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Questions tagged [algebraic-groups]

Algebraic varieties with group operations given by morphisms, or group objects in the category of algebraic varieties, the category of algebraic schemes, or closely related categories.

4
votes
0answers
51 views

Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?

Prove or disprove: Claim: there exists a field $K$ and an integer $n$ such that $\pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $GL_n(K)$? Note that then $G_K \...
-1
votes
0answers
85 views

Specific examples of an algebraic closure of a finite field [closed]

I'm struggling to understand the concept of algebraic closure for finite fields. Are there specific examples I can use to get an intuitive understanding? What sorts of elements do the algebraic ...
2
votes
1answer
122 views

Relative weight lattice

Let $G$ be a reductive group over an algebraically closed field $k$. Let $T$ be a maximal torus, $B$ be a Borel subgroup and $I_G$ is the set of simple roots. Let $P$ be a standard parabolic subgroup, ...
2
votes
0answers
40 views

Maximal tori of a symmetric subgroup

Suppose $G$ is a complex connected reductive algebraic group, $K$ is a symmetric subgroup of $G$ (i.e. the fixed points of an involution $\theta$ of $G$), and $T$ is a $\theta$-stable maximal torus in ...
1
vote
0answers
89 views

Do we have $K \cap P = (K \cap M)(K \cap N)$?

Let $G$ be a connected, reductive group over a $p$-adic field $k$, let $P$ be a parabolic subgroup with Levi $M$ and radical $N$. Let $K$ be a maximal open compact subgroup of $G$ in good position ...
1
vote
0answers
59 views

Homology of SL(2,R) with finite coefficients

Consider the third homology group of a real special linear group $H_3 (SL(2,\mathbb{R}),\mathbb{F}_p)$. It is known$[1]$ that for $p=2$ the third homology group of $SL(2,\mathbb{R})$ vanishes. ...
1
vote
0answers
48 views

When are “square spans” not transversal?

Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...
5
votes
0answers
108 views

Does $G$ act 2-transitively on its Bruhat-Tits building?

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$. Question: If $x,y,x',y'$ are vertices, ...
5
votes
0answers
324 views

Is it true that all smooth group schemes can be deformed?

Consider for instance the map $\mathbb Z/p^2 \to \mathbb Z/p$ and suppose we are given a group smooth scheme $G$ over $\mathbb Z/p$. Is it always possible to lift it to a smooth group scheme $G'$ over ...
7
votes
1answer
395 views

Stabilizer of Sp(n) and U(n) in GL(n)

I would be very grateful for a reference to the following results (which are, I think, true, though I never saw it in the literature). Let $G\subset GL(n,{\Bbb C})$ be $U(n)$, abd $A\in GL(2n,{\Bbb ...
2
votes
1answer
148 views

Symmetric subgroups of simple algebraic groups over finite fields

Let $G$ be a simply connected simple algebraic group over a field $k$. Let $\theta\colon G\to G$ be an involution of $G$ over $k$ (an automorphism of order 2). Let $H=(G^\theta)^0$, the identity ...
1
vote
0answers
57 views

generalization of Bruhat decomposition and $G$-orbits in $(G/B)^n$

Let $G$ be a connected reductive group over an algebraically closed field $k$ and $B$ be one Borel group of $G$. The Bruhat decomposition describes $G$-orbits in $(G/B)^2$ by Weyl group, which is ...
4
votes
1answer
129 views

Constructing algebraic groups of type E6 with split Tits algebras

Let us assume our base field $k$ has characteristic zero. From a series of papers by Borel and Siebenthal it is known that there is an embedding of groups $A_2 \times A_2 \times A_2$ into $E_6$. ...
3
votes
0answers
96 views

Different notions of a stabilizer of a group action in characteristic $p$?

From a paper I'm reading, $G^x$ denotes the stabilizer of a point $x \in X(E')$ in a variety $X$ under the action of an algebraic group $G$. I had a question about what was said at the end. In ...
4
votes
0answers
100 views

Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors

I have a smooth, projective, and rigid Calabi-Yau threefold $X$ defined over $\mathbb{Q}$. Such spaces always have integral models. Let's assume we have an action on $X$ by a finite abelian group $G$...
6
votes
1answer
233 views

“Almost-ideals” in the (simple) Lie algebra of an algebraic group?

