Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

learn more… | top users | synonyms

5
votes
1answer
165 views

Orbit structure of linear representations of complex Lie groups

Let $G$ be a semisimple complex Lie group (or perhaps a reductive algebraic group over $\mathbb{C}$) and $V$ an irreducible finite-dimensional representation of $G$, determined by its highest weight. ...
2
votes
2answers
361 views

Equivariant Cohomology of a Complex Projective Variety

Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...
0
votes
0answers
89 views

Cartan involutions of su(n)

I have a question regarding Cartan involutions of su(n). Some sources say that there is only one up to equivalence (Wikipedia on Cartan Decomposition). Others say there are Types I, II, III. I looked ...
5
votes
1answer
181 views

Haar measure on $O(n)$ reduced to simpler probability space

The background of this question is how a random variable $X$ on the orthogonal group $O(n)$ whose distribution is the normalized Haar measure $\mu$, i.e., $\mu( O(n) ) = 1$, can be realized on a ...
12
votes
3answers
746 views

How can I tell if a group is linear?

The basic question is in the title, but I am interested in both necessary and sufficient conditions. I know the Tits' alternative and Malcev's result that finitely generated linear groups are ...
2
votes
1answer
205 views

A specific question regarding a proof in Knapp's book

I got stuck in an apparently trivial point within the proof of Lemma 3.13 on p. 55 of Knapp's Representation Theory of Semisimple Groups. The author concludes in the first paragraph that $f_v$ must be ...
-1
votes
1answer
223 views

identifying dual of lie algebra of general linear groups

Is there any reference for the following fact? I am looking for a nice and simple proof. Assume that $G=GL(n,C)$, the group of invertible $n\times n$ matrices with complex entries. Why can the dual ...
5
votes
1answer
191 views

Levi decomposition in disconnected linear algebraic group (characteristic 0)?

For algebraic groups or Lie groups, the subject of Levi decompositions tends to be surrounded by some mystery in the literature (and in an older question raised here). While I postpone further my ...
0
votes
0answers
26 views

how to compute $k_0$ from a Vogan diagram(specially exceptional case)

Suppose the Vogan diagram of $E_6$ is given where the root $\alpha_2$ is painted. Now from this diagram how shall I compute the $k_0$ part of $g_0$, Where $g_0 = k_0 + p_0$ is the Crartan ...
5
votes
0answers
223 views

Quotient of 3-sphere by binary octahedral group?

Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...
1
vote
0answers
144 views

Equivariant $K$-theory, singular vectors, and flag manifolds

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...
5
votes
0answers
112 views

Is there an analogue of spin/oscillator representation for the general linear Lie algebra?

(Work over complex numbers) Let $V$ be an orthogonal space. Let $Pin(V)$ be the double cover of the orthogonal group $O(V)$. Then $Pin(V)$ has a basic spin representation which we can think of as the ...
1
vote
2answers
226 views

Finding spherical representations of $GL(n, \mathbb{C})$.

I am looking for literature that might contain the spherical representations of $GL(n, \mathbb{C})$. Here a spherical representation is an irreducible representation $\rho$ of $G$ on $\mathbb{C}$ such ...
5
votes
1answer
226 views

Invariants of a $GL(3,\mathbb{R})$ action

I'm trying to understand the standard $GL(3,\mathbb{R})$ action on the 15-dimensional space of possible values for the derivative of the Riemann curvature tensor of a 3-dimensional manifold $M$ at a ...
9
votes
4answers
720 views

Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
4
votes
2answers
192 views

Homomorphisms of Lie groups preserving regularity

Let $G_1, G_2$ be connected semisimple Lie groups, let us assume for simplicity that both groups are complex (even though, I am interested in the real Lie groups as well). Let $f: G_1\to G_2$ be a ...
2
votes
2answers
118 views

Connectedness of Springer Fibers

Let $G$ be a connected, simply-connected, complex semisimple Lie group with Lie algebra $\frak{g}$. Let $\mu:T^*\mathcal{B}\rightarrow\mathcal{N}$ be the Springer resolution of $\mathcal{N}$. If ...
1
vote
0answers
85 views

Zariski dense subgroup of $SL(3,\mathbb{R})$

Let $\Delta$ be a Zariski dense finitely generated subgroup of $SL(3,\mathbb{R})$. Assume that $\Delta$ contains no element of finite order. Then, does there exist a finite-order element $A \in ...
0
votes
2answers
133 views

quasi-minuscule representations

Wich representations of $F_{4}$, $E_{8}$ and $G_{2}$ are quasi-minuscule?
4
votes
1answer
272 views

Normal forms for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$

Is there a standard normal form for homogeneous cubic polynomials in $\mathbb{R}[x_1, x_2, x_3]$? Or, put another way, is there a nice way to describe the orbit space of the natural (diagonal) action ...
2
votes
1answer
273 views

Reference request - localisation de g-modules

Does anyone have a link to a copy of Beilinson-Bernstein's "Localisation de g-modules", in which they prove the Beilinson-Bernstein theorem? I can't find it anywhere.
2
votes
1answer
278 views

semidirect product

Is it true that the semi direct product of two semi simple Lie groups is semi simple?
3
votes
1answer
136 views

Reduction of antisymmetric complex matrices

Let $E=\mathfrak{so}(n,\mathbb{C})$ be the Lie algebra of antisymmetric complex matrices. We consider the action of the complex orthogonal group $SO(n,\mathbb{C})$ on $E$ by conjugation. Is there a ...
9
votes
1answer
363 views

Borel's Paris Lectures

I am trying to read Harish-Chandra's book on automorphic forms on Semisimple Lie groups, and he keeps referring to Borel's Paris lecture notes. Does anyone have an online version of these notes or ...
2
votes
1answer
153 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 ...
1
vote
1answer
239 views

Thom-Gysin Sequences and Stratifications

Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...
2
votes
0answers
106 views

Automorphism of a Lie group which preserves a maximal torus is necessarily an inner automorphism?

