**5**

votes

**2**answers

179 views

### Centralizers of one parameter subgroups in semi-simple Lie groups

Suppose G is a connected semi-simple Lie group with finite center, and A, B are one parameter subgroups of the same Cartan subgroup. If the connected components of the identity of the centralizers of ...

**2**

votes

**0**answers

61 views

### Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices

I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with ...

**3**

votes

**0**answers

85 views

### Topological obstruction to icosahedral symmetry?

Let $G$ be a compact simple lie group of rank $n$. Then the Poincaré series of $G$ is given by $$P(G,q)=\prod_{i=1}^n (1+q^{2d_i-1}),$$
where the integers $d_1\leq d_2\leq \cdots \leq d_n$ are the ...

**2**

votes

**0**answers

89 views

### Finite subgroups of compact simple Lie groups [duplicate]

The finite subgroups of $SU(2)$ consist of the symmetry groups of the Platonic solids plus the finite subgroups of $O(2)$. I would like to know if there are any results concerning $SU(3)$. In ...

**0**

votes

**1**answer

277 views

### Symplectic structure on $Sym^kG^{\mathbb{C}} $

Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification.
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here ...

**5**

votes

**1**answer

374 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 ...

**2**

votes

**2**answers

117 views

### A property of bi-invarient Finsler metrics on SU(N)

Does the set of bi-invariant Finsler metrics on $SU(N)$ exactly coincide the set of Finsler metrics with the one-parameter subgroups as their geodesics through the identity?
I know that being ...

**4**

votes

**0**answers

206 views

### Local version of a slice (for a Lie group action)

Let $\Upsilon: G \times M \to M$ be a smooth action of a Lie group $G$ on a manifold $M$.
Isenberg and Marsden (1982) define a slice at $m \in M$ as a submanifold $S \subseteq M$ containing $m$ such ...

**9**

votes

**1**answer

167 views

### Are compact simple groups homotopically non-abelian?

Take a compact connected simple centreless Lie group $G$.
Can the commutator map $G\times G\to G$ sending $(x,y)$ to $[x,y]$ be homotopic to a constant map?
I am interested mostly in the case, ...

**3**

votes

**2**answers

175 views

### Moving Between Weight Spaces in Highest-Weight Representations

Let $G$ be a connected, simply-connected complex semisimple linear algebraic group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$ and let $\Delta\subseteq Hom(T,\mathbb{C}^*)$ be the ...

**8**

votes

**1**answer

212 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 ...

**10**

votes

**3**answers

410 views

### orbits of automorphism group for indefinite lattices

I have a question about indefinite lattices.
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily ...

**7**

votes

**2**answers

331 views

### Is it possible to construct a formal group law from a Lie group without choosing coordinates?

There is a three-way correspondence between:
Real (connected and simply connected) Lie groups of dimension $n$;
$\mathbb R$-Lie algebras of dimension $n$;
Formal group laws in $n$ variables over the ...

**8**

votes

**1**answer

502 views

### Local trivializations of the non-trivial $SU(2)$-bundle over $S^5$

It is well known that $SU(3)$ is the unique, non-trivial, principal $SU(2)$- bundle over $S^5$. To my knowledge the way this is proven is by using the following fact:
If $G$ is a Lie group ...

**2**

votes

**1**answer

131 views

### Split real form of $SL(2,\mathbb{C})$ description of the two sphere?

If we denote the parabolic subgroup of $SL(2,\mathbb{C})$ by $P$, then we have the well known isomorphism $SL(2,\mathbb{C})/P \simeq S^2$, where $S^2$ is the two sphere. Now the compact real form of ...

**2**

votes

**1**answer

140 views

### Fixed submanifold of G-manifold

Let $G$ be a compact Lie group, and $M$ be a smooth manifold on which $G$ acts smoothly and effectively. Denote by $F$ the $G$-fixed subset. In general, $F$ has finitely many connected components and ...

**7**

votes

**1**answer

174 views

### Chevalley restriction theorem for exterior algebras

Suppose $G$ is semisimple Lie group, $\mathfrak{g}$ is its Lie algebra, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $W$ is the correspondent Weyl group.
Chevalley restriction theorem ...

**18**

votes

**3**answers

580 views

### The non-simplicity of $SO(4)$ and $A_4$

It is well known that the alternating group $A_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the ...

**2**

votes

**0**answers

151 views

### The fundamental in the tensor square of a complex representation of $SO(N)$

I would like to figure out whether there is an irreducible complex (in the sense non-self-conjugate) representation of a group $SO(N)$, $N>2$, whose tensor square contains the fundamental ...

**0**

votes

**0**answers

93 views

### semisimple conjugacy classes over general bases

Let $k$ be an algebraically closed field, $G$ a connected reductive group, $T$ a maximal torus, $W$ the Weyl group and $\chi:G\rightarrow T/W$ the Steinberg morphism.
We know that if ...

**4**

votes

**1**answer

153 views

### Finding Finite Generators of a Subset of a Quaternion Algebra/Cocompact Lattices

I was wondering if anyone had some ideas (books, papers, experience) on how to explicitly compute generators for the elements of a quaternion algebra, $Q$, with reduced norm $1$. I'm trying to ...

**1**

vote

**1**answer

422 views

### Para-Complexification of Lie Groups

Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...

**12**

votes

**3**answers

734 views

### Why do we need a $G$-universe?

Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$.
Question: Why do we need a $G$-universe?
A $G$-universe is defined to be a countably ...

**2**

votes

**0**answers

114 views

### Manifold with a quasi-positive curvature

As far as I know, in a simply connected compact manifold, still there exists no well-known obstruction for a manifold with a quasi-positive curvature to be a manifold with positive curvature.
But ...

