All Questions
Tagged with rt.representation-theory lie-groups
832 questions
7
votes
0
answers
102
views
Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights
$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
- 6,622
5
votes
1
answer
230
views
Explicit Jacquet-Langlands correspondence for real reductive groups
Let $G$ be a connected reductive group over $\mathbb R$. Let $G'$ over $\mathbb R$ be an inner form of $G$ with ${}^LG={}^LG'$. By local Langlands correspondence over $\mathbb R$, if a $L$-packet of $...
- 6,622
3
votes
1
answer
304
views
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$ as $\operatorname{GL}(n,\mathbb{C})$-modules
Consider the natural representations of $\operatorname{GL}(n,\mathbb{C})$ in the spaces
$\bigwedge^2(\bigwedge^k\mathbb{C}^n)$ and $\operatorname{Sym}^2(\bigwedge^k\mathbb{C}^n)$.
Is it known how to ...
- 21.8k
3
votes
1
answer
102
views
Which compact Lie groups have an upper bound on the dimension of irreducible continuous representations?
To fix notation, if $G$ is a compact Lie group, $Rep(G)$ denotes the set of continuous irreducible unitary representations of G, and $\widehat{G}$ denotes the quotient $Rep(G)/\sim$, which identifies ...
2
votes
3
answers
181
views
Stabilizers of the action of Levi on abelianization of nilpotent radical
$\DeclareMathOperator\Lie{Lie}$Let $G$ be a simple connected reductive group over $\mathbb C$. Consider a parabolic subgroup $P=MU$ of $G$, where $M$ is a Levi of $P$ and $U$ is the unipotent radical ...
- 6,622
1
vote
0
answers
71
views
Component groups of stabilizers for linear representations
Let $G$ be a connected simple reductive group over $\mathbb C$. Let $V$ be a finite-dimensional complex representation of $G$.
Given a vector $v \in V$, it is natural to consider its stabilizer group $...
- 6,622
3
votes
1
answer
111
views
Generalization of a result of Kostant related to Gauss decomposition and Toda lattices
I found myself needing a generalization of a result of Kostant in his famous paper
B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math, Volume 34, 1979, ...
- 173
11
votes
0
answers
283
views
Why are there so few irreducible admissible representations of $\text{GL}(n,\mathbb{R})$ (up to infinitesimal equivalence)?
Studying Langlands's classification of irreducible admissible representations, I have been rather stunned by the following:
Theorem
Up to infinitesimal equivalence, all irreducible admissible ...
- 511
9
votes
2
answers
865
views
Multiplication in Peter-Weyl theorem
$\DeclareMathOperator\SL{SL}$It is known that the coordinate algebra $\mathcal O(\SL_n(\mathbb C))$ decomposes as direct sum of $V \otimes V^*$ for $V$ finite-dimensional irreducible representations ...
- 2,964
11
votes
1
answer
331
views
A question on groups having a subgroup which fixes a vector in every irreducible representations
Given a finite group $G$, I am interested in finding a non-trivial proper subgroup $H$ of $G$ such that $\mathrm{Ind}_H^G\mathbf{1}$ contains all the irreducible representations of $G$, that is, ...
- 251
4
votes
1
answer
101
views
K-types of a representation of the minimal Gelfand-Kirillov dimension
Let $G$ be a noncompact real simple Lie group not of Hermitian type, and $\mathfrak{g}_0$ its Lie algebra. Fix a maximal compact subgroup $K$ in $G$ with its Lie algebra $\mathfrak{k}_0$. Write $\...
- 951
3
votes
1
answer
160
views
Embedding flag manifolds of real semisimple lie group
I want to know given a connected (maybe we can assume it to be simply connected or linear) real semisimple lie group $G$ and one of its maximal parabolic group $P$, how can we embed the flag variety $...
- 175
3
votes
1
answer
129
views
Questions about the quotient of extended Weyl group and the isomorphism of extended Weyl group
When I am reading a paper An Algebraic Characterization of the Affine Canonical Basis by Beck, Chari, and Pressley, and I have some questions about some notations.
In the paper, we assume that $\...
- 137
1
vote
0
answers
72
views
Normalizer of connected subgroup contained in the Weyl group?
Let $ G $ be a simple Lie group. Let $ H $ be a connected subgroup of $ G $ such that $ N(H)/H $ is finite. In such a case, is $ N(H)/H $ always a subgroup of the Weyl group of $ G $?
For $ G=\...
- 4,081
3
votes
1
answer
231
views
Does a representation of the universal cover of a Lie group induce a projective representation of the group itself?
Suppose that $G$ is a connected Lie group, $\tilde{G}$ its universal cover, $p:\tilde{G}\to G$ the covering map. Does a representation $\rho$ of $\tilde{G}$ on a finite-dimensional vector space $V$ ...
