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

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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 ...
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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 ...
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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 ...
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163 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
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1answer
174 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 ...
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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
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140 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 ...
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1answer
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 $G$?
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1answer
181 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
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2answers
342 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 $PSL(2,q)$ .
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1answer
499 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 ...
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55 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 ...
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1answer
361 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 ...
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1answer
378 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
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1answer
168 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 ...
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313 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 ...
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349 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
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1answer
158 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
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1answer
305 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
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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 ...
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82 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 $(s_1,s_2,...,s_n)$ ...
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1answer
302 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 ...
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129 views

Which cocompact subgroups of $G$ do contain a cocompact normal subgroup of $G$?

Let $G$ be a locally compact group and let $H$ be a cocompact (or more generally, a cofinite) subgroup of $G$. Is there any criterion to determine whether $H$ contains a cocompact normal subgroup of ...
2
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1answer
177 views

G-invariant differential forms on homogeneous space of Lie Groups

Let $G$ be a connected Lie Group and $K<G$ a maximal compact subgroup. Denote by $\Omega^q(G/K)^G$ the $G$-invariant real-valued $q$-forms on the manifold $G/K$, i.e. those forms $\omega$ s.t. ...
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1answer
430 views

Are maximal compact subgroups of connected groups connected?

Assume $G$ is a connected locally compact group and $M$ is a maximal compact subgroup of $G$. Is $M$ connected too?
2
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1answer
139 views

Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain

Let $H$ be a bounded symmetric domain. What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?
5
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1answer
134 views

A subgroup of the Weyl group

Let $D$ be a connected Dynkin diagram with an automorphism $\nu$ of order 2. Let $Q=Q(D)$ denote the root lattice of $D$. Let $W=W(D)$ denote the Weyl group, it acts effectively on $Q$ and it is ...
7
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2answers
354 views

Which Lie groups have adjoint representations that are bounded away from zero?

Studying stability of certain non-autonomous dynamical systems on Lie groups I have come across the following question: Exactly which finite-dimensional, real Lie groups have adjoint representations ...
3
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2answers
138 views

Are all orbits semi-Riemannian submanifolds?

Let $M$ be a semi-Riemannian manifold and $G\subset Iso(M)$ a closed connected Lie subgroup which acts properly on $M$. It is known that every orbit of the action is a (closed) submanifold of $M$. My ...
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232 views

Stiefel manifolds and polar decompositions

The real Stiefel manifold $V_{n,k}$ of orthogonal $k$-frames in $\mathbb{R}^n$ can be viewed as the reductive homogeneous space $G/H=O(n)/O(n-k)$. If ${\frak{so}}(n)$ is the Lie algebra of $O(n)$, ...
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2answers
337 views

Discrete subgroups of products of SU(2)

It is known that there are -up to conjugation- 5 classes of discrete subgroups of SU(2). One way to show this is by means of the McKay correspondence. My question is more regarding products of ...
9
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298 views

Extension of the Weyl dimension formula

Let $G$ be a compact semisimple group and let $\Gamma$ be a finite subgroup of $G$. I am interested, for $(\pi,V)\in \widehat G$ (irred rep of $G$), in a formula for $\mathrm{dim} V^\Gamma$, the ...
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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
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317 views

Iwasawa and $KAK$ Decomposition for Diff$(S^1)$

It is well known $\newcommand{\Diff}{\operatorname{Diff}}$ that the group $\Diff(S^1)$ of smooth diffeomorphisms of the circle behaves in many ways like $SL(2,\mathbb R)$. For example, if ...
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1answer
319 views

Iwasawa Decomposition for Matrices [closed]

I was asked to prove that if $$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$ denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ...
3
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1answer
128 views

Infinitesimal equivalence of admissible representations

Let $G_0$ be the $\mathbb{R}$-points of a real reductive group with complexified Lie algebra $\mathfrak{g}$ and maximal compact subgroup $K$. What is the precise relation between the category of ...
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166 views

tensor hierarchy for Lie groups from Maurer-Cartan form

The question is about the family of tensors that are naturally associated to any nice Lie group. Take the Mauer-Cartan form, $\omega=g^{-1} dg$ and I would like to make the covariant index of this ...
6
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1answer
216 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 ...
1
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1answer
78 views

The action of graph automorphism of finite symplectic group on maximal subgroups

Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following: $G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class ...
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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 ...
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561 views

Spin group as an automorphism group

Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with ...
1
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1answer
195 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: ...
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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 ...
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230 views

Equidistant hypersurfaces in symmetric space via exponentiation?

Here's some background and notation: Let $G/K$ be a symmetric space of non-compact type. For concreteness, assume $G$ is in fact a classical simple real Lie group such as SL,SO, or Sp, and $K$ is a ...
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2answers
371 views

How many principal bundles are there over a given base?

I'm currently considering principal bundles and classifying spaces in the context of gauge theory and a crucial question came to my mind: Is there a way to say how many (isomorphism classes of) ...
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2answers
523 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?
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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. ...
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1answer
208 views

Orbits of Root Vectors

Let $G$ be a connected, simply-connected complex semisimple Lie group with Lie algebra $\frak{g}$. Fix a maximal torus $T\subseteq G$, and let ...
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1answer
183 views

Does a free action always induce a diffeomorphism?

Suppose that $G$ is a Lie group with a transitive action on a smooth manifold $M$. The regular theory of Lie groups tells us that $G$ and $M$ are diffeomorphic if the isotropy group is trivial. The ...
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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 ...