All Questions
Tagged with lie-groups reference-request
92 questions with no upvoted or accepted answers
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
13
votes
0
answers
475
views
Singular cohomology of $BG$ and Borel cohomology of $G$
Stasheff, in "Continuous Cohomology of Groups and Classifying Spaces", attributes the following result to Wigner.
For $A$ a discrete abelian group and $G$ a finite dimensional locally compact, $\...
- 9,580
10
votes
0
answers
308
views
Compact Lie groups are rational homotopy equivalent to a product of spheres
According to [1] and [2], it is “well-known” that a compact Lie group $G$ has the same rational homology, and according to [2] is even rational homotopy equivalent, to the product $\mathbb{S}^{2m_1+1} ...
- 32.5k
10
votes
0
answers
269
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differentiating positive energy LG reps
Background:Let $G$ be a cscsc¹ Lie group, and let $\widetilde{LG}$ be the universal central extension (center = $S^1$) of $LG:=Map_{C^\infty}(S^1,G)$, with the topology inherited from the $C^\infty$ ...
- 43.2k
9
votes
1
answer
646
views
Explicit construction of a (the?) dual symmetric space
I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type.
...
- 113
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
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
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
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
8
votes
0
answers
388
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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
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
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
7
votes
0
answers
508
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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
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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
7
votes
0
answers
1k
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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
234
views
Nascent formal group law
$\DeclareMathOperator\FGL{FGL}$The formal group law (cf. Wikipedia, Ex. 1.6 of nLab, Hazewinkel) derived from an analytic function or formal series $f(x) = x + a_2 x^2 + a_3 x^3 + ...$ and its formal ...
- 10.5k
6
votes
0
answers
550
views
Lattices in Lie groups
In the literature, people seem to predominantly look at lattices in nilpotent or reductive groups.
Is there a result that gives a general description of a lattice in an arbitrary Lie group?
Something ...
user130903
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
6
votes
0
answers
1k
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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
6
votes
0
answers
184
views
Reference request: fusion rules for unitary dual of SL(3,R)?
By the fusion rules, I mean: given two unitary irreps of the group, what unitary irreps occur in their tensor product and with what "multiplicity"? (I am guessing that direct integrals ...
- 25.8k
6
votes
0
answers
157
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Are there versions of highly connected covers of Lie groups with highly periodic homotopy groups?
There is much activity around the study of highly connected covers of Lie groups (well, of their "infinite rank" versions like $\displaystyle{\lim_{N\to\infty}} \ O(N)$, say).
Looking at the ...
- 18.4k
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
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
5
votes
0
answers
254
views
Reference on "infinite dimensional Lie algebras" from a mathematical physics point of view
It happens that I stumbled on a class of infinite dimensional Lie algebras that are not Kac-Moody algebras and for which I was not really prepared for. I know some general results on infinite ...
- 295
5
votes
0
answers
345
views
CW-structure on flag manifolds
I want to apologize in advance if my question is too elementary as I am not an expert in Lie theory. I have posted it before on stackexchange without receiving an answer.
Let $G$ be a compact Lie ...
- 12.1k
5
votes
0
answers
276
views
Fundamental group of compact globally symmetric spaces
The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient
$$(*) \quad \pi_1(...
- 1,123
5
votes
0
answers
411
views
Lagrangian subgroup of a nonabelian Lie group
My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.
See a previous post for other background ...
- 10.5k
5
votes
0
answers
99
views
Does there always exist an irreducible representation occurring with multiplicity one when inducing from $M=Z_K(A)$ to $K$?
This question is a more specific version of Does there always exist an irreducible representation occurring with multiplicity one when inducing from a closed subgroup to a compact Lie group? .
Since ...
- 1,942
5
votes
0
answers
304
views
Decompositions of a compact Lie group into "fixed point set types"
Consider a compact Lie group $G$ which acts on a closed Riemannian manifold $M$ by isometries. Then it is well-known that there are only finitely many isotropy types of the $G$-action, i.e. finitely ...
- 1,942
5
votes
0
answers
214
views
Explicit generators for homotopy groups of Lie groups
I would like to know explicit formulas for generators of the infinite cyclic summands in the homotopy groups of Lie groups, in the form of continuous (or smooth if possible) maps $S^n\to G$.
It is ...
- 17.4k
4
votes
0
answers
92
views
Lie bracket of general unipotent matrices
Let $k$ be a field (of characteristic $0$). Let
$$
X:=\begin{pmatrix}1&x_{1,2}&x_{1,3}&\cdots&x_{1,n}\\ &1&x_{2,3}&\cdots&x_{2,n}\\ &&1&\cdots&x_{3,n}\\ ...
- 449
4
votes
0
answers
101
views
Modern reference for a theorem by Bott on the Dolbeault cohomology of compact homogeneous manifolds
I am looking for a modern, maybe shorter or even easier, reference for Theorem II of Homogeneous vector bundles (R. Bott, Annals of mathematics, 1957). This is a theorem where the Dolbeault cohomology ...
4
votes
0
answers
320
views
Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$
Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified.
Is there any characterization of $\Gamma$ such that $\Gamma$...
- 1,441
4
votes
0
answers
552
views
Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$
Edit: Thoughts updated (22/3/2021).
I've come across with the following problem.
Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
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
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
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
3
votes
0
answers
80
views
Can a semisimple orbit always be identified with a cotangent bundle?
Let $H$ be a semisimple element of the Lie algebra $\mathfrak{g}$ of a semisimple Lie group $G$, and let $M:=\mathrm{Ad}_G(H)\subset\mathfrak{g}$ be the corresponding adjoint orbit. If we choose a ...
- 2,527
3
votes
0
answers
94
views
The tangent bundle of $G \times_H M$
Let $G$ be a Lie group with a closed subgroup $H$, and let $M$ be a smooth $H$-manifold. I am searching for a reference where it is proved that the tangent bundle of $G \times_H M$ is isomorphic to ...
- 198
3
votes
0
answers
142
views
Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces
Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
- 954
3
votes
0
answers
50
views
How to construct lattice points in bounded symmetric domain?
Consider the Hermitian bounded symmetric domain for $k \leq m$:
$$
C_{k, m} = \{ Z \in \mathbb{C}^{m\times k} \,|\, Z^*Z < I_k \}
$$
where $I_k$ is the $k\times k$ unit matrix. If I am not mistaken,...
- 8,597
3
votes
0
answers
205
views
Status of RFD groups and $C^*$-algebras
Motivated by this question and its great answers, I become very curious to know what do we know about RFD (residually finite dimensional) groups and $C^*$-algebras, e.g. do we know how these ...
- 1,586
3
votes
0
answers
65
views
Suggested papers or reading for PDE (high dimension) reduction to ODE by symmetries
Could anyone please suggest related papers or article about the topic related to my one question below?
Reduce PDE to ODE by dilation symmetry
I also cite a paper in the link above.
We know that ...
- 345
3
votes
0
answers
58
views
Isoclinism for Lie groups: existing accounts of basic properties?
Philip Hall introduced the relation of isoclinism between two groups. One statement of the definition (not Hall's original statement) is to introduce a category whose objects are the canonical maps
$$...
- 25.8k
3
votes
0
answers
133
views
Weyl group stabilizer of semisimple element in adjoint group
Let $G$ be semisimple group over $\mathbb{C}$ of adjoint type. Let $T$ be a maximal torus, $s\in T$ semisimple element. Let $W$ be a Weyl group and $W(s)$ be a stabilizer of $s$ in $W$. I am ...
- 31
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
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
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
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
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