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.

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Dual root datum and representation category [closed]

Suppose $G$ is the connected reductive group over a field $k$ corresponding to the root datum $(X^*, \Delta, X_*, \Delta^v)$. Let $G^v$ be the connected reductive group over $k$ corresponding to the ...
Peter's user avatar
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3 votes
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The Weil restriction of a simple algebraic group

Let $F$ be a number field, $G$ an $F$-simple affine algebraic group. Then is the Weil restriction $\operatorname{Res}_{F/\mathbb{Q}} G$ $\mathbb{Q}$-simple? I couldn’t find any references.
zom's user avatar
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2 votes
1 answer
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Parabolic subgroup of Weyl group

Let $W$ be the Weyl group of a semisimple algebraic group $G$. $I$ be the simple roots. $J\subset I$ generate a parabolic subgroup of $W$ denote by $W_J$. $w^J$ is the shortest representative of $w$ ...
fool rabbit's user avatar
5 votes
1 answer
193 views

Is the derived group of the G(F) perfect

Let $G$ be a connected reductive group defined over a finite field $F$ of characteristic $> 3$. Is it true that the commutator group of $G(F)$ is perfect? This is true if $G$ is assumed to be ...
Rami's user avatar
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6 votes
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Functions of polynomial growth on linear algebraic groups

$\DeclareMathOperator\GL{GL}$Let $G$ be a complex linear algebraic group, i.e. a subgroup in $\GL_n({\mathbb C})$, defined by a system of polynomial equations $$ p_i(x)=0 $$ (here $p_i$ are ...
Sergei Akbarov's user avatar
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113 views

Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?

Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? Are there some interesting properties ...
Adrien MORIN's user avatar
1 vote
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Why are compact arithmetic surfaces defined through quaternion algebras (usually) only over $\mathbb{Q}$?

As in Chapter 6.2 of "Introduction to arithmetic groups" (by D.W. Morris), compact arithmetic surfaces could be defined through quaternion algebras $\mathbb{H}^{a,b}_F=\big(\frac{a,b}{F}\big)...
Local's user avatar
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9 votes
2 answers
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Difference between $\mathfrak{g}/\!/G$ and $G/\!/G$

I am studying a GIT quotient and I have a question that may be very silly. Let $G$ be a connected reductive group and $\mathfrak{g}$ its Lie algebra. Then are there some differences between $\mathfrak{...
lafes's user avatar
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2 votes
1 answer
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Zeroes of characters of general linear group induced from certain characters of parabolic subgroups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ind{Ind}$My question is about the types of conjugacy classes of $\GL(n,q)$, the general linear group over the finite field with $q$ elements, on which ...
mathseeker's user avatar
1 vote
1 answer
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Free closed group action on varieties

Suppose we are given a reductive group $G$, its closed subgroup $H$ (not necessarily reductive), an affine $G$-variety $X$ and its closed subvariety $Y$ such that (1) The $G$ action on $X$ is free and ...
Randy Ji's user avatar
1 vote
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Orthosymplectic superalgebra

Let $V=V_0 \oplus V_1$ be a $\mathbb Z_2$-graded vector space over $\mathbb C$. Suppose $V$ has an even non-degenerate bilinear form $(-, -)$ which is symmetric on $V_0$, skew symmetric on $V_1$, and ...
jack's user avatar
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7 votes
1 answer
327 views

$N_{G}(E)/C_{G}(E)$ is the Weyl group of $G$?

In the algebraic group $G = \operatorname{PGL}_4(\mathbb{C})$, let $E$ denote the subgroup of elements of order dividing 2 in the diagonal maximal torus; it is generated by the images of the three ...
user488802's user avatar
3 votes
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A quantity computed from weights of representations -- Have you seen it?

