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

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Some questions on analytic vectors and the integrability of Lie-algebra representations

I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
2
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64 views

Lattice model for Affine Grassmannians of non type A

There is a Lattice model for affine Grassmannians of type A, due to Lusztig. It describes affine Grassmannians of type A as the moduli space of certain subspaces in an infinite-dimensional ...
2
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94 views

Semigroup nilpotents and compostional inversion

The integer coefficients of a general partition formula for the compositional inverse of a function are a refined version of the coefficients of the generating series for the number of nilpotents in a ...
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2answers
123 views

Which real Pin groups agree?

In the Lie theory notes on my website it is claimed (Example 7.3.3.5) that $\mathrm{Pin}(4,0)$ and $\mathrm{Pin}(0,4)$ are not isomorphic. As Nigel Ray pointed out to me, this claim is not quite ...
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1answer
310 views

Can an odd map be null homotopic?

Let $G$ be a compact Lie group with invariant measure $\mu$. An odd function is a continuous function, $\phi:G\to \mathbb{C}$, such that $\int_{G} \phi d\mu=0$. An odd map is a continuous map, $f:G\to ...
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369 views

Geodesics in finite groups

It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for ...
5
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3answers
262 views

Poincare duality for (co)homology of Lie algebras?

Let $R$ be a commutative ring and $\mathfrak{g}$ a Lie $R$-algebra that has an $R$-module basis with $n$ elements. In Algebra, Geometry, and Software Systems by Joswig & Takayama on p.200 it ...
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170 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 ...
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1answer
89 views

Writing a complex orthogonal matrix as a conjugation by real orthogonal matrices

Let $Q\in O(n,\mathbb C)$ be a complex orthogonal matrix. I would like to know if $Q$ can always be written as $Q = T^{-1}ST$, where $T\in O(n,\mathbb R)\subset O(n,\mathbb C)$ and $S$ belongs to some ...
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33 views

pro-Lie-groups and the exponential map

Am I right in saying that the image of a one-dimensional subalgebra of the pro-Lie algebra of a pro-Lie group under the exponential map would be abstractly isomorphic to a one-dimensional Lie group? ...
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1answer
68 views

the meaning of “Cauchy filter” for an arbitrary topological group

I was reading a definition of pro-Lie group and it spoke of a "Cauchy filter" on an arbitrary topological group even though there was no mention of a metric. Is there some kind of standard meaning for ...
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66 views

Forming subgroups out of the 15 generators of SU(4)?

Given the 15 generators (last page) of SU(4), is it possible to divide them into 3 "natural" categories of 5 matrices each? By natural, I mean each category could form a group of its own, for example, ...
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3answers
121 views

Any duality between different real forms of a complex Lie group?

A complex Lie group may have several real forms. Are there any duality/trinity... between them? Maybe a trivial question to ask, is $SL(3,\mathbb{C})$ a real form of $SL(3,\mathbb{C})\times ...
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41 views

Can Bruhat cells in semi simple groups be induced from matrices?

Let $G$ be a semisimple Lie group. Embed it as a subgroup into a special linear group of suitable rank, $SL(n)$ (real or complex). The question is: is it always possible to find such an embedding, ...
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107 views

Non invertibility of certain integral arising from group action

Let a compact topological group $G$ with invariant measure $\mu,$ acts on a simply connected compact topological space $X$ and $\rho$ is a $n$-dimensional unitary representation of $G$. ...
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1answer
149 views

The logarith map as a contraction [closed]

Two Questions: (1) Under what conditions(if any) can the logarithm map from a point on a Riemannian manifold, $q_1\in Q$, to the Tangent Space $T_{q_0}Q$, locally, be a contraction mapping? Or more ...
3
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1answer
168 views

Why a nilpotent Lie group must be a matrix group?

