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1
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263 views

Rational structures on the flag variety over a finite field

Some Notions A variety over a field is defined to be a scheme of finite type over this field. An $\mathbb{F}_q$-rational structure of an $\bar{\mathbb{F}}_q$-variety $V$ is a $\mathbb{F}_q$-variety ...
2
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0answers
48 views

Springer Isomorphisms for Adjoint Simple Exceptional Groups

I'm trying to understand explicitly a construction of Springer isomorphisms for adjoint exceptional groups given by Bardsley and Richardson. Their construction is as follows. Let $G$ be an adjoint ...
1
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0answers
68 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space ...
6
votes
3answers
403 views

Does a lisse $\ell$-adic sheaf give rise to an affine group scheme?

Let $k$ be a finitely generated field, $\ell$ a prime different from the characteristic of $k$, $S$ a $k$-variety, and $\mathcal{V}$ a lisse $\ell$-adic sheaf on $S$. Fix an algebraic closure ...
5
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1answer
168 views

Over a finite field, does a torsor under the component group of G lift to a torsor under G?

Let $k$ be a finite field and $G$ a finite type smooth $k$-group scheme. Let $G^0$ and $\Gamma$ be the connected component of identity and the component group of $G$, so there is an exact sequence $1 ...
0
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0answers
102 views

Base extension of the Weil restriction

It is a problem in Waterhouse's book, "Affine group schemes, p 61": (a) Let $B$ be finite dimensional (commutative) $k$-algebra. Let $G$ be an affine group scheme over $B$. Define the Weil ...
3
votes
1answer
180 views

The cardinality of first non-abelian Galois cohomology

Let $G$ be a linear algebraic group over a non-archimedean local field $F$. Let $H^1(F,G)$ be the first non-abelian Galois cohomology. It is known that when $F$ is of characteristic 0, i.e. finite ...
8
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1answer
236 views

Known norm varieties and the Bloch-Kato conjecture

The Bloch-Kato conjecture states that $K_M^n(k)/l \simeq H^n(k,\mu^{\otimes n}_l)$ for every $n,l$,while $l$ is invertible in $k$. A important part in the proof of the Bloch-Kato conjecture is to ...
6
votes
1answer
117 views

About G-modules with good filtrations

Let $k$ be an algebraically closed field of positive characteristic, and let $G$ be a reductive algebraic group over $k$ (for instance a classical group). Let $V$ be a (rational) $G$-module. We say ...
6
votes
1answer
91 views

Number of Richardson orbits in simple Lie algebras of types $E_n$?

This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and ...
6
votes
2answers
182 views

$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$

Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has ...
1
vote
2answers
106 views

Explicit description of algebraic hull

I wonder for an explicit description of the "algebraic hull" (and it's associated Hopf algebra) of a given (discrete) group. I know the answer only for finite groups $G$ which is equal to Spec of the ...
2
votes
0answers
50 views

Cohomology and quotients for the canonical topology

Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example ...
0
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0answers
87 views

torsors on quasi-split groups

Let $\mathbf{G}$ be a split connected reductive group scheme over a scheme $X$. Let $X'\rightarrow X$ an étale Galois cover of group $\Gamma$. We consider $G$ a quasi-split group scheme over $X$ ...
0
votes
0answers
56 views

Group schemes decomposition

Given an abelian group scheme of finite type $(G,+)$ over $\mathbb{F}$ connected, and given two connected closed subgroup schemes of finite type $G$ over $\mathbb{F}$ connected $H$, $N$ of $G$. ...
1
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0answers
205 views

Relationship between algebraic groups and Lie groups? [closed]

In the literature, e.g. in representation theory, there seems to be a passage from Lie groups to (linear) algebraic groups. It is clear, particularly over $\mathbb R$ and $\mathbb C$ that they are ...
2
votes
1answer
83 views

dimensions of strata of Pfaffian varieties

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...
3
votes
2answers
145 views

