# Tagged Questions

The algebraic-groups tag has no wiki summary.

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**1**answer

165 views

### On the Cartan decomposition of unitary group

Hello. I have some question on Cartan decomposition of unitary group, especially $U(2)$.
I am interested in local situation, that is p-adic or archimedian.
Let $F$ be a local field and $E$ be its ...

**5**

votes

**1**answer

387 views

### A question about the proof of Beilinson-Bernstein localisation

I'm trying to understand the proof of the Beilinson-Bernstein localisation theorem at the moment, but there's just one point where I'm having a mental block, and was wondering if anybody could clarify ...

**2**

votes

**1**answer

217 views

### Weyl group of the restriction of scalars of split reductive group

Let $G$ be a connected algebraic group defined over a field $E$ of characteristic $0$. Suppose $G$ reductive $E$-split and let $T \subset G$ a maximal (split) torus defined over $E$.
Set $G' = ...

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vote

**1**answer

239 views

### Thom-Gysin Sequences and Stratifications

Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...

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**2**answers

337 views

### Is there an almost-direct product decomposition for disconnected reductive algebraic groups?

$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...

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vote

**1**answer

130 views

### arithmetic group over function fields and its fundamental domain

Let $G$ be a semi-simple algebraic group defined over a global function field $K$.
Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take ...

**1**

vote

**1**answer

343 views

### About isomorphism of $PGL(2)$ and $SO(3)$ [closed]

I need to prove that $PGL_2(\mathbb{R})\cong SO_3(\mathbb{R})$. Abstract considerations show that both can be identified with the group of projective motions of a conic curve. But maybe there is more ...

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**2**answers

186 views

### a conjugacy question in quasi-split reductive groups

I have a somewhat technical question about conjugacy in
quasi-reductive groups.
Let $k$ be a field (in my main case interest, $k$ is finite), $G$ be a connected
quasi-split reductive group over ...

**2**

votes

**0**answers

86 views

### Integral conjugacy vs. Rational conjugacy

Let $G$ be an algebraic group over a field $F$. Let $g\in G(F)$, and write $C(g)$ for the centralizer of $g$ in $G$. Conjugacy over $F$ is of course not necessarily the same thing as conjugacy over an ...

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**0**answers

113 views

### exponential map for finite group schemes?

Hi,
I am trying to define an exponential map for finite abelian group schemes. The following
looks like it should work, but doesn't (see below). I am putting up this question hoping that someone will ...

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**0**answers

72 views

### Seeking a generalization of group embedding of symmetric varieties

I am looking for generalizations of the following construction.
Let $G$ be a connected, reductive group and let $\theta : G \rightarrow G$ be an involution. Let $H = G^{\theta}$ be the subgroup of ...

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**1**answer

121 views

### A question about $R$-points of an complex reductive group.

I hope somebody can give me a good reference for the following:
Let $G$ be a complex reductive group $H$ be a closed subgroup. Let further $R$ be any $\mathbb{C}$-algebra. Then the canonical map
...

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**0**answers

216 views

### Which p-adic algebraic groups are type I?

It was proved by Jacques Dixmier (Sur les représentations unitaires des groupes de Lie algébriques, Annales de l'institut Fourier, 7 (1957), p. 315-328, doi: 10.5802/aif.73, MR 20 #5820 , Zbl ...

**3**

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**1**answer

282 views

### Centralizers of Nilpotent Elements in Semisimple Lie Algebras

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of ...

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**2**answers

407 views

### Finite subgroups of $PGL(3,K)$

It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...

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**0**answers

71 views

### Base change of affine group schemes with respect to Frobenius map.

Suppose $G$ is an affine group scheme over a perfect field $k$ of characteristic $p>0$. Let $G^{(p)}$ be the base change of $G$ with respect to the Frobenius map of $k$ (i.e. $p$-th power map). Is ...

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**4**answers

717 views

### Simply connected algebraic groups and reductive subgroups of maximal rank

Recall that a connected semisimple algebraic group $G$ over an algebraically closed field $K$ of arbitrary characteristic was defined by Chevalley to be simply connected if the character group $X(T)$ ...

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91 views

### On the explicit formula of the height function occuring on the doubled Weil representation.

Hi! I am wondering the exact formula of height function of $GL(n)$ which occurs in the doubling Weil representation. To be more precise, let me introduce the basic setting for this.
Let $F$ be the ...

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**1**answer

145 views

### Name for a class of parabolic subgroups

This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:
Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, ...

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**3**answers

503 views

### Naive question about the representation theory of algebraic groups and hopf algebras

I have been learning some representation theory and have some questions about the following pattern:
Instance 1: If we have a finite group $G$ and a field $k$, a representation of $G$ over $k$ ...

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**2**answers

393 views

### Morphisms $\mathrm{SL}_n(\mathbb Z) \to \mathrm{SL}_m(\mathbb Z)$

From a result Obtained by O. Schreier and B. L. van der Waerden [Math. Sem. Univ. Hamburg 6, 303- 322 (1928)], one can show that for two fields $\mathbb F$ and $\mathbb G$, and integers $n>m>2$, ...

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**0**answers

69 views

### Degree of a commutator in a hyperalgebra or enveloping algebra

Consider a semisimple algebraic group $G$ over an algebraically closed field of arbitrary characteristic and let $\bar U(G)$ denote its hyperalgebra (ie, the restricted Hopf dual of the coordinate ...

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241 views

### Quotient of a reductive group by a non-smooth subgroup

This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup.
Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic ...

