# Tagged Questions

The algebraic-groups tag has no wiki summary.

**1**

vote

**0**answers

94 views

### The representation theory for the fake Heisenberg groups over non-perfect local field

Let $K$ be a local field of characteristic $p$, where $p$ is a prime number greater than 2. In particular, $(x+y)^p=x^p+y^p$ for $x,y\in K$.
The fake Heisenberg group is defined to be
$$
...

**1**

vote

**2**answers

193 views

### Preimage of a maximal compact open subgroup in the simply connected cover

Let $G$ be a semi-simple algebraic group over $Q_p$ and $K$ in
$G(Q_p)$ a maximal compact open subgroup. Let $\tilde{\pi}\colon \tilde{G}\rightarrow G$
be the simply connected cover. Then ...

**2**

votes

**1**answer

181 views

+50

### Volume of arithmetic quotients of symmetric spaces

Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...

**3**

votes

**2**answers

167 views

### Jacobson-Morozov theorem

Jacobson-Morozov theorem for a semisimple algebraic group $G$ (presumably I am working over algebraically closed field) states that: given a unipotent u, there exists a homomorphism $\phi$ from $SL_2$ ...

**0**

votes

**1**answer

173 views

### generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$.
Let $\mathfrak p$ be a
...

**0**

votes

**1**answer

110 views

### Picard group of a quotient of a group by its maximal parabolic subgroup

Let $G$ be a connected, linear, semi-simple algebraic group over an algebraically closed field of characteristic zero and $P$ be the maximal parabolic subgroup. We know that the quotient $Z=G/P$ is a ...

**3**

votes

**1**answer

180 views

### The stack of group algebraic spaces

The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see ...

**1**

vote

**1**answer

75 views

### Compact form of symplectic groups defined over the rationals

I am a bit confused regarding the possible constructions/realizations of symplectic groups. Basically I am looking for the following:
A linear algebraic group $\mathbb{G}$ defined over $\mathbb{Q}$ ...

**3**

votes

**2**answers

225 views

### Representations of complex semi-simple algebraic group “defined over $\mathbf{Z}$”?

If $G$ is a split semisimple linear algebraic group over $\mathrm{Spec}(\mathbf{Z})$ then does every (algebraic) irrep of $G_{\mathbf{C}}$ extend to a morphism $G\to\mathrm{GL}_n$ over ...

**0**

votes

**1**answer

139 views

### Torsors and Central Extensions

In the setting of algebraic groups:
I understand that a central extension of a group $G$ by an abelian group $A$ is a exact sequence of groups :$0\rightarrow A\rightarrow \tilde{G}\rightarrow ...

**2**

votes

**0**answers

89 views

### From algebraic group actions to group scheme actions

I am trying to understand the basic results of geometric invariant theory. I want to pull off the band aid and use Mumford, but am a neophyte with respect to scheme theory. Thus, I have been trying to ...

**3**

votes

**1**answer

233 views

### Equivariant Derived Category

Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.)
Suppose we have an algebraic group $G$ ...

**6**

votes

**1**answer

556 views

### Chopping up Dynkin diagrams

Suppose I have a simple, simply connected (linear) algebraic group $\mathcal{G}$ over an algebraically-closed field $k$, which could have any characteristic. In fact, to keep things simple, let's ...

**9**

votes

**1**answer

306 views

### Embedding linear algebraic groups of a given dimension into a fixed $\mathrm{GL}_N$

Given $n$, can $n$-dimensional linear algebraic groups over $\mathbb{C}$ be embedded into $\mathrm{GL}(N,\mathbb{C})$ for a uniformly bounded $N$?
Thanks so much for your reply!

**4**

votes

**0**answers

103 views

### Root-theoretic formulation of characteristic polynomial

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra of rank $n$ over $\mathbb{C}$. Let $G$ denote the corresponding simple simply connected algebraic group. By Chevalley's Theorem, ...

**7**

votes

**2**answers

416 views

### Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...

**2**

votes

**2**answers

269 views

### Algebraic groups “generated” by a Lie algebra element

Here is a definition which I invented and which I would like to understand better.
Let $ A $ be a complex affine algebraic group. Let $ X \in \mathfrak g $ be an element in its Lie algebra. We say ...

**1**

vote

**2**answers

207 views

### Reference request: Lusztig's symmetries

Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$,
\begin{align}
...

**5**

votes

**0**answers

184 views

### Hyperplane sections of principal homogeneous spaces

Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...

**2**

votes

**2**answers

387 views

### Reference for Unitary Group attached to $E/k$

Unitary groups are very important objects in the setting of Langland's Conjecture because of the existence of Shimura Variety ( which I don't know) and also because people know how to attach a galois ...

**4**

votes

**0**answers

148 views

### Correspondence between real forms and real structures on complex Lie groups

I asked this in MSE, but without success, so I hope, it will be suitable here.
E.B.Vinberg and A.L.Onishchik in their book give the following two definitions.
For a complex Lie group $G$ its real ...

**16**

votes

**2**answers

707 views

### How bad can $\pi_1$ of a linear group orbit be?

Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...

**9**

votes

**2**answers

523 views

### Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...

**0**

votes

**2**answers

123 views

### Is any F-stable maximal torus contained in some F-stable Borel subgroup? [closed]

Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an affine algebraic group over $\bar{\mathbb{F}}_q$, and let $F$ be a ...

**18**

votes

**1**answer

459 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 ...

