All Questions
2,543 questions
6
votes
1
answer
423
views
Is every complex linear algebraic group a differential Galois group?
Let $ G $ be a complex linear algebraic group. In other words, $ G $ is a subvariety of the space of $ n \times n $ complex matrices and $ G $ is a group under matrix multiplication.
Does there always ...
- 63.9k
3
votes
1
answer
220
views
Why locally algebraic characters of $\text{Gal}(\overline{\mathbb Q}/\mathbb Q)$ are associated to $A_0$ Grossencharacters/algebraic Hecke characters?
$\DeclareMathOperator\Res{Res}\DeclareMathOperator\Gal{Gal}$I am trying to understand lemma 3.1 of "Abelian Varieties over \mathbb Q and modular forms" of Ribet. ArXiv link
Just so everyone ...
- 149k
4
votes
0
answers
116
views
When is the intersection of cosets of a conjugacy class $0$-dimensional?
Let $G = \mathrm{SL}_n$ (say); let $K$ be a field. Let $g$ be a regular semisimple element of $G(K)$, and $\mathrm{Cl}_g$ its conjugacy class, considered as an algebraic variety. Then $\mathrm{Cl}_g$ ...
- 20.2k
5
votes
1
answer
163
views
When is $G(R)\rightarrow G^{\textrm{ad}}(R)$ surjective?
Consider a reductive group $G$ over a field $k$. The adjoint group $G^{\textrm{ad}}$ is defined by the exact sequence $$1\rightarrow Z(G)\rightarrow G\rightarrow G^{\textrm{ad}}\rightarrow 1$$The ...
- 14.2k
2
votes
1
answer
176
views
Isomorphisms $S^d(S^m(V)^*) \cong \Lambda^d(S^{m+d-1}(V)^*)$
$\DeclareMathOperator\SL{SL}$Let $K$ be an algebraically closed field. Let $V$ be a 2-dimensional vector space over $K$. The group $G=\SL_2(K)$ acts naturally on $V$ by left-multiplication on column ...
- 12.9k
2
votes
1
answer
262
views
Connecting homomorphism in non-abelian cohomology
Let $G$ be a simply connected, semisimple algebraic group over $\mathbb{R}$ and let $X$ be a homogeneous space for $G$ with finite commutative stabilizer $\mu$. There is a connecting homomorphism from ...
- 14.2k
8
votes
0
answers
203
views
Word maps from $\mathrm{Cl}^{n+1}$ to $G^n$: quasi-injectivity?
Let $G = \mathrm{SL}_n$ (say). Then, for $g$ regular semisimple, the conjugacy class $\mathrm{Cl}(g)$ has dimension $= n^2-n$ as a variety, and so $\mathrm{Cl}(g)^{n+1}$ and $G^n$ have the same ...
- 20.2k
5
votes
1
answer
133
views
Normal closure of $e_{12}$ in the congruence subgroup $\Gamma_1(p)\subset \mathrm{SL}_2(\mathbb{Z})$
$\DeclareMathOperator\SL{SL}$For an odd prime $p$, let
$$\Gamma_1(p)=\left\{\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in \SL_2(\mathbb{Z}):\begin{pmatrix}a & b \\ c & d\end{pmatrix}\...
- 44.8k
2
votes
0
answers
182
views
GIT quotient and orbifolds
Let $G$ be a connected complex reductive group. Suppose $G$ acts on a smooth complex affine variety $X$. Assume the stabiliser $G_x$ of every point $x\in X$ is finite. Is it true that $X/\!/G$ is an ...
- 2,751
1
vote
0
answers
95
views
Injection of $G(k)/Z(k)$ into $(G/Z)(k)$
In the first answer to the linked question it is mentioned that "the isogeny $G\to G^{ad}$ induces an injection of groups $G(k)/Z(k)\to G^{ad}(k)$". Is there a reference for this result? ...
- 63.9k
4
votes
1
answer
1k
views
How to think about the simple reflection $s_0$ in the affine Weyl group?
Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
- 3,065
5
votes
2
answers
366
views
Simple connectedness of Levi subgroup
Let us consider a connected and simply connected semisimple algebraic group $G$ over $\mathbb{C}$, $B$ a Borel subgroup and $T$ a maximal torus contained in $B$.
Let $P_1$, $P_2$ be two standard ...
- 12.9k
7
votes
1
answer
1k
views
When a free action gives rise to a $G$-principal bundle
When a free action gives rise to a $G$-principal bundle
Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that
$G \backslash X$ is Hausdorff. (equivalently the image of ...
- 178
3
votes
0
answers
97
views
$\mathbb{Z}_p$-points of a $\mathbb{Z}_p$-model of a reductive linear algebraic $\overline{\mathbb{Q}}_p$-group
Let $G$ be a (connected) reductive linear algebraic group over $\overline{\mathbb{Q}}_p$. By definition, this means that $G$ is a closed subgroup of some $\mathrm{GL}_n$. We can always find a ...
