The algebraic-groups tag has no wiki summary.

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### Etale Fundamental group of an algebraic group

I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where ...

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### generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$.
Let $\mathfrak p$ be a
...

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### Examples of quotients by infinitesimal group schemes

I'm looking for examples of explicit actions of the infinitesimal group schemes $\alpha_{p^n}$ on schemes (maybe as simple as the affine plane) in characteristic $p$ or mixed characteristic, and their ...

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### Algebraic Groups of Type H_3 and H_4 [closed]

By coincidence i stumbled over this page
http://www.fields.utoronto.ca/programs/scientific/11-12/exceptional/abstracts.html
, which was installed for a workshop on algebraic groups in 2012.
In the ...

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### Constant group scheme and torsors

Let $X$ be a scheme and $G$ a (commutative) constant group scheme. Consider a $G$-torsor $Y$ for $X$, by which I mean that there is a canonical isomorphism:
$$g_Y \colon Y \times_X Y \cong Y ...

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### Verifying that a differential is surjective

I've been reading "Weakly commensurable arithmetic groups and locally symmetric spaces" (Prasad and Rapinchuk, 2009). I'm having some trouble showing the following fact:
Let $K_v$ be a local field, ...

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### Does every group that satisfies the maximal permutizer condition then satisfy the permutizer condition?

The
permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,
i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...

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### Stabilisers of group actions

Let $G$ be an algebraic group acting on an irreducible algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$.
Suppose there exists some point $x \in X$ whose ...

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### Binary algebra, is it possible to partition the elements in GF(2^12) into 65 subgroups closed under addition?

The set of all binary vectors with 12 components forms a field with 2^12 elements containing 000000000000 and another 65*63 elements. Is it possible to partition these elements into 65 subgroups of 63 ...

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### Is there a generalization of the “characteristic polynomial” to other split/quasi-split algebraic groups?

Let $G = GL_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. ...

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### Solvable Lie algebras: embedded in upper triangular matrices?

Let $K$ be an arbitrary field and $\mathfrak{g}$ a finite-dimensional Lie $K$-algebra.
Let $\mathfrak{nil}_n\leq\mathfrak{sol}_n\leq\mathfrak{gl}_n$ be the Lie algebras of all ((strictly) ...

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### Regular embeddings of reductive groups

A regular embedding of a connected reductive linear algebraic group $G$ defined over $\mathbb{F}_q$ is a morphism $\varphi : G \rightarrow G'$ of algebraic groups which is a closed immersion where ...

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### 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 ...

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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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### Subgroups generated by opposite root groups

Suppose $\mathbf{G}$ is a connected reductive (possibly non-split!) group over a field $F$, $\mathbf{S} \leq \mathbf{G}$ a maximal split subtorus and $\mathbf{Z} \leq \mathbf{G}$ its centralizer. For ...

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### Motives of a variety of type D4

Over the last decade Nikita Semenov, Skip Garibaldi and others have made some progress in the theory of cohomological invariants, (Rost)-motives and motivic decompositions of algebraic groups. For ...

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### 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 ...

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### Connected components of algebraic groups

Let $G$ be an algebraic group, and $G_{Id}$ the connected component of the identity. Then $G_{Id}$ is a normal subgroup of $G$ and $G/G_{Id}$ is the component group of $G$.
Let $G_{c}\subset G$ be ...

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### Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable
extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial?
This follows from the conjecture that the maximal ...

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### Picard group of classifying stack

Suppose $S$ is a scheme, and $G$ a smooth $S$-group scheme.
Then there exists an algebraic stack BG called the classifying stack of $G$, defined as the quotient stack $[S/G]$ where $G$ acts trivially ...

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### Finite field analogue of representations in same packet have equal central character

In Kevin Buzzard's recent question, a warm up question was: if two automorphic representations are nearly equivalent, then are the central characters of their local components equal?
Working my way ...

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### Is the upper boundary of a Schubert variety Cartier?

On $G/B$, the divisor $\bigcup_\alpha X_{r_\alpha}$ is Cartier (where $X_w := \overline{B_- w B}/B$, and $\alpha$ varies over simple roots), not least because $G/B$ is smooth.
Is the same true for ...

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### Explicit equations for Schubert varieties

How can one compute the Schubert variety (by compute I mean having actual polynomials that define it) for SL(n)? If this is well known forgive my ignorance and just point me to the right book/paper.
...

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### uniqueness of quotients of principal congruence subgroups

For each $n \geq 2$, is $\Gamma(2^{n})$ the unique normal subgroup of $\Gamma(2)$ with quotient isomorphic to $\Gamma(2) / \Gamma(2^{n})$ (here we are talking about principal congruence subgroups of ...

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### What is the cohomology of the tangent bundle of a flag variety?

Let $G$ be the general linear group $\operatorname{GL}(n,\mathbb{C})$ and $P$ a parabolic subgroup with Lie algebra $\mathfrak{p}$. Consider the vector bundles
$$
\mathcal{P} = G\times_P ...

