All Questions
Tagged with lie-groups reference-request
298 questions
-1
votes
1
answer
90
views
Reference Request: Carnot Group Not Containing Group of Isometries [closed]
This question is a follow-up to this post, from which I quote:
Let $\mathfrak{e}$ be the 3-dimensional Lie algebra with basis $(H,X,Y)$ and bracket $[H,X]=Y$, $[H,Y]=-X$, $[X,Y]=0$. It is ...
- 5,405
14
votes
5
answers
1k
views
History of the notion of $(G,X)$-structure
I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...
- 403
4
votes
1
answer
194
views
Computing Deligne-Lusztig Characters in General
The goal for this question is to try to find a relatively explicit way of computing the Deligne-Lusztig characters. I understand that the $R_{T,\theta}$ can be computed if we know the values of the ...
- 173
11
votes
1
answer
222
views
Reference requests: Integral cohomology of $G_2$-homogeneous spaces
Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed?
Great computational efforts using representation theory in order to determine the ...
- 2,630
3
votes
0
answers
136
views
Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
5
votes
2
answers
1k
views
Malcev's paper "On a class of homogeneous spaces" in English
I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
- 51
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
5
votes
1
answer
445
views
An easier reference than "On the Functional Equations Satisfied by Eisenstein Series"?
I'd like to learn about Eisenstein series so I started reading "On the Functional Equations Satisfied by Eisenstein Series"by Langlands.
http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/...
- 3,625
5
votes
0
answers
411
views
Lagrangian subgroup of a nonabelian Lie group
My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.
See a previous post for other background ...
- 10.5k
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
15
votes
1
answer
612
views
Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?
Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
- 2,789
7
votes
0
answers
106
views
Normalizers of subsystem subgroups of Lie groups
Let $G$ be a semisimple complex Lie group, and let $H$ be a subgroup corresponding to a subset of the extended Dynkin diagram of $G$ (à la Borel - de Siebenthal). I would like to know if there is a ...
- 71
2
votes
0
answers
406
views
Invariants of the group $SO(2)$
Let $V_d$ be the complex vector space of binary forms of degree $d$ endowed with the natural
action of the special orthogonal group $SO(2).$ Consider the corresponding action of the
group $SO(2)$ on ...
- 656
7
votes
1
answer
237
views
Finite subgroups of $PSU(3)$
I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
- 71
8
votes
0
answers
411
views
Which representations of the Lie algebra of a Lie group come from representations of the group itself?
I think this is very classic mathematics, but I can't find a complete answer in the literature.
Let $G$ be a Lie group, $\mathfrak{g}$ the Lie algebra of $\mathfrak{g}$. Suppose $\rho : \mathfrak{g} \...
- 1,586
5
votes
1
answer
1k
views
Spin groups in terms of matrices and/or linear operators
Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...
- 349
3
votes
0
answers
129
views
Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
5
votes
1
answer
578
views
Lie algebra of a compact Lie group and derivations of the Hopf algebra of representative functions
Let $\mathcal{G}$ be a compact (real) Lie group. We know that the Lie algebra $\mathfrak{g}$ of $\mathcal{G}$ is, by definition, the space of all left-invariant (smooth) vector fields over $\mathcal{G}...
- 718
6
votes
1
answer
169
views
Existence of a real eigenvalue is a necessary condition for the density of all the orbits of a Lie subgroup of $GL(\mathbb{R},d)$
Good morning,
I would like to pose the following (maybe naive) question. Let $\mathfrak{a}\subset \mathfrak{gl}(\mathbb{R},d)$ be any lie subalgebra, and $A$ be the connected, simply connected ...
user17697
8
votes
0
answers
682
views
Convolution theorem on a non-abelian Lie group
Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the ...