- 31
3
votes
2
answers
399
views
Cohomology of the partial flag variety associated with the minimal nilpotent orbit
Let $G$ be a semi-simple group over complex number; for simplicity let us assume that it is simply laced. Let $X$ be the orbit of the highest root line in the adjoint representation of $G$. This is a ...
- 10.7k
3
votes
1
answer
134
views
Connected components of a spherical subgroup from spherical data?
This question is in a similar spirit to this one by Mikhail Borovoi.
Let $G$ be a reductive group over $\mathbb{C}$ and let $X=G/H$ be a homogeneous spherical variety.
Losev proved that the spherical $...
- 15.5k
6
votes
1
answer
396
views
Parabolic subgroups of reductive group as stabilizers of flags
$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
- 149k
2
votes
0
answers
156
views
Classifying stack for finite flat group scheme
Let $G$ be a finite flat non-smooth group scheme over an algebraically closed field $k$, for example, $G$ can be $\operatorname{Spec}(\overline{\mathbb{F}}_p[t]/(t^p))$. Then the classifying stack $\...
- 12.9k
3
votes
0
answers
66
views
Arithmetic lattices are finitely presented
In the book "Kazhdan's Property (T)" by Bekka-de la Harpe-Valette, the following is stated on p.6 of the introduction:
"Of course, it is classical that arithmetic lattices are finitely ...
- 305
14
votes
1
answer
1k
views
Double coset spaces of reductive groups and integral representations of L-functions
Let $G$ be a reductive group over a number field $k$, with center $Z$. Let $P$ be a parabolic subgroup. Let $H$ be a reductive subgroup of $G$. To what extent can we understand the double coset space $...
- 12.9k
10
votes
0
answers
274
views
Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?
Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $
\DeclareMathOperator{\Aut}{Aut}
\Aut(X)$.
Define a torus in $\Aut(X)$ to be a faithful ...
CommunityBot
- 1
4
votes
1
answer
172
views
Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?
Consider a (finite-dimensional) real connected Lie group $G$ with Lie algebra $\frak{g}$. Take a generating set $\mathcal{G} = \{ X_1, \cdots X_n \} $ of $\frak{g}$, i.e. such that any element of $\...
- 63.9k
13
votes
0
answers
509
views
Is there a simple proof that representations of GL(n,k) are determined by their restriction to diagonal matrices?
Let $k$ be a field of characteristic zero. The general linear group $\mathrm{GL}(n,k)$ has a subgroup $\mathrm{D}(n,k)$ consisting of invertible diagonal matrices. These are linear algebraic groups ...
- 22.3k
1
vote
0
answers
80
views
How to know the character table of the twisted group algebra of the symmetric group $S_4$
Given the character table of its Schur cover group, is there a way to obtain the character table of twisted group algebra from that? I am particularly interested in the symmetric group $S_4$.
- 1,774
3
votes
0
answers
66
views
One parameter subgroups of reductive algebraic groups
If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
- 63.9k
4
votes
1
answer
283
views
Fppf or étale extension of group algebraic spaces
Let $S$ be a scheme and let
$$0 \to A \to B \to C \to 0$$
be an exact sequence of abelian sheaves on $(\mathrm{Sch}/S)_\text{fppf}$. Assume that $A$ and $C$ are representable by flat algebraic spaces. ...
- 12.9k
4
votes
0
answers
108
views
Shafarevich conjecture for Abelian varieties over global function fields
Let $S$ be a finite set of places of a global function field $K$. Are there finitely many Abelian varieties over $K$ with good reduction outside $S$? What if we exclude isotrivial families?
- 679
4
votes
0
answers
109
views
Anisotropic semisimple groups with no real compact factor
Let $F$ be a number field, and let $G$ be a semi-simple connected, anisotropic algebraic group over $F$ which is $F$-simple (or almost simple, the question is agnostic to isogenies). Suppose further ...
- 14.2k
2
votes
1
answer
106
views
Does a polynomial $P(X,Y)$ that specializes to a polynomial $P(x_0,Y)$ with distinct roots in $\overline{k}$ have distincts roots in $\overline{k(X)}$
Let $k$ be a field, $P\in k[X][Y]$ be a monic polynomial of degree $n$ in $Y$.
I am looking for a simple proof of the following fact.
"If there exists $x_0\in k$ such that $P(x_0,Y) \in k[Y]$ has ...
- 12.9k
2
votes
0
answers
55
views
Question on generic A-packet
Let $G$ be a classical group and $\phi$ be a generic $A$-parameter of $G$.
I am wondering whether each automorphic representations in the $A$-packet associated to $\phi$ are locally generic at almost ...
- 1,019
0
votes
0
answers
62
views
Involutions in $\operatorname {PSO}(4,K)$
In the algebraic group $G=\operatorname {PSO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there and what are the ...
- 217
3
votes
1
answer
362
views
Does a quasi-split reductive group scheme admit a maximal torus?
