**1**

vote

**1**answer

184 views

### cohomology of orthogonal group of integers

Let
$$
O(\mathbb{Z}^{\oplus k})=GL(\mathbb{Z}^{\oplus k})\cap O(k).
$$
What is $$
H^*(BO(\mathbb{Z}^{\oplus k});\mathbb{Z})?
$$
If it cannot be computed out, can we get
$$
H^*(O(\mathbb{Z}^{\oplus ...

**0**

votes

**0**answers

71 views

### On continuous part of the L^2 spectrum

Suppose $G$ is a real reductive Lie group and $\Gamma$ is a lattice in $G$ (of finite co-volume). I am reading Langlands's paper " On the functional equation satisfied by the Eisenstein series". I ...

**0**

votes

**2**answers

243 views

### Continuous families of finite subgroups of a Lie group

Suppose we have a continuous family of finite subgroups of a compact Lie group G. All the subgroups are necessarily isomorphic. Alternately, we can say we have a continuous family of homomorphisms ...

**2**

votes

**1**answer

183 views

### Relation between Different Definitions of Induced Representation

I've seen two different ways to define induced representation.
One is as in the book Introduction to representation theory: If $G$ is a group, $H$ is a subgroup of it, and $V$ is a representation of ...

**1**

vote

**0**answers

145 views

### rational representation of semisimple algebraic group

Let $G$ be a connected semisimple algebraic group defined over $\mathbb Q$. Could some expert give me a complete classification of finite dimensional $\mathbb Q$-irreducible representations of $G$?
...

**6**

votes

**0**answers

233 views

### What is miraculous about the mirabolic subgroup?

I recently asked this question about Euler subgroups and generalizing the automorphic theory of $\mathrm{GL}_n$ to a more general setting. My question here is more specific.
As mentioned there, the ...

**0**

votes

**0**answers

97 views

### List of irreducible representations whose weights are in a single Weyl group orbit

Let $\mathfrak g$ be a (finite dimensional) simple (nonabelian) Lie algebra
over $\mathbb C$. I need a complete list of irreducible (nontrivial) representations $V$
of $\mathfrak g$ such that the Weyl ...

**1**

vote

**1**answer

193 views

### Volume of a GIT quotient of projective lines

Say $X= \mathbb{P^1}\times \cdots \times \mathbb{P}^1$ is a product of $n\geq3$ lines. Let the group $G=\text{SL}(2)$ act on $X$ diagonally, and let $\mathcal{L} = \mathcal{L}(a_1,\ldots,a_n)$ be the ...

**-3**

votes

**1**answer

166 views

### Fibre bundles and flat connections [closed]

If a fibre bundle can be equipped with a flat connection then it must be necessarily trivial? Let us take for example a real line bundle $L\to M$ with base $M$. If $L$ can be equipped with a flat ...

**0**

votes

**0**answers

67 views

### Wang's C-subgroups and M-manifolds

Let $K$ be a semisimple compact Lie group.
In here H.C. Wang defines a C-subgroup as a closed subgroup $U$ of $K$ such that the semisimple part of $U$ equals the semisimple part of the centralizer ...

**3**

votes

**3**answers

402 views

### classifying space and cohomology of integer general linear group

I have obtained that the classifying space
$$
BGL(\mathbb{R}^n)=BO(\mathbb{R}^n)=G_n(\mathbb{R}^\infty)
$$
is the Grassmannian.
I have also obtained that the mod 2 cohomology is the polynomial ...

**2**

votes

**3**answers

371 views

### Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds

I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I ...

**3**

votes

**2**answers

301 views

### What are cohomology of Lie algebra with coefficients geometrically?

I want to find analog of following two statements.
Let $G$ be a discrete group, $M$ is representation of $G$. Local systems on $BG$ are the same as $G$ representations (because $\pi_1 (BG) =G$). Let ...

**3**

votes

**2**answers

336 views

### SU(2) and differential forms [closed]

I am a physicist with some background in differential geometry and I apologize for any possible unprecise terminology.
Consider the Lie group $SU(2)$ and its tangent space $su(2)$ forming a tangent ...

**4**

votes

**3**answers

359 views

### What is the difference between p-adic Lie groups and linear algebraic groups over p-adic fields?

I thought they were the same, just different names. Let me make question more precise:
Let $G$ be any linear algebraic group over a p-adic field $\mathbb{Q}_p$, is $G$ a p-adic Lie group w.r.t. the ...

