All Questions
Tagged with lie-groups algebraic-groups
112 questions with no upvoted or accepted answers
20
votes
0
answers
763
views
Should the Dynkin diagrams of types $A_1$ and $B_2$ be labelled $C_1$ and $C_2$?
The labels $A$--$G$ attached to connected Dynkin diagrams are of course arbitrary,
the result of historical accidents. In order to avoid repetitions, the four infinite
families $A_\ell, B_\ell, C_\...
- 52.9k
12
votes
0
answers
967
views
What is miraculous about the mirabolic subgroup?
I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...
- 2,516
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
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
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
8
votes
0
answers
145
views
Semisimple Lie groups admitting a free algebra of invariants
Assume we work over an algebraically closed field of characteristic zero.
I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of ...
- 189
8
votes
0
answers
408
views
Connection between two theorems on character values?
In a recent arXiv preprint here, Dipendra Prasad has revisited a 1976 theorem of Kostant (Theorem 2 in the paper On Macdonald's $\eta$-function formula, the Laplacian and generalized exponents, ...
- 52.9k
8
votes
1
answer
382
views
Action of the endomorphism monoid on an irreducible GL-module
Let $G=\mathrm{Gl}_n(\mathbb C)$ and $V$ an irreducible $G$-module on which $G$ acts polynomially. In other words, the algebraic group action of $G$ on the affine space $V$ extends to an algebraic ...
- 4,902
7
votes
0
answers
295
views
Hilbert series for invariant ring
I would like to compute the Hilbert series of the ring of invariants of certain irreducible representations of some groups (namely $SO(5)$ to begin with).
To put it in some broader context, let $G$ ...
- 1,263
7
votes
0
answers
1k
views
Correspondence between real forms and real structures on complex Lie groups
I asked this in MSE, but without success, so I hope, it will be suitable here.
E.B.Vinberg and A.L.Onishchik in their book give the following two definitions.
For a complex Lie group $G$ its real ...
- 7,426
6
votes
0
answers
306
views
Tits construction of algebraic groups of type D₆ and E₇ via C₃
As shown in the Freudenthal magic square, the Tits construction of $D_6$ takes as input
an quaternion algebra and the Jordan algebra of a quaternion algebra (see The Book of Involutions § 41). In ...
- 1,479
6
votes
0
answers
200
views
Why should Serre's conjecture on congruence subgroup property hold?
There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$.
Whether a lattice in the group satisfies the congruence subgroup property,
...
- 1,024
6
votes
0
answers
176
views
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 ...
- 7,426
5
votes
0
answers
122
views
Algebraic groups and formal group laws in characteristic p
In characteristic zero, there is a well-known equivalence between Lie groups, formal group laws and Lie algebras.
Let $p$ be a prime. The equivalence between Lie groups and Lie algebras has an ...
- 413
5
votes
0
answers
156
views
When is a unitary group over a ring of integers dense?
Let $ SU_n(O_d) $ denote an integral unitary group of $ n \times n $ matrices over a totally real number field $ K_d:=\mathbb{Q}(\cos(\frac{2\pi }{d})) $ where $ O_d $ is the ring of integers of $ K_d ...
- 4,081
5
votes
0
answers
92
views
Canonical parabolics vs Levi subgroups
Let $G$ be a reductive group over a field $k$ of characteristic zero. The Jacobson-Morozov theorem gives a method of embedding any unipotent element into an $\mathfrak{sl}_2$ triple, which in turn ...
- 3,799
5
votes
0
answers
298
views
What are the matrix coefficients associated with the irreducible representations of compact real linear algebraic groups?
What are the matrix coefficients associated with the irreducible representations of a compact real linear algebraic group $G$?
Peter-Weyl tells us that $L^2(G)$ is the (closure of) $\bigoplus_\pi A_{\...
- 2,071
5
votes
0
answers
123
views
Conjugacy classes of plane k-jet group
Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(...
- 4,600
5
votes
0
answers
109
views
Restricting projective representations of Lie groups to lattices
Let $G$ be a simple Lie group with trivial center, and let $\Gamma$ be a lattice in $G$. Is it true that an infinite-dimensional projective representation of $G$ restricted to $\Gamma$ can be de-...
- 229
5
votes
0
answers
127
views
What is the minimum number of steps for two elements of a Lie algebra to generate the whole Lie group?
Consider a compact, connected and simply connected Lie Group $G$, and two elements in the corresponding Lie algebra $X$ and $Y$. By successive action of exponential map you can get the following ...
- 101
5
votes
0
answers
140
views
Intermediate subgroups between $H(\mathbb{Z})$ and $H(\mathbb{Z}[\sqrt{2}])$, for anisotropic form of $SL_2$
Let $H$ be an anisotropic $\mathbb{Q}$-form of $SL_2$, with $H(\mathbb{R}) = SL_2(\mathbb{R})$.
Are there any results or conjecture regarding the existence, or non-existence of intermediate subgroups ...
- 474
5
votes
0
answers
208
views
Which groups can occur as the group of units of finite-dimensional noncommutative algebras?
This is a continuation of a previous question: Connectedness of groups of units in finite-dimensional commutative algebras.
Let $k$ be an algebraically closed field of characteristic $0$. Which ...