Let $G \to S$ a reductive group scheme over arbitrary base. Following the conventions from Conrad's Reductive Group Schemes notes, we define a Borel subgroup to be an $S$-subgroup scheme $B \subseteq ...
- 156k
4
votes
0
answers
135
views
Nilpotent orbits in characteristic $0$ vs. positive characteristics
Let $G_\mathbb{C}$ be a connected reductive group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}_{\mathbb{C}}$. For any algebraically closed field $k$, let $G_k$ denote the connected reductive group ...
- 2,751
0
votes
1
answer
173
views
Show that $\mathrm{PSL}_2(C)$ is complex algebraic [closed]
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\M{M}\DeclareMathOperator\im{im}$I meet this problem when reading Artin's book Algebra. ...
- 12.9k
6
votes
0
answers
277
views
Is every free additive action on the affine space conjugate to a translation?
Is every free action of the additive group $\mathbb{G}_a$ on the affine space $\mathbb{A}^3$ conjugate to a translation?
In characteristic zero, the answer is yes, and is due to Kaliman. [Kaliman, S. &...
- 7,722
4
votes
0
answers
310
views
GIT quotient of a reductive Lie algebra by the maximal torus
Let $G$ be a connected complex reductive group with Lie algebra $\mathfrak{g}$. One knows a lot about the GIT quotient $\mathfrak{g}/\!/G$: the invariant ring is a free polynomial algebra on $\mathrm{...
- 2,751
0
votes
1
answer
102
views
An explicit matrix form in the symplectic group
In the algebraic group $G=\operatorname {PCSp}(2^{r},K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[...
- 12.9k
0
votes
1
answer
88
views
An explicit matrix form
In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[
\...
- 12.9k
5
votes
0
answers
152
views
Group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients
What is known about the group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients and what are strategies to compute it (or at least some groups for low degrees)? Here I want to consider ...
- 1,184
2
votes
0
answers
142
views
A possible generalization of "homotopy" to study group actions of various kinds
This is a naive question about abstract homotopy theory by someone who knows nothing about it, except that it involves some generalization of the notion of "homotopy".
If we think of $O(n)$ ...
- 1,433
4
votes
0
answers
183
views
Characters of finite fields - Reference request
Let $\mathbf{F}_q$ ($q=p^f$) be a finite field. We are interested in the characters $\chi: \mathbf{F}_q\rightarrow \mathbf{K}$ ($\chi(0)=0$) where the $ \mathbf{K}$ is an alg.closed field of ...
- 63.9k
10
votes
2
answers
461
views
Presentation of special linear group over localizations of the integers
I am looking (for $n,k\in{\mathbb Z}$) for a presentation (in the best of all worlds concretely, as a list of relators) for the group ${\rm SL}_n(R)$ for $R={\mathbb Z}[\frac{1}{k}]=\{\frac{a}{k^l}\...
6
votes
1
answer
352
views
All surjections onto trivial irrep split equivalent to being reductive
$\DeclareMathOperator\Hom{Hom}$Let $ G $ be linear algebraic group over a field $ k $. Is it true that every short exact sequence of algebraic $ G $-representations
$$
0 \to W \to V \to k \to 0
$$
...
- 4,081
2
votes
1
answer
184
views
What are the Tits algebras of $\mathrm{SO}(A, \sigma)$ if $A$ is split?
Given a connected linear algebraic group $G$ over a field of characteristic zero, there are several constructions of the so called Tits algebras (see Sechin and Semenov - Applications of the Morava K-...
- 2,178
15
votes
1
answer
550
views
Branching rule of $S_n$ and Springer theory
Let $u\in\mathrm{GL}_n$ be a unipotent element, let $\mathcal{B}_u$ be the variety of Borel subgroups containing $u$, and let $d=\dim \mathcal{B}_u$. Then Springer theory tells us that $H^{2d}(\...
- 4,612
2
votes
0
answers
241
views
Action of algebraic group in cohomology of equivariant algebraic vector bundle
Let $X$ be a projective algebraic variety over an algebraically closed field. Let an algebraic group $G$ act algebraically on $X$. Let $\mathcal{F}$ be a $G$-equivariant vector bundle (or, more ...
- 21.8k
14
votes
3
answers
2k
views
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, ...
- 12.9k
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
1
vote
0
answers
64
views
A variation of the dual group of the adjoint group
Let $\mathbf{G}$ be connected reductive group over a $p$-adic field $F$. Denote by $\mathbf{Z}$ the center of $\mathbf{G}$, and $\mathbf{A}$ the maximal split torus of $\mathbf{Z}$ (also called the ...
- 709
1
vote
0
answers
161
views
N(H)/H and the Weyl group
Let $ H $ be a connected subgroup of $ G=\mathrm{SU}(n) $ such that $ N_G(H)/H $ is finite. Is $ N_G(H)/H $ always a subgroup of the symmetric group $ \mathrm{S}_n $?
I just noticed this from the ...
- 4,081