All Questions
Tagged with rt.representation-theory lie-groups
832 questions
4
votes
0
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73
views
a property of the characters for center of universal enveloping algebra
Let $\mathfrak g$ be a complex simple Lie algebra. We fix Cartan subalgebra $\mathfrak h$ and a system of positive roots $\Psi$ for the root system of the pair $(\mathfrak g, \mathfrak h).$ For each $...
- 574
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
114
views
Representation theoretic characterisation of symmetric spaces
Let $G$ be a simple compact Lie group and $H$ a closed subgroup.
Let $\mathfrak{h}\subset \mathfrak{g}$ denote the corresponding Lie algebras. Let $\mathfrak{m}$ be an orthogonal complement to $\...
- 81
4
votes
0
answers
279
views
Cyclic vectors for regular representations
I'm looking for references about the following aspect of cyclic vectors for regular representations.
Let $K$ be a compact Lie group. Let $K$ act on $L^2(K)$ by the left regular representation. Then $...
- 1,903
4
votes
0
answers
91
views
Good range and fair range
Let $G$ be a noncompact simple Lie group with complexified Lie algebra $\mathfrak{g}$. Fix a Cartan involution $\theta$, which defines a maximal compact subgroup $K$ of $G$. Take a $\theta$-stable ...
- 951
4
votes
0
answers
141
views
Langlands dual and integrable representations
Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...
4
votes
0
answers
311
views
Restriction of a representation of a reductive group to the derived subgroup
Let $G$ be a reductive p-adic or Lie group and let $G'$ be its derived subgroup. The main examples I am interested at are $GL_n(F)$ and $SL_n(F)$, similitude groups and classical groups, $GSpin_{n}(F)$...
- 141
4
votes
0
answers
230
views
How to decompose the left regular representation of a real reductive group?
In Dixmier ($C^*$-algebras), the Plancherel theorem states (I will not mention the right regular representation even though the theorem does talk about it):
Let $G$ be unimodular, $\lambda$ be the ...
- 561
4
votes
0
answers
345
views
Representation theory of SU(2) as a discrete group
The representation theory of the Lie group $SU(2)$ is well understood, easy to describe, and in fact central to the classical theory of representation theory of compact Lie groups.
But what happens ...
- 11.9k
4
votes
0
answers
304
views
What is known about the projective representations of $\mathrm{SO}(n_1,n_2)$?
He${}$llo MO.
Let $\mathrm{O}(n_1,n_2)$ be the pseudo-orthogonal group. I am interested in its (continuous, not necessarily unitary, finite-dimensional) irreducible projective representations, for ...
4
votes
0
answers
226
views
Possible dimensions for triples of unitary irreducible representations whose tensor product contains the identity
For which triples $\{A,B,C\}$ of positive integers does there exist a (finite or compact) group $G$ with unitary irreducible representations of dimensions $A$,$B$, and $ C$ whose tensor product ...
- 163
4
votes
0
answers
83
views
Points of failure in definition of X- and A-moduli spaces for arbitrary G
In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. ...
- 1,131
4
votes
0
answers
315
views
Compactly supported distributions as a projective G-module
For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...
- 10.4k
4
votes
0
answers
367
views
Representation theory and associated bundles
I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
- 6,025
4
votes
0
answers
173
views
Ring of SO(n)-invariant differential operators on M_n,m
I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).
There comes a point in the paper (Lemma 2.8) ...
- 41
4
votes
0
answers
154
views
Nilpotent orbits and subspaces
Let ${\mathbb g}$ be a simple complex finite dimensional Lie algebra, $X\subseteq{\mathbb g}$ a nilpotent orbit. Did anyone study maximal vector subspaces of the closure $\overline{X}$?
In particular,...
- 12.3k
4
votes
0
answers
174
views
Number of submodules in $\wedge^2 V$ and $S^2V$ isomorphic to $\mathfrak{g}$
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let
$\mathfrak{g}\subset\mathfrak{so}(V)$ be an orthogonal
irreducible representation. It can be shown that the number of
$\mathfrak{g}$-...
- 59
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
3
answers
997
views
Decomposition of Regular Representation of Non-compact Lie group
Let G be a non-compact Lie group, such as SL(n,R), GL(n,C).
How does the regular representation $L^2(G)$ decompose?
Is there an analogue of Peter-Weyl theorem?
- 3,804
3
votes
1
answer
471
views
Topologie sur l'ensemble des sous-groupes de GL_n(R)
Bonjour,
Est ce qu'il existe une topologie naturelle sur l'ensemble des sous-groupes du groupe général linéaire ?
English translation: "is there a natural topology on the set of subgroups of the ...
- 59
3
votes
2
answers
649
views
Representations of $\mathrm{SO}_n$ versus representations of $\mathrm{Spin}_n$ [duplicate]
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$Since $\Spin_n$ is a compact simply connected simple Lie group, its irreducible representations are equivalent to the irreducible ...
- 327
3
votes
3
answers
629
views
Real representations of SO(n) and U(n)
I would like to get some references where I can find the theory of the real representations of $\mathbf{SO}(n)$ and $\mathbf{U}(n)$.
In particular, I would like to know for which dimensions there ...
- 201
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
1
answer
462
views
R-linear representations of sl(2,C)
Is there some good reference for the classification of finite-dimensional ${\mathbb R}$-linear (as opposed to ${\mathbb C}$-linear) representations of $\mathfrak{sl}_2{\mathbb C}$?
Equivalently, what ...
- 10.4k
3
votes
1
answer
916
views
$SO(N^2-1)$ and the adjoint representation of $SU(N)$
It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...
