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.
2,105
questions
3
votes
1
answer
107
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 $...
5
votes
1
answer
219
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 ...
2
votes
0
answers
116
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 $\...
3
votes
1
answer
308
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 ...
3
votes
0
answers
46
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 ...
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 $...
6
votes
2
answers
237
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 ...
9
votes
0
answers
252
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 ...
4
votes
1
answer
127
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 $\...
13
votes
0
answers
438
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 ...
1
vote
0
answers
66
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 $S_4$.
3
votes
0
answers
47
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 ...
4
votes
1
answer
239
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. ...
2
votes
1
answer
496
views
Irreducible action of an algebraic group
Is the following claim true?:
Let $G$ be an algebraic group such that $G^\circ$ is reductive. Suppose $G$ acts irreducibly on $V$. Is it true that $V$ is decomposed (written as direct sum) into $G^\...
1
vote
0
answers
83
views
Question on the geometric lemma in $p$-adic representation theory
$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B} $
Let $F$ be a $p$-adic field and $\Sp_{2n}$ the symplectic group over a $2n$-...
4
votes
0
answers
85
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 ...
2
votes
0
answers
51
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 ...
0
votes
0
answers
56
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 ...
3
votes
1
answer
302
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 ...
3
votes
0
answers
100
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 ...
0
votes
1
answer
143
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. ...
2
votes
1
answer
106
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 ...
6
votes
0
answers
257
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. &...
1
vote
0
answers
43
views
Openness of strong irreducibility
Let $\Gamma$ be a finitely generated group.
A linear representation of $\Gamma$ is irreducible if it does not preserve a proper subspace, and strongly irreducible if it does not preserve a finite ...
4
votes
0
answers
216
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{...
0
votes
1
answer
94
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:
$$
e=\left[...
0
votes
1
answer
83
views
An explicit matrix form
In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[
\...
5
votes
0
answers
135
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 ...
8
votes
1
answer
779
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
2
votes
0
answers
135
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 $O(n)$ ...
10
votes
2
answers
413
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
343
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
$$
0 \to W \to V \to k \to 0
$$
...
2
votes
1
answer
178
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-...
15
votes
1
answer
502
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}(\...
2
votes
0
answers
228
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 ...
13
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, ...
1
vote
0
answers
60
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 ...
1
vote
0
answers
149
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 ...
2
votes
1
answer
221
views
Question on the modulus character of classical p-adic group
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$It is well known for the formula of the computation of modulus character of general linear groups. For example, for the standard Borel subgroup $...
3
votes
1
answer
181
views
Regular embeddings of a reductive groups with induced center
Let $G$ be a reductive group over the finite field $\mathbb{F}_q$. Then a regular embedding of $G$ is an $\mathbb{F}_q$-rational embedding $\iota \colon G \rightarrow G'$ into a second reductive group ...
2
votes
0
answers
116
views
The double quotient of SU(N) by its diagonal maximal torus
$\DeclareMathOperator\SU{SU}$The special unitary group $\SU(N)$ contains $T^{N-1}$ as a maximal torus, which we take to be the diagonal subgroup of $\SU(N)$. Can we describe the double quotient space
$...
3
votes
1
answer
142
views
Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$
Let $V_i$ be the $(i+1)$-dimensional representation of the special unitary group $\mathrm{SU}(2)$ with the highest weight $i$. Is there any uniform way to compute the irreducible decomposition for the ...
0
votes
0
answers
59
views
What does it mean for a linear algebraic group to act reductively
I was reading this paper by Baues and on page 918 he mention that $S$ acts reductively on the cochain complex and on page 919 again he mention the word "Since $T$ acts reductively on the complex.....
3
votes
1
answer
250
views
Which algebraic groups are generated by (lifts of) reflections?
$\DeclareMathOperator\SL{SL}$The Cartan–Dieudonné theorem
states that each element $g \in \operatorname{O}(V)$, where $V$ is a quadratic space of dimension $n$ over a field of characteristic $\neq 2$, ...
1
vote
0
answers
121
views
A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?
Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...
1
vote
2
answers
296
views
Finding lectures PDF "Four lectures on simple groups and singularities"
I would be very interested to find the PDF "Four lectures on simple groups and singularities" by Peter Slodowy, especially the lecture 4. I used to print them but lost it. Does anyone has ...
9
votes
2
answers
510
views
Quotients of schemes by connected groups
Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$.
By the Keel-Mori theorem, the quotient $X/G$ is ...
4
votes
1
answer
218
views
Question regarding the definition of linearization of line bundles
I'm reading Dolgachev's book 'Lectures on invariant theory'. In Chapter 7, the linearization of a group action is discussed. Let $G$ be a linear algebraic group acting on a quasi-projective variety $X$...
3
votes
0
answers
57
views
Anisotropic kernel of groups of type A
I'm studying the results of classification of reductive groups using Tits index and anisotropic kernel.
It is known that simple groups with Tits index $^1 A_{n,r}^{(d)}$ are of the form $SL_{r+1}(D)$, ...
2
votes
0
answers
66
views
Arbitrary base change of a parahoric subgroup in split case
Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...