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3
votes
1answer
160 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, ...
4
votes
3answers
514 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 $k$ ...
5
votes
2answers
401 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 $n>m>2$, ...
0
votes
0answers
73 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 ...
5
votes
2answers
246 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 ...
1
vote
1answer
198 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 ...
1
vote
0answers
101 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 ...
3
votes
0answers
215 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, ...
1
vote
1answer
174 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 ...
0
votes
1answer
214 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 ...
2
votes
1answer
178 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 ...
2
votes
0answers
134 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 ...
8
votes
2answers
941 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 ...
15
votes
3answers
952 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 ...
2
votes
2answers
226 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 ...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
7
votes
4answers
613 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 ...
1
vote
0answers
112 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?
1
vote
1answer
172 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?
1
vote
1answer
83 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 ...
0
votes
0answers
160 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 ...
3
votes
2answers
348 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 $G$ ...
3
votes
2answers
298 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 $E_7$.
2
votes
3answers
374 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 ...
9
votes
2answers
371 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 ...
6
votes
1answer
541 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 ...
1
vote
1answer
309 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 ...
5
votes
1answer
237 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 ...
1
vote
1answer
270 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? ...
3
votes
0answers
162 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 ...
3
votes
2answers
372 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 ...
1
vote
3answers
296 views

How we obtain information about a variety from an algebraic group acting on it

Let $G$ be an algebraic group acting on a variety $V$. Which information can be obtained by looking the action of $G$, and subgroups of $G$ that fixes points of $V$?. In other words how we obtain $V$ ...
3
votes
2answers
165 views

On local parameters at the origin in an algebraic group

Let $k$ be an algebraically closed field and $G$ an algebraic group over $k$ which is also a $k$-variety (so $G$ is integral, etc). Let $I$ be the ideal defining the identity $e \in G$ and let $\{ ...
2
votes
2answers
436 views

Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...
9
votes
2answers
521 views

Rational orthogonal matrices

``everybody knows'' that an integral orthogonal matrix is a signed permutation matrix, so there are exactly $2^n n!$ such matrices in $O(n).$ Now, what if we ask for the enumeration of elements of ...
4
votes
2answers
369 views

Basic question about affine group schemes

I've been reading Waterhouse's book "Introduction to affine group schemes", in part to help prepare myself for an (oral) advanced topic exam in algebraic geometry. There is one exercise in chapter 1 ...
3
votes
1answer
288 views

Higgs bundle and stable bundle

Let $(E,\phi)$ be a $G$-Higgs bundle $\phi\in H^{0}(X,ad(E)\otimes D)$ where $D$ is a divisor on X. I suppose that $(E,\phi)\in \mathcal{M}^{ani}$ the anisotropic locus. In particuler, this bundle ...
5
votes
0answers
139 views

On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity. ...
0
votes
0answers
164 views

tangent bundle of the toric variety of the wonderful compactification.

Let G be a adjoint group over $k$,algebraically closed of caracteristic zero. Let $\overline{G}$ be its wonderful compactification. I denote by $\overline{T}$ the closure of the torus $T$ in ...
5
votes
2answers
367 views

Normal subgroups of $SL_2$ of a polynomial ring

What is known about normal subgroups of $SL_2(\mathbb{C}[X])$? Can one hope for a congruence subgroup property, i.e. that every (non-central) normal subgroup contains the kernel of the reduction ...
8
votes
1answer
385 views

What is the universal deformation of the formal additive group $\widehat{\mathbb{G}}_a$ over $\mathbb{F}_p$?

Lubin and Tate show in their paper Formal moduli for one-parameter formal Lie groups that for any formal group over a field $k$ of characteristic $p>0$ with height $h<\infty$, the functor of ...
1
vote
1answer
183 views

How we characterize a subgroup of finite group of Lie type with unipotent elements.

Let $G$ be a finite group of Lie type. Let $H$ be a subgroup of $G$ which contains unipotent elements. I want to find a 'nice' subgroup of $G$ that contains $H$, for example a Levi subgroup of $G$ ...
0
votes
1answer
141 views

subgroups of a $p$-solvable group and complete reducibility

1. Let $G$ be a $p$-solvable group and $V$ be a finite dimensional faithful $kG$-module, where the characteristic of $k$ is $p$. But $V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...
0
votes
0answers
56 views

decomposition lemma in adelic groups II

Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$. Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$. On each point, we have an évaluation morphisme ...
3
votes
2answers
291 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ stabilizing an isotropic subspace. Is the Borel subgroup of $G$ maximal soluble in $P$ and is there ...
0
votes
0answers
95 views

on a decomposition lemma in adelic groups

Let X a curve over an algebraically closed k. Fix $x$ and $y$ two distinct closed points of X. Let G be a connected reductive group over k. We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal ...
3
votes
2answers
301 views

Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers

Consider $2n$ coordinates $x_1,\ldots,x_n,y_1,\ldots,y_n$ and the quadratic form $q = \sum_{i=1}^n x_i y_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with ...
10
votes
4answers
837 views

Is the normalizer of a reductive subgroup reductive?

Let $G$ be a reductive algebraic group over an algebraically closed field (of characteristic zero if it matters) and $H \subset G$ a subgroup, also reductive. Is the identity component of the ...
0
votes
1answer
192 views

For which semisimple element $s$ in finite group of lie type centralizer $C_{G}(s)$ of $s$ is a Levi subgroup ?

Let $G$ be a finite simple group of lie type. Let $s$ be a semisimple element lying in maximal torus $T_{w}$ for $w\in W$ where $W$ is the Weyl group of $G$. Can we say that $C_{G}(s)$ is Levi ...