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4
votes
3answers
254 views

What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields?

I thought they were the same, just different names. Let me make question more precise: Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the ...
3
votes
1answer
172 views

line bundle descends?

Let the permutation group $S_4$ act on $\mathbb C^4$ by permuting the coordinates. Consider the categorical quotient $\mathbb P(\mathbb C^4)/S_4$. It is a projective variety by a theorem of Mumford. ...
0
votes
0answers
59 views

Bruhat order in homogeneous spaces

Let $R$ be a root system with simple roots $\Delta$. For all $\alpha\in\Delta$ let $\omega_\alpha$ be the fundamental weight associated to $\alpha$. Is then the length of $\alpha$ equal to the length ...
-4
votes
0answers
52 views

What are “small” finite groups with “exponentially” large expansion? [closed]

Given an integer $k$ suppose one wants the group to be polynomial in size in $k$ but the expansion to be exponential in size in $k$. I am not sure I am asking a precise question. Kindly add in ...
3
votes
1answer
167 views

Special linear groups over function fields

Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements. What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as ...
4
votes
0answers
103 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 ...
4
votes
1answer
314 views

Tannakian fundamental group of two explicit tensor categories

Let $K/k$ is a field extension and $G$ an affine group scheme over $K$. What are the Tannakian fundamental groups of these two $k$-tensor categories (with trivial fiber functors over $k$): 1. The ...
1
vote
0answers
119 views

Zariski dense subgroups and conjugates

Let $H \leq \mathrm{SL}_3(\mathbb{Z})$ be a finitely generated subgroup which is not Zariski dense, and let $g \in \mathrm{SL}_3(\mathbb{Z})$. Must there be some $a \in \mathrm{SL}_3(\mathbb{Z})$ such ...
2
votes
1answer
189 views

twists of algebraic groups

If $k$ is some field - for convenience, of characteristic 0 -, $\bar{k}$ is an alg. closure of $k$, and $G$ is some $k$-algebraic group, one can define a twist of $G$ to be some $k$-algebraic group ...
5
votes
1answer
159 views

Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel. Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and ...
2
votes
1answer
182 views

highest weight representations inside tensor product

Let $G$ be a semisimple simply connected group over an algebraically closed field $k$ of characteristic zero, $B$ a Borel and $T$ a maximal torus. Let $\lambda,\mu,\nu$ be dominant characters of $T$. ...
0
votes
0answers
56 views

adjoint quotient and points in DVRs

Let $G$ be a connected reductive group over an algebraically closed field $k$, $T$ a maximal torus and $W$ its Weyl group. We have a Steinberg map $\chi:G\rightarrow \mathfrak{C}:=T/W$ if we have a ...
1
vote
0answers
77 views

Need information about particular kind of quotients of semisimple algebraic groups by free abelian discrete subgroups

Let me start with the simplest version of the question since already there I don't know anything. For a complex number $q$, consider the quotient space $X_q:=\mathrm{SL}_2(\mathbb ...
6
votes
1answer
156 views

Extensions of an abelian variety by a torus vs. extensions of their $\ell$-adic Tate modules

Let $K$ be a number field, let $A$ be an abelian variety over $K$, and let $H$ be a torus over $K$. For a prime $l$, we have the natural map $$\mathrm{Ext}^1(A, H) \otimes_{\mathbb{Z}} \mathbb{Z}_l ...
1
vote
0answers
68 views

Exponential map on a unipotent group

Let $G$ by a unipotent linear algebraic group defined over a field of characteristic $0$, with Lie algebra $\mathfrak{g}$. The exponential map $\mathfrak{g}\to G$ is bijective, and we can recover the ...
2
votes
1answer
85 views

Rational Points of a Quotient of a Reductive Group by a Parabolic Subgroup

Let $G$ be a reductive group and let $P$ be a parabolic subgroup of $G$ all defined over $\mathbb{Z}$. Also, let $F$ be a number field, is it true (and if so, please provide a reference) that $$ ...
3
votes
0answers
313 views

A Step in the Proof of the Drinfeld-Simpson theorem

I hope that this is the appropriate place for asking about a step I don't understand in a proof which I think is due to a lack of knowledge. This is a step in Drinfeld-Simpson's paper: ``$B$ ...
3
votes
1answer
156 views

How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...
5
votes
1answer
173 views

Closure order on nilpotent orbits in exceptional Lie algebras

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent ...
7
votes
3answers
218 views

Polarizations generate the ring of invariants?

