0
votes
1answer
91 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 ...
2
votes
0answers
107 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 ...
2
votes
1answer
169 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 ...
8
votes
4answers
389 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 ...
16
votes
3answers
354 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 ...
5
votes
3answers
295 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 $(SL_2(\mathbb{C}), PGL_2(O_F))$ ...
0
votes
0answers
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 ...
0
votes
0answers
88 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 ...
0
votes
0answers
110 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 $f: U \rightarrow \mathbb{R}$ ...
1
vote
4answers
201 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 $W$ ...
4
votes
1answer
95 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
118 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
45 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
421 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
107 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
164 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
177 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
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 ...
2
votes
1answer
137 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
120 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
413 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
366 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
208 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
203 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 ...
3
votes
3answers
288 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
337 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
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
6
votes
1answer
215 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
181 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
193 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
415 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
521 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
321 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
97 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
57 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
194 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 $C^\infty$ ...
14
votes
2answers
900 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
204 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
303 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
371 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
167 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
195 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
295 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
174 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
249 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
543 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
461 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
179 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 ...