- 3,115
3
votes
0
answers
75
views
Non-vanishing of a "push-forward" Fourier–Harish-Chandra transform on a compact set
Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ ...
- 81
3
votes
1
answer
162
views
Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
4
votes
0
answers
87
views
Doubling constructions beyond classical groups: general principles?
The doubling method for constructing integral representations of L-functions has been successfully applied to classical groups, as demonstrated in this paper. However, extending this method to a wider ...
- 111
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
2
votes
2
answers
87
views
Computation of ideal of functions, given by explicit quadratic equations, vanishing on $G/P$ for the exceptional Lie group $G_2.$
In Section 10.6.6 of Procesi's "Lie Groups" he writes that a theorem due to Kostant tells us that for an algebraic group $G$ and a parabolic subgroup group $P,$ the ideal of functions ...
- 201
2
votes
1
answer
144
views
Paper request: Graev's classification of SU(2,2) irreducible unitary representations
I am interested in Graev's paper in "M. L. Graev:Dokl. Akad. Nauk SSSR,98, 517 (1954); Amer. Math. Soc. Transl.,66, 1 (1968)." in which the irreducible unitary representations of SU(2,2) are ...
- 679
1
vote
1
answer
102
views
Multiplicities and double and triple tensor products of simple $\frak{g}$-modules
Given a complex simple Lie algebra $\frak{g}$ and a simple module $V_{\lambda}$ for some dominant weight $\lambda$. Consider the tensor product decomposition
$$
V_{\lambda} \otimes V_{\lambda} \simeq ...
- 417
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
6
votes
2
answers
794
views
Tensor algebra and universal enveloping algebra
Let $\mathfrak g$ be a Lie algebra which is not reductive. Let $T(\mathfrak g)$ and $U(\mathfrak g)$ be the tensor algebra and universal enveloping algebra of $\mathfrak g$ respectively. We have a ...
- 673
2
votes
0
answers
59
views
Measure rigidity for higher-rank subgroup actions on homogeneous space
Let $G$ be a semisimple Lie group, $\Gamma$ a lattice in $G$, and $H$ a higher-rank subgroup of $G$ (e.g., a non-split Cartan subgroup). Let $\mu$ be an $H$-invariant and ergodic probability measure ...
16
votes
0
answers
188
views
Representation theory of Pin groups
I am (still) thinking about branching rules from $\mathfrak{so}(n+m)$ to $\mathfrak{so}(n) \oplus \mathfrak{so}(m)$, using Proctor's paper as the starting point.
Proctor describes this rule for $m = 2$...
- 1,574
11
votes
1
answer
383
views
Is there a comprehensive survey of the discrete series representation of a real reductive group?
Vague form of the question: where can one find a comprehensive and possible modern account of the discrete series representations of a real reductive group?
Motivation:
I am a master's student trying ...
- 511
10
votes
0
answers
225
views
Third homology of simply connected Chevalley–Demazure group schemes
I’ve been studying the group of $\mathbb{Z}$-points of the simply connected Chevalley–Demazure group scheme of type $E_7$, denoted $G_{\text{sc}}(E_7,\mathbb{Z})$; see Vavilov and Plotkin - Chevalley ...
- 545
1
vote
1
answer
114
views
Source for highest weight vectors for $\text{SL}_n(\mathbb{C})$ representations
I have a list of a lot of irreducible $\text{SL}_n(\mathbb{C})$ representations and need to know what all of their highest weight vectors are. For example, using the notation from Fulton and Harris, I ...
- 181
1
vote
0
answers
108
views
Schur Weyl Duality for Maximal Torus
I wanted to know if there's some version of Schur Weyl Duality for the maximal torus $T \subset \operatorname{GL}(V)$? Is there also some combinatorial object which might be useful for the same?
2
votes
1
answer
244
views
Decomposition of an $\text{SL}_n(\mathbb{C})$ representation
Let $W = V \oplus V^*$, where $V$ is the standard $\text{SL}_n(\mathbb{C})$ rep and $V^*$ is its dual. I'm ultimately trying to decompose the space $(W \otimes \bigwedge^2 W) / {\bigwedge^3 W}$.
This ...
- 181
6
votes
2
answers
236
views
What is the highest weight of the representation of special orthogonal group $SO(n)$ on the space of harmonic polynomials $\mathcal H_m(\mathbb R^n)$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\diag{diag}$
Let
$$
\mathcal{H}_m(\mathbb{R}^n)=\left\{P\in \mathbb{C}[x_1,\dotsc ,x_n]\left| \begin{align}
P\text{ is homogeneous of degree }m\text{ ...
