All Questions
86 questions
3
votes
0
answers
362
views
Unitary representation of finite-dimensional unitary group
the question is the following. Let n,m be integers, $U(n)$ be the unitary group of $M_n(\mathbb C)$, and $\phi\colon U(n)\to U(m)$ be a continuous group homomorphism, that is moreover irreducible as a ...
1
vote
0
answers
70
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$\Gamma$ cohomology of principal series
Let $G$ be a noncompact connected real semisimple Lie group with finited center. Let $\Gamma$ be a cocompact discrete subgroup of $G$, and let $P$ be a parabolique subgroup
with Langlands ...
- 1,111
10
votes
1
answer
354
views
Derivation of Blattner's conjecture in the Beilinson-Bernstein picture
On the last page of Schmid's article "Discrete Series", he says
"In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb ...
CommunityBot
- 1
2
votes
1
answer
280
views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 23k
5
votes
2
answers
1k
views
Borel–Weil theorem - reference request
I am asking about good references (both books and papers) for the well-known Borel–Weil theorem. Thank you very much!
- 6,210
6
votes
4
answers
477
views
Topological properties of $K$ orbits in $G/B$
I'll be working over the complex numbers.
Let $G$ be a connected reductive group, $\theta\colon G\to G$ an involution. Let $K=G^{\theta}$ be the fixed point subgroup. I am trying to track down ...
- 2,449
6
votes
1
answer
255
views
Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky
The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, \...
- 4,612
4
votes
2
answers
341
views
reference help indecomposable representations of SL(2,R)
Let $\mathfrak{g}$ be the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, $K=SO(2)$ the maximal compact subgroup of $SL_2(\mathbb{R})$. Then the classification of irreducible admissible $(\mathfrak{g},K)$-...
- 2,554
3
votes
0
answers
214
views
Unitary dual of $Sp_4(\mathbb{R})$
Do we know the unitary dual of $Sp_4(\mathbb{R})$? If so, can someone provide me any references? How about other rank 2 real groups? Thank you!
- 52.9k
7
votes
1
answer
389
views
Lie group actions with only one orbit type, but not defining a principal bundle
Searched-for situation: A compact connected Lie group acts effectively on a closed Riemannian manifold by isometries, such that there is only one orbit type of dimension strictly less than that of the ...
CommunityBot
- 1
5
votes
0
answers
304
views
Decompositions of a compact Lie group into "fixed point set types"
Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...
- 1,942
6
votes
0
answers
455
views
Cohomology of Bott-Samelson varieties?
How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...
- 1,719
3
votes
1
answer
461
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 ...
- 31.5k
6
votes
3
answers
757
views
Decomposition of $L^2(\Gamma \backslash G)$
Let $G$ be a semisimple Lie group, and $\Gamma$ be an lattice (arithmetic) - typical examples I am thinking about would be $(SL_2(\mathbb{R}), SL_2(\mathbb{Z})$, or $(SL_2(\mathbb{C}), PGL_2(O_F))$ (...
CommunityBot
- 1
4
votes
1
answer
256
views
Weyl group action on complexified Iwasawa decomposition
Let $G$ be a complex, reductive, algebraic group and let $G=KB$ be the complexified Iwasawa decomposition of $G$, see also [SW02]. Let $T$ be a maximal torus of $B$, therefore a maximal torus of $G$. ...
- 27.8k
6
votes
0
answers
244
views
Zariski closure of orbits of real groups on complex flag manifolds
Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...
- 27.8k
1
vote
0
answers
196
views
Reference Help: Matsuki duality Orbits
I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...
- 111
12
votes
3
answers
2k
views
What is a good introduction to branching rules in representation theory?
I'm looking for a book or introductory article, that explains branching rules in representation theory of real Lie groups.
When a Lie group has a set of irreducible representations, I'd like to know ...
- 23k
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 ...
- 2,639
1
vote
1
answer
308
views
Holomorphic representations of complex reductive Lie groups and the boundary of orbits (Reference request)
I have difficulties finding an appropriate reference for the following question (which I hope that it to be true).
