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
3 votes
1 answer
278 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
45 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
230 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 ...
8 votes
0 answers
251 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
126 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
433 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
45 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
234 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
495 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
76 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
84 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
47 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
299 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
97 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
140 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
100 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
256 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
210 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
404 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
342 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
501 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
226 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
148 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
216 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
115 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
249 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
289 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
509 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
214 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 $R$ ...
7 votes
0 answers
125 views

Quasisplit forms of wonderful varieties

I will assume that $k$ is a characteristic $0$ non-archimedean field. A classical result of Tits [T] states that a quasisplit connected reductive group $G$ over $k$ is classified up to strict isogeny ...
23 votes
2 answers
906 views

Solvable groups that are linear over $\mathbb{C}$ but not over $\mathbb{Q}$?

Let $\Gamma$ be a finitely generated finitely presented virtually solvable group. Assume that there exists an injective representation $\Gamma \to \operatorname{GL}_n(\mathbb{C})$. Is it true that ...
2 votes
0 answers
133 views

Unipotent closure in classical groups

Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...

1
2 3 4 5
43