Elementary abelian $p$-subgroups of maximal rank in finite groups of Lie type

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G$ be a reductive group defined over $\mathbb{F}_p$. For any $d\in\mathbb{Z}^+$, let $C_d(G)$ be the set of conjugacy ...
0
votes
0answers
68 views

Commutative algebraic groups endowed with a ring action

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of k-algebraic abelian connected groups $$ 1\rightarrow K\rightarrow G \rightarrow ...
16
votes
3answers
357 views

Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...
3
votes
1answer
357 views

Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...
2
votes
0answers
78 views

surjective homomorphism with compact kernel (Milne's note on Shimura varieties)

I'm reading Milne's Introduction to Shimura varieties (http://www.jmilne.org/math/xnotes/svi.pdf) and there is something I don't get. Let $G$ be a connected semisimple algebraic group $G$ over ...
1
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1answer
66 views

when the derived group of the group of $k$-rational points has nonempty interior in the strong topology

Suppose that $G$ is an absolutely quasi-simple algebraic group defined over a non-archimedean local field $k$ of positive characteristic. Would there be any kind of reasonable sufficient condition for ...
3
votes
4answers
336 views

Reference for an algebraic group preserving a cubic form

Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...
1
vote
1answer
122 views

Tables of data associated to reductive algebraic groups?

I am looking for a reference that contains lots of calculations for specific examples of various objects one can associate to a reductive algebraic group. For example, given a (specific) linear ...
3
votes
2answers
164 views

an algebraic group where the function field is not separable over the ground field

Suppose we have an algebraic group $G$ defined over a field $k$. Suppose we consider the fraction field of $k[G]$. Is it possible to get a situation where this field is not a separable extension of ...
9
votes
2answers
516 views

Is GL2( R ) - > PGL2( R ) surjective?

Consider $GL_2$ as the affine group scheme with coordinate ring ${\mathbb Z}[x_1,x_2,x_3,x_4,y]/(\det\left(\begin{array}{cc}x_1& x_2\\ x_3& x_4\end{array}\right)y-1)$. The group scheme ...
3
votes
1answer
201 views

Quasi-coherent sheaves on classifying stacks

Let $G$ be a smooth group scheme over some base $S$. Then we have the $S$-stack $BG$ whose $T$-points are the $G$-torsors on $T$. Under which conditions do we have $\mathsf{Qcoh}(BG) \simeq ...
12
votes
3answers
572 views

Does every reductive group scheme admit a maximal torus?

A theorem of Grothendieck states that any smooth reductive algebraic group over a field $k$ admits a maximal torus over $k$. My question concerns what happens for schemes. Let $S$ be a scheme and ...
1
vote
1answer
132 views

$\Gamma$-action on maximal tori in Borel-Tits

This is about section 6.2 in Borel-Tits' Groupes réductifs where they define a certain $\Gamma$-action on maximal split tori, denoted as $_\Delta \gamma$, distinct from the "usual" one. (If I am not ...
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0answers
125 views

Reductive Group Actions and Completion

Suppose that $A$ is a Noetherian (not necessarily commutative) $\mathbb{C}[h]$-algebra equipped with a rational action of an affine reductive group $G$, i.e., $A$ is a $\mathbb{C}[G]$-comodule and ...
1
vote
1answer
94 views

Iwasawa decomposition of the pseudo-orthogonal group

This is a soft-question, but I haven't found an answer anywhere: do the factors of the Iwasawa decomposition of the pseudo-orthogonal group SO(p, q) have a simple form, in the same way that the ...
0
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0answers
59 views

What algebras does the hidden subgroup problem for finite abelian groups apply to?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, of which factoring and the discrete logarithm problem for integers belong to. Apparently the HSP for finite ...
6
votes
3answers
384 views

$SL(n) \times SL(n)$-invariants of $m$-tuples of matrices

I work over field of complex numbers. Let $G=SL(n) \times SL(n)$, and $(A,B) \in G$ acts on $m$-tuples of matrices $M_{n \times n}(\mathbb{C})^{\oplus m}$ as follows $$ (A,B) \cdot (M_1, \ldots, M_m) ...
1
vote
4answers
202 views

Bruhat order and Schubert cycles

I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ ...
2
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0answers
89 views

Is the group of rational points of an anisotropic absolutely quasi-simple algebraic group over a non-archimedean local field known to be perfect?