Let $G<\mathrm{GL_n}$ be a simple linear algebraic group defined over a finite field $K$. Let $\mathfrak{g}$ be its Lie algebra. Assume $\mathfrak{g}$ is simple. Is it necessarily the case that ...
2
votes
0answers
72 views

Local factoriality of moduli space of semistable G-Higgs bundles on curve

Let $X$ be an irreducible smooth complex projective curve of genus $g \geq 2$. Let $G$ be a connected reductive affine algebraic group over $\mathbb{C}$. Let $\mathcal{M}_{G, Higgs}^{\delta}$ be the ...
3
votes
0answers
143 views

Tamagawa number of GL(n)

Weil's conjecture, proved by Kottwitz, states that the Tamagawa number of a semisimple, simply connected algebraic group (over a number field) is 1. For example, $SL(n)$ and induced tori. Is the ...
5
votes
1answer
205 views

$G(k)/H(k)$ as a submanifold of $G/H(k)$

Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an ...
20
votes
2answers
438 views

$GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class

It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...
1
vote
1answer
105 views

The quotients of double cosets $P_\theta \backslash P_\theta w P_\Omega$ are algebraic varieties over $k$

Let $k$ be a $p$-adic field, $G$ a connected reductive group over $k$ with minimal parabolic $P_0$ containing a maximal split torus $A_0$. Let $W = N_G(A_0)(k)/Z_G(A_0)(k)$ be the Weyl group, and $S \...
6
votes
0answers
155 views

Degree of the graph of the product map

Let $G<GL_n\subset M_n\sim \mathbb{A}^{n^2}$ be a linear algebraic group defined over an algebraically closed field $K$. Consider the graph of the product map: $$V = \{(g_1,g_2,g_1 g_2) : g_1,g_2\...
2
votes
0answers
76 views

Alternatives to the ring of invariants depicting the orbit closures?

Let $G$ be an affine algebraic group over $\mathbb{C}$ and $V=\textrm{Spec}(A)$ an affine $G$-variety. Assume that $G$ is reductive. My understanding is that a central object in studying the action of ...
5
votes
1answer
346 views

What is $\mathrm{O}_q/\mathrm{SO}_q$ if $q$ is a quadratic $\mathbb{Z}$-form which is degenerate?

Any binary quadratic $\mathbb{Z}$-form $q$ induces a symmetric bilinear form $$ B_q(u,v) = q(u+v) - q(u) -q(v) \ \ \forall u,v \ \in\mathbb{Z}^2 $$ and it is considered non-degenerate (over $\mathbb{...
0
votes
0answers
66 views

Is a tempered representation globally generic?

I know there is an general belief that globally generic representation is tempered. I am wondering whether the converse is known, that is tempered representation is generic? If it is not known, is ...
3
votes
2answers
163 views

Representability of Hom of two finite flat group schemes

I am reading a note at Page 63 ftp://ftp.math.ethz.ch/users/pink/FGS/CompleteNotes.pdf It says whenever $G$ is finite and flat over $S$ the functor ${\rm Hom}(G,H)$ is representable. But it does not ...
4
votes
0answers
106 views

Group scheme with isomorphic fibers

Let $X$ be a smooth irreducible algebraic curve over $\mathbb C$. Let $\mathcal G\rightarrow X$ be a smooth affine group scheme over $X$ such that for any closed points $p\in X$, we have $\mathcal G_p\...
2
votes
0answers
44 views

Comparison of length functions on Weyl groups

Let $G$ be a connected reductive group over an algebraically closed field $k$ (with nice enough characteristic), and let $\sigma:G\to G$ be a finite order automorphism of $G$. The connected component $...
1
vote
0answers
50 views

Does Hom functor preserve restricted tensor product?

Let $\pi$ is an automorphic representation of reductive group $G$. Then we can decompose $\pi = \otimes \pi_v$ as restricted tensor product by Flath theorem. I am wondering whether $\text{Hom}_G(\pi,...
2
votes
0answers
117 views

tangent space to a (not necessarily algebraic/Lie/..) group

Are there some standard ways to define the tangent space to a group $G$ at its unit element, $e$, when the group is not (pro)algebraic/(pro)Lie, not necessarily over a field, does not have the ``...
8
votes
0answers
316 views

What is the mirror of an algebraic group?

Background: Kontsevich's homological mirror symmetry conjecture posits the existence of pairs $(X,\check X)$ with an equivalence of dg/$A_\infty$-categories $$\mathcal F(X)=\mathcal D^b(\check X)$$ ...
8
votes
3answers
387 views

How should I think about the Grothendieck-Springer alteration?