The question may be a little naive (or even appear as a duplicate) as I guess the result is well known. I saw on the other thread that " c) A solvable Lie group G is linear iff its commutator ...
5
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90 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 ...
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62 views

What Lie groups have a discrete set of order two elements? [migrated]

We know that the set of order two elements of $R^n$, tori and $S^3$ are discrete. Are there others examples of Lie groups with such property? Are there some characterization of such class?
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107 views

Fixed sets of orbit spaces

I've run across something that surprises me, so I'm wondering (1) Is it true? and (2) Is it well known? (And if the answers are affirmative, why didn't I know this already?) Let $G$ be a compact Lie ...
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1answer
153 views

When is the Ad (Adjoint Representation) Morphism a Closed Map

Given a Lie group $\mathfrak{G}$ with finite centre and with Lie algebra $\mathfrak{g}$, I am looking at a simple proof that negative definite Killing form implies compactness. This proof is given ...
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81 views

Left invariant connection on Lie groups

Let $M^n$ be a Riemannian manifold, $G^{n + k}, \: k\geq 1 $ be a connected Lie group equipped with a left invariant metric and $f:M^n \to G^{n + k}$ one isometric immersion. Let $X, Y$ left invariant ...
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40 views

Fixed Point Algebras of Adjoint Actions of Banach Lie groups

I have the following question: Let a be an element in a connected Banach Lie group G (over K, where K is the reals or the complex numbers). We assume that G is not trivial, that has more than one ...
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2answers
63 views

A full dim. subvector space of $\chi^{\infty}(M)$ which all non zero elements are nonvanishing vec.field

What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?: There is an $n$ dimensional sub vector space ...
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1answer
110 views

How to “lift” a transitive group action on a manifold?

Let $M=G/H$ be a homogeneous manifold, with $G$ connected Lie group. Suppose that $\widetilde{M}$ is a covering of $M$. QUESTION: is there a general prescription to obtain a Lie group ...
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179 views

Is the centralizer $Z_G(A)=\{g\in G| a g= g a\}$ of a finite $A\subset G$ connected for a connected compact Lie group?

Let $G$ be a connected compact Lie group, consider the left/right action on itself. For any finite $A\subset G$, consider the centralizer $Z_G(A):=\{g\in G| a g= g a\}$. Q: is $Z_G(A)$ a connected ...
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1answer
181 views

Basics on lattice in classical groups

as a beginner,I am not sure whether this question is too basic to post here./-\。 Many textbook will talk about the prototypical example SL(n,Z)\SL(n,R), which can be identified with the space of ...
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182 views

Derivation of Blattner's conjecture in the Beilinson-Bernstein picture

On the last page of Schmid's article "Discrete Series", he says "In the Beilinson-Bernstein picture, discrete series modules are attached to closed $K$-orbits in $X$... the $K_{\mathbb R}$-structure ...
3
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0answers
119 views

What do we know about isospectral Cayley graphs?

Let $\Gamma_1=\Gamma_1(G_1,S_1)$ (respectively $\Gamma_2=\Gamma_2(G_2,S_2)$) be a Cayley graph for the group $G_1$ and the finite generating set $S_1$ (respectively $G_2$ and the finite generating set ...
9
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2answers
353 views

on the center of a Lie group

I'm trying to set straight my various pieces of knowledge about the center of a compact Lie group, and I'm running in circles... First some definitions: • Let $G$ be compact, simple, and simply ...
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1answer
116 views

Maximal subgroups of indefinite special orthogonal group SO(p,q)

Can someone answer the following question: Is there any classification of maximal proper Zariski-closed real subgroups of $SO(p,q)$ which are not parabolic, and satisfy the following conditions: ...
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1answer
101 views

Does the homogeneous spaces $K^{\mathbb{C}}/{{Z(k)}^{\mathbb{C}}}$ have a natural Kähler or sympletic structure?

