# Tagged Questions

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71 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 ...

**3**

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238 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 ...

**3**

votes

**1**answer

181 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 ...

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155 views

+50

### 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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50 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 ...

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**3**answers

458 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 ...

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**1**answer

214 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**

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**1**answer

142 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 ...

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68 views

### Splitting for Subsequence of Automorphism Sequence for Algebraic Groups

Let $G$ be a split reductive algebraic group over an arbitrary field $k$ Suppose we have a split maximal torus $T$. There is a short exact sequence of groups
$$
1\to \mathrm{Inn}(G)\to ...

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**1**answer

157 views

### Is a semisimple conjugacy class closed?

Let $G$ be any algebraic subgroup of $\mathrm{GL}_n$ over an algebraically closed field of any characteristic.
If $s$ is a semisimple element of $G$, can the $G$-conjugacy class of $s$ fail to be ...

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**1**answer

228 views

### rationality question while dealing with an isogeny

I don't think that the following is known, but before going to other things, I would like to know what can be said about it. Thanks in advance for any relevant comment !
So here is the situation. Let ...

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**1**answer

171 views

### fundamental group and torus action

Let $T$ be the complex torus acting on a complex connected algebraic variety $X$
and let $p \colon X\rightarrow Y$ be a good quotient for this action.
For any $y\in Y$ we have a sequence $p^{-1}(y) ...

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vote

**2**answers

183 views

### Duality for group variety

For any abelian variety $A$, there is a dual abelian variety $\hat{A}$ which parametrizes degree zero line bundles.
Is it possible to expect similar duality for group varieties (suppose over ...

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**1**answer

193 views

### About the pro-algebraic group structure of $G(\mathbb{C}[[t]])$

I hope this is not too elementary!
Let $G$ be a algebraic reductive group over $\mathbb{C}$.
The group $G(\mathbb{C}[[t]])$ can be given the structure of a pro algebraic group as follows.
Let $l\in ...

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votes

**1**answer

185 views

### Quotient of an algebraic group by a closed algebraic subgroup

Let $G$ be a complex, linear algebraic group and $H\subseteq G$ a closed and normal subgroup. Then, the quotient $G/H$ has the structure of a affine variety. I am looking for the most "modern" ...

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**1**answer

519 views

### automorphisms of fat points

Let $k$ be an algebraically closed field. I am looking for an
easily quotable description of automorphism groups of $\mathrm{Spec} k[x]/(x^n)$. I could compute explicit matrix representations for ...

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**2**answers

224 views

### Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...

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214 views

### conjugacy classes in anisotropic semisimple groups

You have an anistropic semisimple algebraic group $G$ defined over a non-archimedean local field $k$. When can you say that the $k$-rational part of the conjugacy class of a $k$-rational point is ...

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**0**answers

97 views

### Enveloping algebras of map algebras as hyperalgebras of algebraic groups

This is a continuation of various questions about Chevalley groups over rings, cf these two questions (and a rather bad question of mine here). Consider a semisimple Lie algebra $\mathfrak g$ over ...

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198 views

### Open cell decomposition after applying a Weyl group element

Let $G=\operatorname{GL}(n,\mathbb C)$. What follows can be put into a more general context, but I would like to first understand it for this case, the generalization is a second step.
For ...

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**2**answers

267 views

### Quotient of a rational variety by a finite group

Let $X$ be a rational variety and let $G$ be a finite group acting on $X$. Let us consider the diagonal action of $G$ over the product $X^{h} = X\times...\times X$,
$$G\times(X\times...\times ...

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**0**answers

79 views

### Properties of algebraic vector fields which generates a $\mathbb{C^*}$ action

My question is rather vague and I apologize. Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. I am interested in whether there are homological properties which distinguish algebraic ...

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107 views

### centralisers of maximal split tori

Suppose that $G$ is a reductive group defined over a field $k$ which is not quasisplit. Suppose that $S$ is a maximal $k$-split torus. Let $\mathcal{L}(S)$ be the centraliser of $S$ and ...

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252 views

### isogeny and congruence subgroup

Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By ...

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**1**answer

306 views

### Orbits of group scheme action

I am interested in orbits of the action of a group scheme on a scheme and I'm particularly interested in the following special case: Let $k$ be an algebraically closed field, let $G$ be an affine ...

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173 views

### Chevalley groups over $k[t]/t^n$

This question is motivated partly by a recent question on Chevalley groups over arbitrary commutative rings (and see also this older question). The answers to that question point to a large and ...

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**1**answer

156 views

### Mumford-Tate groups of products of Hodge structures

Let $V_1$, $V_2$ be two polarised simple $Q$-Hodge structures which are non-isomorphic.
I am assuming that the Mumford-Tate groups of $V_1$ and $V_2$ are semi-simple adjoint.
Is it true in this case ...