The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "...
Avi Steiner's user avatar
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Generic Zariski density after reduction mod p

Let $G$ be a simple algebraic group over $R:=\mathbb{Z}[\frac{1}{N}]((t))$. Let $\rho:\Gamma\to G(R)$ be a representation where $\Gamma$ is a finitely generated group. Then $\rho$ induces a ...
Higgs-Boson's user avatar
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References about automorphism group of a smooth projective variety

We work over the field of complex numbers. Let $X$ be a smooth projective variety such that $\operatorname{Pic}(X)\simeq \mathbb Z$. Suppose that $X$ admits a non trivial $\mathbb C^*$-action, that is ...
Bobech's user avatar
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Reference request & more: compute vector bundles for homogeneous $G$-varieties

We work over the field of complex numbers $\mathbb C$. Let $G$ be a simple linear algebraic group and let $P,Q$ be standard maximal parabolic subgroups of $G$ containing the same Borel subgroup $B$. ...
Bobech's user avatar
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3 votes
1 answer
126 views

Reference request: Criterion for a subgroup of $\mathrm{GL}_{n}(\mathbb{C})$ being reductive in terms of the trace

Let $G$ be a complex algebraic group embedded into $\operatorname{GL}_{n}(\mathbb{C})$. A criterion for $G$ to be reductive is the following. Let $\mathfrak{g}$ be the Lie algebra of $G$, and let $B$ ...
unknownymous's user avatar
3 votes
0 answers
156 views

Dimension of the $G$-orbit $\mathcal O_{I,J}(w)$ given by Bruhat decomposition in $G/P_I \times G/P_J$

Let $G$ be a reductive group over an algebraically closed field. Fix a maximal torus $T$ and a Borel subgroup $B$ containing $T$. Let $(W,S)$ be the Coxeter system associated to $(B,T)$, where $S$ ...
Suzet's user avatar
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Action of $T_p$ on automorphic forms, and error in Gelbart's "Automorphic forms on adele groups"?

Let $f\in\mathcal{S}_k(\Gamma_0(N),\chi)$ be a cuspidal modular form, and $\phi_f\in\mathcal{A}_0(\text{GL}_2(\mathbb{Q})\backslash\text{GL}_2(\mathbb{A}_\mathbb{Q}),\omega)$ be its corresponding ...
klein4's user avatar
  • 111
3 votes
0 answers
112 views

Richardson variety over arbitrary field

Let $G$ be a split semisimple algebraic group. Let $B$ and $T$ be borel subgroup ,maximal torus respectively. In this case the Weyl group $W(B,T)$ is a constant finite group scheme. Let $P$ be a ...
fool rabbit's user avatar
1 vote
1 answer
205 views

Zariski density preserved under $p$-adic completion?

Let $G$ be an almost simple group defined over $\mathbb{Q}$. Assume that $\Gamma$ is a subgroup of $G(\mathbb{Q})$ which is Zariski dense. Consider now the $p$-adic completion $\mathbb{Q}_p$ of $\...
Higgs-Boson's user avatar
4 votes
0 answers
138 views

Is the homogeneous coordinate ring of a flag variety a UFD?

I was wondering if $G$ is a semisimple complex algebraic group, then is the homogeneous coordinate ring of a flag variety a UFD or not?
F.H.A's user avatar
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0 answers
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Explicit description of wonderful compactification for PGL_3

Let $k$ be an algebraically closed field of positive characteristics. Let $X$ be the wonderful compactification of $PGL_3$ (see for example section 6 of "Frobenius Splitting Methods in Geometry ...
Asav's user avatar
  • 153
1 vote
1 answer
160 views

$\sigma$-compactness of some locally compact Hausdorff topological groups

Is the topological group $(\mathbf{Q}_p/\mathbf{Z}_p)^{\oplus k}$, $k\ge 1$, a $\sigma$-compact topological group when endowed with its natural $p$-adic topology? More generally, I'm looking for a ...
user avatar
3 votes
0 answers
178 views

Faithfulness of parabolic induction

I've only recently begun to study the representation theory of $p$-adic groups, so the following question might be quite silly. Let $F$ be a non-archimedean local field of residue characteristic $p$, $...
user avatar
4 votes
1 answer
352 views

May Schubert cell intersection with opposite big cell polynomial count?