The symmetric group $S_n$ acts on $\mathbb R^n$ by permuting the coordinates and the ring of polynomial invariants is generated by the elementary symmetric polynomials. If we restrict the action to ...
2
votes
1answer
164 views

Lifting one parameter subgroups of algebraic groups

Let $G$ be a linear algebraic group over an algebraically closed field $\mathbb C$ of characteristic zero and $U$ its unipotent radical, then $H:=G/U$ is a reductive group. Assume that I have a one ...
7
votes
1answer
250 views

what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...
1
vote
0answers
118 views

The definition of $SK_1$ for an arbitrary ring

Let $R$ be a unitary associative ring. If $R$ is commutative, then one defines $SK_1(R)$ as the quotient $$SK_1(R)=SL(R)/E(R)$$ (Definition 2.8 of ...
4
votes
0answers
130 views

An extension of group schemes admitting Neron models

Let $R$ be a discrete valuation ring, $K$ its field of fractions, and $$ 0 \rightarrow G_K' \rightarrow G_K \rightarrow G_K'' \rightarrow 0$$ a short exact sequence of smooth $K$-group schemes of ...
4
votes
1answer
238 views

Geometrically connected components of an algebraic group

Suppose that $G$ is an algebraic group over a field $k$. Let $G^o$be the connected component of the identity. Since $G^0$ contains a $k$-rational point (the identity) therefore it is geometrically ...
0
votes
0answers
127 views

Comparison of two Chevalley basis

Let $G$ be a connected reductive group over an algebraically closed field and $T$ a maximal torus. Let $H$ be a pseudo-Levi subgroup, say the neutral component of a centralizer of a semisimple element ...
5
votes
0answers
199 views

Torsors and twists of algebraic groups

Let $G/S$ be an affine group scheme. Then the automorphism group of every $G$-torsor over $S$ is a twist of $G$, but it this functor isn't essentially surjective in general (It may be not full nor ...
1
vote
1answer
112 views

Decomposing quasi-finite separated group schemes

Let $U$ be a punctured disk, and let $G\to U$ be a quasi-finite separated group scheme. (Assume $K$ of char zero if it helps) Why is $G = G_1\sqcup G_2$, where $G_1 \to U$ is finite and $G_2\to U$ ...
4
votes
1answer
124 views

orders of maximal abelian subgroups

What are the orders of maximal abelian subgroups of the simple groups $F_4(q)$ and $C_4(q)$, where $F_4(q)$ is an exceptional group and $C_4(q)$ is a symplectic group?
2
votes
1answer
172 views

Quotient of a (non-linear) algebraic group by a closed subgroup

Let G be a (not necessarily linear) algebraic group and H a closed algebraic subgroup. Does the, say categorical, quotient G/H exist? If yes, where do I find a proof? If no, do you have a ...
3
votes
1answer
206 views

Representation of GL(n, F_p) over F_p, for n small

The question is related to this post Representation theory of the general linear group over a finite prime field However, I am asking for more detailed references for n small, for example, for n=2, ...
1
vote
0answers
122 views

Analogies, Riemann surfaces and Algebraic groups

Let G be a complex simple Lie group of adjoint type. Then, it is well known that every such $G$ contains, unique up to conjugacy, an irreducibly embedded copy of $PSL(2,\mathbb{C}).$ This fact seems ...
1
vote
0answers
695 views

Picard functor of an algebraic group

Let $K$ be a field, $G$ a $K$-group scheme of finite type, and $X$ a $G$-torsor. Is the Picard functor $\mathrm{Pic}_{X/K}$ representable? I recall that $\mathrm{Pic}_{X/K}$ is the fppf sheafification ...
2
votes
0answers
105 views

$\mathbb{Q}$-forms of $\mathrm{SL}_2\times \mathrm{SL}_2$

I am learning something about lattices in algebraic groups. Consider the algebraic group $\mathrm{SL}_2\times \mathrm{SL}_2$. What are the $\mathbb{Q}$-forms of such groups?
2
votes
2answers
188 views

Connected unipotent algebraic groups

Let $G$ be a connected unipotent algebraic (affine) group defined over a perfect field $k$. Then $G$ is $k$-isomorphic as an algebraic $k$-variety to the affine space $\mathbb A^n_k$. Is there any ...
12
votes
2answers
695 views

Is there a scheme parametrizing the closed subgroups of an algebraic group?

