The algebraic-groups tag has no wiki summary.

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### Examples of non-split algebraic groups

I am interested in knowing various examples of non-split (added hypothesis reductive) reductive linear algebraic groups. In particular, I would like to collect the following examples in my ...

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246 views

### generators for derived category

Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...

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199 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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### Is there any way to determine the “effeciency” of Jantzen's sum formula?

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a reductive algebraic group over $k$.
In order to determine the structure of the Weyl modules $V(\lambda)$, ...

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233 views

### Suslin's Stability Theorem for Chevalley Groups

I am looking for a version of Suslin's Stability Theorem for Chevalley groups.
The version of the theorem for $G=SL_n({\mathbb Z}[x_1, \dots , x_m])$ states that the if $n\ge m+2$, the elementary ...

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### Indefinite orthogonal groups over p-adics

Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...

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65 views

### Regular or elliptic elements in the multiplicative group of central division algebra

For an element $g$ of a connected reductive group $G$ over a field $F$,
$g$ is called $regular$ if the dimension of the centralizer of $g$ is equal to the rank of the algebraic group $G$,
$g$ is ...

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204 views

### What is “special” maximal compact subgroup of algebraig group over local field?

Learning the theory of Langlands correspondence, I met the notion of "special" maximal compact subgroup of a (reductive) algebraic group over a local field.
Here, I think the word "compact" is used ...

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143 views

### Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field),
$g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...

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### Earliest use of the term “linearly reductive”?

Recently a number of MO questions have referred to a "linearly reductive group", usually in a way that is out of focus. It's unclear to me why this terminology is so popular, since over a field of ...

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537 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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### Algebraic characters and quasi-characters of reductive algebraic group over non-archimedean local field

Let $G$ be a reductive algebraic group over $F$, where $F$ is a non-archimedean local field.
Then $G(F)$ is a p-adic group.
Let $\Psi(G)$ be the lattice of algebraic characters.
Let $\Lambda_G$ be the ...

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169 views

### Weyl group invariants in a maximal torus

Suppose G is a semi-simple adjoint group over complex numbers. Suppose T is a
maximal torus in G. Does one know what are the W invariant (non-trivial) elements in T? Perhaps I might give a few ...

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248 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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### Restriction of the Steinberg representation

Let $G$ the group $GL(n,F)$ where F is a locally compact non Archimedean field, and $G^{0}$ the subgroup of $G$ consists of elements $g$ in $G$ such that $\det(g)$ in $\mathcal{O}_{F}^{\times}$, where ...

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254 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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### radical unipotent of a parahoric

Let $G$ a split connected reductive group over $\mathbb{C}$. $F=\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers.
Let $B$ a Borel subgroup and $I$ the corresponding Iwahori.
Let ...

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88 views

### intersection of Borel in wonderful compactification

Let $\overline{G}$ the wonderful compactification of an adjoint $G$ over an algebraically closed field $k$.
Let $B$ a Borel of $G$ and $w\in W$ an element of the Weyl group and $\overline{B}$ the ...

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403 views

### Getting the story of Dynkin and Satake diagrams straight

I've been trying to teach myself the theory of Lie groups. The sources I've been reading reference Lie algebras in the context of Dynkin and Satake diagrams, but not Lie groups (which I am more ...

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### Reference request: Lusztig's symmetries

Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$,
\begin{align}
...

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### 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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### The centralizer $Z_G(X)$ of a nilpotent element in a real simple Lie group

I am looking for the description of the centralizer $Z_G(X)$ , where $G$ is a real simple Lie Group and $X\in \ Lie (G) $ such that $X^d=0,\ X^{d-1}\neq 0 $. It is is helpful to me any references or ...

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### Non-abelian group from affine hermitian curve

I was playing with the Hermitian curve $y^q + y = x^{q+1}$ over the field $GF(q^2)$ and chanced upon the following (non Abelian) group law on the points of the affine curve:
$(a,b) * (c,d) = ...

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### Form of a block upper triangular matrix of finite order

If I take a block upper triangular matrix and the matrix is known to be diagonalizable and the leading blocks in the diagonal are each of finite order, is it true that away from the leading block ...

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### 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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### Which subgroups of a finite reflection group have distingushed coset representatives?

Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of ...

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306 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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### rational conjugacy classes

Say that $G$ is a reductive group over a local field $F$ and $g\in G(F)$. Can we show that the conjugacy class of $g$ in $G(F)$ has finite index in the $F$-rational part of the conjugacy class of $g$ ...

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### 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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### 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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### 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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### 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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### faithful modules of algebraic group

Let $G$ be a linear algebraic group over a field $k$. $k[G]$ is the
coordinate ring of $G$. $k[G]^{*}$ is the dual algebra of the
coalgebra $k[G]$. $H=k[G]^{\circ}$ is the finite dual of the Hopf
...

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### Preimage of a maximal compact open subgroup in the simply connected cover

Let $G$ be a semi-simple algebraic group over $Q_p$ and $K$ in
$G(Q_p)$ a maximal compact open subgroup. Let $\tilde{\pi}\colon \tilde{G}\rightarrow G$
be the simply connected cover. Then ...

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### When does the derived subgroup of $G(F)$ contains the $F$-points of unipotent subgroups of $G$

Let $F$ be a local field of characteristic $0$ and $G$ a connected split reductive group over $F$.
Let's look at the derived groups. We have $(G(F),G(F)) \subset (G,G)(F)$ and this inclusion is of ...

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### Root system automorphisms as inner automorphisms of extended Chevalley group

For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...

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### How to decompose tensor products of simple modules for algebraic groups in GAP (or similar) [closed]

Is it possible to decompose tensor products for algebraic groups (in characteristic zero) in GAP?
I know that GAP has a Littlewood-Richardson rule function and is very good for character tables of ...

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80 views

### Cohen-Macaulayness of the scheme of centralizer

Let $G$ be a simply connected group over an algebraically closed field $k$, and
$I:=\{(g,\gamma)\in G\times G\vert~ g\gamma=\gamma g\}$
the scheme of centralizer.
Is $I$ a Cohen-Macaulay scheme ...

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176 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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### 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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### 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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### Zariski dense subgroups of linear algebraic groups

The theorem of Matthews, Vaserstein and Weisfeiler asserts that if $G$ is a simply connected absolutely almost simple groups over $Q$ and $\Gamma$ a finitely generated subgroup of $G(Q)$ which is ...

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### 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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### Subgroup containing a given torus as a maximal torus

Let T be a non-trivial torus in a semisimple group G of adjoint type defined over Q.
I was wondering whether there exists a reductive subgroup H of G (defined over Q !) such that T is a maximal torus ...

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### Centralizer of a subtorus in a reductive group is Levi?

Questions a bit similar to this one have already appeared I think on the forum but I couldn't find the answer to my question using those answers. I must say from the beginning that my knowledge of ...

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### Higher-dimensional generalization of Pink's theorem

Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...

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### 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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### 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 ...

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### 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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### 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 ...