# Tagged Questions

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### 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 ...

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85 views

### adjoint representation of 2-Lie groups

Baez and Crans in their paper on Lie 2-algebras refer to adjoint representations of Lie 2-groups but don't say much, as far as I can tell, except to say that such a representation acts on a 2-Lie ...

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62 views

### Reference request: How you can reach any point in the vector space of vector fields generated by Lie brackets

By a Theorem of Chow, you can reach any point in the vector space of vector fields generated by Lie brackets.
Do you know any reference for this theorem?

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**1**answer

133 views

### Maximal compact subgroups of a semisimple Lie group are conjugate

I'm trying to go through the proof that all maximal compact subgroups of a semisimple Lie group $G$ are conjugate. I know that a possible proof follows the following steps:
Take one maximal compact ...

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**3**answers

567 views

### nth term in the Baker-Campbell-Hausdorff formula

I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...

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127 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 ...

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**1**answer

172 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 ...

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395 views

### Action of a Lie group with finitely many orbits

EDIT: Let a real Lie group $G$ act on a smooth manifold $M$ with finitely many orbits such that each orbit is locally closed ($M$, but not $G$, may be assumed to be compact in my case). Let ...

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370 views

### Real Lie groups versus real linear algebraic groups: differences in connexity and fundamental group

There are many introductory texts on real Lie groups, and many on linear algebraic groups in general, but fewer on the specific case of linear algebraic groups over the reals, and even fewer that try ...

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303 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 $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...

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69 views

### Unions of orbits of dimension $\leq n$

Let $G$ be a complex linear algebraic group acting on a smooth complex projective variety $X$ with finitely many orbits. Note that each $G$-orbit is a smooth locally closed subvariety of $X$.
For a ...

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89 views

### Reps of a compact connected Lie group are equivalent iff they are equivalent as reps of a maximal torus

Let $G$ be a compact connected Lie group, $T$ a maximal torus in $G$ and $V$, $W$ finite-dimensional $G$-representations. Using characters and the fact that every element of $G$ can be conjugated into ...

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117 views

### Jacobian change of basis matrix for different dimensions

I am considering a real Lie group $G$ acting transitively on an open set $U$ in a real Euclidean space of lower dimension. Given a smooth, compactly supported function $f: U \rightarrow \mathbb{R}$ ...

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205 views

### Bruhat order and Schubert cycles

I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ ...

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98 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 $G$. ...

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121 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 $G=SL_2(\mathbb C)$, ...

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47 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 ...

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106 views

### Reference to definition of matrix log with domain SO(3) which is Borel measurable

I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...

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428 views

### Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...

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110 views

### Connection between degree of growth and return probabilities of random walks on Lie groups

Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...

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166 views

### A question on Lie algebras

To what extent, the following types of Lie algebras are classified :
Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.

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182 views

### Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation

Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:
as a quotient of a semisimple real Lie group $G$ ...

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89 views

### Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...

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141 views

### Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...

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122 views

### Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...

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420 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 ...

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372 views

### Geometric structure of flag manifolds, Borel -Weil-Bott theorem

I want to know if there is proof of Borel Weil Bott theorem, that is as geometric as it can be.
Let $G$ be a semisimple compact Lie group and $T$ be a maximal torus. We know that $G/T$ is a ...

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**1**answer

210 views

### The Image of the Mod 2 Homology of BSp in the Homology of BSO

I'm essentially trying to figure out exactly what the title asks for. I've been scouring old Seminaires Henri Cartan and books by Stong to try to see exactly how to do this, but the combination of ...

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212 views

### Characterisation of Q-rank 1

I'm looking for a reference and/or the original source for the following fact:
An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...

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290 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 ...

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340 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 ...

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84 views

### Citation for positive Ricci curvature

Does anyone know a citeable source where it is shown that the Ricci curvature of SU(n) is strictly positive? I can sketch the proof but I need to shorten my notes.
Thanks,
Stefan

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223 views

### Easy argument for “connected simple real rank zero Lie groups are compact”?

Let $G$ be a connected simple Lie group. It is known that if $G$ has real rank zero, then $G$ is compact.
Background: every connected (semi)simple Lie group $G$ (with Lie algebra $\mathfrak{g}$) has ...

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187 views

### spherical buildings for non-split groups

I am looking for references to explicit descriptions of Tits buildings for semisimple (classical) Lie groups via language of incidence geometry. Such descriptions are well-documented in the case of ...

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197 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 complexiļ¬cation and $\tau: ...

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420 views

### Conditions for a topological group to be a Lie group

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157):
Let $G$ be a locally compact ...

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535 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?

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322 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. ...

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99 views

### Hilbert's Finiteness Theorem for connected semisimple Lie groups in Weyl's “Classical Groups”

First of all, sorry for using this account. Somehow I can't login to my previous one anymore and am thus using the account associated to my MSE one. Also, I already asked the question on MSE, but ...

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58 views

### Hilbert's Finiteness Theorem for connected semisimple Lie groups over $\mathbb{C}$ in Weyl's “Classical Groups” [duplicate]

In Nagata's "Lectures on the 14th problem of Hilbert" I found a reference to Weyl's "Classical Groups". Nagata writes that Weyl gives a positive answer to the original problem
If ...

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197 views

### differentiating positive energy LG reps

Background:Let $G$ be a cscscĀ¹ 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 $C^\infty$ ...

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921 views

### Where did Sophus Lie write the group commutator for two one parameter groups

If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the ...

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220 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 ...

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316 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.

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378 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 ...

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174 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 ...

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196 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 ...

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304 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 ...

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176 views

### Lie groups and NSS+LC group

Let $G$ be a locally compact group without small subgroups. Is $G$ a "finite" dimensional Lie group? (i.e, $G$ is not infinite dimensional Lie group.)
Are Lie groups precisely the locally Euclidean ...

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250 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 ...