**2**

votes

**1**answer

97 views

### Bounds on Hilbert-Schmidt norm of difference of products of matrices

I suspect the following is well-known, but don't know of a reference (and it is not close to the area I normally work in).
I have two sequences of matrices $Q_{1},\ldots,Q_{k}$ and ...

**1**

vote

**1**answer

128 views

### $Spin(7)$ as stabilizer of a $4$-form revisited

For a better understanding of this question, please see the question and answer here.
In $Spin(8)$ there are plenty of copies of $Spin(7)$; consider, for instance, the antiimage of $SO(7)<SO(8)$ ...

**2**

votes

**0**answers

66 views

### Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...

**0**

votes

**1**answer

221 views

### generalization of highest weight theorem for semisimple lie algebras

Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition
$\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$.
Let $\mathfrak p$ be a
...

**2**

votes

**1**answer

145 views

### Unitary representation with fixed Casimir

Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of ...

**5**

votes

**0**answers

115 views

### Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B

This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...

**5**

votes

**0**answers

86 views

### How to characterize the class of $(\mathfrak{g},K)$-modules with a fixed lowest K-type in the framework of D-modules?

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be ...

**-1**

votes

**1**answer

171 views

### Reductive space & Reductive Lie algebra

If $M=G/H$ is a reductive space and $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition, then are $\mathfrak{g}$ or $\mathfrak{h}$ or both reductive lie algebras? (in this case, ...

**2**

votes

**1**answer

222 views

### Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows:
$$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...

**2**

votes

**1**answer

322 views

### Canonical representation of $\operatorname{SL}(2,\mathbb{R})$ on $L^2(\mathbb{R}^2)$

As a unimodular subgroup of the group of automorphisms of $\mathbb{R}^2$, $\operatorname{SL}(2,\mathbb{R})$ can be represented as a subgroup of $\mathcal{U}(L^2(\mathbb{R}^2))$ (the group of unitary ...

**3**

votes

**1**answer

366 views

### Is $M=E_{7(7)}/SU(7)\times\mathbb{R}^{+}$ a (pseudo)Kähler-Hodge manifold? Open problem

I have been told that the following is an open problem in mathematics, but I am pretty sure that experts in the topic surely know the answer.
Is the manifold
$$M=\frac{E_{7(7)}}{SU(7)}\times ...

**1**

vote

**1**answer

207 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

91 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

283 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

234 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

172 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

264 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

124 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

213 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

220 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

71 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

481 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

447 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

327 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

419 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

524 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

102 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

312 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

193 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

195 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 ...

**18**

votes

**8**answers

1k 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

164 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

228 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

198 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

390 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

84 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

**1**answer

104 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

90 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

226 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

90 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

124 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

66 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

92 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

194 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

216 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

218 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

**1**answer

211 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 $\mathcal{G}$ ...

**5**

votes

**2**answers

121 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

119 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

252 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 ...