Suppose that $G$ is an algebraic group defined over a non-archimedean local field $k$ which is absolutely quasi-simple and anisotropic over $k$. Is it known whether the group $G(k)$ is necessarily ...
1
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0answers
38 views

abstract simplicity results for anisotropic quasi-simple algebraic groups defined over a non-archimedean local field

I was wondering if anyone has some references I can look at that deal with results that identify some abstractly quasi-simple normal subgroup of the group of rational points of an anistropic ...
8
votes
1answer
525 views

What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense: Conjecture ...
8
votes
3answers
540 views

Tensor products of two irreducible representations of reductive groups and their inclusions

Let $G$ be a reductive group and $\lambda$, $\mu$ and $\nu$ be dominant weights of $G$. Denote by $V_\lambda$ the irreducible representation of $G$ of highest weight $\lambda$. It seems to be true ...
4
votes
1answer
208 views

Is the big cell a principal open set?

Let $G$ be a complex affine reductive algebraic group, $B\subseteq G$ a Borel with maximal torus $T$ and unipotent radical $U$. Let $w\in\operatorname N_G(T)$ be a representative of the longest Weyl ...
1
vote
1answer
308 views

What is the current status of representations of $GL_n(F)$ (and other algebraic groups)?

What is the current status of representations of $GL_n(F)$ (and other algebraic groups)? When $F$ is a local field, the representations of $GL_n(F)$ are classified by Bernstein and Zelevinsky in ...
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0answers
191 views

Pullback of a sheaf associated to a divisor

I am reading a paper Desingularisation des varietes de Schubert generalisees by Demazure. I am interested in Lemma 3 on page 58. In particular, I would like to know whether the lemma is true and how ...
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0answers
248 views

Connection between two theorems on character values?

In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
4
votes
2answers
162 views

Does a spherical building embeds in a building of type $A_n$?

I'm interested in the question in the title. Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$? By embedding I mean an isometric embedding with respect ...
2
votes
0answers
57 views

Homomorphisms between groups of Hermitian type and Hodge type and orthogonality

By a group of Hermitian type I mean a real group $H$ with a maximal compact subgroup $K$ such that $H/K$ is Hermitan Symmetric domain. A real group $W$ is called of Hodge type if the associatd ...
3
votes
3answers
480 views

Non existence of cyclic infinite linear algebraic groups

Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of ...
6
votes
1answer
228 views

Is $G_{\operatorname{red}}$ normal in $G$?

Let $G$ be an affine group scheme of finite type over a field $k$. It is well known that the associated reduced subscheme $G_{\operatorname{red}}$ of $G$ is a subgroup if $k$ is perfect. So let us ...
7
votes
1answer
147 views

Complexity of rational $\mathrm{GL}_{n(r)}$-modules

Let $k$ be an algebraically closed field of characteristic $p>0$, and let $G=\mathrm{GL}_n(k)$ for some natural number $n$. For any integer $r\ge 1$, let $G_{(r)}$ denote the $r$th Frobenius ...
5
votes
1answer
92 views

ubiquity of free subgroups of special linear groups

I have a proof that if $n$ is an integer such that $n>1$ and $k$ is any field, then if $g$ is an element of $\mathrm{SL}(n,k)$ of infinite order then the set of all $h$ with the property that $g$ ...
1
vote
4answers
225 views

About structure of parabolic subgroups of finite classical algebraic groups

Dear Members of Mathoverflow, I am interested about a Fact (if it is right) of the structure of parabolic subgroups of finite classical algebraic groups: Let G be a classical algebraic group over ...