent to giving a corresponding pullback ...
- 805
6
votes
2
answers
743
views
Measuring how far from being cocompact a lattice is
Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by left-...
- 9,486
1
vote
1
answer
234
views
Describing a matrix group (with integer coefficients) through conditions on the coefficients.
I'm wondering if there's always a (not too complicated?) way to characterize a matrix group by conditions on the coefficients.
I know if I'm dealing with matrix groups over a field, then it's sort of ...
- 10.7k
5
votes
1
answer
782
views
Representations of reductive groups over local fields through parahoric induction
Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...
- 805
3
votes
1
answer
765
views
Are extensions of linear algebraic groups (over a field) themselves linear algebraic?
The title says it all.
A very similar question was asked and answered about linear groups, but none of the counterexamples are algebraic:
Are extensions of linear groups linear?
If $A$, $B$ are ...
- 1,366
4
votes
1
answer
1k
views
Involution of the Fermat quartic
Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by
$$x^4-y^4-z^4+w^4 = 0$$
and consider the involution
$$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$
The surface $X$ can be seen ...
- 8,998
9
votes
1
answer
890
views
On q-Demazure operators
Setup
Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements ...
- 3,637
3
votes
3
answers
815
views
Irreducibility of fundamental Weyl modules
It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corresponding to the ...
- 115
1
vote
0
answers
192
views
"Higher" Tangent spaces in char-p geometry - definition?
Hi, everyone!
I have some construction that requires exact definition.
I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
- 1,422
4
votes
1
answer
1k
views
Closed subgroups of a $p$-adic algebraic group
Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G$ the group of rational points of an algebraic group defined over $F$, endowed with the natural topology. Any Zariski closed subgroup $H \subset G$...
- 3,605
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
2
votes
1
answer
1k
views
Are certain simple Lie groups linear algebraic groups?
Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)
Such a group should automatically ...
- 196
11
votes
3
answers
1k
views
Regular elements in the torus of a group of Lie type
Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...
- 11.2k
11
votes
2
answers
2k
views
Mostow's theorem on algebraic groups
In his classical 1956 paper
Fully reducible subgroups of algebraic groups
Mostow proves the following theorem:
Theorem 7.1.
Let $G$ be an algebraic group over a field $K$ of characteristic 0,
...
- 14.2k
14
votes
1
answer
1k
views
Frobenius splitting of Fano varieties
Dear MO,
Question 1. Do you know of an example of a Fano variety which is not Frobenius split?
Background
(1) A variety $X$ in characteristic $p$ is called Frobenius split if there is a "$p$-th ...
- 16.2k
15
votes
6
answers
3k
views
Characteristic zero and characteristic $p$ in algebraic geometry
Are there non-trivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known ...
user16974
11
votes
2
answers
3k
views
Why are $S$-arithmetic groups interesting?
Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$.
Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion ...
- 349
8
votes
2
answers
497
views
When is an orbit spherical?
I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here:
Let's assume we have an affine, reductive, ...
- 4,902
7
votes
2
answers
429
views
Separating subspaces in an irreducible representation
Suppose $G$ is a semisimple $\mathbb{R}$-algebraic group with finite center, and suppose $G$ acts irreducibly on a vector space $V$. Suppose $U \subset V$ and $W \subset V$ are subspaces.
$\mathbf{...
- 3,201
5
votes
1
answer
422
views
Rational automorphisms of semisimple algebraic groups
Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. ...
- 583
3
votes
0
answers
929
views
On the structure of commutative group schemes
The structure of a commutative affine algebraic group $G$ over a field $k$ is understood (SGA 3): $G$ has a maximal subgroup of multiplicative type $M$ and the quotient $G/M$ is unipotent.
I am ...
- 31
6
votes
2
answers
2k
views
a question on TITS' note "Reductive groups over local fields"
This note appears in "Proceedings of Symposia in pure mathematics" vol.33 1979 part 1 pp. 26-69.
The question will be about materials on page 31-32.
Let $G$ be a reductive algebraic group (not ...
- 853
5
votes
5
answers
2k
views
Commutator of algebraic subgroups is connected
Let $G$ be an algebraic group over an
algebraically closed field. If $H$ and
$K$ are closed subgroups and one of
them is connected, then their
commutator $[H,K]$ is also connected.
Is there ...
- 899
8
votes
3
answers
1k
views
Failure of Jacobson-Morozov in positive characteristics
The Jacobson-Morozov theorem that any nilpotent element $e$ in the Lie algebra of a simple algebraic group $G$ can be embedded in an $\mathfrak{sl}_2$-triple, has a restriction (in terms of the ...
- 115
22
votes
3
answers
2k
views
Is SL(2,C)/SL(2,Z) a quasi-projective variety?
Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).
Is $SL(...
- 18.7k
7
votes
2
answers
999
views
Kostant's theorem on invariant polynomials in positive characteristic
Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of ...
- 3,637
0
votes
1
answer
375
views
For an algebraic group acting on a variety, why are orbits representable?
I suspect this is really obvious, but I'm not seeing it.
For an algebraic group $G$ acting on a variety $V$, and for a point $x \in \text{hom}(\text{Spec}(K),V)$, we define the orbit $G(x)$ to be $G(...
26
votes
1
answer
2k
views
Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?
This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
- 52.9k
3
votes
0
answers
420
views
When is a subgroup the Weil restriction of another subgroup?
I asked this question on Math.StackExchange, to no avail. I try my chance on this one.
Let $M/F$ be an extension of fields. Let $G$ be an algebraic group over $F$, and consider the $F$-group $H$ ...
- 407
7
votes
3
answers
1k
views
homomorphism into reductive groups
Let $k$ be an algebraically closed field with char($k$)$= p > 0$.
Let $P$ be a finite $p$-group. For any homomorphism
$\rho : P \rightarrow GL(n,k)$ we know that the image $im(\rho)$ can be
put ...
- 135
4
votes
0
answers
203
views
The Killing form on quantized enveloping algebras and reduction to the classical case
Let $U_q$ be the quantized enveloping algebra associated to a semisimple Lie algebra $\mathfrak g$. It is a result due to Tanisaki (see here; also see Chapter 6 of Jantzen's book Lectures on Quantum ...
- 3,637
6
votes
0
answers
181
views
On an interesting subalgebra of the functions on the cotangent bundle of the flag variety
Setup
Let $G$ be a semisimple algebraic group over a field $k$ of characteristic $p$ where $p = 0$ or $p > 0$ is a good prime for $G$. Fix a Borel subgroup $B \subseteq G$ corresponding to the ...
- 3,637
5
votes
2
answers
700
views
Why $G\to G/H$ is faithfully flat?
Some questions about algebraic groups.
Let $G$ be an affine algebraic group over algebraically closed field $k$.
Questions: Let $H$ be a closed subgroup of $G$, then (as I learnt from some paper) ...
- 157
2
votes
2
answers
400
views
Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups
Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak R=\mathrm{Hom}(\Gamma,G)...
- 1,211