- 7,127
5
votes
0
answers
291
views
Hyperplane sections of principal homogeneous spaces
Let $P_i$ denote the $i$-th vertex in the Dynkin diagramm of an algebraic group. It symbolizes a parabolic subgroup of $G$ corresponding to the other vertices, meaning $G/P_i$ is a smooth, projective, ...
- 1,479
4
votes
0
answers
100
views
Embedding of a nilpotent algebraic group in upper triangular matrices
Suppose we have a polynomial group law on $G=\mathbb{R}^n$ which gives it a structure of a nilpotent algebraic group.
Is it true that there exists an embedding of $G$ into the group of upper-...
- 728
4
votes
0
answers
147
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?
- 201
4
votes
0
answers
93
views
Homomorphism from a product of spin groups to a bigger spin group
In the paper "Essential dimension of spinor and clifford groups" by Chernousov and Merkurjev, it says that there is a natural homomorphism
$\operatorname{Spin}(n)\times \operatorname{Spin}(m)...
- 321
4
votes
0
answers
180
views
Zariski density for certain subsemigroups
$\DeclareMathOperator\GL{GL}$Suppose that we have a Zariski-dense subgroup $\Gamma$ of $\GL(d,\mathbb{R})$. Let $\delta$ be the abscissa of convergence of the series
$$
\sum_{x \in \Gamma} e^{-s \log\|...
- 89
4
votes
0
answers
211
views
Books on integration on semisimple Lie groups
Can anyone suggest me some good books where I can find integration theory on semisimple Lie groups (using KAK, KAN and other type of decompositions)?
I have read Knapp's book "Lie groups beyond ...
- 1,879
4
votes
0
answers
68
views
The weak restriction of the Jacquet module
Let $P= MN$ be a parabolic subgroup of a reductive p-adic group $G$, and $(\pi, V)$ is an irreducible, admissible representation of $G$. The Jacquet module is the representation $(\pi_N, V_N)$, where $...
- 41
4
votes
0
answers
156
views
Reference request - conjugacy classes over local fields
Is there a nice reference for reductive groups over local fields, which for example contains discussion of things such as: Given a semisimple element in $G(F)$, its $G(F)$-conjugacy class is closed in ...
- 5,562
4
votes
0
answers
300
views
Number of connected components of the centre of a Levi subgroup
Let $G$ be a connected complex semisimple algebraic group and $T\subset B\subset G$ a choice of maximal torus and Borel subgroup. Let $\Phi$ be the root system and $\Pi\subset\Phi$ the set of simple ...
- 41
4
votes
0
answers
105
views
Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$
Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\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
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
4
votes
0
answers
449
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 $...
- 453
4
votes
0
answers
197
views
$G$-Invariant Differential Operators
Let $G$ be a complex algebraic group, $K$ a closed subgroup so that $X=G/K$ is a homogeneous space.
Let $\mathcal{D}(X)$ denote the algebra of differential operators on $X$. The group $G$ acts on $\...
- 1,077
4
votes
0
answers
184
views
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 ...
- 18.7k
3
votes
0
answers
194
views
A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 673
3
votes
0
answers
200
views
Theorem of highest weight of semisimple Lie algebras: what fails precisely for reductive case
Let $\mathfrak{g}$ a complex semisimple Lie algebra. It is well known that by the theorem of the highest weight, all finite-dimensional complex irreps of $\mathfrak{g}$ are (up to iso) classified by ...
- 6,018
3
votes
0
answers
65
views
One parameter subgroups of reductive algebraic groups
If I have a reductive algebraic group $G$ defined over a non-archimedean local field $F$. We can define a one-parameter subgroup to be a group homomorphism from $G_{m}$ to $G$. I was wondering, if I ...
- 63
3
votes
0
answers
237
views
Centralizers and algebraic groups
Suppose $G$ is a linear algebraic group - I am also happy to assume $G$ is a simple algebraic group over an algebraically closed field of characteristic zero, but the question won't require this.
The ...
- 359
3
votes
0
answers
119
views
Describing the outer automorphism of a special unitary group in terms of the Hermitian form
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$Let $h$ be a non-degenerate Hermitian form on $\mathbb{C}^n$ with signature $(p,q)$. Let $\U_h$ denote the associated ...
- 7,527
3
votes
0
answers
101
views
Character formula for real representations
For an irreducible representation of a complex semisimple Lie algebra the Weyl character formula is well known. The real representations of a real semisimple Lie algebra are classified using their ...
- 103
3
votes
0
answers
202
views
The group of fixed points of an involution of a Weyl group
Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$.
Let $W=W(R)$ denote its Weyl group.
Let $S\subset R$ be a basis of $R$ (a system of simple roots).
Let $D=D(R,S)$ denote the ...
- 14.1k
3
votes
0
answers
138
views
density of unipotent flows in algebraic groups
Let $\mathcal{G}$ be a reductive algebraic group over $\mathbb{Q}$ with a model $G$ over $\mathbb{Z}$ such that $G(\mathbb{R})$ is compact modulo centre. Let $T$ be a maximal torus of $\mathcal{G}$. ...
- 81
3
votes
0
answers
190
views
Harmonic analysis on reductive groups over $\mathbb{R}$
A common way of doing harmonic analysis on (the $\mathbb{R}$-points of) reductive groups over $\mathbb{R}$ seems to be to use results from semisimple groups and "see what happens on the center&...
- 131
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
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
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
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