Let $SL(n)$ be algebraic group defined over finite field $\mathbb{F}_{p^n}$, $B$ be Borel subgroup consist of upper triangular matrices and $T$ be maximal torus consist of diagonal matrices. Let $W$ ...
fool rabbit's user avatar
12 votes
1 answer
856 views

Pointless groups III

This question is a sequel to Pointless groups, to which @DanielLitt produced an elegant and easy-to-understand counter-example, and Pointless groups II, where @R.vanDobbendeBruyn pointed out that my ...
LSpice's user avatar
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7 votes
1 answer
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Pointless groups II

This question is a sequel to Pointless groups, where I asked for a certain kind of counterexample. @DanielLitt produced an elegant and easy-to-understand counterexample, but also suggested a sense in ...
LSpice's user avatar
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11 votes
1 answer
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Pointless groups

This question now has two sequels, Pointless groups II (to which @R.vanDobbendeBruyn gave a counterexample for an infinite, imperfect field) and Pointless groups III, both using revised wording ...
LSpice's user avatar
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3 votes
1 answer
154 views

Affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$

I saw the following results on affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$ without giving any references, where $\mathrm{SU}_3$ is the quasi-split inner form of special ...
Yachen Liu's user avatar
5 votes
1 answer
330 views

Characters of tori in finite reductive group

Let $G$ be a connected split reductive group over a finite field $k$. Suppose $G$ has connected centre. Let $T$ be a maximal split torus with Weyl group $W$. Note that $W$ acts on the finite group $T(...
Dr. Evil's user avatar
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1 vote
1 answer
258 views

Plus and minus Białynicki-Birula decomposition for normal variety

We work over $\mathbb{C}$. Let $X$ be a normal projective irreducible variety, and let $\mathbb{C}^*$ act nontrivially on $X$. The fixed point locus of $X$, namely $X^{\mathbb{C}^*}$, can be ...
Mago's user avatar
  • 43
11 votes
1 answer
298 views

Galois cohomology class of a reductive group not coming from a torus

Let $G$ be a (connected) reductive group over a perfect field $k$, and let $\xi\in H^1(k,G)$ be a cohomology class. By a theorem of Steinberg (Serre, Galois cohomology, Appendix 1 to Chapter III, ...
Mikhail Borovoi's user avatar
1 vote
0 answers
121 views

Bruhat decomposition and standard Frobenius

Let $G$ be a linear algebraic group define over $\overline{\mathbb{F}_p}$, consider it as a subgroup of $\operatorname{GL}(n)$. Let $F_p$ be the standard Frobenius. Let $B$ and $Q$ be an $F_p$-stable ...
fool rabbit's user avatar
5 votes
1 answer
274 views

Reductive groups over positive characteristics

Let $G$ be a connected split reductive group over a field $k$ of characteristic $p$. Let $\mathfrak{g}:=T_e(G)$ denote its Lie algebra. Let $T$ be a maximal split torus and $W$ the Weyl group (of the ...
Dr. Evil's user avatar
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2 votes
0 answers
97 views

Intersection of certain parabolic subgroups in $G$

Let $G$ be a simple (linear) algebraic group over $\mathbb C$. Let us fix a maximal torus $H \subset G$ and let $w_0 \in N_G(H)=\{g \in G: gH=Hg\}$ be such that the class of $w_0$ in $N_G(H)/H$ is ...
Bobech's user avatar
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1 vote
1 answer
197 views

action of the extra-special group

I'm reading a paper which has this line: A direct computation shows that $P\Omega_8$($\mathbb K$) has an elementary abelian subgroup $X = 2^2$ such that $C_{P\Omega_8(\mathbb K)}(X) = T_4.2^{1+4}_+$. ...
user477707's user avatar
4 votes
0 answers
225 views

Reference/list of reductive subgroups of reductive groups?

