All Questions
2,543 questions
1
vote
1
answer
208
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 ...
user480741
3
votes
0
answers
196
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 $p$, $...
user484137
2
votes
0
answers
147
views
Automorphism groups of "reductive" Lie algebras in positive characteristic
I put "reductive" in quotes because, of course, in positive characteristic one should speak of Lie algebras of reductive groups, not of reductive Lie algebras.
Let $G$ be a reductive group ...
- 12.9k
4
votes
1
answer
446
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 $W$ ...
- 653
12
votes
1
answer
879
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 ...
- 12.9k
7
votes
1
answer
614
views
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 ...
- 12.9k
11
votes
1
answer
1k
views
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 ...
- 12.9k
3
votes
1
answer
184
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 ...
- 83
5
votes
1
answer
350
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(...
- 2,751
1
vote
1
answer
293
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 ...
11
votes
1
answer
328
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, ...
- 14.2k
1
vote
0
answers
131
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 ...
- 653
5
votes
1
answer
309
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 ...
- 2,751
2
votes
0
answers
120
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 ...
- 381
1
vote
1
answer
202
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 $C_{P\Omega_8(\mathbb K)}(X) = T_4.2^{1+4}_+$. ...
- 43
5
votes
0
answers
264
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 ...
- 3,799
2
votes
1
answer
104
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 ...
- 3,472
2
votes
1
answer
361
views
Lie algebroid in algebraic geometry
When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
- 145
2
votes
0
answers
133
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 ...
- 231
4
votes
0
answers
149
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 ...
- 61
2
votes
0
answers
97
views
Non-noetherian Cartier Isomorphism
A result in positive characteristic is that if $R/\mathbb{F}_p$ is a smooth ring, then we have a Cartier isomorphism
$$\Omega_{R}^\bullet\cong H^\bullet(\Omega_R^\bullet)$$
which is essentially ...
- 4,428
1
vote
0
answers
176
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 ...
- 27
3
votes
0
answers
86
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 ...
- 10.5k
2
votes
0
answers
92
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 $T$-...
- 61
4
votes
1
answer
230
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 ...
- 12.9k
4
votes
1
answer
394
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 ...
- 241
2
votes
0
answers
253
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 $0$, ...
- 121
9
votes
2
answers
783
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 ...
- 1,969
2
votes
1
answer
204
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 ...
- 233
2
votes
1
answer
224
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 ...
- 479
1
vote
0
answers
80
views
Intersection of open orbits in homogeneous space
Let $G$ be a simple complex algebraic group. Let $P(\alpha_i),P(\alpha_k)$ be maximal standard parabolic subgroups of $G$ associated to simple roots $\alpha_i,\alpha_k$ in the root system associated ...
- 381
1
vote
1
answer
185
views
General centralizer of algebraic group
Perhaps there is a simple answer, but I'm very puzzled by the following question:
Question: Does there exist a (smooth, connected) algebraic group $G$ such that the general centralizer (i.e. the ...
- 507
3
votes
0
answers
133
views
Classification of semisimple algebraic groups which act transitively on a projective space
Let $k$ be an algebraically closed field of characteristic 0, and $V$ be a vector space on $k$ of dimension $>1$.
In this situation, is there a classification of connected semisimple groups (up to ...
user163485
2
votes
1
answer
391
views
Existence of regular semisimple elements in linear group over local field
Let $ L $ be a finite extension of $p$-adic numbers $ \mathbb{Q}_p $. Let $ \text{GL}_{n}(L) $ denote the general linear group $ \text{GL}_{n}(L) $ over $L$ equipped with the topology induced from the ...
- 863
5
votes
1
answer
262
views
Group scheme with an isotrivial maximal torus
Let $G$ be a reductive group scheme over a normal ring $A$. Then, we know that Zariski locally it admits a maximal torus.
Let us assume that it admits a maximal torus after a finite surjective (resp. ...