- 315
3
votes
3
answers
791
views
Peter-Weyl theorem as proven in Cartier's Primer
I'm reading Pierre Cartier's A primer of Hopf algebras to educate myself. In its subsection 3.3 (which doesn't need any Hopf algebra theory), he sketches a proof why compact Lie groups are algebraic. ...
- 33.8k
3
votes
2
answers
492
views
Pairing a root with the half-sum of positive roots
Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...
3
votes
1
answer
484
views
Relationship between the representation theory of $\operatorname{Spin}(n)$ and $\operatorname{SO}(n)$
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\Spin{Spin}$What is the exact relationship between the finite dimensional representations of the group $\SO(n)$ and its covering group $\Spin(n)$? More ...
- 393
3
votes
3
answers
582
views
Reductive group with simply connected derived group has all root groups $\mathrm{SL}_2$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}$Motivation: I am trying to understand why the Deligne-Langlands conjectures are only stated for $p$-adic reductive groups with connected ...
- 143
3
votes
1
answer
279
views
Peter–Weyl decomposition for compact Lie groups with isomorphic Lie algebras
Let $G$ and $H$ be two compact Lie groups with isomorphic Lie algebras $\frak{h} \simeq \frak{g}$, but which are non-isomorphic as topological spaces. From the isomorphism assumption it (should) ...
- 145
3
votes
2
answers
978
views
The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
- 131
3
votes
2
answers
594
views
How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents?
I'm trying to understand the proof of the Oppenheim conjecture using Ratner's theorem, and I don't immediately see why $SO(2,1)$ is generated by unipotents. Why is $SO(2,1)$ generated by unipotents? ...
- 93
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 $\...
3
votes
1
answer
280
views
Decomposition of tensor powers of the vector representation of $\frak{sl}_n$
Let $V(\pi_1)$ be the usual vector/matrix representation of the Lie algebra $\frak{sl}_n$, for $n > 2$. A basic fact is the tensor product $V(\pi_1) \otimes V(\pi_1)$ decomposes as
$$
V(\pi_1) \...
- 257
3
votes
1
answer
147
views
Does every nilpotent orbit have an element supported on the simple root spaces?
Let $G$ be a connected reductive algebraic group (over $\mathbb{C}$) and $\mathfrak{g}$ its Lie algebra. Let $O \subset \mathfrak{g}$ be an orbit of a nilpotent element. Let $\Pi = \{\alpha_1, \dots ,\...
- 57
3
votes
1
answer
503
views
Is the representation of finite simple groups fully understood?
Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...
- 203
3
votes
1
answer
294
views
Lie group GL(4) representation decomposition
Let $V$ be the defining representation of $GL(4,\mathbb C)\to GL(V)$ with $V=\mathbb{C}^4$.
Let $Ext\;V$ be the exterior square of $V$ which is a 6-dim repsentation.
My question: How does $$V\otimes ...
- 131
3
votes
1
answer
3k
views
Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $
My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...
user21574
3
votes
1
answer
152
views
Defining a family of rotations with certain properties
Let $d \ge 2$, and consider the sphere $S^{d-1}$ embedded in $\mathbb R^d$. Does there exist a family of rotations $\{\mathcal O_v\}_{v \in S^{d-1}}$ which satisfies:
$\mathcal O_v e_1 = v$, and
$\...
- 8,512
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
3
votes
1
answer
195
views
A representation of $\frak{sl}_n$ as partial derivatives on polynomials
As is known to all, the Lie algebra $\frak{sl}_2$ admits a very nice representation on
$$
\mathbb{K}[X,Y]
$$
the polynomials in two variables, given by
$$
E \mapsto X\frac{\partial }{\partial Y}, ~~ F ...
- 1,144
3
votes
1
answer
572
views
Dimension of the zero weight space in $V_{2\rho}$
Let $\rho$ be the half sum of the positive roots (also the sum of fundmental weights) for $SL_n(\mathbb C)$. Then $2ρ$ is in the root lattice. Then what is the dimension of the zero weight space for ...
- 31
3
votes
2
answers
704
views
Closure relations between Bruhat cells on the flag variety
Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...
- 1,719
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
2
answers
180
views
Algorithm for finding the symmetries of a linear operator
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}$Let $V, W$ be finite dimensional complex vector spaces and $M\in \Hom(V, W)$ a full rank linear map. I want to see if there exists a Lie group ...
- 111
3
votes
1
answer
221
views
Cartan subspace of graded Lie algebras
Suppose $\mathfrak{g}$ is a complex reductive Lie algebra and $\theta$ is an automorphism of order $2$. Let $\mathfrak{g} = \mathfrak{g_0} \oplus \mathfrak{g}_1$ be the corresponding $\mathbb Z_2$-...
- 673
3
votes
1
answer
355
views
The normalizer of SU(n) in U(m)?
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\Spin{Spin}$Consider the special unitary group $\SU(5)$ and the unitary group $\U(16)$.
Below I specify a specfic way to embed $...
- 10.5k
3
votes
1
answer
298
views
Littlewood Richardson Rule for general linear group over finite field
I just finished reading Green's 1955 paper on characters of general linear groups and have also been reading Macdonald's Symmetric Functions and Hall Polynomials. I see that there is a recursive ...
- 173
3
votes
1
answer
364
views
tensor product of massless Poincare representations
Consider two massless representations of the connected Poincare group $ISO_0(1,3)$ with helicities $s$ and $t$. What is the decomposition of their tensor product into irreducibles?
Massless ...
- 3,873
3
votes
1
answer
136
views
symmetric group of regular polyhedrons
Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let $c:=c(...
- 543