- 605
1
vote
1
answer
114
views
A correspondence between projective representations of $G$ with those of its universal cover
Let $G$ be a connected Lie group and $\mathcal{H}$ be a Hilbert space. Let $U(\mathcal{H})$ denote the the group of all unitary operators on $\mathcal{H}$ with function composition (i.e., $\hat{U}:\...
- 287
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
0
votes
0
answers
68
views
A reference for this statement (representations of universal central extensions)
Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact:
"Every projective unitary ...
- 287
6
votes
1
answer
317
views
Which Lie groups are covers of matrix groups?
I would like to ask a variation on a question (not yet answered) I previously asked on math.SE, namely:
Which Lie groups are covers of matrix Lie groups? That is, which Lie groups $G$ admit discrete ...
- 3,115
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
4
votes
0
answers
105
views
Irreducible representation of $\mathrm{GL}(2,\mathbb{R})$ that is not admissible
It is a basic theorem of Harish-Chandra that an irreducible unitary representation $\pi$ of a reductive group $G$ over $\mathbb{R}$ on a Hilbert space is admissible, meaning that every irreducible ...
- 413
3
votes
1
answer
107
views
Does the reproducing property of the unitary group Poisson kernel require a multiple of the identity?
The Poisson kernel of the unitary group is
$$ P(Z,U)=\frac{\det(1-ZZ^\dagger)^N}{\det(1-ZU^\dagger)^N\det(1-UZ^\dagger)^N}.$$
It has a reproducing property, $\int dU P(Z,U)f(U)=f(Z)$, where $dU$ is ...
- 2,552
3
votes
0
answers
107
views
Representations of a reductive Lie group vie central character and K-types
Let $G$ be a real reductive group, let $\widehat G$ denote the unitary dual and $\widehat G_{adm}\supset\widehat G$ be the admissible dual, i.e., the set of isomorphy classes of irreducible Harish-...
- 460
2
votes
0
answers
133
views
What can be said about these tensor representations of $\mathrm{SL}(2)$?
Let $W = V \otimes \dots \otimes V$, the product of $n$ copies of $V = \mathbb{C}^2$. Let $G$ and $H$ be two subgroups of the symmetric group $S_n$ and let $\chi$ be a character of $G$.
Associated to $...
- 5,215
6
votes
1
answer
173
views
Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
Let $\mathfrak{sp_n}$ be the symplectic Lie algebra, that is, the $C_n$ complex simple Lie algebra. Is it true that the first fundamental, which is to say the vector space, representation $V_1$ of $\...
- 417
2
votes
1
answer
165
views
Trivial representation of a maximal torus
Let G be a connected compact Lie group and $T\subset G$ a maximal torus. For an irreducible representation $V_\lambda$ of $G$, the multiplicity of the trivial representation of $T$ in $V_\lambda$ is ...
- 128
1
vote
1
answer
240
views
Is the irreducible $ \mathrm{SU}(3) $ subgroup of $ \mathrm{SU}(6) $ maximal?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\S{S}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSL{PSL}\...
- 4,081
4
votes
0
answers
143
views
Representation theory of spinors - Understanding how $\mathrm{SO}_3$ acts in particle physics
$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}$I have started to study particle physics, beginning with wikipedia and I am now reading David ...
- 141
6
votes
1
answer
221
views
Does every Lie algebra appear as centralizer of an element in a semisimple Lie algebra?
Given a finite dimensional, complex, semisimple (fcss) Lie algebra $\mathfrak{g}$ and an element $x\in\mathfrak{g}$, denote by $\mathfrak{g}^x$ the centralizer of $x$ in $\mathfrak{g}$ i.e. the set $\{...
- 188
9
votes
0
answers
366
views
Mappings of the sphere (to itself) defined by homogeneous polynomials
Preamble
$\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that
If $G$ is a subgroup of $\SO(m+1)$ ...
- 39k
5
votes
0
answers
86
views
Spherical functions in the space of functions on real Grassmannians
Let $G=O(n)$ be the orthogonal group. Let $K=S(O(k)\times O(n-k))$ be the subgroup of $O(n)$.
Then the pair $(G,K)$ is symmetric, and the homomogeneous space $G/K$ is the Grassmannian of $k$-...
- 21.8k
0
votes
0
answers
74
views
A question on projective unitary representation of a Lie group
$\DeclareMathOperator\GL{GL}$Let $\mathcal{H}$ be a Hilbert space and $\GL(\mathcal{H})$ denote the group of invertible linear transformations of $\mathcal{H}$. Assume that $G=\{ f:\mathbb{P}\mathcal{...
- 287
14
votes
0
answers
527
views
Is the monster group maximal in SO(196883)?
$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
- 4,081