- 3,472
9
votes
1
answer
425
views
Abelianization of $\mathrm{GL}_2(R)$
$\DeclareMathOperator\GL{GL}$Let $R$ be a number ring. Are there known lower bounds for $H_1(\GL_2(R);\mathbb Q)$ or $H_1(\GL_2(R),\GL_1(R);\mathbb Q)$ in terms of properties of $R$ (class number, ...
- 965
2
votes
1
answer
236
views
Compactifications of group varieties
Let $V$ be a nonempty, irreducible, smooth projective variety over $\mathbf{C}$.
Is there a smooth projective variety $X$ over $\mathbf{C}$, a surjective map $X\to V$ of varieties over $\mathbf{C}$, ...
user335418
4
votes
0
answers
215
views
Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence
My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
- 485
4
votes
0
answers
236
views
Number of homomorphisms from a group to $\mathrm{GL}_n(\mathbb{F}_q)$
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Fix a group $\Gamma$ and a positive integer $n$. Let $c(q):=\lvert\Hom(\Gamma, \GL_n(\mathbb{F}_q)\rvert$ denote the number of homomorphisms ...
- 2,751
6
votes
1
answer
256
views
Which Lie groups are a central extension of an algebraic group?
Suppose $G$ is a connected real Lie group. The quotient $G/Z(G)$ is the image of the adjoint representation, so a linear group. Is it known for which groups this quotient is Lie isomorphic to an ...
- 161
2
votes
1
answer
141
views
An upper bound on the dimension of a subalgebra of $\mathfrak{so}(p,q)$ with non-trivial centre
Let $\mathfrak{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathfrak{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(...
- 23
6
votes
1
answer
445
views
Is every finite subgroup the integer points of a linear algebraic group?
Cross Posting this from MSE since it's been there for almost a month and it got a couple upvotes but no answers. MSE link Is every finite subgroup the integer points of a linear algebraic group?
Let $ ...
- 4,081
2
votes
0
answers
173
views
Understanding the proof of a theorem by Van Den Bergh
I'm trying to understand the proof of a theorem by Van Den Bergh, which is Proposition 6 in the paper Bessenrodt, Christine and Lieven Le Bruyn. “Stable rationality of certain PGLn-quotients.” ...
- 839
3
votes
0
answers
399
views
Analogies to the chromatic layers of the sphere spectrum
Is there an analogy between the chromatic layers the sphere spectrum $\mathbb{S}$ and the ramification groups of the absolute Galois group $G(\mathbb{Q}^{\mathrm{sep}}/\mathbb{Q})$?
- 705
1
vote
1
answer
383
views
$ S_4 $ subgroups and $ \operatorname{SO}_3(\mathbb{R}) $
$\DeclareMathOperator\SO{SO}$I posted this on MSE 10 days ago and it got 3 upvotes but no answers or comments, so I'm cross-posting to MO.
Background: The group of rotations $ \SO_3(\mathbb{R}) $ has ...
- 4,081
3
votes
0
answers
149
views
What direction does the derivation of an inseparable algebraic variable point in?
I've been thinking about the geometry of inseparable field extensions lately, since I'm studying smoothness in commutative rings in an advanced topics course this semester. I've generally come to the ...
- 1,969
5
votes
1
answer
524
views
Is there a non-split algebraic torus (over a finite field) satisfying the following properties?
Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties?
$T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
- 1,833
6
votes
1
answer
507
views
Degree-2 étale covers of curves in characteristic 2 vs torsion points on the Jacobian
It can be found, at Hartshorne exercise 4.2.7 for example, that in the case where $\operatorname{char}(k) \neq 2$ we have a nice correspondence between etale degree 2 covers of a curve $C$ and 2-...
- 679
4
votes
1
answer
257
views
Question regarding semistability of a point of GIT quotient
$\DeclareMathOperator\SL{SL}$I am currently looking at the paper titled "$\SL(2,\mathbb{C})$ quotients de $(\mathbb{P^1})^n$" by Marzia Polito. The author has considered diagonal action of $\...
- 585
2
votes
0
answers
69
views
Abelian category for $(\mathfrak{g},T)$ modules with nontrival Grothendieck group
Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\...
- 1,077