Skip to main content

Questions tagged [algebraic-groups]

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.

Filter by
Sorted by
Tagged with
4 votes
1 answer
150 views

Extending Tannakian "dictionary" to gerbes

The following is Proposition 2.21 in Deligne and Milne's "Tannakian Categories". Let $f: G \to G'$ be a homomorphism of affine group schemes over a field $k$ and let $\omega^f$ be the ...
bsbb4's user avatar
  • 333
1 vote
0 answers
45 views

Cartan decomposition over a not-necessarily-discretely-valued field

Let $K$ be a valued field, and let $R$ be the valuation ring of $K$. Let $G$ be a split reductive group over $K$ and $T$ a maximal torus of $G$. On page 107 Berkvoich's book "Spectral theory and ...
Dcoles's user avatar
  • 61
2 votes
0 answers
85 views

What is the natural module?

Lemma 2.9 of [1]: Let $\operatorname{char}(K) \neq 2 $ and let $G$ be $\operatorname{Spin}(m,K)$, $n=\operatorname{rank} G$, and let $V$ be the natural $m$-dimensional module. Suppose $f\in G$ and $f^...
scsnm's user avatar
  • 165
5 votes
0 answers
126 views

Classification of visible actions for *reducible* representations?

Is there a classification of the pairs $(G,V)$ such that $G$ is reductive [and connected, if you like], and $G$ acts faithfully and visibly on $V$ - crucially, including all cases where $V$ is ...
Joshua Grochow's user avatar
1 vote
0 answers
153 views

Computer computation of the first Galois cohomology of a $p$-adic torus?

Let $T\subset {\rm GL}(N,{\mathbb Q})$ be an $n$-dimensional ${\mathbb Q}$-torus given by its Lie algebra $\mathfrak{t}\subset \mathfrak{gl}(N,{\mathbb Q})$. I want to compute, in some sense ...
Mikhail Borovoi's user avatar
6 votes
1 answer
235 views

Extensions of algebraic groups and extensions of fpqc sheaves

There are at least two ways to define $\operatorname{Ext}^p(F,G)$ when $F$ and $G$ are commutative algebraic groups over a field $k$: Pass to the associated fppf sheaves and use an injective ...
Jonathan Wise's user avatar
4 votes
0 answers
142 views

Spaces of fixed points

I am reading the paper Space with $\mathbb{G}_{m}$-action, hyperbolic localization and nearby cycles by Timo Richarz and I am having some troubles in understanding the proof of Lemma 1.10. The setting ...
Alexey Do's user avatar
  • 823
8 votes
0 answers
278 views

Has the notion of a unipotent group scheme been studied?

The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...
Gabriel's user avatar
  • 1,139
3 votes
1 answer
79 views

Restriction of scalar commutes with taking maximal subtorus for semisimple group G

I was wondering such a question: for a semisimple complex Lie group $G$, whether it is true that the maximal subtorus of $\mathrm{Res}_{\mathbb{C}/\mathbb{R}}(G)$ is $\mathrm{Res}_{\mathbb{C}/\mathbb{...
mhahthhh's user avatar
  • 435
1 vote
0 answers
69 views

Decomposition of $BwBw^{-1}B$

Let $(B,N,W,S)$ be a Tits system with $W$ a finite coxeter group. Let $w\in W$, consider $BwBw^{-1}B$, then by Bruhat decomposition, it is a disjoint union of some $BxB$, $x\in W$. My question: Let $X\...
fool rabbit's user avatar
2 votes
0 answers
66 views

Irreducibility of Białynicki-Birula cells

Let $X\subset \mathbb{P}^n$ be a smooth complex projective variety, and consider a non-trivial action of $\mathbb{C}^*$ on $X$. For any connected fixed component $Y$ of the fixed point locus, we may ...
YetAnotherPhDStudent's user avatar
3 votes
0 answers
169 views

A property of an irreducible root system

Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
jack's user avatar
  • 611
2 votes
1 answer
262 views

Prodiscreteness of rational points of algebraic groups

Let $F$ be a field of characteristic 0 complete for a discrete non-archimedean valuation. Let $G$ be a commutative smooth algebraic group over $F$. Let us put on $G(F)$ the topology induced by the ...
rtwo's user avatar
  • 95
3 votes
0 answers
49 views

Algebraicity of the group of equivariant automorphisms of an almost homogeneous variety

The base field is the field of complex numbers. Let $G$ be a connected linear algebraic group. Let $X$ be an almost homogeneous algebraic variety, i.e. $G$ acts on $X$ with a dense open orbit $U \...
sabrebooth's user avatar
7 votes
2 answers
824 views

Hilbert's Satz 90 for real simply-connected groups?