- 914
4
votes
1
answer
183
views
Multiplicities in Plancherel theorem for SL2(R)
The usual formulation of the Plancherel theorem one writes $f(1)$ as an integral over the dual $\widehat G$. The support of the measure is the set of representations which weakly occur in $L^2(G)$. ...
user1688
3
votes
3
answers
1k
views
Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
- 6,023
5
votes
3
answers
849
views
Weyl's Branching Rule for $SU(N)$-Setting
On the Wikipedia page for restricted representations
https://en.wikipedia.org/wiki/Restricted_representation
there is presented a number of explicit "branching rules". In particular, there is the ...
- 835
3
votes
1
answer
96
views
The role of semisimple factors in transitive group actions on manifolds
I am trying to remember a result stating that under certain assumptions,
given a transitive smooth action of a (compact?) Lie group on a smooth manifold, also the action of the semisimple factor is ...
- 4,729
5
votes
0
answers
99
views
Does there always exist an irreducible representation occurring with multiplicity one when inducing from $M=Z_K(A)$ to $K$?
This question is a more specific version of Does there always exist an irreducible representation occurring with multiplicity one when inducing from a closed subgroup to a compact Lie group? .
Since ...
- 1,942
8
votes
1
answer
673
views
Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
- 193
8
votes
0
answers
129
views
Is there a splitting rule for the restriction of a $GL(23, \mathbb{Q})$-representation to $O(23, \mathbb{Q})$?
I am interested in a $23$-dimensional $\mathbb{Q}$-vector space $V$ which I am viewing as a GL$_{23}(\mathbb{Q})$ representation. Schur functors can be defined over $\mathbb{Q}$, so we get ...
- 233
4
votes
1
answer
189
views
cohomology of finite groups of lie type with coefficients in the adjoint module
Let $\mathbb G$ be a connected, semisimple, split group over a finite field $\mathbb F_q$ and let $G = \mathbb G(\mathbb F_q)$. Let $\mathfrak g$ be its Lie algebra, an $\mathbb F_q$-vector space with ...
- 391
3
votes
0
answers
170
views
Finding generators of equivariant cohomology
Let $(M,\omega)$ be a symplectic manifold with symplectic form $\omega$, carrying a Hamiltonian action of a compact connected Lie group $G$ with moment map $\mu:M\to \mathfrak{g}^\ast$, where $\...
- 1,942
2
votes
0
answers
81
views
Fourier transform in the complex motion group
I am looking for a reference that deals with the unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$ i.e., the semi-direct product of $\mathbb C^2$ with the special unitary group $K=...
- 650
9
votes
1
answer
874
views
Proofs that the conformal group in dimension $\ge 3$ is a Lie group
Let $M$ be a smooth manifold of dimension $\ge 3$, equipped with a conformal structure (or a Riemannian metric). Then, the group of conformal diffeomorphisms is a finite dimensional Lie group.
A ...
- 6,741
8
votes
2
answers
617
views
Reference request: Models of cuspidal representations of GL(n,k) where k is a finite field
Let $k=\mathbb{F}_q$ where $q$ is a prime power of odd cardinality.
Where could I find explicit models of all irreducible cuspidal (complex) representations of $GL_n(k)$ for $n\ge 3$?
I understand ...
- 960
15
votes
1
answer
951
views
Duistermaat and Kolk's lost chapters on Lie groups
In Duistermaat and Kolk's book Lie Groups, it is written in the preface that "the text contains references to chapters belonging to a future volume". I could not find this second volume anywhere. Has ...
- 151
9
votes
0
answers
470
views
Branching rules for compact Lie groups
Let $G$ be a compact connected Lie group, and let $H\subset G$ be a closed subgroup. For an irreducible representation $\pi:G\to\mathrm{End}_\mathbb{C}(V)$ of $G$ ($\dim\pi<\infty$) I want to know ...
- 1,959
6
votes
0
answers
227
views
Origins of the generalized shift operator exp(t*g(z)d/dz)
Charles Graves in the 1850s investigated iterated operators of the form $g(x) \frac {d}{dx}$ (see page 13 in The Theory of Linear Operators ... (Principia Press, 1936) by Harold T. Davis). Graves ...
- 10.5k
17
votes
1
answer
1k
views
References for Langlands classification
I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.