Let $K$ be a connected compact Lie group, $K^{\mathbb{C}}$ be complexified Lie group of $K$. Denote $Z(k)$ by the centralizer of k∈K and $Z^{\mathbb{C}}(k) $ be the complexified Lie group of $Z(k)$ ...
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51 views

Localization on orbit type submanifolds (generalization of Atiyah-Bott-Berline-Vergne)

In equivariant cohomology, the Atiyah-Bott-Berline-Vergne localization theorem says roughly speaking that the integral of an equivariant cohomology class on the $G$-manifold $M$ has only contributions ...
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372 views

A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?

The following is my first question here on mathoverflow. Let $M$ be a closed connected Riemannian manifold with an isometric effective action of a compact connected Lie group $G$. Consider the ...
5
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1answer
218 views

Harish-Chandra isomorphism for compact symmetric spaces

I would be interested to have an explicit description of the algebra of invariant differential operators on functions on a compact symmetric space $G/K$. A reference would be especially useful. ...
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1answer
105 views

Large and Small Conformal Groups

It's well-known that on a Riemannian manifold $(M,g)$ with dimension larger than 2, the dimension of its conformal group $\text{conf}(M,g)$ is bounded above by ${n+2\choose 2}$. A Riemannian manifold ...
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1answer
157 views

When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple?

This is in a sense a follow up on the popular question Induction and Coinduction of Representations, where this particular question is one of several points, and it is neglected. It seems that the ...
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191 views

Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book

This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course. Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
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65 views

First Variation of Dyson Series/Magnus Expansion

Given the matrix differential equation $\frac{dU_t}{dt}=A_t U_t$ there are at least two ways to write a formal solution. Both the Dyson series: $U_t = \mathcal{T} e^{\int_{0}^{t} A_t dt}$ and the ...
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2answers
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$H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free. ...
3
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1answer
112 views

Two matrix Fisher distributions on SO(3)?

After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to ...
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88 views

Is every irreducible unitary class one representation induced?

Let $G$ be a connected semi simple Lie group with finite center. Fix a maximal compact subgroup $K$. An irreducible representation $(\pi,V)$ of $G$ is called a "class-one representation", if it ...
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299 views

Do compact groups acting irreducibly have finite subgroups which do the same?

Let $G$ be a closed subgroup of $U(n,{\bf C})$, not necessarily connected. Regard ${\bf C}^n$ as a complex $G$-module $M$. Q. Suppose $M$ is irreducible as a $G$-module (equivalent, I think, to ...
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1answer
136 views

Rational homogenous spaces and symmetric spaces

What are the complex rational homogenous spaces $G/P$ ($G$ a semi-simple complex Lie group, $P$ a parabolic subgroup) such that the set of real points $(G/P)(\mathbb R)$ is a (compact) riemannian ...
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63 views

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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1answer
69 views

equation for geodesics of a right invairant Finsler metric on $SU(n)$ which are parallel to a linear affine distribution

I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. ...
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382 views

Who originated the standard symbols for Lie groups GL, SL, SU, etc.?

Who was first to use symbols GL, SL, O, SO, U, SU, Sp and their projective versions, and how did this notation become standard? The notation appears in fairly modern form in Weyl's "The Classical ...
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1answer
43 views

Converting Dirichlet boundary conditions for E-L equations on a Lie group into an equivilent condition for EP equaiton

I need to find a specific geodesic of a right invariant Finsler geodesic on a Lie group ($SU(n)$) that connects $I$ to some desired $O$. These are Dirichlet boundary conditions for the E-L equations ...
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378 views

Characterising the adjoint representation of SU(N)

One can show that the adjoint representation of $\mathrm{SU}(n)$, the image of the map $\mathrm{Ad}:\mathrm{SU}(n) \rightarrow \mathrm{Aut}(\mathrm{su(n)})\subset \mathrm{GL}(\mathrm{su}(n))$, is an ...
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1answer
48 views

Bi-invariant one forms on compact Lie groups

I'm hoping to learn of a criterion for the existence of a bi-invariant one form on a Lie group $G$. I'm looking for a reference that there are no such one forms son $SU(n)$ (as long as this is in fact ...