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $K/k$ be a Galois extension. Then one generalisation of Hilbert's Satz 90 states that $H^1(\Gal(K/k),\GL_n(K))=...
user148212's user avatar
  • 1,606
3 votes
0 answers
117 views

Smooth unipotent algebraic groups over $\mathbb A^n$

Let $G\to \mathbb A^n_{\mathbb C}$ be a smooth morphism whose fibers at any point of $\mathbb A^n$ are unipotent groups. Can we conclude that $G\simeq \mathbb A^{n+N}_{\mathbb C}$ for some $N$, as a ...
W. Rether's user avatar
  • 435
4 votes
1 answer
590 views

An algebraic group has how many representations?

Let $G$ be a connected, linear algebraic group over $\mathbb{C}$. Let $\rho:G\to \mathrm{GL}(V)$ be a rational representation. Does $\rho$ have at most countably many subrepresentations (up to ...
Doug Liu's user avatar
  • 545
0 votes
0 answers
113 views

Induced action on infinitesimal thickenings by an algebraic group

Let $X$ be an irreducible locally noetherian $k$-scheme (for $k$ any field), $G$ an algebraic group acting on $X$ via $a:G \times X \to X$ and $x \in X$ a closed point, which is by Zariski's lemma ...
user267839's user avatar
  • 5,770
3 votes
0 answers
209 views

Action of an algebraic group $G$ on a scheme $X$ with fixed rational point

Let $X$ be an irreducible locally noetherian $k$-scheme ($k$ any field) and $G$ an algebraic $k$-group acting on $X$. Proposition 3.1.6 in these notes by M. Brion claims Let $a : G \times X \to X$ be ...
user267839's user avatar
  • 5,770
3 votes
0 answers
150 views

Centers and conjugacy classes of groups relative to a pair of group homomorphisms

$\newcommand{\defeq}{\mathbin{\overset{\mathrm{def}}{=}}}$Given a group $G$, its center $\mathrm{Z}(G)$ and set of conjugacy classes $\mathrm{Cl}(G)$ are defined by \begin{align*} \mathrm{Z}(G) &\...
Emily's user avatar
  • 11.5k
4 votes
0 answers
189 views

Questions about the fixed point functor $X^G$ of a $G$-scheme

Let $X$ a (locally Noetherian; but not sure if that's really matter) $k$-scheme, $G$ a $k$-group scheme acting on $X$ via morphism $a:X \times G \to X$. The fixed point functor of $X$ (where $X$ is ...
user267839's user avatar
  • 5,770
1 vote
1 answer
103 views

Torsor of finite presentation and surjectivity of map of $\overline{k}$-valued points

I have a question about the content of remark 2.6.6. (i) (p 18) from M. Brion's notes on structure of algebraic groups. Let $G$ be a group scheme over certain fixed base field $k$ (as all other ...
user267839's user avatar
  • 5,770
2 votes
0 answers
71 views

Number of rational points of a connected reductive group in a compact subset

Let $G$ be a connected reductive $\mathbb{Q}$-group. Let $\mathbb{A}$ denote the ring of adèles of $\mathbb{Q}$. Let $B \subset G(\mathbb{A})$ be a compact, let $x \in G(\mathbb{A})$ and consider the ...
Sentem's user avatar
  • 51
2 votes
1 answer
239 views

Normalizer of Levi subgroup

Let $G$ be a reductive group (we can work on an algebraically closed field if needed) and let $L$ be a parabolic subgroup, i.e. the centralizer of a certain torus $T \subseteq G$. Associated with this ...
a_g's user avatar
  • 53
2 votes
0 answers
183 views