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### Rational group scheme

Suppose $G$ is a group scheme over a field $k$, i.e., $G$ is a functor from the category $\text{Alg}_k$ of unital commutative, associative $k$-algebras to the category of $\text{Groups}$. Suppose that ...

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### Converse to Weil Restriction of Scalars

Let $k$ be a field of characteristic zero (I'm only interested in number fields), and let $\mathbb{G}_{/k}$ be a linear algebraic group defined over $k$ which is almost $k$-simple (all normal ...

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### Torsors under the group scheme over projective line

Consider the group scheme $\mathcal T$ over $\mathbf P^1$ given locally (variable $t$) by the equation
$x^2 - f(t)y^2 = 1$
where $f(t)$ is a polynomial of degree $r$ with distinct roots (assume that ...

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### Stalks of higher direct image under open embedding

Let $U$ be an open subset of $\mathbb P^1$ without two points (say $t=0$ and $t=\infty$) and $j: U\to \mathbb P^1$ be an open immersion. Ground field $k$ is algeraically closed. Let $G$ be the group ...

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### Is an algebraic space group always a scheme?

Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme?
My feeling is that the answer is "yes" ...

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### Invariant subalgebra and dual torus for symmetric group

Given permutation module with three generators and corresponding Galois action of symmetric group $\mathfrak S_3$ I am interested in computing corresponding dual torus $T$ (which should be of ...

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### Are Zariski connected and closed semisimple subgroups of semisimple and simply connected algebraic groups again simply connected? [closed]

Let $G$ be a semisimple and simply connected linear algebraic group over $\mathbb{C}$.
Let $H$ be a connected, Zariski closed and semisimple linear algebraic $\mathbb{Q}$-subgroup of $G$.
Is $H$ a ...

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### A bijection between Lusztig series induced by inflation

Context:
Let $\pi: \widehat{G} \rightarrow G$ be a surjective morphism between connected reductive groups defined over $\mathbb{F}_q$ whose kernel is a central torus. Then $\pi : \widehat{G}^F ...

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### When are toral orbits in buildings the difference of fixed-sets?

Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: ...

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### Geometric interpretation of Cusps for general groups?

Let $\mathrm{G}$ be a reductive group over a number field $F$, but for simplicity we can think about $\mathrm{G}=\mathrm{GL_n}$ for $n>2$ and $F =\mathbb{Q}$.
Then for an automorphic form,
...

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### $G_\mathbb{Z}$-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 ...

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### Automorphisms of a quotient variety

Let $X$ be a variety, and $G\subset Aut(X)$ a subgroup of the automorphism group of $X$. Assume that the quotient $Y = X/G$ is a variety. Does there exist some simple relation between $Aut(X)$, $G$ ...

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### A question on the effective cone

Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$.
I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In ...

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### Exactness on rational points of algebraic groups

Let $k$ be a finite extension of the p-adic number field $Q_p$ and G be a connected algebraic (not affine) group over $k$. It is well-known (see e.g. [1] Proposition 3.1) that G decomposes as
...

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### Symplectic K-theory

For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...

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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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### Equivariant Derived Category

If $G$ is an connected unipotent group over $k$,and $X$ a scheme of finite type over $k$, (an algebraic closed field of positive characteristic) then we can define the bounded derived categorie of ...

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### Dimension of affine Springer fiber and its functor of points as an ind-scheme

Let $k$ be a finite field and let $F = k( (t))$ with ring of integers $\mathfrak{o} = k[ [t]]$. Let $G$ be a connected linear algebraic $k$-group with Lie algebra $\mathfrak{g}$. Suppose that ...

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### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

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### rational representation of semisimple algebraic group

Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$?
...

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### Hyperspecial parahoric group schemes/Chevalley groups

Let $G$ be a simple group over $k=\mathbb{C}$, $A=k[[t]]$, $K=k((t))$, and consider the group $G(K)$. This group is a split reductive group over a local field, and therefore the results of Bruhat and ...

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### LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...

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### Derived subgroup of rational points versus rational points of derived subgroup

Let $\mathbf G$ be a connected algebraic group defined over a field $\mathbb F_p$. If $q=p^n$, then the groups $\mathbf G^\prime (\mathbb F_q)$ and $\mathbf G (\mathbb F_q)^\prime$ are not always ...

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### Bruhat decomposition of $G/Q$

Let $G$ be a semisimple algebraic group over $\mathbb C$, $T$ be a maximal torus and $B$ be a Borel subgroup of $G$ containing $T$. Let $R^+$ be the set of positive roots with respect to $B$. Let $Q$ ...

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### Left U_n-invariants of SL_n - an exercise in Kraft-Procesi

I am sorry for spamming MO with questions I have not thought about for more than 3 hours, but currently I am quite busy with preparing a talk on representations of $S_n$, and I don't want these to get ...