All Questions
Tagged with lie-groups algebraic-groups
349 questions
2
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0
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212
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Compute the discriminant for reductive groups
Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...
- 3,472
3
votes
0
answers
90
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on Neron defect of smoothness for groups schemes
Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...
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5
votes
1
answer
453
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A subgroup of the Weyl group
Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2.
Let $Q=Q(D)$ denote the root lattice of $D$.
Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is ...
- 52.9k
5
votes
2
answers
758
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Equivariant Cohomology of a Complex Projective Variety
Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...
- 27.8k
3
votes
2
answers
409
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Connectedness of Springer Fibers
Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If $G=\...
- 1,201
2
votes
1
answer
1k
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Thom-Gysin Sequences and Stratifications
Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
- 40.2k
3
votes
2
answers
2k
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Is there an almost-direct product decomposition for disconnected reductive algebraic groups?
$\textbf{Some definitions:}$
Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
CommunityBot
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11
votes
2
answers
1k
views
Finite subgroups of $PGL(3,K)$
It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
CommunityBot
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4
votes
1
answer
282
views
Name for a class of parabolic subgroups
This is a reference request for a (the) name of the following class of parabolic subgroups of a complex simple Lie group $G$:
Recall that parabolic subgroups of $G$, containing fixed Borel subgroup, ...
- 1,219
1
vote
1
answer
326
views
A Criterion for Reductivity of Lie Subgroups
Let $G$ be a connected, simply-connected, complex, semisimple Lie group. Suppose that $H$ is a Zariski-closed subgroup of $G$ with reductive Lie algebra $\frak{h}$. Under what conditions may one ...
- 52.9k
1
vote
1
answer
376
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on z-extensions
Let $G$ a group split over a local field $F$.
We call a $z$-extension a group $G'$ such that $G'_{der}$ is simply connected, $G'$ is a central extension of $G$ by a central torus $Z$.
Can we find a $...
- 14.2k
4
votes
2
answers
1k
views
Dimension of Unipotent Radicals
A parabolic subgroup of a linear algebraic group $G$ defined over a field $k$ is a subgroup $P\subseteq G$, closed in the Zariski topology, for which the quotient space $G/P$ is a projective algebraic ...
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0
votes
0
answers
272
views
minuscule representations and classical groups
Let $G$ a semisimple group over an algebraically closed field $k$.
We assume that $G$ is classical.
We call a $z$-extension, a group $\tilde{G}$ such that $\tilde{G}$ is a central extension of $G$ by ...
- 3,472
11
votes
2
answers
972
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Rational orthogonal matrices
``everybody knows'' that an integral orthogonal matrix is a signed permutation matrix, so there are exactly $2^n n!$ such matrices in $O(n).$ Now, what if we ask for the enumeration of elements of $O(...
- 1,363
4
votes
1
answer
1k
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is connected complex Lie group with a trivial center linear?
There is a theorem of Rosenlicht ("Some basic theorems on algebraic groups", 1956, Theorem 13) asserting that a quotient of a connected algebraic group by its center is linear. So a connected ...
- 11.2k
1
vote
0
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82
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decomposition lemma in adelic groups II
Let $X$ a curve on a field $k=\bar{k}$. G a connected reductive group over $k$.
Let fix $d$ closed points $(x_{1},...,x_{d})$ of $X$.
On each point, we have an évaluation morphisme $ev_{x}:G(k[[t_{x}...
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6
votes
2
answers
740
views
Measuring how far from being cocompact a lattice is
Let $G$ be a locally compact group and $\Gamma$ a lattice (=discrete
subgroup of $G$ such that $G/\Gamma$ carries a probability measure $\mu$
that is invariant under the action of $G$ by left-...
- 3,201
2
votes
1
answer
1k
views
Are certain simple Lie groups linear algebraic groups?
Assume you have an almost connected simple Lie group G with trivial center. (In particular excluding non-algebraic examples such as the universal cover of SL_2(R).)
Such a group should automatically ...
- 13.8k
1
vote
0
answers
268
views
how to determine the Weyl group of a diagonalizable subgroup?
Assume that $G$ is a connected compact Lie group (or a connected complex reductive group), and $K$ is a diagonalizable subgroup of $G$. It is known that the Weyl group $W_G(K)$ of $K$, defined as $N_G(...
CommunityBot
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22
votes
3
answers
2k
views
Is SL(2,C)/SL(2,Z) a quasi-projective variety?
Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold).
Is $SL(...
- 8,828
2
votes
2
answers
399
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Infinitesimal deformations of a discrete group inside Lie groups vs. algebraic groups
Let $G$ be an algebraic group with Lie algebra $\mathfrak g$ and let $\Gamma$ be any finitely generated (discrete) group. One can consider the representation variety $\mathfrak R=\mathrm{Hom}(\Gamma,G)...
- 31.2k
6
votes
1
answer
643
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question about equivariant embeddings of riemannian symmetric domains
Here by riemannian symmetric domain is understood an riemannian symmetric space with only factors of non-compact types. Such domains are realized as quotients of the form $D=G/K$, where $G$ is a ...
CommunityBot
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2
votes
2
answers
757
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Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
- 8,493
14
votes
1
answer
1k
views
Lie groups vs. algebraic groups and deformations
I am interested in deformations of (discrete subgroups of) Lie groups. But, as I understand it, deformation theory, as a theory, prefers to speak schemes.
At least the classical Lie groups can be ...
- 31.2k
0
votes
2
answers
212
views
A kind of orthogonal subtorus
Here $\mathbb{T}^n := (\mathbb{R} / \mathbb{Z})^n$ is the topological group of the n-dimensional torus and $k \in \mathbb{Z}^n$ is a non-null vector, I'm working about the subgroup
$S = \{x \in \...