Theorem of highest weight of semisimple Lie algebras: what fails precisely for reductive case

Let $\mathfrak{g}$ a complex semisimple Lie algebra. It is well known that by the theorem of the highest weight, all finite-dimensional complex irreps of $\mathfrak{g}$ are (up to iso) classified by ...
user267839's user avatar
  • 5,770
3 votes
1 answer
111 views

Involution of $\text{GL}_{m+n}(\mathbb{C})$ fixing Levi and exchanging parabolic subgroups

Is there any involution of $\text{GL}_{m+n}$ which is the identity on $\text{GL}_m\times\text{GL}_n\subset\text{GL}_{m+n}$ and that exchanges the positive and negative associated parabolic subgroups $...
jrg's user avatar
  • 33
1 vote
0 answers
127 views

Question on two types of Frobenius theorem in $p$-adic groups

Let $G$ be a $p$-adic classical group and let $P_0$ be a minimal parabolic subgroup of $G$. Let $P=MN$ be a standard parabolic subgroup containing $P_0$. Let $\text{Ind}$ and $\text{Jac}$ be the ...
Andrew's user avatar
  • 969
6 votes
1 answer
408 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 ...
Ian Gershon Teixeira's user avatar
3 votes
1 answer
182 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 ...
JoseCanseco's user avatar
2 votes
0 answers
141 views

What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$

In Table 22.1 on Page 193 of Malle & Testerman's book "Linear algebraic groups and finite groups of Lie type", the fixed point subgroup $G^F$ (where $F$ is a Steinberg endomorphism) of ...
scsnm's user avatar
  • 165
5 votes
1 answer
159 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 ...
X. DOR's user avatar
  • 53
2 votes
1 answer
172 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 ...
Jon Elmer's user avatar
  • 175
7 votes
1 answer
323 views

Nilpotent orbits of a parabolic subgroup

Suppose $G$ is a reductive group over an algebraically closed field of characteristic $0$ with parabolic $P$, Levi quotient $M$, and unipotent radical $U$. We denote the nilpotent elements of $\mathrm{...
Alexander's user avatar
  • 953
4 votes
2 answers
146 views

Interpretation of the algebra of natural endomorphisms of the fiber functor of $\operatorname{Rep}(G)$

Let $G$ be a connected algebraic group over an algebraically closed field $k$ of characteristic zero (I'm mostly interested in the case of a reductive group). By the Tannakian formalism, $G(k)$ can be ...
Antoine Labelle's user avatar
4 votes
0 answers
110 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 $\mathrm{Cl}_g$ ...
H A Helfgott's user avatar
  • 19.3k
8 votes
0 answers
199 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 ...
H A Helfgott's user avatar
  • 19.3k
5 votes
1 answer
124 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}\...
Max's user avatar
  • 95
1 vote
0 answers
93 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? ...
Μάρκος Καραμέρης's user avatar
2 votes
0 answers
152 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 ...
Dr. Evil's user avatar
  • 2,711
5 votes
2 answers
324 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 ...
fool rabbit's user avatar
3 votes
0 answers
90 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 ...
Otto's user avatar
  • 31
5 votes
1 answer
302 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 ...
a_g's user avatar
  • 497
2 votes
0 answers
143 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 $\...
mhahthhh's user avatar
  • 435
3 votes
2 answers
361 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 ...
Alexander Braverman's user avatar
3 votes
0 answers
54 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 ...
studiosus's user avatar
  • 295
6 votes
2 answers
317 views

Reference for Langlands dual homomorphisms

I am looking for a reference that explains in detail the existence of Langlands dual homomorphisms. It seems that in the literature two references are given most often. The first is Borel's article ...
user449595's user avatar
4 votes
1 answer
152 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 $\...
Another User's user avatar
1 vote
0 answers
72 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 $S_4$.
Wenxia Wu's user avatar
13 votes
0 answers
484 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 ...
John Baez's user avatar
  • 21.8k
3 votes
0 answers
57 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 ...
Ekta's user avatar
  • 63

1
2 3 4 5
43