**3**

votes

**0**answers

90 views

### Maximal compact subgroup of SL(2,|H)

The exceptional isomorphism $Spin(5,1)\simeq SL(2,\mathbb{H})$ is well-known, and I can find references that say the maximal compact of $Spin(5,1)$ is $Spin(5) \simeq Sp(2)$. So I know the answer to ...

**4**

votes

**0**answers

240 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**2**

votes

**1**answer

186 views

### A Krull-Schmidt Theorem for Lie groups?

I wondered whether there is an analogue of the Krull-Schmidt theorem for real Lie-groups. More precisely, what conditions do you have to impose on a connected finite dimensional Lie group $G$ so that ...

**1**

vote

**0**answers

188 views

### A (possible) equivalent relation on the space of vector bundles

Edit: According to the essential comment of Alex Degtyarev, we revise the question as follows;
Assume that $\alpha$ and $\beta$ are two oriention preserving automorphism of Lie groups $O(n)$ and ...

**16**

votes

**8**answers

899 views

### Examples of categorical adjunctions in analysis, Lie theory, and differential geometry?

In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis, Lie theory, ...

**-1**

votes

**1**answer

128 views

### cartan killing metric [closed]

I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...

**4**

votes

**2**answers

208 views

### An algorithm to compare two representations of a simple Lie algebra?

I have two representations of a simple (complex or real) finite-dimensional Lie algebra $S$, both given in terms of their structure constants on a given basis.
the first one is the adjoint ...

**2**

votes

**0**answers

191 views

### The geometry of the holomorph of a Lie group

Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left ...

**2**

votes

**3**answers

313 views

### What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...

**1**

vote

**0**answers

76 views

### Are lattices in the special real linear group subgroup seperable?

Let $G \leq SL_2(\mathbb{R})$ be a lattice, let $H \leq G$ be a finitely generated subgroup of infinite index, and let $n \in \mathbb{N}$. Must there be some $H \leq U \leq G$ such that $n \leq [G : ...

**1**

vote

**0**answers

128 views

### representation of $SO(p)\times SO(q)$ with $p,q$ odd

Assume $p,q$ odd. We denote by $\sigma_p$ the standard representation of $SO(p)$, that is the representation of $SO(p)$ acting on $\mathbf{R}^p$ as matrix. So is $\sigma_q$.
Take $K=SO(p)\times ...

**1**

vote

**1**answer

85 views

### Existence of a fixed-point free map in a manifold [closed]

I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map.
I know ...

**1**

vote

**0**answers

84 views

### homomorphisms from one Lie group to another

I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie"
http://www.ams.org/mathscinet-getitem?mr=0379749
and I was wondering if I could ask for help with understanding one ...

**5**

votes

**1**answer

202 views

### Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$

Let $G = SL_n(\mathbb{C})$, $B$ be a Borel subgroup, and $B^-$ be the opposite Borel.
Both the $B$ and $B^-$ orbits on the flag variety $G/B$ are indexed by the Weyl group $W$. Let $S_{w_1}$ and ...

**0**

votes

**1**answer

87 views

### query about Jacques Tits' “Homorphismes `abstraits' de groupes de Lie”

I am reading Jacques Tits' paper "Homomorphismes `abstraits' de groupes de Lie" and he seems to be making a claim that if you have a simply connected Lie group then the derived subgroup is always ...

**2**

votes

**0**answers

77 views

### Orthosymplectic group, matrix representations

We have the orthosymplectic $osp(n,m|2k)$. The bosonic part is $so(n,m)\times sp(2k)$. The lie algebra generators are given in eg
http://cds.cern.ch/record/524737/files/0110257.pdf$
where the group ...

**0**

votes

**0**answers

59 views

### adjoint quotient and points in DVRs

Let $G$ be a connected reductive group over an algebraically closed field $k$, $T$ a maximal torus and $W$ its Weyl group.
We have a Steinberg map $\chi:G\rightarrow \mathfrak{C}:=T/W$ if we have a ...

**4**

votes

**0**answers

81 views

### Connected compact Lie groups with Lie algebra so(4n, R)

I am trying to write a complete list of connected compact simple Lie groups (or of connected complex simple Lie groups, both tasks are equivalent). I am missing just one case.
Consider the Lie ...