CommunityBot
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4
votes
2
answers
578
views
Proper compact connected subgroup of $Spin(n)$
What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am ...
- 108k
7
votes
1
answer
561
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How does the right regular of GL(n, R) and GL(n,Qp) decompose?
The question is contained in the title. I would guess that this question is already answered in the literature.
Given the reductive group $GL(n)$ over a complete local field, how does the right ...
- 11.2k
3
votes
0
answers
289
views
Conjugation of faces in root systems / of parabolic subgroups having same Levi in split reductive groups
If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is ...
- 31
1
vote
0
answers
189
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Exotic Chains for Group Homology of a Complex Lie Group
Related Question: Exotic Chains for Group Cohomology of a Complex Lie Group
Let's take the group homology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
CommunityBot
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4
votes
0
answers
184
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Exotic Chains for Group Cohomology of a Complex Lie Group
Related Question: Exotic Chains for Group Homology of a Complex Lie Group
Let's take the group cohomology of a affine algebraic group over $\mathbb C$ (with its discrete topology). The natural free ...
CommunityBot
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3
votes
1
answer
559
views
Springer isomorphisms and parabolics
Let $G$ be a semisimple, simply-connected algebraic group over an algebraically closed field $k$ of positive characteristic. Fix a Borel subgroup $B \subseteq G$ with unipotent radical $U$. Also let $...
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1
vote
0
answers
157
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On closed abelian reductive subgroups of Real reductive groups
Hello everybody. I would first like to apologise for the basic question; I'm not expert on Lie Theory. Can someone please help me with the following questions
Let $\mathrm{G}=\mathrm{K} \exp(\...
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9
votes
1
answer
903
views
Principal congruence subgroups in higher rank
I don't seem to have seen any explicit generators for the principal congruence subgroups of $SL(n, \mathbb{Z}),$ for $n>2,$ although it is known (Sury+Venkataramana) is that the number of ...
- 96.4k
3
votes
1
answer
121
views
Lower bound on the degree of a product of elements in a hyperalgebra/enveloping algebra
Background:
Fix a linear algebraic group $G$ over an algebraically closed field $k$ of arbitrary characteristic and let $B \subseteq G$ be a Borel subgroup with unipotent radical $N$. Let $\Delta^+$ ...
- 3,702
7
votes
2
answers
734
views
What does a homogeneous space of a linear algebraic group know about the group?
Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers
and $H\subset G$ is an algebraic subgroup.
In general, we can write the algebraic variety $X$...
- 52.9k
2
votes
2
answers
503
views
Lie Algebras and Simple Connectivity for general algebraic groups
In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
- 3,246
5
votes
1
answer
384
views
Conjugacy classes with elliptic limit points
Let $G$ be a reductive algebraic group over $\mathbb R$ and $K$ a maximal compact subgroup. Then we refer to the conjugacy class in $G$ of some $k \in K$ as an elliptic conjugacy class.
Question: ...
- 8,866
1
vote
0
answers
418
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Centralizers and Cartan involutions
This should be an easy question about centralizers in reductive lie groups, but I wonder if it is already available from the literature.
Consider $G$ a connected non-compact semi-simple Lie group, ...
- 12.6k
6
votes
1
answer
2k
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how to recognize subgroups through Dynkin diagram?
Fix $\mathbb{C}$ as the base field, and reductive groups are assumed to be connected.
Consider the example $SO_N\subset SL_N$. $SO_N$ is its own normalizer in $SL_N$, and the rank is much smaller ...
CommunityBot
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3
votes
0
answers
374
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a question about centralizers in semi-simple groups
I have a question concerning centralizers in real reductive groups. I'd like to know if the following property is available in any references.
Let $L\subset H\subset G$ be an inclusion chain of ...
- 1,305
4
votes
1
answer
677
views
An identity for sheaf cohomology of flag varieties
Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ denote the projection.
Given a representation ($\theta,V$) ...
- 52.9k
8
votes
4
answers
3k
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"Why" is every polynomial representation of SL(2) selfdual?
Given a field $K$ of characteristic $0$. It seems to me that every finite-dimensional polynomial representation of $\mathrm{SL}_2\left(K\right)$ is self-dual (i. e., isomorphic to its dual). In fact, ...
- 52.9k
13
votes
4
answers
5k
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Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
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2
answers
1k
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Is there a way to see a topological group as the "Cayley graph" of its "infinitesimal generators"?
At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most ...
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6
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3
answers
2k
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What's the classification of the algebraic subgroups of Sp(4,R)?
Hi! I would like to know if there is an explicit classification of the algebraic (i.e., Zariski closed) subgroups of the symplectic group Sp(4,R) and/or more generally Sp(2n,R) somewhere in the ...
- 9,389
10
votes
4
answers
1k
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Algebraicity of holomorphic representations of a semisimple complex linear algebraic group
Let $G$ be a complex linear algebraic group, given to us as a closed subgroup of some $\mathrm{GL}(n,\mathbb{C})$. Suppose moreover that $G$ is semisimple. Then it's a fact that every finite-...
- 4,540
0
votes
0
answers
700
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Questions on orbit properties of group action on varieties
Let $F$ be a p-adic field or $\mathbb{R},\mathbb{C}$, $G$ a group(not necessarily reductive) over $F$, $X$ an algebraic variety defined over $F$, and $G$ acts on $X$. Now we have several questions ...
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0
votes
1
answer
470
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Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
CommunityBot
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8
votes
2
answers
1k
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number of irreducible representations over general fields
For a finite group, there are finitely many irreducible representations of complex numbers.
What if the field is changed to some other fields? Like real numbers, p-adic field, finite field?
In ...
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