All Questions
Tagged with lie-groups algebraic-groups
349 questions
4
votes
0
answers
211
views
Books on integration on semisimple Lie groups
Can anyone suggest me some good books where I can find integration theory on semisimple Lie groups (using KAK, KAN and other type of decompositions)?
I have read Knapp's book "Lie groups beyond ...
- 1,879
5
votes
1
answer
729
views
The normalizer of block diagonal matrices
Let $G=\mathrm U_n$ or $\mathrm{GL}_n(\mathbf C)$ and $H$ the subgroup of block diagonal matrices respecting a partition $n=n_1+\dots+n_k$. Is the normalizer $N=N_G(H)$ computed anywhere in the ...
- 31.5k
3
votes
0
answers
71
views
Conjugacy classes in reductive group under adjoint action of parabolic subgroup
Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results ...
- 1,846
3
votes
1
answer
578
views
Conjugacy of elements in a parabolic subgroup
Let $G$ be a complex connected reductive group, and let $P \subseteq G$ be a parabolic subgroup. My question is the following: if $g$ and $h$ are elements of $P$ which are conjugate as elements of $G$,...
- 491
7
votes
1
answer
1k
views
When a free action gives rise to a $G$-principal bundle
When a free action gives rise to a $G$-principal bundle
Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that
$G \backslash X$ is Hausdorff. (equivalently the image of ...
- 2,639
6
votes
1
answer
541
views
Is the connected centralizer of a semisimple element in a connected reductive group also a centralizer?
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field and let $g\in G$ be semisimple. Write $C=\mathrm{C}_G(g)$ and $C^\circ=\mathrm{C}_G(g)^\circ$ for the ...
- 993
0
votes
0
answers
120
views
Commensurability of arithmetic, irreducible, nonuniform lattices
Let $n \in \mathbb{Z}_{\geq 2}$ be arbitrary. Let $r_1$ and $r_2$ be arbitrary elements of $\mathbb{Z}_{\geq 0}$ that satisfy $r_1 + r_2 > 0.$ Let $G := {\rm SL}_n(\mathbb{R})^{r_1} \times {\rm SL}...
- 255
4
votes
0
answers
68
views
The weak restriction of the Jacquet module
Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...
- 41
6
votes
1
answer
395
views
Explicit fundamental domain for the action of $\operatorname{O}(n,1)(\mathbb{Z})$ on $\operatorname{O}(n,1)(\mathbb{R})$
Minkowski computed explicit fundamental domains for the action of $\operatorname{SL}_n(\mathbb{Z})$ on $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$ for each $n \leq 6$. In the ...
1
vote
0
answers
77
views
simple Lie groups over C [closed]
For an affine algebraic group over $\mathbb{C}$ which is simple, in the sense of no normal subgroups closed in the Zariski topology except for finite central subgroups and the whole thing, I'm trying ...
- 2,125
4
votes
1
answer
196
views
The Lie algebra of the subgroup of $GL(n)$ preserving a given variety
Let $V=k^n$ for an algebraically closed field $k$ of characteristic 0, and let $W \subseteq V$ a subspace. Let $G_W\subseteq GL(V)$ be the set of invertible linear maps that preserve $W$, i.e.
$$
G_W=\...
- 980
4
votes
1
answer
441
views
Is the Jordan decomposition for reductive groups algebraic?
Let $G$ be a connected affine algebraic group over $\mathbb{C}$. It's a known fact that elements of $G$ admit a decomposition into semisimple and unipotent elements. Namely, choose a faithful ...
- 491
4
votes
0
answers
156
views
Reference request - conjugacy classes over local fields
Is there a nice reference for reductive groups over local fields, which for example contains discussion of things such as: Given a semisimple element in $G(F)$, its $G(F)$-conjugacy class is closed in ...
- 5,562
3
votes
0
answers
160
views
Orbit representatives for the action of the maximal compact subgroup
Let $F$ be a non-Archimedean local field and $O$ be the ring of integers in $F$. Take $G=GL(2,F)$ and $K=GL(2,O)$. Consider the action of $K$ on $G$ by conjugation. Is it possible to explicitly write ...