**2**

votes

**1**answer

361 views

### A question about flag variety of $SL(n,\mathbb{C})$

We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of ...

**3**

votes

**1**answer

118 views

### On matrices conjugated in a faithful representation

Let $k$ an algebraically closed field.
Let $O=k[[\pi]]$ and $F=k((\pi))$ and $G\rightarrow GL_{n}$ a faithful representation of a semisimple group.
Let $A, B\in G(O)\cap G(F)^{rs}$ (rs for regular ...

**6**

votes

**1**answer

215 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 ...

**2**

votes

**0**answers

83 views

### integral stable conjugacy classes

Let $G$ be a semisimple simply connected group over $k$ algebraically closed field .
Let $\gamma,\gamma'\in G(k[[\pi]])$ that are generically regular semisimple on $G(k((\pi)))$.
We assume that ...

**1**

vote

**0**answers

140 views

### on the open bruhat cell

Let $G$ a connected reductive group and $S=U^{-}TU$ the open cell.
Do we have $G=\bigcup\limits_{g\in G}gSg^{-1}$?
And also if I assume that $G$ is adjoint and $\overline{G}$ is the de ...

**3**

votes

**3**answers

292 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 ...

**2**

votes

**0**answers

94 views

### Can class in $H^4(BT)$ be realized as the second Chern class of a principal SU(2) bundle?

The question in the title, to which I add some clarification. Can every class in $H^4(B\mathbb{T})$ be realized as the second Chern class of a principal $SU(2)$ bundle?
$B\mathbb{T}$ is the ...

**4**

votes

**0**answers

170 views

### How to find the unitary matrices in this exponential matrix representation

In the following post
Representing a product of matrix exponentials as the exponential of a sum
there is a statement regarding the result of the multiplication of two matrix exponentials:
if $A$ and ...

**4**

votes

**1**answer

177 views

### How does one calculate homotopy classes for group coset spaces?

Inspired by Witten's Wess-Zumino term arguments, I'm curious to know how one calculates homotopy classes more generally for coset spaces. In the above example the coset is $G/H=(SU(3)_L\times ...

**8**

votes

**1**answer

341 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

**0**answers

142 views

### Compute the discriminant for reductive groups

Consider $G=GL_{2}$ and $F=k((\pi))$, and a diagonal matrix $t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right)$.
The characteristic polynomial of $t$ is $X^{2}-(a+b)X+ab$, and the ...

**0**

votes

**1**answer

105 views

### question about twisted group of Lie type A_n

Let $G=PSU_3(q)$ and $q=p^n$, where $n$ is odd. Can we conclude that $PSU_3(p)$ is a subgroup of $G$?

**0**

votes

**1**answer

187 views

### Coadjoint orbits and homogeneous symplectic $G$-manifolds

We know this important fact from A.A.Kirillov that :
Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a central ...

**5**

votes

**2**answers

355 views

### The number of conjugacy classes of the simple group PSL(2,q)

If $q=p^a$ , where $p$ is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$ and $2$ in the simple group $PSL(2,q)$ .

**2**

votes

**1**answer

528 views

### Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $

My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...

**3**

votes

**0**answers

56 views

### on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$.
Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group ...

**12**

votes

**1**answer

367 views

### Is there a category of representations of a simple Lie algebra on which its Weyl group naturally acts?

For any simple Lie algebra $\mathfrak{g}$, is there a category $C$ of (possibly infinite-dimensional) representations of $\mathfrak{g}$ such the Weyl group $W$ of the corresponding root system acts in ...

**0**

votes

**1**answer

401 views

### fiber bundle on an orbit of $\mathfrak{g}\oplus\mathfrak{g^*}$

Let $G$, be a Lie Group and $\mathfrak{g}$ be its Lie algebra ,i.e, $Lie(G)=\mathfrak{g}$. Let $\zeta=(\ X,F)\ \in \mathfrak{g}\oplus\mathfrak{g^*}$. Here $X\in \mathfrak{g} $ and $F\in ...

**0**

votes

**1**answer

169 views

### when $g^*$ is invariant under $Ad(G)$?

Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^*$ (dual of $\mathfrak{g}$)is invariant under ...

**6**

votes

**1**answer

318 views

### Representation ring and induced representation

Let $i:H \to G$ be a homomorphism of compact Lie groups. The induced representation $\iota_*V := \mathrm{Map}^H(G,V)$ of an $H$-representation $V$ does not give an element of the representation ring ...

**9**

votes

**3**answers

353 views

### Diffeomorphisms of the sphere conjugate to a rotation

What are sufficient condition on a given diffeomorphism of the sphere (say, given explicitly with formulas) that can ensure that it is conjugate to a rotation, in the group of diffeomorphism of the ...

**3**

votes

**1**answer

161 views

### The Gysin Sequence for an Associated Bundle over a Partial Flag Variety

Let $G$ be a connected, simply-connected complex semisimple Lie group, and let $P\subseteq G$ be a parabolic subgroup. Suppose that $V$ is a $1$-dimensional complex $P$-representation and consider the ...

**4**

votes

**1**answer

310 views

### Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)

Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that.
Cheers

**2**

votes

**0**answers

67 views

### Free S^1 action on a symmetric space of compact type

Consider a symmetric space G/H of compact type where rank(G) is greater than rank(H). The Euler characteristic of this space is known to be zero. What can be said about the existence of a fixed point ...

**2**

votes

**0**answers

85 views

### The condition of maximality in branching rules of $SO$ group representations

Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...

**4**

votes

**1**answer

313 views

### About using the character formula for $SO(2n)$.

I have known of the following equation for characters of a $SO(2n)$ representation with highest weights $(h_1,...,h_n)$ and for $(t_1,t_2,..,t_n,t_1^{-1},t_2^{-1},..,t_n^{-1})$ being the eigenvalues ...