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.

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6
votes
1answer
287 views

Do representations of real semisimple algebraic group have to be algebraic?

If $G$ is the real points of a semisimple algebraic group and $\rho:G\to GL(n,\mathbb R)$ is continuous representation. Is $\rho$ an algebraic morphism?
2
votes
1answer
178 views

Is it possible to describe the action of the Weyl group on the cohomology of the fibers of the Grothendieck-Springer resolution?

I am confused about the following: can one describe the action of the Weyl group on the cohomology of each fiber of the Grothendieck-Springer resolution? I only need the case of ${\mathfrak sl}_n$. ...
1
vote
0answers
151 views

Does the functor of taking invariants commute with tensor products? [closed]

Suppose that $G$ is a group acting on a commutative ring $R$, inducing an action on each $R$-module. For any $R$-module $M$, let $M^G$ denote the collection of elements of $M$ invariant under the $G$-...
3
votes
1answer
336 views

Maximal ideal of group ring

Let $R$ be a finite commutative ring with identity and $G$ an finite abelian group. Is there any more conditions (on $R$ or on $G $) under which we can characterize maximal ideals of group ring $RG $, ...
8
votes
2answers
314 views

Parahoric Group Scheme

I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference! thanks
6
votes
1answer
138 views

Pullback of line bundles and existence of divisors representing line bundles

I am studying the proof of Lemma 7.2. on page 108 in Dolgachev's "Lectures on invariant theory". It states (everything is done over the field $k = \overline{k}$): Let $X$ be a normal affine ...
2
votes
0answers
135 views

Form over $ \mathbb{Z} $ of non-split simple algebraic groups over non-archimedean local fields

Here is a basic observation : On page 68 of his article Reductive groups over local fields, Jacques Tits writes: All types of groups listed (...) exist over an arbitrary [non-archimedean local] ...
4
votes
2answers
175 views

Questions about $\mathbb{C}[G/U^-]$ and $\mathbb{C}[B]$

Let $G = GL_n$. By algebraic Peter-Weyl theorem, we have $$ \mathbb{C}[G] = \bigoplus_{\lambda} V_{\lambda} \otimes V_{\lambda}^*, $$ where $\lambda$'s are dominant weights. Let $U^-$ be the ...
2
votes
1answer
272 views

Any representation is a subrepresentation of a direct sum of the regular representation

I need a reference for the following statement: Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $...
6
votes
1answer
322 views

Is there a non-smooth algebraic group scheme in char $p$, all of whose defining relations have degree less than $p$?

Let $k$ be an algebraically closed field of characteristic $p>0$. All the examples of non-smooth algebraic group schemes over $k$ that I have seen (apart from "artificial" examples; see below) have ...
1
vote
1answer
319 views

Classification of finite group schemes over a field

What is known about the classification of finite group schemes over a field? By a finite group scheme I mean $Spec A$ where $A$ is a finite-dimensional algebra over a field. Is there a full ...
6
votes
2answers
360 views

How simple does a $\mathbb{Q}$-simple group remain after base change to $\mathbb{Q}_{\ell}$?

Of course the general answer to the question in the title is: not very simple. I could not think of a better title, so let me explain my question in more detail. I have a number field $E/\mathbb{Q}$, ...
0
votes
0answers
54 views

Relation between $\Gamma$-percuspidal parabolic subgroups and split parabolic subgroups of real semisimple Lie groups

Let $G$ be a reductive algebraic group defined over $\mathbb{Q}$. Let $\Gamma$ be a lattice in $\mathcal{G}:= G(\mathbb{R})$. I am interested in knowing under what conditions on either of $\mathcal{G}...
1
vote
1answer
251 views

Generalization of a theorem of Steinberg

Steinberg has a beautiful theorem counting $F$-stable maximal tori in a reductive group. Here's the version of the result that you can find as Theorem 3.4.1 of Carter's Finite groups of Lie type: ...
2
votes
0answers
73 views

Connected components of a certain real homogeneous space

Let $m>0$ be a natural number. Consider the following semisimple algebraic groups over ${\mathbb{R}}$: $$ G={\mathrm{SU}}(2m,4m),\ \ H={\mathrm{SU}}(2m,2m)\times{\mathrm{SU}}(2m). $$ We embed $H$ ...
4
votes
1answer
160 views

Matrix from a homomorphism of simply connected groups

Let $H$ be a simple algebraic group of type $\mathbf{G}_2$ over $\mathbb{C}$. Let $\rho$ be the standard 7-dimensional complex representation $$ \rho\colon H=\mathbf{G}_2\to \mathrm{SO}_7.$$ We ...
2
votes
1answer
109 views

How to associate a proper parabolic subgroup of a real s.s Lie group $G$ to a non-trivial unipotent element in a non uniform lattice in $G$?

Let $G$ be a real semi-simple Lie group. Let $\Gamma$ be a non-uniform lattice in $G$. Then it is known that $\Gamma$ contains a non-trivial unipotent element. When $\mathbb{R}$-rank of $G$ is 1, it ...
3
votes
2answers
216 views

degeneration of reductive group

If $A$ is a mixed characteristic complete DVR (I'm only actually interested in $\mathbf{Z}_p$) and $G/A$ is a closed subgroup scheme of $GL(n)$ whose generic fibre is connected reductive and split, is ...
2
votes
0answers
120 views

Hasse diagrams of G/P_1 and G/P_2

in the Paper http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.5052&rep=rep1&type=pdf at the end, we can see Hasse diagrams for several projective, homogeneous $G$-varieties for $G$ ...
9
votes
1answer
255 views

Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$. Is the set of $S$-isomorphism classes of $G$-torsors ...
4
votes
1answer
245 views

Lifting torsors in characteristic $p$ to characteristic zero

Let $R$ be a local integral domain with residue field $k$ such that $R$ is of characteristic zero and $k$ is of characteristic $p>0$. Let $G$ be a smooth finite type affine group scheme with ...
3
votes
2answers
150 views

generalization of result on K_1 of $SL(n,R)$

Let R be a "nice" ring with 1 (e.g. Euclidean domain). Then the subgroup E(n,R) generated by the elements $I+te_{i,j}$ is equal to $SL(n,R)$. My question is as follows: Instead of $SL(n,R)$ I look ...
2
votes
0answers
155 views

Reference request: proofs of the theorems in the paper “On the representation of the group GL(n, K) where K is a local field”

In the paper On the representation of the group $GL(n, K)$ where $K$ is a local field by Gelfand and Kazhdan, it is said that the proofs of the theorems in the paper are published in some other papers....
1
vote
0answers
187 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 $$ G=\{\begin{...
3
votes
3answers
312 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 $SL_2$ ...
3
votes
1answer
259 views

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 ...
1
vote
1answer
134 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 ...
4
votes
1answer
213 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
1answer
112 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
2answers
308 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 $\mathrm{Spec}(\...
0
votes
1answer
182 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 G\...
2
votes
0answers
138 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
1answer
283 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 $G$ ...
10
votes
1answer
340 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!
6
votes
0answers
140 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, $\mathbb{...
2
votes
3answers
426 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 ...
5
votes
0answers
223 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, ...
4
votes
0answers
196 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 ...
0
votes
2answers
156 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 ...
17
votes
2answers
1k 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 ...
8
votes
1answer
411 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
1answer
125 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
0answers
97 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
1answer
115 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 ...
5
votes
2answers
201 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 ...
3
votes
1answer
175 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
2answers
224 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
1answer
86 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 ...
1
vote
1answer
100 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, $...
4
votes
1answer
142 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 \...