Given a simple complex Lie algebra $\mathfrak{g}$, recall the Springer resolution of its nilpotent cone $\widetilde{\mathcal{N}}\to \mathcal{N}$. Several times I have seen someone explaining Springer ...
4
votes
1answer
82 views

Reference for parabolic root systems

Let $G$ be a connected reductive group with maximal split torus $A_0$, and $P = MN$ a parabolic subgroup with Levi $M$ containing $A_0$. Let $A_M$ be the split component of $\mathfrak a_M^{\ast} = X(...
2
votes
0answers
30 views

Adjacent parabolic subgroups and proportionality to $\alpha^{\vee}$

Let $P = MN$ be a parabolic subgroup of a $p$-adic reductive group $G$ with split component $A_M$. There is bijection from the set of parabolic subgroups of $G$ with Levi $M$ and the chambers of $\...
3
votes
1answer
201 views

Definition of simple linear algebraic group

Why is it that many sources define simple (or almost-simple) linear algebraic group $G/k$ to be a connected, semisimple linear algebraic group such that every proper connected normal subgroup is ...
7
votes
1answer
279 views

Subtori of groups of type E6

Let $G$ be a semisimple algebraic group of type $E_6$, defined over a perfect field $k$ (so $G$ is a group scheme over $k$ and $G_{\bar{k}}$ is a semisimple algebraic group in the usual sense), and ...
4
votes
2answers
225 views

How does multiplication affect degrees?

Let $M(n) \sim \mathbb{A}^{n^2}$ be the space of $n$-by-$n$ matrices, seen as an affine space over a field $K$, and endowed with the usual matrix multiplication. Let $V$ and $W$ be subvarieties of $M(...
6
votes
1answer
246 views

Cokernel of map of étale sheaves

Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...
4
votes
0answers
246 views

Bézout and products in algebraic groups

Let $G$ be a linear algebraic group -- be it a Lie group or a group of Lie type. Let $V$, $W$ be subvarieties of $G$. Of course, $V\cap W$ is also a variety (not necessarily irreducible) and $V\cdot W^...
3
votes
0answers
169 views

Representations of GL(n,2) over a field of characteristic 2

I would appreciate very much if you can point to me some references on the following: 1) Representations of the linear group $GL(n,2)$ over $F_2$. 2) Representations of $GL(n,2)$ over an algebraic ...
1
vote
0answers
237 views

Some questions on linear algebraic groups and their eigenvalues

Let $G$ be a connected linear algebraic group (a torus, for example) of finite type over a field $K$. I have several, I suppose rather base, questions concerning the theory of such groups. Suppose ...
3
votes
0answers
91 views

Normalizer of a split torus

Let $G$ be a connected reductive group split over a field $k$. Let $T$ be a maximal split torus of $G$. Consider $N_G(T)$, the normalizer of $T$ in $G$, we have $N_G(T)/T \cong W$, the Weyl group of $...
5
votes
0answers
125 views

Tannakian theory for Lie algebras

Let $G$ be a reductive (just in case) linear algebraic group over $\mathbb{C}$ and let $\mathfrak{g}$ be the Lie algebra of $G$. Consider the category $\operatorname{Rep}(G)$ of finite dimensional ...
4
votes
1answer
158 views

The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators

Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$). How is the embedding $\mathfrak{g}...
12
votes
0answers
268 views

Escaping from a centralizer

Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$. By Stirling's ...
5
votes
0answers
93 views

$\mathbb{F}_q$-rational elements in unipotent classes of a finite group of Lie-type

I've tried posting this question on MSE, but didn't manage to get an answer there, so I'm trying again here. Sorry in advance if this question is trivial or trivially false. I haven't managed to find ...
3
votes
1answer
90 views

Automorphisms of homogeneous space $F_4/P_{\{\beta_2\}}$ over the exceptional group $F_4$

Let $F_4$ be the connected, simply connected, simple, complex, linear algebraic group of type $\mathsf{F}_4$, with Dynkin diagram $$ \beta_1-\beta_2\Rightarrow\beta_3-\beta_4\,. $$ Let $P_{\{\beta_2\}}...
3
votes
1answer
187 views

Reference Request: Anisotropic Algebraic Groups Have No Unipotent Elements

I have found the following fact stated in a number of places: If $k$ is any field, a connected reductive group $G$ is anisotropic if and only if its only unipotent element is $e$ and $\mathrm{Hom}_k(...
3
votes
1answer
594 views

The generalized Kronecker delta and three sets of 16 tetrahedra defined by 192 of the 240 roots of E8 (vertices of Gosset's 8-polytope 4_21)

Original question (without additional information from Wendy): Using 192 of the 240 roots of E8 (vertices of 4_21), Wendy Krieger has defined 48 disjoint tetrahedra this way: Taking the E8 as {128,...
2
votes
0answers
67 views

Classical reductive group schemes vs. unitary groups of separable algebras with involution — reference request

Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-...