**3**

votes

**1**answer

177 views

### Are norm-continuous representations smooth?

Let $G$ be a real Lie group and $A$ a unital Banach algebra. Let us call a map $\varphi:G\to A$ a (norm-)continuous representation, if it is continuous
$$
x_i\to x\quad\Longrightarrow\quad ...

**3**

votes

**1**answer

171 views

### How to think about the simple reflection s_0 in the affine Weyl group?

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight ...

**1**

vote

**2**answers

167 views

### Local structure of the quotient of a Lie group action

Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure.
Are there any results for the general case? (a) If the ...

**1**

vote

**0**answers

131 views

### Integrating Poisson groups

Recall that a symplectic groupoid (http://projecteuclid.org/euclid.bams/1183553676) is a Lie groupoid $\mathcal{G}\rightrightarrows X$ together with a symplectic structure on $\mathcal{G}$ ...

**4**

votes

**2**answers

114 views

### Measurable representations of semi simple Lie groups

Let $G$ be a semi simple Lie group. I'm particularly interested in $SL(n,\mathbb{R})$. It is proved in
I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple
Lie groups, ...

**4**

votes

**0**answers

99 views

### Isometries of Compact Semisimple Lie Groups

In this delightful question, the poster mentioned that the isometry group of a compact Lie group $G$, equipped with the metric from the Killing form, is $G\times G/Z(G)$, where $Z(G)$ is the center of ...

**3**

votes

**1**answer

232 views

### Weyl groups of $E_6$ and $E_7$

The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order ...

**4**

votes

**1**answer

149 views

### Questions about the $\mathbf{i}$-trails of Berenstein and Zelevinsky

The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, ...

**3**

votes

**1**answer

223 views

### Lifting one parameter subgroups of algebraic groups

Let $G$ be a linear algebraic group over an algebraically closed field $\mathbb C$ of characteristic zero and $U$ its unipotent radical, then $H:=G/U$ is a reductive group. Assume that I have a one ...

**8**

votes

**1**answer

267 views

### what is the intersection of all congruence subgroups of the profinite completion of SL(2,Z)?

Let $\widehat{SL(2,\mathbb{Z})}$ be the profinite completion of $SL(2,\mathbb{Z})$. Let $\Gamma(N)$ denote the typical principal congruence subgroup of $SL(2,\mathbb{Z})$ (ie, all matrices congruent ...

**7**

votes

**2**answers

405 views

### Do geodesics in SL2R map to geodesics in the hyperbolic plane?

I am looking for a reference/proof/disproof of the following statement.
Equip the Lie group $SL_2(\mathbb{R})$ with the left-invariant Riemannian metric, whichis given on the Lie algebra by $\langle ...

**2**

votes

**1**answer

149 views

### Fourier transform of a matrix represented compact lie group

In physics, I come across this kind of integration (in the nonlinear sigma model):
\begin{equation}
S[g] = \frac{1}{\lambda} \int d^dr\ \text{tr}[\triangledown g\triangledown g^{-1}]
\end{equation}
...

**5**

votes

**0**answers

175 views

### Approximating Lie groups by finite groups

How can one approximate compact Lie groups by finite groups?
My wish is something like this:
Let $G$ be a compact Lie group.
There is a sequence of nested finite subgroups $G_n$ so that $G_n\to ...

**1**

vote

**1**answer

140 views

### Equation for non-invertible elements in Clifford algebras

Suppose we have a Clifford algebra $Cl(V,q)$, $V\simeq \mathbb{R}^n$ and $q$ non-degenerate bilinear form. Then every non-zero element of $V\subset Cl(V,q)$ invertible, but they are not the only ones ...

**0**

votes

**0**answers

133 views

### Comparison of two Chevalley basis

Let $G$ be a connected reductive group over an algebraically closed field and $T$ a maximal torus.
Let $H$ be a pseudo-Levi subgroup, say the neutral component of a centralizer of a semisimple element ...

**1**

vote

**0**answers

134 views

### Analogies, Riemann surfaces and Algebraic groups

Let G be a complex simple Lie group of adjoint type. Then, it is well known that every such $G$ contains, unique up to conjugacy, an irreducibly embedded copy of $PSL(2,\mathbb{C}).$ This fact seems ...

**2**

votes

**2**answers

244 views

### Closure relations between Bruhat cells on the flag variety

Given a Lie group $G$ over $\mathbb{C}$ and a Borel subgroup $B$. There is this famous Bruhat decomposition of the flag variety $G/B$.
How do we prove the closure relations between the cells, which ...