Let $G$ be a connected Lie group with a maximal torus $T$. Suppose $\sigma$ is an automorphism of $G$ so that $\sigma(T)=T$. Then can we conclude that $\sigma$ is an inner automorphism of $G$? (i.e. ...
3
votes
1answer
160 views

Normal subgroup of the identity component of a linear Lie group is normal in the whole group?

Suppose $G$ is a linear Lie group (i.e. $G$ admits a finite dimensional faithful representation) and $G$ has finitely many connected components. Let $G_0$ be the identity component of $G$. If $N$ is a ...
2
votes
2answers
337 views

Is there an almost-direct product decomposition for disconnected reductive algebraic groups?

$\textbf{Some definitions:}$ Let $G$ be an algebraic group (for me that is the complex points of an affine algebraic group). We say $G$ is reductive if its unipotent radical (maximal connected normal ...
4
votes
1answer
295 views

Transitive action on the sphere

Hello! From the book "Einstein manifolds" by Arthur L. Besse (at section 7.B), Lie groups $Sp(n)$, $Sp(n)\cdot U(1)$, $SU(2n)$ and $U(2n)$ constitute the complete list of Lie subgroups of $U(2n)$ ...
4
votes
4answers
189 views

Stratifications and Cohomology Computations

I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...
1
vote
2answers
132 views

Cross section for closed Lie subgroup in a Lie group

Let $G$ be a Lie group and $H$ a closed Lie subgroup. Is there an explicit way to construct a local cross section of $H$ in $G$ so that $\pi: G\to G/H$ is a fiber bundle?
10
votes
5answers
435 views

What are the invariant Pseudo-differential operators on a Lie group?

It is well-known that (left) $G$-invariant differential operators on a Lie group $G$, has an algebraic description, i.e. universal enveloping algebra of the Lie algebra of the group. On the other ...
4
votes
0answers
264 views

Which orbits of a separable representation of the infinite unitary group are closed?

Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following: Is it true that all ...
8
votes
3answers
685 views

HIgher Homotopy Groups and Representation Theory

Let $G$ be a compact Lie group, and $g$ its associated Lie algebra. In what ways do the higher homotopy groups $\pi_{n}(G)$ with $n>1$ appear in the representation theory of $G$? As an example, ...
7
votes
3answers
405 views

Homotopy classes of maps to Lie groups

In Physics one often encounters maps from a certain manifold $M$ to a Lie group $G$. For example, in gauge theories, this gives a gauge transformation, wich is a symmetry of a theory. It is then ...
0
votes
2answers
236 views

Extensions of Groups

I believe there is a reasonable notion of $\text{Ext}^1(G,H)$ in the category of groups (where $G$ and $H$ are groups). Is there a decent reference describing this? My particular situation involves a ...
3
votes
1answer
282 views

Centralizers of Nilpotent Elements in Semisimple Lie Algebras

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$, and let $\xi\in\frak{g}$ be a nilpotent element. I am interested in understanding the structure of ...
0
votes
1answer
127 views

A question about G-Manifolds

I am looking for a clear reason for following fact:Is there any reference ? Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...
3
votes
1answer
170 views

Maximal subgroups of semisimple Lie groups

The problem of finding and classifying the maximal subgroups of simple Lie groups like $SU(3)$ is well known and solved in the literature. What about maximal subgroups of semisimple groups like $SU(3) ...
0
votes
1answer
390 views

Is the unitary matrix group path-connected? [closed]

Is the group $\mathcal{U} (n)$ of all $n\times n$ unitary matrices over $\mathbb{C}$ a (local) path-connected space? If so, what are the connected components of the unitary matrix group ...
4
votes
2answers
249 views

classification for coadjoint orbits of lower or upper triangular matrices

Is there any classification for coadjoint orbits of lower or upper triangular matrices in general case $n\times n$. Is there any reference?
1
vote
1answer
73 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
1
vote
1answer
181 views

Torsion-free $G$-Structures

I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of ...
2
votes
2answers
272 views

Parallel forms and cohomology of symmetric spaces

Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then $$ (\alpha \text{ is induced by an ...
3
votes
2answers
178 views

Kostant's Theorem on Principal TDS

I have a few questions concerning Kostant's work on principal three-dimensional subalgebras (TDS). Let $\frak{g}$ be a finite-dimensional complex semisimple Lie algebra, and ...
10
votes
2answers
407 views

Finite subgroups of $PGL(3,K)$

It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). These facts are ...
10
votes
0answers
485 views

Definition of a uniformly bounded dual of a group

The unitary dual of a group $G$ is the set of equivalence classes of irreducible unitary representations of $G$ with the Fell topology. (This topology is defined using convergence of positive definite ...
1
vote
1answer
247 views

The (-)-Connection on a Lie Group

Is the geodesic exponential map for a Lie group with the (-)-connection a diffeomorphism? This connection is one of two flat connections introduced by Cartan and Schouten on a Lie group and has ...
2
votes
1answer
260 views

Can this Lie group written as a direct product?

Let $G=G_1.G_2$ be a Lie Subgroup of $SO(k) \times SO(2) \subset SO(k,2)$, where $G_1=SU(k/2) \subset SO(k)$ and $G_2$ is a Lie subgroup of $SO(k) \times SO(2)$ isomorphic to $SO(2)$. Let $G_1 \cap ...