- 185
9
votes
0
answers
234
views
Non-algebraic representations of $\text{SL}_n(\mathbb{R})$
My question is easily stated: are all continuous finite-dimensional real representations of $\text{SL}_n(\mathbb{R})$ algebraic representations?
This is false if you drop the word "continuous" (e.g. ...
- 383
3
votes
0
answers
87
views
Recovering a $G$-valued representation/parameter
Number theoretic phrasing
Let $G$ be a connected reductive group over a characteristic $0$ field $F$. Associated to $G$ is its Langlands dual group $^{L}G$. For every dominant cocharacter $\mu$ of $...
- 953
3
votes
0
answers
204
views
Miraculous Parahorics
Let $G$ be a connected simple group over a local field $k$. Let $I\subset G(\mathcal{O})$ denote an Iwahori subgroup of $G(k)$ with Lie algebra $\mathfrak{i}$. Let $P\supseteq I$ be any other ...
- 2,751
10
votes
1
answer
262
views
What is the hidden symmetry behind four generic planes in $\mathbb{R}^4$?
Consider the action of $\operatorname{GL}(\mathbb{R}^4)$ on the Grassmannian of 2-dimensional subspaces of $\mathbb{R}^4$. In experiments, I observe that four randomly drawn points in this space are ...
- 7,633
2
votes
0
answers
73
views
Does $\mathfrak{g}^*$ split off from the augmentation ideal
(Note: I had this posted on MSE for a while but didn't get much of a response... so I'm posting it here now.)
Let $G$ be an affine algebraic group over an algebraically closed field $k$ of ...
- 1,077
3
votes
0
answers
261
views
Do these Zariski-dense subgroups of $\operatorname{SO}_{6}(\mathbb C)$ have non-empty intersection with this subset?
Let $G\leq \operatorname{SO}_{6}(\mathbb Z)$ be a finite-index normal subgroup, so it's a Zariski dense subgroup of $\operatorname{SO}_{6}(\mathbb C)$; and let $H$ be the subset of $\operatorname{SO}_{...
- 71
2
votes
1
answer
304
views
The simple reflections of the Weyl group in $\operatorname{SO}_{2n}(\mathbb C)$
Let $W$ be the Weyl group corresponding to the maximal torus $diag(t_1, . . . , t_{n}, t^{−1}_n, . . . , t^{−1}_1)$ in a Borel group of $\operatorname{SO}_{2n}(\mathbb C)$.
What are the matrices ...
- 332
6
votes
1
answer
658
views
Anti-holomorphic involutions of a complex linear algebraic group
Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$.
Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$.
Let $\sigma$ be an anti-...
- 14.1k
4
votes
0
answers
300
views
Number of connected components of the centre of a Levi subgroup
Let $G$ be a connected complex semisimple algebraic group and $T\subset B\subset G$ a choice of maximal torus and Borel subgroup. Let $\Phi$ be the root system and $\Pi\subset\Phi$ the set of simple ...
- 41
4
votes
0
answers
105
views
Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$
Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\Delta = \{e_1 - e_2, ... , e_{n-1} - e_n, e_n\}$$ be a set of simple roots of $T$ corresponding to a ...
- 6,180
1
vote
0
answers
112
views
Nontrivial relations of the irreducible root systems
For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$, we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,...
- 332
2
votes
0
answers
173
views
Characterize an element of $\operatorname{SL}_n(\mathbb Z)$
I'm trying to generalize a theorem on $\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$.
In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where
$$e_{1,n}(m)=
...
- 332
12
votes
1
answer
684
views
Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group?
Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?
I was told some fact along this line is true but could not find any ...
- 511
1
vote
0
answers
370
views
Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$
Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...
- 332
3
votes
1
answer
276
views
For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?
I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
- 332
1
vote
0
answers
153
views
Is the Bruhat cell Zariski open in a connected algebraic group $G$? [closed]
Is the Bruhat cell Zariski-open in a connected algebraic group $G$?
Specifically, is the big Bruhat cell Zariski-open (and maybe Zariski-dense)?
Is it true for all the Bruhat cells?
- 332
2
votes
1
answer
216
views
How to prove that Chevalley groups over $\mathbb R$ have no compact factors
I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem.
I've been told in another thread ...