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**1**answer

251 views

### A question on algebraic loop groops

Setup:
Let $\mathcal{K}=\mathbb{C}((t))$, $\mathcal{O}:= \mathbb{C}[[t]]$ and $G$ be a reductive algebraic group (over $\mathbb{C}$). Let further $\mathcal{K}_n$ denote the $\mathcal{O}$-ideal in ...

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195 views

### Bruhat decomposition for reductive groups in characteristic zero?

Let $G$ be a reductive, linear algebraic group (variety) over an algebraically closed field $\Bbbk$ of characteristic zero. If $G$ is connected, I know from Humphrey's book that for any Borel subgroup ...

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**1**answer

146 views

### regular semisimple elements on spherical varieties

Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$).
What can ...

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39 views

### points with small U stabilizer on a spherical variety

Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of ...

**4**

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262 views

### When does a group action on a k-algebra induce an algebraic action on the spectrum?

This question arose from my last question, which I considered answered - from the comments, however, it is obvious that the answer is only complete in characteristic zero, and I am trying to ...

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87 views

### semisimple conjugacy classes over general bases

Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism.
We know that if ...

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votes

**2**answers

359 views

### Equivariant normalization?

Let $G=\mathrm{Gl}_n\mathbb C$ and let $X$ be an affine $G$-variety. Let $\phi:\tilde X\to X$ be the normalization of $X$, i.e. the spectrum of the integral closure of $\mathbb C[X]$ in its fraction ...

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225 views

### Smooth affine algebraic subgroups as complete intersections

Let's fix an algebraically closed field $k$ of arbitrary characteristic and a connected nonsingular affine algebraic $k$-group $G$. Under what conditions can I assume that a connected nonsingular ...

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**1**answer

143 views

### Equivariant fibre product

Let $G$ be an algebraic group. Let $X$ and $Y$ be $S$-schemes such that $X$, $Y$ and $S$ are $G$-schemes and the structural morphisms are equivariant. My question is: Can the fiber product ...

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**1**answer

331 views

### Action of the endomorphism monoid on an irreducible GL-module

Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...

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**1**answer

209 views

### Existence of quotient variety for group implies existence of quotient for normal subgroups

Let $G=G_1\times G_2$ be a product of two linear algebraic groups over an algebraically closed field. Assume that $G$ acts on a variety $X$ such that the quotient $X\rightarrow X/G$ exists in the ...

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126 views

### determine if a toric variety is Gorenstein

Let $G$ a simply connected group over $k$ and $car(k)=0$.
Let $T_{+}=(T\times T)/Z_{G}$ we consider the closure $\overline{T}_{+}$ of the torus $T_{+}$ in $\prod ...

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55 views

### on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...

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**2**answers

312 views

### Partial (or complete) flag varieties as GIT quotients of affine spaces

I am looking for presentations of partial or complete flag varieties as GIT quotients of affine varieties spaces. That is, for a choice of of dimensions $0=d_1<d_2<\dots<d_k = n$, I would ...

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83 views

### Is a proconstructible subsemigroup of $M_n(\mathbb{C})$ an intersection of constructible subsemigroups?

Let $S$ be a proconstructible subsemigroup of $M_n(\mathbb{C})$, that is a subsemigroup which is an intersection of constructible sets. Is $S$ an intersection of constructible subsemigroups?
The ...

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159 views

### K-theory of categories of group schemes and abelian varieties

Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group ...

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153 views

### Compactifications of group schemes

Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...

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226 views

### Is SL_n/S(GL_k x GL_n-k) symmetric?

Background: a symmetric variety is a homogeneous space $G/H$ associated to an involution $\theta$ of a semisimple algebraic group $G$ and $\{g | \theta(g) = g\} = G^\theta \subset H \subset ...

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**1**answer

347 views

### Deformation theory of octonion algebras?

In Grothendieck's Brauer group papers, he uses deformation theory to bootstrap the theory of central simple algebras over a field to the theory of Azumaya algebras over rings (and schemes). I am ...

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225 views

### Are principal bundles isotrivial?

Let $U$ be a $k$-scheme, where $k$ is a field. Let $G$ be a smooth affine $k$-group. Recall that a principal $G$-bundle over $U$ is a smooth surjective $U$-scheme $E$ with an action of $G$ on $E$ such ...

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**1**answer

255 views

### Group scheme over a DVR whose special fibre is the image of points under reduction mod p

Let $R$ be a complete discrete valuation ring with maximal ideal
$\mathfrak{p}$ and algebraically closed residue field $k$. Denote
the field of fractions of $R$ by $F$. Let $G$ be an affine flat
group ...

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237 views

### For what fields is $GL_n(k)$ a rational variety?

I know that every linear algebraic group is rational over algebraically closed fields. To what extent is that true for other fields? For example: is $GL_n(\mathbb{Q}_p)$ a rational variety? Are there ...

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**2**answers

361 views

### Equivariant Cohomology of a Complex Projective Variety

Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...