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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
387 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
513 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
342 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
275 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
133 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
157 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 ...
4
votes
2answers
353 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
373 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
181 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
55 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
280 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
287 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
806 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
182 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 ...
7
votes
1answer
279 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 ...
6
votes
1answer
580 views

Groups becoming algebraic groups

Let $G$ be an algebraic variety over an algebraically closed field $k$ (any characteristic). Suppose that: (1) the set of $k$-points has the structure of a group. (2) for any $g\in G$ the ...
5
votes
1answer
369 views

Confusing Point in Proof: Semisimple Automorphism Fixes Torus

I am reading a proof on p.51 of Robert Steinberg in his book "Endomorphisms of Algebraic Groups" and I am having a bit of difficulty understanding one point in the proof. The setting is as follows. ...
2
votes
0answers
184 views

finite etale group scheme over a field

Could I have an example of a finite etale group scheme over a field k which is not a constant group scheme? I just know that the category of etale group schemes over a k is equivalent to the category ...
0
votes
1answer
292 views

On the Steinberg section

Let $\chi:G\rightarrow T/W$ the Steinberg map. I assume that G is simply connected. Then $T/W=\mathbb{A}^{r}$ and Steinberg constructed a section to this map given by ...
8
votes
1answer
424 views

Recovering classical Tannaka duality from Lurie's version for geometric stacks

In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects $$ f \colon X \to Y$$ is equivalent to giving a corresponding pullback ...
6
votes
2answers
303 views

Measuring how far from being cocompact a lattice is

Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$ that is invariant under the action of $G$ by ...
1
vote
1answer
173 views

Describing a matrix group (with integer coefficients) through conditions on the coefficients.

I'm wondering if there's always a (not too complicated?) way to characterize a matrix group by conditions on the coefficients. I know if I'm dealing with matrix groups over a field, then it's sort of ...
4
votes
1answer
335 views

Representations of reductive groups over local fields through parahoric induction

Let me take $G$ to be a simple (connected) split reductive group over a local field $K$. One way I might go about constructing a (smooth, admissible) complex representation $\sigma$ of $G$ is as ...
3
votes
1answer
400 views

Are extensions of linear algebraic groups (over a field) themselves linear algebraic?

The title says it all. A very similar question was asked and answered about linear groups, but none of the counterexamples are algebraic: Are extensions of linear groups linear? If $A$, $B$ are ...
5
votes
1answer
320 views

Involution of the Fermat quartic

Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by $$x^4-y^4-z^4+w^4 = 0$$ and consider the involution $$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$ The surface $X$ can be seen ...
8
votes
1answer
268 views

On q-Demazure operators

Setup Let $G$ be a semisimple algebraic group over an algebraically closed field of arbitrary characteristic with Borel subgroup $B$. Let $\Lambda$ denote the weight lattice of $G$; we write elements ...
3
votes
3answers
383 views

Irreducibility of fundamental Weyl modules

It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corresponding to the ...
1
vote
0answers
146 views

“Higher” Tangent spaces in char-p geometry - definition?

Hi, everyone! I have some construction that requires exact definition. I considered polynomial homomorphisms from $\Bbbk$ to $U_n(\Bbbk)$ (unitriangular matrices), where $char \Bbbk = p>0$ (more ...
0
votes
1answer
318 views

Closed subgroups of a $p$-adic algebraic group

Let $F$ be a finite extension of $\mathbb{Q}_p$ and $G$ the group of rational points of an algebraic group defined over $F$, endowed with the natural topology. Any Zariski closed subgroup $H \subset ...
2
votes
1answer
415 views

Are certain simple Lie groups linear algebraic groups?

Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).) Such a group should automatically ...
7
votes
2answers
397 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 ...
6
votes
2answers
741 views

Mostow's theorem on algebraic groups

In his classical 1956 paper Fully reducible subgroups of algebraic groups Mostow proves the following theorem: Theorem 7.1. Let $G$ be an algebraic group over a field $K$ of characteristic 0, ...
5
votes
1answer
513 views

Why are $S$-arithmetic groups interesting?

Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$. Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion ...
7
votes
2answers
315 views

When is an orbit spherical?

I asked the following question over at math.stackexchange, but got no answers. Maybe it's less well-known than I thought, but I still wanted to ask here: Let's assume we have an affine, reductive, ...
7
votes
2answers
275 views

Separating subspaces in an irreducible representation

Suppose $G$ is a semisimple $\mathbb{R}$-algebraic group with finite center, and suppose $G$ acts irreducibly on a vector space $V$. Suppose $U \subset V$ and $W \subset V$ are subspaces. ...
3
votes
1answer
235 views

Rational automorphisms of semisimple algebraic groups

Suppose $G$ is a semisimple algebraic group defined over a field $k$. Let $\mathrm{Aut}(G)$ and $\mathrm{Inn}(G)$ denote the groups of automorphisms and inner automorphisms (respectively) of $G$. ...
2
votes
0answers
317 views

On the structure of commutative group schemes

The structure of a commutative affine algebraic group $G$ over a field $k$ is understood (SGA 3): $G$ has a maximal subgroup of multiplicative type $M$ and the quotient $G/M$ is unipotent. I am ...
5
votes
2answers
717 views

a question on TITS' note “Reductive groups over local fields”

This note appears in "Proceedings of Symposia in pure mathematics" vol.33 1979 part 1 pp. 26-69. The question will be about materials on page 31-32. Let $G$ be a reductive algebraic group (not ...
5
votes
5answers
548 views

Commutator of algebraic subgroups is connected

Let $G$ be an algebraic group over an algebraically closed field. If $H$ and $K$ are closed subgroups and one of them is connected, then their commutator $[H,K]$ is also connected. Is ...
2
votes
0answers
175 views

The fibers of a flat quotient morphism.

Let $G$ be an affine algebraic group, $G_0$ be its derived subgroup, and $S$ be an algebraic monoid with $G$ as its unit group. It can be shown that $k[S]\hookrightarrow k[G]$ as $G\times G$ modules. ...
4
votes
2answers
442 views

Failure of Jacobson Morozov in positive characteristics

The Jacobson-Morozov theorem that any nilpotent $e$ in the lie algebra of a simple algebraic group $G$ can be embedded in an $sl_2$-triple, has a restriction (in terms of the coxeter number) on the ...
19
votes
3answers
882 views

Is SL(2,C)/SL(2,Z) a quasi-projective variety?

Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold). Is ...
5
votes
2answers
471 views

Kostant's theorem on invariant polynomials in positive characteristic

Let $k$ be an algebraically closed field and let $G$ be a reductive linear algebraic group over $k$ with Lie algebra $\mathfrak g$. If the characteristic of $k$ is $0$ then, by a classical result of ...
0
votes
1answer
277 views

For an algebraic group acting on a variety, why are orbits representable?

I suspect this is really obvious, but I'm not seeing it. For an algebraic group $G$ acting on a variety $V$, and for a point $x \in \text{hom}(\text{Spec}(K),V)$, we define the orbit $G(x)$ to be ...
16
votes
1answer
910 views

Example of non-projective variety with non-semisimple Frobenius action on etale cohomology?

This question was motivated by a more general question raised by Jan Weidner here. In general one starts with a variety $X$ (say smooth) over an algebraic closure of a finite field $\mathbb{F}_q$ of ...
3
votes
0answers
175 views

When is a subgroup the Weil restriction of another subgroup?

I asked this question on Math.StackExchange, to no avail. I try my chance on this one. Let $M/F$ be an extension of fields. Let $G$ be an algebraic group over $F$, and consider the $F$-group $H$ ...
7
votes
3answers
557 views

homomorphism into reductive groups

Let $k$ be an algebraically closed field with char($k$)$= p > 0$. Let $P$ be a finite $p$-group. For any homomorphism $\rho : P \rightarrow GL(n,k)$ we know that the image $im(\rho)$ can be put ...