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So one may prove the second equality of the question (the so-called Weyl integral formula) in the following way: For every $H\in\mathfrak{h}=Lie(T)$ we denote $$ Q(H):=\prod_{\alpha\in\Delta^+}(e^{\ …

So this is a comment to my question. In general if $G/k$ ($k$ is any field) is a connected linear algebraic $k$-group then one can show that it is generated over $\bar{k}$ by its Cartan subgroups defi …

So let $G$ be a compact real Lie group. Let $\rho:G\rightarrow GL_n(\mathbb{C})$ be an irreducible representation of $G$ and let $\chi_{\rho}$ be the character associated to $\rho$. Let $\Lambda_{\rho …

Let $G$ be be a connected real algebraic reductive Lie group. Is it always possible to find finitely many maximal algebraic $\mathbf{R}$-tori $\{T_i\}_{i=1}^n$ such that the group generated by the $T_ …

Let $G$ be a topological group. One general approach to show that $G$ is connected is the following: For every subgroup $H\leq G$ (not necessarily closed) we have a projection map: $$ \pi: G\rightar …

So the following statement seems to be obvious but I don't see how to prove it: Q: How does one prove that a closed totally disconnected subgroup of a connected real Lie group is discrete? Note that …

Let $F:\mathbf{R}^n\rightarrow\mathbf{R}$ be a non-degenerate quadratic forms. Let $$ O(F):=\{g\in GL_n(\mathbf{R}):F(gv)=F(v),\forall v\in \mathbf{R}^n\} $$ be the isotropy group of $F$. Q: So how d …

Let $G$ be a semi-simple real Lie group such that $|\pi_0(G)|<\infty$ and let $K$ be a maximal compact subgroup of $G$. Q1: How does one prove that $N_G(K)=K$? So I know a nice (and low-tech) proof …

Let $X$ be a complete Riemannian space. Let us denote by $Iso(X)$ the group of isometries of $X$. It is a well-known fact that the group $Iso(X)$, when endowed with the compact-open topology, is a Lie …

How does one construct a non-continuous representation $\rho:\operatorname{SL}_2(\mathbf{R})\rightarrow G$ for some connected (finite dimensional) Lie group $G$?

Is there a simple proof (short and low-tech) of the following fact: (E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to $K\times\mathbb{R}^n$ where $K$ is a maximal compac …