**0**

votes

**0**answers

83 views

### Surjectivity of ring homomophism induced by Frobenius endomorphism

Denote by $F_q$ the finite field with $q$ elements, and denote by $\bar{F_q}$ its algebraic closure. Let $V$ be an affine $\bar{F_q}$-variety and $F$ be the Frobenius endomorphism corresponding to an ...

**19**

votes

**4**answers

1k views

### algebraic group G vs. algebraic stack BG

I've gathered that it's "common knowledge" (at least among people who think about such things) that studying a (smooth) algebraic group G, as an algebraic group, is in some sense the same as studying ...

**8**

votes

**1**answer

330 views

### Group schemes, adeles, double cosets, and étale cohomology

Let $K$ be a number field, $R$ the ring of integers of $K$,
${\mathbf{A}^f}$ the ring finite adeles of $K$, and ${\widehat{R}}\subset {\mathbf{A}^f}$ the ring of integral adeles.
Let $G$ be an affine ...

**1**

vote

**1**answer

98 views

### about subgroup of general linear group [closed]

Thanks for any comments
Let $G=GL_n(F)$ be general linear group over finite field $F$. Consider two isomorphic subgeoup $H_1,H_2$ of $G$ such that $H_i\cong GL_k(\bar{F})$, where $\bar{F}$ is an ...

**2**

votes

**0**answers

83 views

### Extension of the Hilbert-Mumford Criterion

Let $X$ be a smooth variety, $L$ a line bundle on $X$ and $G$ a reductive group actin on $X$ with a linearization of the action to $L$. Say we are over the complex numbers.
Both the concept of GIT ...

**1**

vote

**1**answer

99 views

### Finite groups normalizing a torus

Let $G$ be a semi-simple linear algebraic group over the complex numbers, e.g. the special linear group. Can you find an example of a finite sub-group $H$ of $G$ which does not normalize any maximal ...

**1**

vote

**2**answers

339 views

### Regular embeddings of reductive groups

A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where ...

**4**

votes

**1**answer

102 views

### Lindel's theorem for semisimple simply connected G

Let $k$ be a field.
$G/k$ be a simply connected semisimple algebraic group.
Let $X/k$ be a smooth affine $k$-scheme.
Question: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back ...

**2**

votes

**1**answer

117 views

### Etale Fundamental group of an algebraic group

I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where ...

**1**

vote

**2**answers

185 views

### Examples of quotients by infinitesimal group schemes

I'm looking for examples of explicit actions of the infinitesimal group schemes $\alpha_{p^n}$ on schemes (maybe as simple as the affine plane) in characteristic $p$ or mixed characteristic, and their ...

**-1**

votes

**1**answer

70 views

### Algebraic Groups of Type H_3 and H_4 [closed]

By coincidence i stumbled over this page
http://www.fields.utoronto.ca/programs/scientific/11-12/exceptional/abstracts.html
, which was installed for a workshop on algebraic groups in 2012.
In the ...

**2**

votes

**0**answers

373 views

### Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y ...

**1**

vote

**1**answer

87 views

### Verifying that a differential is surjective

I've been reading "Weakly commensurable arithmetic groups and locally symmetric spaces" (Prasad and Rapinchuk, 2009). I'm having some trouble showing the following fact:
Let $K_v$ be a local field, ...

**2**

votes

**0**answers

89 views

### Does every group that satisfies the maximal permutizer condition then satisfy the permutizer condition?

The
permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,
i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...

**6**

votes

**2**answers

255 views

### Stabilisers of group actions

Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$.
Suppose there exists some point $x \in X$ whose ...

**2**

votes

**1**answer

94 views

### Binary algebra, is it possible to partition the elements in GF(2^12) into 65 subgroups closed under addition?

The set of all binary vectors with 12 components forms a field with 2^12 elements containing 000000000000 and another 65*63 elements. Is it possible to partition these elements into 65 subgroups of 63 ...

**4**

votes

**3**answers

227 views

### Is there a generalization of the “characteristic polynomial” to other split/quasi-split algebraic groups?

Let $G = GL_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. ...

**3**

votes

**2**answers

165 views

### Solvable Lie algebras: embedded in upper triangular matrices?

Let $K$ be an arbitrary field and $\mathfrak{g}$ a finite-dimensional Lie $K$-algebra.
Let $\mathfrak{nil}_n\leq\mathfrak{sol}_n\leq\mathfrak{gl}_n$ be the Lie algebras of all ((strictly) ...

**9**

votes

**2**answers

477 views

### Examples of non-split algebraic groups

I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my ...

**1**

vote

**1**answer

129 views

### Subgroups generated by opposite root groups

Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For ...

**4**

votes

**1**answer

233 views

### Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...

**9**

votes

**2**answers

365 views

### Differential/difference algebraic groups as “group schemes”

While the common approach to algebraic groups is via representable functors, it seems that there is no such for differential algebraic groups (defined by differential polynomials). Neither the book by ...

**1**

vote

**2**answers

197 views

### Connected components of algebraic groups

Let $G$ be an algebraic group, and $G_{Id}$ the connected component of the identity. Then $G_{Id}$ is a normal subgroup of $G$ and $G/G_{Id}$ is the component group of $G$.
Let $G_{c}\subset G$ be ...

**10**

votes

**0**answers

338 views

### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

**2**

votes

**1**answer

134 views

### Picard group of classifying stack

Suppose $S$ is a scheme, and $G$ a smooth $S$-group scheme.
Then there exists an algebraic stack BG called the classifying stack of $G$, defined as the quotient stack $[S/G]$ where $G$ acts trivially ...