In the following, let $G=\operatorname{GL}_n(\mathbb{C})$ or $G=\operatorname{\mathbb PGL}_n(\mathbb{C})$, depending on whichever has a better chance of yielding an affirmative answer. One could more ...
0
votes
0answers
26 views

conjugacy classes of proper semisimple subgroups in parabolic groups

Is it true that there are only finitely many conjugacy classes of proper semisimple subgroups in a parabolic subgroup which in turn is contained in an semisimple algebraic group? Let $G$ be a ...
3
votes
1answer
110 views

On Serre's problem regarding the injectivity of Albert-Algebra cohomological invariants

In these Lecture Notes http://molle.fernuni-hagen.de/~loos/jordan/archive/cohinv/cohinv.pdf from 2006 by Garibaldi on page 21. 7.5 there is the following open problem mentioned: Is the map $g_3 ...
1
vote
1answer
89 views

Reducible reductive Lie subalgebras of so(p,q)

Is it true that $S(O(p) \times O(q))$ is the only proper subgroup of $SO(p,q)$ of full rank acting on the natural representation $\mathbb{R}^{p+q}$ of $SO(p,q)$ that stabilizes a $p$-dimensional ...
0
votes
1answer
78 views

Reductive subgroup and its derived subgroup with an irreducible represenation

Could you please answer the following question: Let $V$ be a faithful irreducible representation of a connected reductive group $H$ defined over $\mathbb{R}$ Is it true that the derived group of $H$, ...
5
votes
1answer
347 views

Representation theory of the general linear group over a finite prime field

I am re-posting a question I asked on math.se here because I am unsatisfied with the answers I obtained. The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely ...
1
vote
0answers
45 views

exponential and anisotropic torus

Let $F$ be a local p-adic field and $G$ a semisimple simply connected group over $F$, $\mathfrak{g}$ its Lie algebra. Let $T$ a maximal anisotropic torus of $G$, split over an etale extension of $F$ ...
5
votes
1answer
167 views

Optimal definition of “paving by affine spaces”?

Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine ...
7
votes
1answer
208 views

Meaning of fibration in Kazhdan and Lusztig's paper on affine flag manifolds

Kazhdan and Lusztig's paper "Fixed point varieties on affine flag manifolds" has the following definition on p.143: define inductively a variety $Z$ to be an "almost affine space" if $Z$ is affine or ...
0
votes
1answer
90 views

on lifting extensions

Let $G$ be a connected reductive group with $G_{der}$ simply connected and $T$ a maximal torus over an algebraically field $k$. We consider a extension $\tilde{T}$ of the maximal torus $T$ by a torus ...
4
votes
2answers
250 views

Is a “central” extension of $\mathbb{Z}/m\mathbb{Z}$ by $\mathrm{GL}_n$ necessarily split?

Let $m \ge 1$ be an integer, let $k$ be a field of characteristic $0$, and let $$ 1 \rightarrow \mathrm{GL}_n \rightarrow E \rightarrow \mathbb{Z}/m\mathbb{Z} \rightarrow 1 $$ be an extension of ...
0
votes
0answers
83 views

endoscopy and simply connectedness

Let $G$ be a connected reductive group with $G_{der}$ simply connected over a local field $F$. Let $\hat{G}$ be its Langlands dual, $s$ a semisimple element in $\hat{G}$ and $\hat{H}=\hat{G}_{s}$. ...
2
votes
0answers
96 views

Generators of the algebra of invariant polynomials on a Lie algebra and the root-space decomposition

Let $G$ be a connected, simply-connected complex semisimple group with Lie algebra $\mathfrak{g}$. Fix a pair $T\subseteq B\subseteq G$ of a maximal torus and Borel subgroup, and let $\mathfrak{t}$ ...
3
votes
1answer
190 views

Is $G \rightarrow G/P$ surjective on $K$-points over a local field?

Let $K$ be a local field, $G$ a (connected) reductive $K$-group, and $P \le G$ a parabolic subgroup. Is the map $G(K) \rightarrow (G/P)(K)$ necessarily surjective, and, if so, then why?
1
vote
2answers
247 views

A question from the proof of affine algebraic group is a linear

In (some version of) the proof of the fact that any affine algebraic group is a linear algebraic group, there is an important step as follows (for example in Borel's book "Linear Algebraic Groups", ...