My ...
- 5,215
7
votes
0
answers
508
views
Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
- 1,942
7
votes
0
answers
218
views
Freudenthal geometries for exceptional simple Lie groups
When reading answer to this question I recall Freudenthal, Lie groups and foundations of geometry, 1964. In chapter 4 he describes 2-dim elliptic geometry, 2-dim projective geometry, 5-dim symplectic ...
user21230
5
votes
1
answer
245
views
Equivariant cohomology ring is an integer domain
Let $G$ be a connected compact Lie group and let $V$ be a complex $G$-representation. Denote by $\mathbb{P}(V)$ the projectivization of the vector space $V$. I would like to ask a couple of questions ...
3
votes
0
answers
156
views
How large is the intersection of the root system of a subalgebra of a compact Lie algebra with the original root system?
Let $\mathfrak{g}$ be a finite-dimensional real compact Lie algebra and $\mathfrak{t}\subset \mathfrak{g}$ a maximal abelian subalgebra. Let $\Delta(\mathfrak{g}_\mathbb{C},\mathfrak{t}_\mathbb{C})\...
- 1,942
20
votes
4
answers
3k
views
Online References for Cartan Geometry
I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization...", which would take a long time ...
- 5,215
5
votes
1
answer
978
views
Existence proof of Bourbaki, Differentiable and Analytic Manifolds
I am reading through Chapter III of Bourbaki, Lie Groups and Lie Algebras, and many proofs cite the Bourbaki volume Differentiable and Analytic Manifolds. I can't find this book anywhere. Does it ...
- 6,180
3
votes
1
answer
396
views
Choosing canonical representatives of Weyl group elements, some questions
Let $G$ be a connected, reductive group which is quasisplit over a field $k$ of characteristic zero. Let $B$ be a Borel subgroup defined over $k$, containing a maximal torus $T$ defined over $k$. ...
- 6,180
3
votes
0
answers
184
views
Mackey Obstruction Class with Integral Coefficients
Consider an exact sequence of groups
\begin{equation}
1\rightarrow H\rightarrow K\rightarrow G \rightarrow1~.
\end{equation}
Mackey theory enables us to understand representations of $K$ in terms of ...
- 2,087
6
votes
0
answers
1k
views
Definition of Admissible Representation
Let $G$ be a connected, reductive group over a number field $k$. Let $v$ be a place of $k$.
If $v$ is finite, an admissible representation of $G(k_v)$ is defined to be an abstract representation of $...
- 6,180
2
votes
0
answers
571
views
Clebsch–Gordan(CG) coefficients for SO(N) and Sp(N) group
I know how to calculate the CG coefficients for $SU(N)$, but there are other simple Lie group like $SO(N)$ and $Sp(N)$. But up to now I can't find any textbook tells me how to calculate these and I ...
- 249
5
votes
1
answer
270
views
Is there any work on quantization of distributions?
Let $G$ be a Lie group and consider the space $C_c^\infty(G)$ of compactly supported complex-valued smooth functions on $G$ and $D'(G) = (C_c^\infty(G))'$ the topological dual linear space of $C_c^\...
- 561
6
votes
1
answer
242
views
Imbedding of a representation of a compact subgroup
Let $G$ be a compact subgroup of $O(n)$. Let $\rho$ be a continuous finite dimensional representation of $G$.
Question Is it true that there exists a continuous finite dimensional representation $\...
- 21.8k
1
vote
0
answers
955
views
Iwasawa decomposition and compact subgroups
Let $G$ be the $k$-points of a connected, reductive group $\mathbf G$ over a local field $k$. I have heard several statements about compact subgroups and Iwasawa decomposition, mostly in the context ...
- 6,180
2
votes
1
answer
833
views
Which Lie algebras are realised as vector fields on a group?
Fix a Lie group $G$. The vector fields on $G$ form an (infinite-dimensional) Lie algebra with the commutator of vector fields as Lie bracket. What (finite-dimensional) Lie algebras can I realise as ...
- 503