- 332
1
vote
0
answers
140
views
Describing compact Lie groups in purely topological terms
Compact Lie groups are a very special type of compact group, namely those which admit a differentiable structure. Is it possible to describe compact Lie groups in purely topological terms, that is, ...
- 993
2
votes
1
answer
259
views
Character of a semisimple connected Lie groups [closed]
I'm trying to see why the Chevalley groups over $\mathbb C$ have no nontrivial character?
I know that a compact connected semisimple Lie group has no nontrivial character but is the compactness ...
- 332
7
votes
1
answer
467
views
Finite index subgroup of $\mathrm{GL}_n(\Bbb C)$ and Chevalley groups
I'm trying to show that if $G$ is a Chevalley group, then every finite index subgroup of $G(\Bbb Z)$ is Zariski dense in $G(\Bbb C)$. ($G(\Bbb Z)$ is the Chevalley group over $\Bbb Z$ and similarly ...
- 332
4
votes
1
answer
119
views
Reference for nonquasi-split groups of type $E_6$ and $E_7$ over local fields
The semisimple groups over a local field have been classified by Tits, cf. [1] "Classification of algebraic semisimple groups" in Boulder and [2] "Reductive groups over local fields" in Corvallis.
In ...
- 991
7
votes
1
answer
852
views
Connected components of real Lie groups
(This is a follow-up to this question of mine.)
Is there an example of a connected reductive algebraic group $G$ over $\mathbb{R}$ such that:
$G$ is not isomorphic to a product $G_1 \times G_2$ of ...
- 37k
7
votes
1
answer
521
views
Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$
Let $X$ be a scheme over an algebraically closed field $k$ and let $\operatorname{Aut}(X)$ denote the functor sending a $k$-scheme $T$ to the group $\operatorname{Aut}_T(X \times_k T)$ of ...
- 749
2
votes
1
answer
217
views
On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not
In the paper: Pattern Avoidance and Rational Smoothness of
Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), https://www.sciencedirect.com/science/article/pii/...
- 235
1
vote
1
answer
506
views
On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya
I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. ...
- 235
2
votes
1
answer
113
views
Nice Form of Vector Field
Let $G$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $k$ of characteristic zero. Let $X$ be a homogeneous affine $G$-variety, i.e. $X=G/K$ for ...
- 1,077
4
votes
3
answers
681
views
Real points of reductive groups and connected components
Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...
- 6,180
9
votes
3
answers
576
views
Reference Request: Structure constants for G2
Let $G$ be a split semisimple real Lie group in characteristic zero, and let $B=TU$ be a Borel subgroup with unipotent radical $U$ and Levi $T$. Fix an ordering on the roots $\Phi^+$ of $T$ in $U$, ...
- 6,180
9
votes
0
answers
161
views
Can semisimple orbits be written $\exp(\mathfrak{g})\cdot x$?
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $G$ be its adjoint group. If $x\in\mathfrak{g}^{rs}$ is a regular semisimple element, is its orbit
$$G\cdot x=\{\mathrm{Ad}_gx:g\in G\}$$
...
- 1,383
3
votes
1
answer
159
views
Is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ a unitary group?
Let $F$ be an p-adic field, and $E$ be a quadratic extension of $F$, then is $({F^{\times})^{ diag}}\backslash(GL_2 \times E^{\times})_{det=\mathbb{N}}$ isomorphic to some unitary group $U_{E/F}(2)$? ...
- 467
8
votes
0
answers
381
views
Significance of half sum of non-simple positive roots
In representation theory, there are plenty of places that a $\rho$-shift makes an appearance, where $\rho$ is the half sum of positive roots. See, for instance, this post for some discussions of the ...
- 2,751
9
votes
1
answer
543
views
Polynomial invariants for simple algebraic groups
Let $G$ be a simple complex algebraic group. Let $V$ be a finite-dimensional algebraic representation of $G$. Thus, we can write $V=V_1\oplus \cdots \oplus V_n$ where $V_i$'s are irreducible ...
- 2,751
2
votes
1
answer
236
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, ...
user100841
7
votes
1
answer
466
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 ...
- 9,237
9
votes
2
answers
419
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 ...
- 6,180
4
votes
0
answers
280
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^...
- 20.2k