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vote

**1**answer

195 views

### Fixed points of group action

Let us consider the group $PGL(2,\mathbb{R})$ as the group of automorphisms of real projective line and $H\subset PGL(2,\mathbb{R})$ is a subgroup of prime order $> 2$. Is it true that there always ...

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100 views

### on geometric Satake and functions

Let $G(F)/G(O)$ the affine grassmanian with $F=k((t))$ where $k$ is a finite field.
For $\lambda$ a dominant cocharacter, we have by Cartan decomposition the schubert strata ...

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202 views

### Reductive Lie Groups and Complexification

Let $G$ be a complex Lie group (not necessarily connected) with reductive Lie algebra $\frak{g}$. (We may assume that $G$ has finitely many connected components and is linear-algebraic.) Of course, ...

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167 views

### A Criterion for Reductivity of Lie Subgroups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...

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**1**answer

198 views

### Decomposing Semisimple Perverse Sheaves

So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do ...

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**1**answer

168 views

### S-arithmetic subgroup question

I've been reading a proof concerning S-arithmetic subgroups of algebraic groups and I'm having trouble determining why the following step should be true. First, the setup:
Let $G$ be a connected ...

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129 views

### Do there exist pseudo-reductive (but not reductive) groups of small dimension?

I am working on questions in linear algebraic groups $k$, where $k$ is a local field of positive characteristic $p$. I would like to exclude some bad behaviour using the assumption that $p$ is ...

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809 views

### Simply connected simple algebraic groups

Before asking the question I should say that I don't know much about algebraic groups and I'm not sure if the question has the right level for MO. If not, please let me know and I will delete the ...

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**3**answers

918 views

### Small index subgroups of SL(3,Z)

I would like to know the smallest index subgroups of SL(3,Z).
The smallest I could find has even entries $a_{3,1}$ and $a_{3,2}$,
along the bottom row. I could not figure out whether there are
...

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**2**answers

218 views

### Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (ie. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...

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58 views

### reduced group covers of a curve

Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a ...

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44 views

### on degree zero elements in adelic groups

Let $G$ a split connected reductive group and $G(\mathbb{A})$ his points in the ring of adeles.
We have a degree map $G(\mathbb{A})\rightarrow X_{*}(Z)$ where $Z$ is the center of $G$.
Let ...

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**4**answers

573 views

### Classification of Tori of GL2, up to conjugation

Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers ...

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110 views

### Frobenius kernel for unipotent algebraic groups

Let $G$ be a connected algebraic group in positive characteristic $p$. If the Frobenius kernel $G_{(p)}=ker (F:G\to G^{(p)})$ is unipotent, do we have $G$ also unipotent?

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159 views

### Extension of unipotent algebraic groups

Let $G$ be an algebraic group with closed normal subgroup $N$. Suppose that $N$ and $G/N$ are both unipotent. Does it imply that $G$ is also unipotent?

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80 views

### on z-extensions

Let $G$ a group split over a local field $F$.
We call a $z$-extension a group $G'$ such that $G'_{der}$ is simply connected, $G'$ is a central extension of $G$ by a central torus $Z$.
Can we find a ...

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153 views

### minuscule representations and classical groups

Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...

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**2**answers

336 views

### The Lang isogeny

Let $G$ be a connected commutative algebraic group over $\mathbb{F}_q$. If $\text{Fr}_q : G \to G$ denotes the $q$-Frobenius morphism, we define the Lang isogeny $L_q$ to be the endomorphism of $G$ ...

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291 views

### degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$

How can we find the degrees of the invariants for the action of $SL(V)$ on $\wedge^4V$ , $dimV=8$ by the model in the Lie algebra $E_7$.

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352 views

### Dimension of Unipotent Radicals

A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...

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366 views

### Automorphisms of $SL_n$ as a variety

What are the automorphisms of $SL_n$ as an algebraic variety?
In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of ...

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**1**answer

483 views

### General Bruhat decomposition (with parabolic not necessarily Borel)

Here is the general Bruhat decomposition (which I have seen in various paper but never with a proof or a complete reference).
Let $G$ be a split reductive group, $T$ a split maximal torus and $B$ a ...

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**1**answer

294 views

### center of the centralizer of semisimple element

Let $G$ be an adjoint group over an algebraically closed field $k$ and $s\in G$ a semisimple element.
Let $H=C_{G}(s)^{0}$ the neutral component of the centralizer of $s$. Do we have that the center ...

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228 views

### Conjugation of homogeneous spaces

Let $X$ be a smooth irreducible algebraic variety
over the field of complex numbers ${\mathbb{C}}$.
Let $x\in X({\mathbb{C}})$.
Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily ...

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vote

**1**answer

268 views

### Questions about multiplicative homomorphism of $\mathbb{R}$

Regard $K=\mathbb{R}-\lbrace{0\rbrace}$ as a multiplication group. Let $f:K\to K$ be a multiplication homormorphism.
Question 1. Whether that $f$ is surjective implies that $f$ is injective?
...

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**0**answers

151 views

### Classify cross-sections of the adjoint quotient for a semisimple algebraic group?

[This question arises from trying to understand an incompletely formulated earlier question
here.]
Let $G$ be a semisimple algebraic group over an algebraically closed field $K$ of
characteristic ...

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votes

**2**answers

362 views

### Center of the algebraic group G_{\mathbb{R}} for a centerless G

This must be an easy question but I don't have a good argument for it and have not found a counterexample: Let $G$ be a connected semisimple algebraic group over $\mathbb{Q}$ such that the center of ...