1
vote
4answers
167 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 $W$ ...
-3
votes
0answers
54 views

Reference request for the irreducibility of the adjoint representation of $SU(n)$ [closed]

I would like to quote the following fact: "The adjoint representation of $SU_n$ on its Lie algebra $\mathfrak{su}_n$ is irreducible." Do you know any standard reference?
4
votes
1answer
83 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 $G$. ...
6
votes
0answers
106 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 $G=SL_2(\mathbb C)$, ...
1
vote
0answers
39 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 ...
3
votes
1answer
104 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 ...
6
votes
5answers
367 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 ...
4
votes
1answer
95 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 $G$. ...
0
votes
2answers
158 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.
6
votes
1answer
145 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 $G$ ...
3
votes
0answers
84 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 ...
1
vote
0answers
147 views

Characterization of Differentiability via Lie Derivatives [closed]

As the starting point of a project I intend to propose the proof of the statement below, but I want to make sure that it is not already proved somewhere else before. Let $G$ be a Lie group, and $f$ a ...
2
votes
1answer
134 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 ...
6
votes
0answers
110 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 ...
5
votes
3answers
378 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 ...
5
votes
1answer
274 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 ...
8
votes
1answer
206 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 ...
6
votes
1answer
191 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 ...
4
votes
3answers
280 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 ...
8
votes
1answer
331 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
votes
1answer
79 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
6
votes
1answer
203 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 ...
5
votes
1answer
177 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 ...
1
vote
1answer
190 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: ...
7
votes
2answers
388 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 ...
8
votes
2answers
484 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?
6
votes
3answers
314 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. ...
2
votes
1answer
92 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 ...
2
votes
2answers
55 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 ...
9
votes
0answers
192 views

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 $C^\infty$ ...
15
votes
2answers
824 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 ...
5
votes
1answer
189 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 ...
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.
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
152 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 ...
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 ...
2
votes
2answers
271 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 ...
1
vote
2answers
170 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 ...
4
votes
2answers
239 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 ...
5
votes
3answers
512 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 ...
1
vote
3answers
413 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 ...
1
vote
2answers
175 views

Copies of ax+b inside the AN part of an Iwasawa decomposition?

As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
2
votes
0answers
132 views

Reference request: injective homomorphisms between unitary groups

Let $U(n)$ be the group of unitary $n\times n$ matrices over $\mathbb{C}$. Is there a classification of the continuous, injective group homomorphisms $U(m)\to U(n)$? If so, is there a modern account ...
7
votes
2answers
389 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 ...
1
vote
1answer
197 views

Reference request: Calculation in exceptional Lie groups

Let $G$ be a compact connected simple exceptional Lie group. Let $G$ be contained in a unitary group ${\rm U}(n)$ by some standard (low dimensional) unitary representation. For example in the case of ...
4
votes
2answers
242 views

comprehensive presentation of the unitary dual of $SO_0(n,1)$

The unitary dual (unitary irreducible represenations) is determined for every connected noncompact semisimple Lie group of real rank one. I would like to have a reference for the particular case ...
10
votes
5answers
644 views

About the intrinsic definition of the Weyl group of complex semisimple Lie algebras

It may be a easy question for experts. The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra ...
5
votes
1answer
303 views

Spin and SO groups associated to a degenerate symmetric bilinear form

In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But the rest of the book ...
3
votes
2answers
227 views

Does every embedding of one unipotent group (over R) in another extend to an embedding of the respective upper triangular matrix groups?

Let $T^*$ denote upper triangular matrices (of the appropriate size) with positive diagonal entries and $\mathrm{UT}$ upper triangular matrices with all diagonal entries equal to 1. Does every ...
5
votes
1answer
182 views

Reference request for the list of maximal subgroups of SU(3,1)

Is there a reference with the list of maximal subgroups of SU(p,q) for "small" values of p and q? (such as SU(3,1) as suggested in the title of the question)