Let $G$ be a (say, connected) reductive group over an algebraically closed field of characteristic zero (say, $\mathbb C$). I am looking for simple examples of (ideally) complete characterizations of ...
Tian An's user avatar
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2 votes
1 answer
84 views

conjugacy in adjoint representation

Let $G$ be an adjoint algebraic group over $\mathbb{C}$, $\mathfrak{g}$ its Lie algebra. Let $\rho:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let $g,g'\in G$ be two semisimple ...
prochet's user avatar
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2 votes
0 answers
120 views

Hasse principle for $H^2$ of a maximal torus of a connected quasisplit group?

Let $k$ be a number field and let $G$ be a quasisplit reductive algebraic group over $k$. Does there exist a maximal torus in $G$ such that the Hasse principle in dimension $2$ holds, i.e., such that ...
cgb5436's user avatar
  • 221
4 votes
0 answers
134 views

Centraliser of a maximal $k$-split torus of a reductive $k$-group

Let $G$ be a connected reductive group defined over a field $k$, and let $S$ be a maximal $k$-split $k$-torus of $G$. Then the centraliser $\mathscr Z_{G}(S)$ is defined over $k$. In fact, it is a ...
Lisa's user avatar
  • 61
1 vote
0 answers
161 views

When the action of reductive group on algebraic variety is not equidimensional?

I saw the question When is an almost geometric quotient flat? which said "The quotient $\pi$ is flat if and only if $\pi$ is equidimensional and $X$ is smooth". I am curious is there an ...
Mary Susy's user avatar
2 votes
0 answers
66 views

Cohomology of compact open subgroups of semisimple groups over local fields

Let $E$ be a local field, $\mathcal{O}$ its ring of integers, $k$ its residue field, and $G$ a split semisimple group over $\mathcal{O}$. Let $K$ be an open subgroup of $G(\mathcal{O})$; more ...
naf's user avatar
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2 votes
0 answers
87 views

Conjugates of relative root groups by an element of the Weyl group

Let $G$ be a reductive group (over an algebraically closed field), $T$ a maximal torus, and $\Phi$ the root system of $(G,T)$. Then for each root $\alpha \in \Phi$ there is a unique connected $T$-...
Lisa's user avatar
  • 61
4 votes
1 answer
208 views

Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group?

Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of Tits building of a linear algebraic group. I ...
LSpice's user avatar
  • 11.2k
4 votes
1 answer
348 views

Reductive subgroups of $\mathrm{GL}_2$ over an algebraically closed field of characteristic zero

I am reading a very nice paper of Newton and Thorne, Symmetric power functoriality for holomorphic modular forms, and there is an argument concerning the (Zariski-closure of) image of certain $p$-adic ...
babu_babu's user avatar
  • 229
2 votes
0 answers
192 views

The Jacquet module of the Steinberg Representation

I have also posted this question also on Math Stack Exchange, please inform me if the level is too low for this forum. Let $G=GL_2(F)$ where $F$ is a non-Archimedean local field of characteristic $0$, ...
carraig's user avatar
  • 121
9 votes
2 answers
704 views

Can we use formal groups to recover Lie-theoretic representation theory in characteristic p?

In differential geometry, Lie's theorems allow us to integrate any Lie algebra representation to a Lie group representation. The algebraic version of this is more complicated (and I'm not terribly ...
Doron Grossman-Naples's user avatar
2 votes
1 answer
181 views

Algebraic groups acting on affine varieties with finite-dim orbits in the coordinate ring

Let $K$ be an algebraically closed field of characteristic zero, and $X$ be an affine $K$-variety (identify $X$ with its set of $K$-points). Let $G$ be group acting "abstractly" on $X$, by ...
Joshua Ruiter's user avatar
2 votes
1 answer
213 views

Parahoric subgroup over a local field

$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...
M masa's user avatar
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