Let $U$ be a compact Lie group, $G:=U^{\mathbb{C}}$ its complexification and $\tau: U^...
CommunityBot
- 1
15
votes
2
answers
2k
views
Isomorphism between Spin(3,2) and Sp(4, R)
I've been using the fact that Spin(3,2) is isomorphic to Sp(4, R) for a while, but I've never seen a proof. Can anyone point me in the direction of a good reference?
- 45.7k
9
votes
3
answers
894
views
Representation rings of exceptional Lie groups
Let $G$ be a compact Lie group and let $R(G)$ denote its complex representation ring. If $G$ is simply connected, such as $G_2$, $F_4$ or $E_8$, then it is known that $R(G)$ is a polynomial ring [F. ...
- 5,191
4
votes
1
answer
713
views
Criterion for nilradical of a maximal parabolic subalgebra to be abelian?
This question has some overlap with previous ones but doesn't seem to have a well-documented answer. I recall some literature (mostly involving Lie groups and hermitian symmetric pairs, etc.) which ...
CommunityBot
- 1
4
votes
3
answers
2k
views
Lie subgroups of SU(4)
Other than subgroups of SU(3), what are the Lie subgroups of SU(4)? Assume that the subgroup is closed but not necessarily connected.
Additionally, which of these subgroups admit four dimensional ...
- 51
5
votes
2
answers
452
views
"geometric" description of the algebra of central functions on a Lie group
I am looking for a a description of the algebra of continuous central functions on a group, say a compact simple Lie group $G$, as the algebra of all continuous functions on a "nice" compact Hausdorff ...
- 52.9k
7
votes
2
answers
595
views
Representation theory of Discrete Subgroups of Lie groups
My question is the following. Which representations of $Sp(2g, \mathbb Z)$ are extendable to representations of $Sp(2g, \mathbb C)$ or $Sp(2g, \mathbb R)$. Is there a general theory and a good ...
- 11.2k
6
votes
3
answers
1k
views
Good book on representation theory of GL(n)
I am interested in a recommendation for a good book which discuses representation theory of GL(n)(say over field of complex numbers).
I know only a basic representation theory.
The question I am ...
- 14.5k
6
votes
1
answer
2k
views
A request for suggestions of advanced topics in representation theory
Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast ...
- 5,984
8
votes
0
answers
388
views
Reference Request - Spaces of Smooth Vectors
I was recently looking for examples of non-nuclear spaces of smooth vectors of representations of Lie groups. I'll recall the basic definitions. Let $\pi$ be a unitary irreducible representation of a ...
- 431
6
votes
3
answers
466
views
Representation of double cover of $U(n)$ on eigenspaces of harmonic oscillator
Consider the metaplectic representation of $Mp(n)$ on $L^2(\mathbb R^n)$. We can view $U(n)$ as a subgroup of $Sp(n)$ and so inside $Mp(n)$ is a double cover $\tilde U(n)$ of $U(n)$. The restriction ...
- 23k
19
votes
3
answers
1k
views
Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
- 965
6
votes
1
answer
1k
views
Decomposition of semisimple Lie group into almost simple factors
Can anyone suggest a reference that defines or explains that a semisimple real Lie group can be decomposed into a product of almost simple factors? In some papers that I read recently people keep talk ...
- 52.9k
6
votes
2
answers
921
views
Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved)
I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference ...
- 799
8
votes
2
answers
572
views
reference containing the list of irreducible finite dimensional representation of real general linear group
It seems that it is not easy to find a reference containing a classification and construction of finite dimensional irreducible representations of $GL_n(\mathbb{R})$. One way to look at it is via $(\...
- 52.9k
6
votes
1
answer
2k
views
How to calculate partition function of a QFT by summing over irreducible representations of the symmetry group?
By definition computing the partition function of a QFT amounts to doing a Feynman Path Integral exactly. At a schematic level I can see why this can become a question of summing/integrating over ...
- 4,577
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
$\...
- 44.4k